# Why 23 People Make a Shared Birthday Likely > Why can only 23 people make a shared birthday more likely than not? This lecture replaces the misleading count of people with the rapidly growing count of pairs, computes the probability by multiplying the shrinking chances that every birthday remains different, and follows the probability curve through its 50 percent crossing. It then separates the birthday paradox from the much harder question of matching one specified birthday, before carrying the same collision logic into hash functions and the birthday bound. - Canonical watch page: [Why 23 People Make a Shared Birthday Likely](https://academa.ai/lectures/the-birthday-paradox) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Statistics - Published: 2026-08-29T01:31:14.000Z - Updated: 2026-08-29T01:31:14.000Z - Duration: PT592S (9 minutes 52 seconds) - Chapters: 4 - Views: 1 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14V09JNY5H01VZKTHC75907/2/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14V09JNY5H01VZKTHC75907/2/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14V09JNY5H01VZKTHC75907/2/dark/poster.jpg) ## Description A visual explanation of why 23 people cross 50%, why one fixed birthday needs 253 others, and how the same logic predicts hash collisions. ## Chapters - [00:00–01:49.17 · Count the Pairs, Not the People](https://academa.ai/lectures/the-birthday-paradox?t=0) - [01:49.17–04:32.904 · Multiply the Shrinking Fractions](https://academa.ai/lectures/the-birthday-paradox?t=109.17004166666666) - [04:32.904–06:54.903 · Someone Who Matches Your Birthday](https://academa.ai/lectures/the-birthday-paradox?t=272.9036666666666) - [06:54.903–09:52 · The Same Collision in a Hash Function](https://academa.ai/lectures/the-birthday-paradox?t=414.90341666666654) ## Transcript ### [00:00 · Count the Pairs, Not the People](https://academa.ai/lectures/the-birthday-paradox?t=0) Twenty three people sounds far too small for a shared birthday to be more likely than not. That reaction is sensible, but it is counting the wrong thing. The birthdays are not compared with the calendar one person at a time. They are compared with one another, pair by pair. Start with three people. Each blue dot is one person, and each yellow line will represent one question: do the two people at its ends share a birthday? The first person can be compared with the second. The first can also be compared with the third. Finally, the second can be compared with the third. Three people already make three distinct pairs. Now add a fourth person. That newcomer creates three fresh comparisons, one with everyone already present. Add a fifth person and the newcomer creates four more. We added two people, but we added seven pairs. Five people therefore make ten pairs. In general, n people each appear in n minus one comparisons. That counts every pair twice, once from each end, so we divide by two. Now put twenty three people around the circle. The dots still look sparse. That is the picture our intuition notices. But every dot must be compared with every other dot. The formula gives twenty three times twenty two, divided by two. That is two hundred fifty three. Here is every comparison the birthday question is actually making. So the experiment is not twenty three lonely attempts to hit one special day. It is two hundred fifty three opportunities for some pair to collide. We have not proved the fifty percent claim yet, but now the scale of it should no longer feel absurd. ### [01:49.17 · Multiply the Shrinking Fractions](https://academa.ai/lectures/the-birthday-paradox?t=109.17004166666666) To keep the calculation clean, suppose there are three hundred sixty five equally likely birthdays, ignore leap day, and treat people's birthdays as independent. Real birthdays are not perfectly uniform, but this model is close enough to expose the main effect. Counting every possible kind of match directly is messy. Two people might match, three might match, or two separate pairs might match. So turn the question around. What is the chance that every birthday is different? The first person can have any birthday. For the second person to avoid a match, only one day is forbidden. That leaves three hundred sixty four safe days out of three hundred sixty five. Bring in a third person. Now two occupied birthdays must be avoided, so only three hundred sixty three days are safe. Both conditions must hold, which is why the fractions multiply. Each new person contributes another fraction, with one fewer safe day in the numerator. For n people, multiply from three hundred sixty four over three hundred sixty five down to three hundred sixty six minus n over three hundred sixty five. At twenty three people, that product is about zero point four nine two seven. So the chance that everyone differs is just under one half. Subtract from one, and the chance of at least one shared birthday is about zero point five zero seven three, just over one half. Now view every group size on one graph. The horizontal axis is the number of people, and the vertical axis is the chance that some pair shares a birthday. The dashed line marks one half. With ten people, the chance is only about eleven point seven percent. That feels unsurprising, and it gives us a low starting point on the curve. Move to twenty two people. The chance has climbed to about forty seven point six percent, still below the dashed half line. Add just one more person. That newcomer does not create one new chance to match. They create twenty two new pair comparisons. At twenty three, the probability reaches fifty point seven percent and crosses the line. The rise continues quickly. At thirty people the chance is about seventy one percent. At fifty people it is about ninety seven percent. The curve accelerates because every new arrival is compared with everyone already present. That is why twenty three is enough. It is not large compared with three hundred sixty five days, but it creates two hundred fifty three pairs. The product calculation and the graph now say the same thing: a shared birthday is more likely than not. ### [04:32.904 · Someone Who Matches Your Birthday](https://academa.ai/lectures/the-birthday-paradox?t=272.9036666666666) Now ask a question that sounds almost identical. How many people are needed before someone probably shares my birthday specifically? The word my changes the comparison structure completely. With seven people in the room, only six comparisons matter for my birthday. Each yellow line joins me to one other person. Birthdays shared entirely among the blue people do not answer this question. For the original birthday problem, every pair matters. The same seven people create twenty one comparisons, because every person is checked against every other person. The left count grows one at a time as people arrive. The right count grows by the whole number already in the room. That is why the original any-pair question reaches one half so much sooner. Let n mean the number of other people, not counting me. One person fails to match my birthday with probability three hundred sixty four over three hundred sixty five. For nobody to match me, all n people must fail. A match becomes more likely than not when the no-match probability falls below one half. So we set the shrinking power below one half. Take logarithms to bring n down from the exponent. Because the logarithm of three hundred sixty four over three hundred sixty five is negative, dividing by it reverses the inequality. The result is n greater than two hundred fifty two point six. Since n must be a whole number, we need two hundred fifty three other people. Including me, that is two hundred fifty four people in the room. Put both questions on one graph. The horizontal axis now counts everyone in the room, including me. The dashed line is again a probability of one half. The blue curve asks whether any pair matches. It crosses one half at twenty three people because those people create two hundred fifty three pairs. The green curve asks whether someone matches one fixed birthday. It rises far more slowly and crosses one half at two hundred fifty four people, meaning two hundred fifty three comparisons with me. So the intuition that twenty three feels too small was not foolish. It was quietly answering the green question while the birthday paradox asks the blue one. Always ask which pairs actually count. ### [06:54.903 · The Same Collision in a Hash Function](https://academa.ai/lectures/the-birthday-paradox?t=414.90341666666654) The birthday idea appears anywhere many objects are assigned to a limited set of labels. A hash function turns a file, password, or message into one fixed-size output. Two different inputs that receive the same output create a collision. Here are sixteen possible outputs. Send in one item, then a second, a third, and a fourth. So far every item has landed in a different slot. Now send in the fifth. It lands in the same slot as the second item. Sixteen slots sounded generous for five items, but five items already create ten pairs that might collide. Suppose a hash has N equally likely outputs and we try k inputs. The number of input pairs is k times k minus one over two, exactly the same pair count as before. Each pair has about one chance in N of sharing an output. Combining all those small risks gives this standard approximation for no collision. Subtracting from one gives the chance of at least one collision. Set that collision chance to one half and solve. The result is about one point one eight times the square root of N. The constant matters for an exact estimate, but the memorable fact is the square root. The birthday example checks the scale. Three hundred sixty five possible days have a square root near nineteen, and the more accurate crossing is twenty three. A sixty four bit hash has two to the sixty fourth possible outputs, but collisions become likely after roughly two to the thirty second inputs. That is about four billion, far below the number of available outputs. For a one hundred twenty eight bit hash, the birthday scale is about two to the sixty fourth trials. For a two hundred fifty six bit hash, it is about two to the one hundred twenty eighth. Collision resistance therefore provides roughly half as many security bits as the output length. That matters when hashes identify files, support digital signatures, or detect duplicates. A real design either chooses enough output bits to make accidental collisions fantastically unlikely, or includes a clear plan for handling collisions when they occur. Four ideas now hold the whole lecture together. Count comparisons rather than objects. Compute the easier no-collision probability. Check whether every pair counts or only comparisons with one fixed target. And expect collisions near the square root of the number of possible outcomes. Twenty three people are enough because they do not create twenty three chances to match. They create two hundred fifty three pairs. Once that hidden web is visible, the birthday result stops being a trick and becomes an ordinary collision problem. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14V09JNY5H01VZKTHC75907/2/semantic.json) Record version: 1. Render attempt: 2. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. 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full\_dots\_23: a Point \[blue\] drawn in full (location=(-0.26979677115702516, 0.962917287347799)) - full\_dots\_3: a Point \[blue\] drawn in full (location=(0.5195839500354336, 0.8544194045464886)) - full\_dots\_4: a Point \[blue\] drawn in full (location=(0.7308359642781241, 0.6825531432186541)) - full\_dots\_5: a Point \[blue\] drawn in full (location=(0.8878852184023752, 0.46006503773115215)) - full\_dots\_6: a Point \[blue\] drawn in full (location=(0.9790840876823229, 0.2034560130526337)) - full\_dots\_7: a Point \[blue\] drawn in full (location=(0.9976687691905392, -0.06824241336467095)) - full\_dots\_8: a Point \[blue\] drawn in full (location=(0.9422609221188205, -0.3348796121709862)) - full\_dots\_9: a Point \[blue\] drawn in full (location=(0.8169698930104421, -0.5766803221148671)) - heading\_full: a Heading that says "Twenty Three People in Full" - heading\_small: a Heading that says "Count the Pairs, Not the People" - pair\_formula: a Math \[text\] that says "$upright("pairs") = frac(n(n - 1), 2)$" - punch: a Tex \[text\] that says "Twenty three people create two hundred fifty three pairs." - small: a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)) - small\_chords: a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(0.9510565162951535, 0.3090169943749474)) - small\_chords\_10: a Line \[yellow\] drawn in small (start=(-0.587785252292473, -0.8090169943749475), end=(-0.9510565162951536, 0.3090169943749473)) - small\_chords\_2: a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(0.5877852522924731, -0.8090169943749475)) - small\_chords\_3: a Line \[yellow\] drawn in small (start=(0.9510565162951535, 0.3090169943749474), end=(0.5877852522924731, -0.8090169943749475)) - small\_chords\_4: a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(-0.587785252292473, -0.8090169943749475)) - small\_chords\_5: a Line \[yellow\] drawn in small (start=(0.9510565162951535, 0.3090169943749474), end=(-0.587785252292473, -0.8090169943749475)) - small\_chords\_6: a Line \[yellow\] drawn in small (start=(0.5877852522924731, -0.8090169943749475), end=(-0.587785252292473, -0.8090169943749475)) - small\_chords\_7: a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(-0.9510565162951536, 0.3090169943749473)) - small\_chords\_8: a Line \[yellow\] drawn in small (start=(0.9510565162951535, 0.3090169943749474), end=(-0.9510565162951536, 0.3090169943749473)) - small\_chords\_9: a Line \[yellow\] drawn in small (start=(0.5877852522924731, -0.8090169943749475), end=(-0.9510565162951536, 0.3090169943749473)) - small\_dots: a Point \[blue\] drawn in small (location=(6.123233995736766e-17, 1.0)) - small\_dots\_2: a Point \[blue\] drawn in small (location=(0.9510565162951535, 0.3090169943749474)) - small\_dots\_3: a Point \[blue\] drawn in small (location=(0.5877852522924731, -0.8090169943749475)) - small\_dots\_4: a Point \[blue\] drawn in small (location=(-0.587785252292473, -0.8090169943749475)) - small\_dots\_5: a Point \[blue\] drawn in small (location=(-0.9510565162951536, 0.3090169943749473)) - three\_count: a Math \[text\] that says "$3 thin upright("people") arrow.r 3 thin upright("pairs")$" #### Beats ##### [00:00](https://academa.ai/lectures/the-birthday-paradox?t=0) Narration: Twenty three people sounds far too small for a shared birthday to be more likely than not. That reaction is sensible, but it is counting the wrong thing. The birthdays are not compared with the calendar one person at a time. They are compared with one another, pair by pair. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/the-birthday-paradox?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/the-birthday-paradox?t=1.5): card: enter:write-left-to-right. - [00:16.126](https://academa.ai/lectures/the-birthday-paradox?t=16.126): card is hidden from the screen — left the board. ##### [00:17.326](https://academa.ai/lectures/the-birthday-paradox?t=17.326) Narration: Start with three people. Each blue dot is one person, and each yellow line will represent one question: do the two people at its ends share a birthday? Board: Unchanged from the preceding beat in this scene. Actions: - [00:17.326](https://academa.ai/lectures/the-birthday-paradox?t=17.326): heading\_small is shown on the screen, written out. - [00:17.326](https://academa.ai/lectures/the-birthday-paradox?t=17.326): small is shown on the screen, written out. - [00:18.185](https://academa.ai/lectures/the-birthday-paradox?t=18.185): small\_dots is shown on the screen, written out. - [00:18.305](https://academa.ai/lectures/the-birthday-paradox?t=18.305): small\_dots\_2 is shown on the screen, written out. - [00:18.425](https://academa.ai/lectures/the-birthday-paradox?t=18.424999999999997): small\_dots\_3 is shown on the screen, written out. ##### [00:28.306](https://academa.ai/lectures/the-birthday-paradox?t=28.305500000000002) Narration: The first person can be compared with the second. The first can also be compared with the third. Finally, the second can be compared with the third. Three people already make three distinct pairs. Board: small — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); heading\_small — a Heading that says "Count the Pairs, Not the People"; small\_dots — a Point \[blue\] drawn in small (location=(6.123233995736766e-17, 1.0)); small\_dots\_2 — a Point \[blue\] drawn in small (location=(0.9510565162951535, 0.3090169943749474)); small\_dots\_3 — a Point \[blue\] drawn in small (location=(0.5877852522924731, -0.8090169943749475)) Actions: - [00:30.685](https://academa.ai/lectures/the-birthday-paradox?t=30.685000000000002): small\_chords is shown on the screen, written out. - [00:33.657](https://academa.ai/lectures/the-birthday-paradox?t=33.657): small\_chords\_2 is shown on the screen, written out. - [00:34.934](https://academa.ai/lectures/the-birthday-paradox?t=34.934): small\_chords\_3 is shown on the screen, written out. - [00:38.139](https://academa.ai/lectures/the-birthday-paradox?t=38.138999999999996): small moves to a new place on the board. - [00:38.139](https://academa.ai/lectures/the-birthday-paradox?t=38.138999999999996): three\_count is shown on the screen, written out. ##### [00:41.398](https://academa.ai/lectures/the-birthday-paradox?t=41.3975) Narration: Now add a fourth person. That newcomer creates three fresh comparisons, one with everyone already present. Add a fifth person and the newcomer creates four more. We added two people, but we added seven pairs. Board: three\_count — a Math \[text\] that says "$3 thin upright("people") arrow.r 3 thin upright("pairs")$"; small — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); heading\_small — a Heading that says "Count the Pairs, Not the People"; small\_dots — a Point \[blue\] drawn in small (location=(6.123233995736766e-17, 1.0)); small\_dots\_2 — a Point \[blue\] drawn in small (location=(0.9510565162951535, 0.3090169943749474)); small\_dots\_3 — a Point \[blue\] drawn in small (location=(0.5877852522924731, -0.8090169943749475)); small\_chords — a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(0.9510565162951535, 0.3090169943749474)); small\_chords\_2 — a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(0.5877852522924731, -0.8090169943749475)); small\_chords\_3 — a Line \[yellow\] drawn in small (start=(0.9510565162951535, 0.3090169943749474), end=(0.5877852522924731, -0.8090169943749475)) Actions: - [00:42.268](https://academa.ai/lectures/the-birthday-paradox?t=42.268): small\_dots\_4 is shown on the screen, written out. - [00:44.95](https://academa.ai/lectures/the-birthday-paradox?t=44.949999999999996): small\_chords\_4 is shown on the screen, written out. - [00:45.07](https://academa.ai/lectures/the-birthday-paradox?t=45.06999999999999): small\_chords\_5 is shown on the screen, written out. - [00:45.19](https://academa.ai/lectures/the-birthday-paradox?t=45.19): small\_chords\_6 is shown on the screen, written out. - [00:49.362](https://academa.ai/lectures/the-birthday-paradox?t=49.361999999999995): small\_dots\_5 is shown on the screen, written out. - [00:51.393](https://academa.ai/lectures/the-birthday-paradox?t=51.392999999999994): small\_chords\_7 is shown on the screen, written out. - [00:51.513](https://academa.ai/lectures/the-birthday-paradox?t=51.51299999999999): small\_chords\_8 is shown on the screen, written out. - [00:51.633](https://academa.ai/lectures/the-birthday-paradox?t=51.632999999999996): small\_chords\_9 is shown on the screen, written out. - [00:51.753](https://academa.ai/lectures/the-birthday-paradox?t=51.75299999999999): small\_chords\_10 is shown on the screen, written out. - [00:54.981](https://academa.ai/lectures/the-birthday-paradox?t=54.980999999999995): five\_count is shown on the screen, written out. ##### [00:56.765](https://academa.ai/lectures/the-birthday-paradox?t=56.765) Narration: Five people therefore make ten pairs. In general, n people each appear in n minus one comparisons. That counts every pair twice, once from each end, so we divide by two. Board: three\_count — a Math \[text\] that says "$3 thin upright("people") arrow.r 3 thin upright("pairs")$"; five\_count — a Math \[text\] that says "$5 thin upright("people") arrow.r 10 thin upright("pairs")$"; small — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); heading\_small — a Heading that says "Count the Pairs, Not the People"; small\_dots — a Point \[blue\] drawn in small (location=(6.123233995736766e-17, 1.0)); small\_dots\_2 — a Point \[blue\] drawn in small (location=(0.9510565162951535, 0.3090169943749474)); small\_dots\_3 — a Point \[blue\] drawn in small (location=(0.5877852522924731, -0.8090169943749475)); small\_chords — a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(0.9510565162951535, 0.3090169943749474)); small\_chords\_2 — a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(0.5877852522924731, -0.8090169943749475)); small\_chords\_3 — a Line \[yellow\] drawn in small (start=(0.9510565162951535, 0.3090169943749474), end=(0.5877852522924731, -0.8090169943749475)); small\_dots\_4 — a Point \[blue\] drawn in small (location=(-0.587785252292473, -0.8090169943749475)); small\_chords\_4 — a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(-0.587785252292473, -0.8090169943749475)); small\_chords\_5 — a Line \[yellow\] drawn in small (start=(0.9510565162951535, 0.3090169943749474), end=(-0.587785252292473, -0.8090169943749475)); small\_chords\_6 — a Line \[yellow\] drawn in small (start=(0.5877852522924731, -0.8090169943749475), end=(-0.587785252292473, -0.8090169943749475)); small\_dots\_5 — a Point \[blue\] drawn in small (location=(-0.9510565162951536, 0.3090169943749473)); small\_chords\_7 — a Line \[yellow\] drawn in small (start=(6.123233995736766e-17, 1.0), end=(-0.9510565162951536, 0.3090169943749473)); small\_chords\_8 — a Line \[yellow\] drawn in small (start=(0.9510565162951535, 0.3090169943749474), end=(-0.9510565162951536, 0.3090169943749473)); small\_chords\_9 — a Line \[yellow\] drawn in small (start=(0.5877852522924731, -0.8090169943749475), end=(-0.9510565162951536, 0.3090169943749473)); small\_chords\_10 — a Line \[yellow\] drawn in small (start=(-0.587785252292473, -0.8090169943749475), end=(-0.9510565162951536, 0.3090169943749473)) Actions: - [00:56.765](https://academa.ai/lectures/the-birthday-paradox?t=56.765): pair\_formula is shown on the screen, written out. - [01:5.925](https://academa.ai/lectures/the-birthday-paradox?t=65.925): pair\_formula (the "n(n - 1)" part) is emphasized. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): pair\_formula moves to a new place on the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): five\_count is hidden from the screen — left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): heading\_small is hidden from the screen — left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small is hidden from the screen — left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_dots is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_dots\_2 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_dots\_3 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_2 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_3 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_dots\_4 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_4 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_5 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_6 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_dots\_5 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_7 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_8 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_9 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): small\_chords\_10 is hidden from the screen — small left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): three\_count is hidden from the screen — left the board. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): full is shown on the screen, written out. - [01:9.594](https://academa.ai/lectures/the-birthday-paradox?t=69.594): pair\_formula (the "n(n - 1)" part) is no longer emphasized. ##### [01:10.794](https://academa.ai/lectures/the-birthday-paradox?t=70.794) Narration: Now put twenty three people around the circle. The dots still look sparse. That is the picture our intuition notices. But every dot must be compared with every other dot. Board: pair\_formula — a Math \[text\] that says "$upright("pairs") = frac(n(n - 1), 2)$"; full — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)) Actions: - [01:10.794](https://academa.ai/lectures/the-birthday-paradox?t=70.794): heading\_full is shown on the screen, written out. - [01:11.572](https://academa.ai/lectures/the-birthday-paradox?t=71.572): full\_dots is shown on the screen, written out. - [01:11.602](https://academa.ai/lectures/the-birthday-paradox?t=71.602): full\_dots\_2 is shown on the screen, written out. - [01:11.632](https://academa.ai/lectures/the-birthday-paradox?t=71.632): full\_dots\_3 is shown on the screen, written out. - [01:11.662](https://academa.ai/lectures/the-birthday-paradox?t=71.662): full\_dots\_4 is shown on the screen, written out. - [01:11.692](https://academa.ai/lectures/the-birthday-paradox?t=71.69200000000001): full\_dots\_5 is shown on the screen, written out. - [01:11.722](https://academa.ai/lectures/the-birthday-paradox?t=71.72200000000001): full\_dots\_6 is shown on the screen, written out. - [01:11.752](https://academa.ai/lectures/the-birthday-paradox?t=71.75200000000001): full\_dots\_7 is shown on the screen, written out. - [01:11.782](https://academa.ai/lectures/the-birthday-paradox?t=71.782): full\_dots\_8 is shown on the screen, written out. - [01:11.812](https://academa.ai/lectures/the-birthday-paradox?t=71.812): full\_dots\_9 is shown on the screen, written out. - [01:11.842](https://academa.ai/lectures/the-birthday-paradox?t=71.842): full\_dots\_10 is shown on the screen, written out. - [01:11.872](https://academa.ai/lectures/the-birthday-paradox?t=71.872): full\_dots\_11 is shown on the screen, written out. - [01:11.902](https://academa.ai/lectures/the-birthday-paradox?t=71.902): full\_dots\_12 is shown on the screen, written out. - [01:11.932](https://academa.ai/lectures/the-birthday-paradox?t=71.932): full\_dots\_13 is shown on the screen, written out. - [01:11.962](https://academa.ai/lectures/the-birthday-paradox?t=71.962): full\_dots\_14 is shown on the screen, written out. - [01:11.992](https://academa.ai/lectures/the-birthday-paradox?t=71.992): full\_dots\_15 is shown on the screen, written out. - [01:12.022](https://academa.ai/lectures/the-birthday-paradox?t=72.022): full\_dots\_16 is shown on the screen, written out. - [01:12.052](https://academa.ai/lectures/the-birthday-paradox?t=72.052): full\_dots\_17 is shown on the screen, written out. - [01:12.082](https://academa.ai/lectures/the-birthday-paradox?t=72.08200000000001): full\_dots\_18 is shown on the screen, written out. - [01:12.112](https://academa.ai/lectures/the-birthday-paradox?t=72.11200000000001): full\_dots\_19 is shown on the screen, written out. - [01:12.142](https://academa.ai/lectures/the-birthday-paradox?t=72.142): full\_dots\_20 is shown on the screen, written out. - [01:12.172](https://academa.ai/lectures/the-birthday-paradox?t=72.172): full\_dots\_21 is shown on the screen, written out. - [01:12.202](https://academa.ai/lectures/the-birthday-paradox?t=72.202): full\_dots\_22 is shown on the screen, written out. - [01:12.232](https://academa.ai/lectures/the-birthday-paradox?t=72.232): full\_dots\_23 is shown on the screen, written out. ##### [01:22.174](https://academa.ai/lectures/the-birthday-paradox?t=82.1745) Narration: The formula gives twenty three times twenty two, divided by two. That is two hundred fifty three. Here is every comparison the birthday question is actually making. Board: pair\_formula — a Math \[text\] that says "$upright("pairs") = frac(n(n - 1), 2)$"; full — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); heading\_full — a Heading that says "Twenty Three People in Full"; full\_dots — a Point \[blue\] drawn in full (location=(6.123233995736766e-17, 1.0)); full\_dots\_2 — a Point \[blue\] drawn in full (location=(0.2697967711570242, 0.9629172873477994)); full\_dots\_3 — a Point \[blue\] drawn in full (location=(0.5195839500354336, 0.8544194045464886)); full\_dots\_4 — a Point \[blue\] drawn in full (location=(0.7308359642781241, 0.6825531432186541)); full\_dots\_5 — a Point \[blue\] drawn in full (location=(0.8878852184023752, 0.46006503773115215)); full\_dots\_6 — a Point \[blue\] drawn in full (location=(0.9790840876823229, 0.2034560130526337)); full\_dots\_7 — a Point \[blue\] drawn in full (location=(0.9976687691905392, -0.06824241336467095)); full\_dots\_8 — a Point \[blue\] drawn in full (location=(0.9422609221188205, -0.3348796121709862)); full\_dots\_9 — a Point \[blue\] drawn in full (location=(0.8169698930104421, -0.5766803221148671)); full\_dots\_10 — a Point \[blue\] drawn in full (location=(0.631087944326053, -0.7757112907044197)); full\_dots\_11 — a Point \[blue\] drawn in full (location=(0.39840108984624134, -0.917211301505453)); full\_dots\_12 — a Point \[blue\] drawn in full (location=(0.13616664909624704, -0.9906859460363306)); full\_dots\_13 — a Point \[blue\] drawn in full (location=(-0.13616664909624646, -0.9906859460363308)); full\_dots\_14 — a Point \[blue\] drawn in full (location=(-0.3984010898462416, -0.9172113015054529)); full\_dots\_15 — a Point \[blue\] drawn in full (location=(-0.6310879443260529, -0.7757112907044198)); full\_dots\_16 — a Point \[blue\] drawn in full (location=(-0.816969893010442, -0.5766803221148672)); full\_dots\_17 — a Point \[blue\] drawn in full (location=(-0.9422609221188204, -0.33487961217098633)); full\_dots\_18 — a Point \[blue\] drawn in full (location=(-0.9976687691905393, -0.0682424133646704)); full\_dots\_19 — a Point \[blue\] drawn in full (location=(-0.9790840876823229, 0.20345601305263336)); full\_dots\_20 — a Point \[blue\] drawn in full (location=(-0.8878852184023754, 0.46006503773115165)); full\_dots\_21 — a Point \[blue\] drawn in full (location=(-0.7308359642781239, 0.6825531432186542)); full\_dots\_22 — a Point \[blue\] drawn in full (location=(-0.5195839500354336, 0.8544194045464886)); full\_dots\_23 — a Point \[blue\] drawn in full (location=(-0.26979677115702516, 0.962917287347799)) Actions: - [01:22.54](https://academa.ai/lectures/the-birthday-paradox?t=82.53999999999999): count\_23 is shown on the screen, written out. - [01:29.668](https://academa.ai/lectures/the-birthday-paradox?t=89.66799999999999): full\_chords is shown on the screen, written out. - [01:29.674](https://academa.ai/lectures/the-birthday-paradox?t=89.67399999999999): full\_chords\_2 is shown on the screen, written out. - [01:29.68](https://academa.ai/lectures/the-birthday-paradox?t=89.67999999999999): full\_chords\_3 is shown on the screen, written out. - [01:29.686](https://academa.ai/lectures/the-birthday-paradox?t=89.68599999999999): full\_chords\_4 is shown on the screen, written out. - [01:29.692](https://academa.ai/lectures/the-birthday-paradox?t=89.692): full\_chords\_5 is shown on the screen, written out. - [01:29.698](https://academa.ai/lectures/the-birthday-paradox?t=89.698): full\_chords\_6 is shown on the screen, written out. - [01:29.704](https://academa.ai/lectures/the-birthday-paradox?t=89.704): full\_chords\_7 is shown on the screen, written out. - [01:29.71](https://academa.ai/lectures/the-birthday-paradox?t=89.71): full\_chords\_8 is shown on the screen, written out. - [01:29.716](https://academa.ai/lectures/the-birthday-paradox?t=89.716): full\_chords\_9 is shown on the screen, written out. - [01:29.722](https://academa.ai/lectures/the-birthday-paradox?t=89.722): full\_chords\_10 is shown on the screen, written out. - [01:29.728](https://academa.ai/lectures/the-birthday-paradox?t=89.728): full\_chords\_11 is shown on the screen, written out. - [01:29.734](https://academa.ai/lectures/the-birthday-paradox?t=89.734): full\_chords\_12 is shown on the screen, written out. - [01:29.74](https://academa.ai/lectures/the-birthday-paradox?t=89.74): full\_chords\_13 is shown on the screen, written out. - [01:29.746](https://academa.ai/lectures/the-birthday-paradox?t=89.746): full\_chords\_14 is shown on the screen, written out. - [01:29.752](https://academa.ai/lectures/the-birthday-paradox?t=89.752): full\_chords\_15 is shown on the screen, written out. - [01:29.758](https://academa.ai/lectures/the-birthday-paradox?t=89.758): full\_chords\_16 is shown on the screen, written out. - [01:29.764](https://academa.ai/lectures/the-birthday-paradox?t=89.764): full\_chords\_17 is shown on the screen, written out. - [01:29.77](https://academa.ai/lectures/the-birthday-paradox?t=89.77): full\_chords\_18 is shown on the screen, written out. - [01:29.776](https://academa.ai/lectures/the-birthday-paradox?t=89.776): full\_chords\_19 is shown on the screen, written out. - [01:29.782](https://academa.ai/lectures/the-birthday-paradox?t=89.782): full\_chords\_20 is shown on the screen, written out. - [01:29.788](https://academa.ai/lectures/the-birthday-paradox?t=89.788): full\_chords\_21 is shown on the screen, written out. - [01:29.794](https://academa.ai/lectures/the-birthday-paradox?t=89.794): full\_chords\_22 is shown on the screen, written out. - [01:29.8](https://academa.ai/lectures/the-birthday-paradox?t=89.8): full\_chords\_23 is shown on the screen, written out. - [01:29.806](https://academa.ai/lectures/the-birthday-paradox?t=89.806): full\_chords\_24 is shown on the screen, written out. - [01:29.812](https://academa.ai/lectures/the-birthday-paradox?t=89.812): full\_chords\_25 is shown on the screen, written out. - [01:29.818](https://academa.ai/lectures/the-birthday-paradox?t=89.818): full\_chords\_26 is shown on the screen, written out. - [01:29.824](https://academa.ai/lectures/the-birthday-paradox?t=89.824): full\_chords\_27 is shown on the screen, written out. - [01:29.83](https://academa.ai/lectures/the-birthday-paradox?t=89.83): full\_chords\_28 is shown on the screen, written out. - [01:29.836](https://academa.ai/lectures/the-birthday-paradox?t=89.836): full\_chords\_29 is shown on the screen, written out. - [01:29.842](https://academa.ai/lectures/the-birthday-paradox?t=89.842): full\_chords\_30 is shown on the screen, written out. - [01:29.848](https://academa.ai/lectures/the-birthday-paradox?t=89.848): full\_chords\_31 is shown on the screen, written out. - [01:29.854](https://academa.ai/lectures/the-birthday-paradox?t=89.854): full\_chords\_32 is shown on the screen, written out. - [01:29.86](https://academa.ai/lectures/the-birthday-paradox?t=89.85999999999999): full\_chords\_33 is shown on the screen, written out. - [01:29.866](https://academa.ai/lectures/the-birthday-paradox?t=89.86599999999999): full\_chords\_34 is shown on the screen, written out. - [01:29.872](https://academa.ai/lectures/the-birthday-paradox?t=89.87199999999999): full\_chords\_35 is shown on the screen, written out. - [01:29.878](https://academa.ai/lectures/the-birthday-paradox?t=89.87799999999999): full\_chords\_36 is shown on the screen, written out. - [01:29.884](https://academa.ai/lectures/the-birthday-paradox?t=89.88399999999999): full\_chords\_37 is shown on the screen, written out. - [01:29.89](https://academa.ai/lectures/the-birthday-paradox?t=89.88999999999999): full\_chords\_38 is shown on the screen, written out. - [01:29.896](https://academa.ai/lectures/the-birthday-paradox?t=89.89599999999999): full\_chords\_39 is shown on the screen, written out. - 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[01:31.15](https://academa.ai/lectures/the-birthday-paradox?t=91.14999999999999): full\_chords\_248 is shown on the screen, written out. - [01:31.156](https://academa.ai/lectures/the-birthday-paradox?t=91.15599999999999): full\_chords\_249 is shown on the screen, written out. - [01:31.162](https://academa.ai/lectures/the-birthday-paradox?t=91.16199999999999): full\_chords\_250 is shown on the screen, written out. - [01:31.168](https://academa.ai/lectures/the-birthday-paradox?t=91.16799999999999): full\_chords\_251 is shown on the screen, written out. - [01:31.174](https://academa.ai/lectures/the-birthday-paradox?t=91.17399999999999): full\_chords\_252 is shown on the screen, written out. - [01:31.18](https://academa.ai/lectures/the-birthday-paradox?t=91.17999999999999): full\_chords\_253 is shown on the screen, written out. ##### [01:33.612](https://academa.ai/lectures/the-birthday-paradox?t=93.612) Narration: So the experiment is not twenty three lonely attempts to hit one special day. It is two hundred fifty three opportunities for some pair to collide. We have not proved the fifty percent claim yet, but now the scale of it should no longer feel absurd. Board: pair\_formula — a Math \[text\] that says "$upright("pairs") = frac(n(n - 1), 2)$"; count\_23 — a Math \[text\] that says "$frac(23 dot.op 22, 2) = 253$"; full — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); heading\_full — a Heading that says "Twenty Three People in Full"; full\_dots — a Point \[blue\] drawn in full (location=(6.123233995736766e-17, 1.0)); full\_dots\_2 — a Point \[blue\] drawn in full (location=(0.2697967711570242, 0.9629172873477994)); full\_dots\_3 — a Point \[blue\] drawn in full (location=(0.5195839500354336, 0.8544194045464886)); full\_dots\_4 — a Point \[blue\] drawn in full (location=(0.7308359642781241, 0.6825531432186541)); full\_dots\_5 — a Point \[blue\] drawn in full (location=(0.8878852184023752, 0.46006503773115215)); full\_dots\_6 — a Point \[blue\] drawn in full (location=(0.9790840876823229, 0.2034560130526337)); full\_dots\_7 — a Point \[blue\] drawn in full (location=(0.9976687691905392, 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count\_23 is indicated — a transient flash. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): count\_23 is hidden from the screen — left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full is hidden from the screen — left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_2 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_3 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_4 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_5 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_6 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_7 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_8 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_9 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_10 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_11 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_12 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_13 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_14 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_15 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_16 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_17 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_18 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_19 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_20 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_21 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_22 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_dots\_23 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_2 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_3 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_4 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_5 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_6 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_7 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_8 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_9 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_10 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_11 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_12 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_13 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_14 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_15 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_16 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_17 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_18 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_19 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_20 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_21 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_22 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_23 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_24 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_25 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_26 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_27 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_28 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_29 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_30 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_31 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_32 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_33 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_34 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_35 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_36 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_37 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_38 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_39 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_40 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_41 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_42 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_43 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_44 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_45 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_46 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_47 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_48 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_49 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_50 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_51 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_52 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_53 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_54 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_55 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_56 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_57 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_58 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_59 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_60 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_61 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_62 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_63 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_64 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_65 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_66 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_67 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_68 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_69 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_70 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_71 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_72 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_73 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_74 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_75 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_76 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_77 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_78 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_79 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_80 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_81 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_82 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_83 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_84 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_85 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_86 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_87 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_88 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_89 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_90 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_91 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_92 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_93 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_94 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_95 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_96 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_97 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_98 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_99 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_100 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_101 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_102 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_103 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_104 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_105 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_106 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_107 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_108 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_109 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_110 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_111 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_112 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_113 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_114 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_115 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_116 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_117 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_118 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_119 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_120 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_121 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_122 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_123 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_124 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_125 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_126 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_127 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_128 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_129 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_130 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_131 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_132 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_133 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_134 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_135 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_136 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_137 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_138 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_139 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_140 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_141 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_142 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_143 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_144 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_145 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_146 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_147 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_148 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_149 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_150 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_151 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_152 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_153 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_154 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_155 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_156 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_157 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_158 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_159 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_160 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_161 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_162 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_163 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_164 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_165 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_166 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_167 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_168 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_169 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_170 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_171 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_172 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_173 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_174 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_175 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_176 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_177 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_178 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_179 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_180 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_181 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_182 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_183 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_184 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_185 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_186 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_187 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_188 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_189 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_190 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_191 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_192 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_193 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_194 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_195 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_196 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_197 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_198 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_199 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_200 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_201 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_202 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_203 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_204 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_205 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_206 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_207 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_208 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_209 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_210 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_211 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_212 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_213 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_214 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_215 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_216 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_217 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_218 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_219 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_220 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_221 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_222 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_223 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_224 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_225 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_226 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_227 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_228 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_229 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_230 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_231 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_232 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_233 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_234 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_235 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_236 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_237 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_238 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_239 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_240 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_241 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_242 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_243 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_244 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_245 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_246 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_247 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_248 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_249 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_250 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_251 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_252 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): full\_chords\_253 is hidden from the screen — full left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): heading\_full is hidden from the screen — left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): pair\_formula is hidden from the screen — left the board. - [01:48.128](https://academa.ai/lectures/the-birthday-paradox?t=108.12837499999999): punch is hidden from the screen — left the board. ### Scene 2: [Multiply the Shrinking Fractions](https://academa.ai/lectures/the-birthday-paradox?t=109.17004166666666) Span: 01:49.17–04:32.904 (109.17004166666666s–272.9036666666666s). #### Objects - assumption: a Panel that says "Use 365 equally likely birthdays, ignore leap day, and treat different people's birthdays as independent." - crossing: a Point \[green\] labelled "(23, 0.507)" drawn in plot (location=(23.0, 0.5073)) - curve: a FunctionPlot \[blue\] drawn in plot (function=\, x\_range=(1.0, 60.0)) - group: a VariableNumber (initial\_value=10.0, format\_spec='.0f') - half: a Line \[gray\] labelled "0.5" drawn in plot (start=(1.0, 0.5), end=(60.0, 0.5), dashed=True) - heading\_curve: a Heading that says "Where the Chance Crosses One Half" - heading\_model: a Heading that says "The Probability Model" - heading\_product: a Heading that says "Ask for No Match First" - plot: an Axes (x\_range=(1.0, 60.0), x\_ticks\_every=10.0, y\_ticks\_every=0.25) - probe: a PlotPoint \[yellow\] labelled "10" drawn in plot (target='curve', x=\) - readout: a Table \[text\] that says "People Chance of a match 10 11.7% 22 47.6% 23 50.7% 30 70.6% 50 97.0%" (rows=(('People', 'Chance of a match'), ('10', '11.7%'), ('22', '47.6…, header=True) - work: a Derivation \[text\] that says "$Q(n) = P(upright("all n birthdays differ")) \\ Q(2) = frac(364, 365) \\ Q(3) = frac(364, 365) dot.op frac(363, 365) \\ Q(n) = frac(364, 365) dot.op frac(363, 365) dots.c frac(366 - n, 365) \\ Q(23) approx 0.4927 \\ P(upright("a match")) = 1 - Q…$" #### Beats ##### [01:49.17](https://academa.ai/lectures/the-birthday-paradox?t=109.17004166666666) Narration: To keep the calculation clean, suppose there are three hundred sixty five equally likely birthdays, ignore leap day, and treat people's birthdays as independent. Real birthdays are not perfectly uniform, but this model is close enough to expose the main effect. Board: Empty. Actions: - [01:49.17](https://academa.ai/lectures/the-birthday-paradox?t=109.17004166666666): heading\_model is shown on the screen, written out. - [02:2.22](https://academa.ai/lectures/the-birthday-paradox?t=122.22004166666666): assumption is shown on the screen, written out. - [02:5.099](https://academa.ai/lectures/the-birthday-paradox?t=125.09854166666666): assumption is hidden from the screen — left the board. - [02:5.099](https://academa.ai/lectures/the-birthday-paradox?t=125.09854166666666): heading\_model is hidden from the screen — left the board. ##### [02:6.299](https://academa.ai/lectures/the-birthday-paradox?t=126.29854166666667) Narration: Counting every possible kind of match directly is messy. Two people might match, three might match, or two separate pairs might match. So turn the question around. What is the chance that every birthday is different? Board: Unchanged from the preceding beat in this scene. Actions: - [02:6.299](https://academa.ai/lectures/the-birthday-paradox?t=126.29854166666667): heading\_product is shown on the screen, written out. - [02:16.782](https://academa.ai/lectures/the-birthday-paradox?t=136.78204166666666): work is shown on the screen, written out. ##### [02:19.542](https://academa.ai/lectures/the-birthday-paradox?t=139.54154166666666) Narration: The first person can have any birthday. For the second person to avoid a match, only one day is forbidden. That leaves three hundred sixty four safe days out of three hundred sixty five. Board: heading\_product — a Heading that says "Ask for No Match First" Actions: - [02:22.874](https://academa.ai/lectures/the-birthday-paradox?t=142.87404166666664): work is shown on the screen, written out. ##### [02:31.938](https://academa.ai/lectures/the-birthday-paradox?t=151.93754166666668) Narration: Bring in a third person. Now two occupied birthdays must be avoided, so only three hundred sixty three days are safe. Both conditions must hold, which is why the fractions multiply. Board: Unchanged from the preceding beat in this scene. Actions: - [02:32.982](https://academa.ai/lectures/the-birthday-paradox?t=152.98204166666665): work is shown on the screen, written out. - [02:37.754](https://academa.ai/lectures/the-birthday-paradox?t=157.75404166666667): work (the "frac(363, 365)" part) is emphasized. - [02:44.453](https://academa.ai/lectures/the-birthday-paradox?t=164.45304166666665): work (the "frac(363, 365)" part) is no longer emphasized. ##### [02:45.053](https://academa.ai/lectures/the-birthday-paradox?t=165.05304166666667) Narration: Each new person contributes another fraction, with one fewer safe day in the numerator. For n people, multiply from three hundred sixty four over three hundred sixty five down to three hundred sixty six minus n over three hundred sixty five. Board: Unchanged from the preceding beat in this scene. Actions: - [02:50.788](https://academa.ai/lectures/the-birthday-paradox?t=170.78804166666666): work is shown on the screen, written out. ##### [03:0.247](https://academa.ai/lectures/the-birthday-paradox?t=180.24654166666664) Narration: At twenty three people, that product is about zero point four nine two seven. So the chance that everyone differs is just under one half. Subtract from one, and the chance of at least one shared birthday is about zero point five zero seven three, just over one half. Board: Unchanged from the preceding beat in this scene. Actions: - [03:2.185](https://academa.ai/lectures/the-birthday-paradox?t=182.18504166666665): work is shown on the screen, written out. - [03:8.931](https://academa.ai/lectures/the-birthday-paradox?t=188.93104166666666): work is shown on the screen, written out. - [03:15.92](https://academa.ai/lectures/the-birthday-paradox?t=195.92004166666663): A box is drawn around work. - [03:17.151](https://academa.ai/lectures/the-birthday-paradox?t=197.15104166666666): heading\_product is hidden from the screen — left the board. - [03:17.151](https://academa.ai/lectures/the-birthday-paradox?t=197.15104166666666): work is hidden from the screen — left the board. - [03:17.151](https://academa.ai/lectures/the-birthday-paradox?t=197.15104166666666): plot is shown on the screen, written out. ##### [03:18.351](https://academa.ai/lectures/the-birthday-paradox?t=198.35104166666667) Narration: Now view every group size on one graph. The horizontal axis is the number of people, and the vertical axis is the chance that some pair shares a birthday. The dashed line marks one half. Board: plot — an Axes (x\_range=(1.0, 60.0), x\_ticks\_every=10.0, y\_ticks\_every=0.25) Actions: - [03:18.351](https://academa.ai/lectures/the-birthday-paradox?t=198.35104166666667): plot moves to a new place on the board. - [03:18.351](https://academa.ai/lectures/the-birthday-paradox?t=198.35104166666667): heading\_curve is shown on the screen, written out. - [03:20.742](https://academa.ai/lectures/the-birthday-paradox?t=200.74204166666667): curve is shown on the screen, drawn. - [03:23.424](https://academa.ai/lectures/the-birthday-paradox?t=203.42404166666665): readout is shown on the screen, written out. - [03:30.623](https://academa.ai/lectures/the-birthday-paradox?t=210.62304166666667): half is shown on the screen, written out. ##### [03:32.036](https://academa.ai/lectures/the-birthday-paradox?t=212.03554166666666) Narration: With ten people, the chance is only about eleven point seven percent. That feels unsurprising, and it gives us a low starting point on the curve. Board: plot — an Axes (x\_range=(1.0, 60.0), x\_ticks\_every=10.0, y\_ticks\_every=0.25); heading\_curve — a Heading that says "Where the Chance Crosses One Half"; curve — a FunctionPlot \[blue\] drawn in plot (function=\, x\_range=(1.0, 60.0)); half — a Line \[gray\] labelled "0.5" drawn in plot (start=(1.0, 0.5), end=(60.0, 0.5), dashed=True) Actions: - [03:32.627](https://academa.ai/lectures/the-birthday-paradox?t=212.62704166666666): probe is shown on the screen, written out. - [03:34.543](https://academa.ai/lectures/the-birthday-paradox?t=214.54304166666665): readout is shown on the screen, written out. ##### [03:41.436](https://academa.ai/lectures/the-birthday-paradox?t=221.43554166666667) Narration: Move to twenty two people. The chance has climbed to about forty seven point six percent, still below the dashed half line. Board: plot — an Axes (x\_range=(1.0, 60.0), x\_ticks\_every=10.0, y\_ticks\_every=0.25); heading\_curve — a Heading that says "Where the Chance Crosses One Half"; curve — a FunctionPlot \[blue\] drawn in plot (function=\, x\_range=(1.0, 60.0)); half — a Line \[gray\] labelled "0.5" drawn in plot (start=(1.0, 0.5), end=(60.0, 0.5), dashed=True); probe — a PlotPoint \[yellow\] labelled "10" drawn in plot (target='curve', x=\) Actions: - [03:42.202](https://academa.ai/lectures/the-birthday-paradox?t=222.20204166666667): probe is redrawn as the numbers it depends on change. - [03:42.202](https://academa.ai/lectures/the-birthday-paradox?t=222.20204166666667): group ticks to 22.0. - [03:45.116](https://academa.ai/lectures/the-birthday-paradox?t=225.11604166666666): readout is shown on the screen, written out. ##### [03:49.652](https://academa.ai/lectures/the-birthday-paradox?t=229.65154166666667) Narration: Add just one more person. That newcomer does not create one new chance to match. They create twenty two new pair comparisons. At twenty three, the probability reaches fifty point seven percent and crosses the line. Board: Unchanged from the preceding beat in this scene. Actions: - [03:50.545](https://academa.ai/lectures/the-birthday-paradox?t=230.54504166666666): probe is redrawn as the numbers it depends on change. - [03:50.545](https://academa.ai/lectures/the-birthday-paradox?t=230.54504166666666): group ticks to 23.0. - [03:59.868](https://academa.ai/lectures/the-birthday-paradox?t=239.86804166666664): readout is shown on the screen, written out. - [04:1.285](https://academa.ai/lectures/the-birthday-paradox?t=241.28504166666667): crossing is shown on the screen, written out. - [04:1.807](https://academa.ai/lectures/the-birthday-paradox?t=241.80704166666666): crossing is indicated — a transient flash. ##### [04:3.023](https://academa.ai/lectures/the-birthday-paradox?t=243.02254166666665) Narration: The rise continues quickly. At thirty people the chance is about seventy one percent. At fifty people it is about ninety seven percent. The curve accelerates because every new arrival is compared with everyone already present. Board: plot — an Axes (x\_range=(1.0, 60.0), x\_ticks\_every=10.0, y\_ticks\_every=0.25); heading\_curve — a Heading that says "Where the Chance Crosses One Half"; curve — a FunctionPlot \[blue\] drawn in plot (function=\, x\_range=(1.0, 60.0)); half — a Line \[gray\] labelled "0.5" drawn in plot (start=(1.0, 0.5), end=(60.0, 0.5), dashed=True); probe — a PlotPoint \[yellow\] labelled "10" drawn in plot (target='curve', x=\); crossing — a Point \[green\] labelled "(23, 0.507)" drawn in plot (location=(23.0, 0.5073)) Actions: - [04:5.379](https://academa.ai/lectures/the-birthday-paradox?t=245.37904166666667): probe is redrawn as the numbers it depends on change. - [04:5.379](https://academa.ai/lectures/the-birthday-paradox?t=245.37904166666667): group ticks to 30.0. - [04:6.83](https://academa.ai/lectures/the-birthday-paradox?t=246.83004166666666): readout is shown on the screen, written out. - [04:8.653](https://academa.ai/lectures/the-birthday-paradox?t=248.65304166666667): probe is redrawn as the numbers it depends on change. - [04:8.653](https://academa.ai/lectures/the-birthday-paradox?t=248.65304166666667): group ticks to 50.0. - [04:9.849](https://academa.ai/lectures/the-birthday-paradox?t=249.84904166666666): readout is shown on the screen, written out. ##### [04:17.136](https://academa.ai/lectures/the-birthday-paradox?t=257.13604166666664) Narration: That is why twenty three is enough. It is not large compared with three hundred sixty five days, but it creates two hundred fifty three pairs. The product calculation and the graph now say the same thing: a shared birthday is more likely than not. Board: Unchanged from the preceding beat in this scene. Actions: - [04:18.088](https://academa.ai/lectures/the-birthday-paradox?t=258.08804166666664): crossing is indicated — a transient flash. - [04:30.209](https://academa.ai/lectures/the-birthday-paradox?t=270.2090416666666): readout is indicated — a transient flash. - [04:31.862](https://academa.ai/lectures/the-birthday-paradox?t=271.86199999999997): heading\_curve is hidden from the screen — left the board. - [04:31.862](https://academa.ai/lectures/the-birthday-paradox?t=271.86199999999997): plot is hidden from the screen — left the board. - [04:31.862](https://academa.ai/lectures/the-birthday-paradox?t=271.86199999999997): curve is hidden from the screen — plot left the board. - [04:31.862](https://academa.ai/lectures/the-birthday-paradox?t=271.86199999999997): half is hidden from the screen — plot left the board. - [04:31.862](https://academa.ai/lectures/the-birthday-paradox?t=271.86199999999997): probe is hidden from the screen — plot left the board. - [04:31.862](https://academa.ai/lectures/the-birthday-paradox?t=271.86199999999997): crossing is hidden from the screen — plot left the board. - [04:31.862](https://academa.ai/lectures/the-birthday-paradox?t=271.86199999999997): readout is hidden from the screen — left the board. ### Scene 3: [Someone Who Matches Your Birthday](https://academa.ai/lectures/the-birthday-paradox?t=272.9036666666666) Span: 04:32.904–06:54.903 (272.9036666666666s–414.90341666666654s). #### Objects - answer\_others: a Math \[text\] that says "$n = 253 thin upright("other people")$" - answer\_total: a Math \[text\] that says "$N = 254 thin upright("people including me")$" - both: an Axes (x\_range=(1.0, 300.0), x\_ticks\_every=50.0, y\_ticks\_every=0.25) - calculation: a Derivation \[text\] that says "$P(upright("no one matches me")) &= (frac(364, 365))^n \\ (frac(364, 365))^n &\< frac(1, 2) \\ n ln frac(364, 365) &\< ln frac(1, 2) \\ n &\> frac(ln(1 slash 2), ln(364 slash 365)) \\ n &\> 252.6$" - curve\_any: a FunctionPlot \[blue\] labelled "upright("any pair")" drawn in both (function=\, x\_range=(1.0, 300.0)) - curve\_mine: a FunctionPlot \[green\] labelled "upright("matches me")" drawn in both (function=\, x\_range=(1.0, 300.0)) - half: a Line \[gray\] drawn in both (start=(1.0, 0.5), end=(300.0, 0.5), dashed=True) - heading\_calc: a Heading that says "How Many People Match One Fixed Birthday" - heading\_compare: a Heading that says "Two Questions That Sound Similar" - heading\_curves: a Heading that says "The Two Questions on One Graph" - left\_label: a Tex \[text\] that says "Everyone compared with me" - mark\_any: a Point \[blue\] labelled "23" drawn in both (location=(23.0, 0.5073)) - mark\_mine: a Point \[green\] labelled "254" drawn in both (location=(254.0, 0.5005)) - mesh: a Figure (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), aspect=(1.0, 1.0)) - mesh\_dots: a Point \[blue\] drawn in mesh (location=(6.123233995736766e-17, 1.0)) - mesh\_dots\_2: a Point \[blue\] drawn in mesh (location=(0.7818314824680298, 0.6234898018587335)) - mesh\_dots\_3: a Point \[blue\] drawn in mesh (location=(0.9749279121818236, -0.2225209339563144)) - mesh\_dots\_4: a Point \[blue\] drawn in mesh (location=(0.4338837391175582, -0.9009688679024191)) - mesh\_dots\_5: a Point \[blue\] drawn in mesh (location=(-0.43388373911755806, -0.9009688679024191)) - mesh\_dots\_6: a Point \[blue\] drawn in mesh (location=(-0.9749279121818236, -0.2225209339563145)) - mesh\_dots\_7: a Point \[blue\] drawn in mesh (location=(-0.7818314824680299, 0.6234898018587334)) - mesh\_lines: a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(0.7818314824680298, 0.6234898018587335)) - mesh\_lines\_10: a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(-0.9749279121818236, -0.2225209339563145)) - mesh\_lines\_11: a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(-0.7818314824680299, 0.6234898018587334)) - mesh\_lines\_12: a Line \[yellow\] drawn in mesh (start=(0.9749279121818236, -0.2225209339563144), end=(0.4338837391175582, -0.9009688679024191)) - mesh\_lines\_13: a Line \[yellow\] drawn in mesh (start=(0.9749279121818236, -0.2225209339563144), end=(-0.43388373911755806, -0.9009688679024191)) - mesh\_lines\_14: a Line \[yellow\] drawn in mesh (start=(0.9749279121818236, -0.2225209339563144), end=(-0.9749279121818236, -0.2225209339563145)) - mesh\_lines\_15: a Line \[yellow\] drawn in mesh (start=(0.9749279121818236, -0.2225209339563144), end=(-0.7818314824680299, 0.6234898018587334)) - mesh\_lines\_16: a Line \[yellow\] drawn in mesh (start=(0.4338837391175582, -0.9009688679024191), end=(-0.43388373911755806, -0.9009688679024191)) - mesh\_lines\_17: a Line \[yellow\] drawn in mesh (start=(0.4338837391175582, -0.9009688679024191), end=(-0.9749279121818236, -0.2225209339563145)) - mesh\_lines\_18: a Line \[yellow\] drawn in mesh (start=(0.4338837391175582, -0.9009688679024191), end=(-0.7818314824680299, 0.6234898018587334)) - mesh\_lines\_19: a Line \[yellow\] drawn in mesh (start=(-0.43388373911755806, -0.9009688679024191), end=(-0.9749279121818236, -0.2225209339563145)) - mesh\_lines\_2: a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(0.9749279121818236, -0.2225209339563144)) - mesh\_lines\_20: a Line \[yellow\] drawn in mesh (start=(-0.43388373911755806, -0.9009688679024191), end=(-0.7818314824680299, 0.6234898018587334)) - mesh\_lines\_21: a Line \[yellow\] drawn in mesh (start=(-0.9749279121818236, -0.2225209339563145), end=(-0.7818314824680299, 0.6234898018587334)) - mesh\_lines\_3: a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(0.4338837391175582, -0.9009688679024191)) - mesh\_lines\_4: a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(-0.43388373911755806, -0.9009688679024191)) - mesh\_lines\_5: a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(-0.9749279121818236, -0.2225209339563145)) - mesh\_lines\_6: a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(-0.7818314824680299, 0.6234898018587334)) - mesh\_lines\_7: a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(0.9749279121818236, -0.2225209339563144)) - mesh\_lines\_8: a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(0.4338837391175582, -0.9009688679024191)) - mesh\_lines\_9: a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(-0.43388373911755806, -0.9009688679024191)) - narrator: a Point \[red\] labelled "upright("me")" drawn in star (location=(6.123233995736766e-17, 1.0)) - note\_any: a Tex \[text\] that says "Any pair: above one half at 23 people." - note\_mine: a Tex \[text\] that says "My birthday: above one half at 254 people." - right\_label: a Tex \[text\] that says "Everyone compared with everyone" - star: a Figure (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), aspect=(1.0, 1.0)) - star\_lines: a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.7818314824680298, 0.6234898018587335)) - star\_lines\_2: a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.9749279121818236, -0.2225209339563144)) - star\_lines\_3: a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.4338837391175582, -0.9009688679024191)) - star\_lines\_4: a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.43388373911755806, -0.9009688679024191)) - star\_lines\_5: a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.9749279121818236, -0.2225209339563145)) - star\_lines\_6: a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.7818314824680299, 0.6234898018587334)) - star\_others: a Point \[blue\] drawn in star (location=(0.7818314824680298, 0.6234898018587335)) - star\_others\_2: a Point \[blue\] drawn in star (location=(0.9749279121818236, -0.2225209339563144)) - star\_others\_3: a Point \[blue\] drawn in star (location=(0.4338837391175582, -0.9009688679024191)) - star\_others\_4: a Point \[blue\] drawn in star (location=(-0.43388373911755806, -0.9009688679024191)) - star\_others\_5: a Point \[blue\] drawn in star (location=(-0.9749279121818236, -0.2225209339563145)) - star\_others\_6: a Point \[blue\] drawn in star (location=(-0.7818314824680299, 0.6234898018587334)) #### Beats ##### [04:32.904](https://academa.ai/lectures/the-birthday-paradox?t=272.9036666666666) Narration: Now ask a question that sounds almost identical. How many people are needed before someone probably shares my birthday specifically? The word my changes the comparison structure completely. Board: Empty. Actions: - [04:32.904](https://academa.ai/lectures/the-birthday-paradox?t=272.9036666666666): heading\_compare is shown on the screen, written out. - [04:32.904](https://academa.ai/lectures/the-birthday-paradox?t=272.9036666666666): star is shown on the screen, written out. - [04:36.7](https://academa.ai/lectures/the-birthday-paradox?t=276.6996666666666): star\_others is shown on the screen, written out. - [04:36.8](https://academa.ai/lectures/the-birthday-paradox?t=276.7996666666666): star\_others\_2 is shown on the screen, written out. - [04:36.9](https://academa.ai/lectures/the-birthday-paradox?t=276.8996666666666): star\_others\_3 is shown on the screen, written out. - [04:37](https://academa.ai/lectures/the-birthday-paradox?t=276.9996666666666): star\_others\_4 is shown on the screen, written out. - [04:37.1](https://academa.ai/lectures/the-birthday-paradox?t=277.0996666666666): star\_others\_5 is shown on the screen, written out. - [04:37.2](https://academa.ai/lectures/the-birthday-paradox?t=277.1996666666666): star\_others\_6 is shown on the screen, written out. - [04:38.872](https://academa.ai/lectures/the-birthday-paradox?t=278.8716666666666): narrator is shown on the screen, written out. ##### [04:44.464](https://academa.ai/lectures/the-birthday-paradox?t=284.4636666666666) Narration: With seven people in the room, only six comparisons matter for my birthday. Each yellow line joins me to one other person. Birthdays shared entirely among the blue people do not answer this question. Board: star — a Figure (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), aspect=(1.0, 1.0)); heading\_compare — a Heading that says "Two Questions That Sound Similar"; narrator — a Point \[red\] labelled "upright("me")" drawn in star (location=(6.123233995736766e-17, 1.0)); star\_others — a Point \[blue\] drawn in star (location=(0.7818314824680298, 0.6234898018587335)); star\_others\_2 — a Point \[blue\] drawn in star (location=(0.9749279121818236, -0.2225209339563144)); star\_others\_3 — a Point \[blue\] drawn in star (location=(0.4338837391175582, -0.9009688679024191)); star\_others\_4 — a Point \[blue\] drawn in star (location=(-0.43388373911755806, -0.9009688679024191)); star\_others\_5 — a Point \[blue\] drawn in star (location=(-0.9749279121818236, -0.2225209339563145)); star\_others\_6 — a Point \[blue\] drawn in star (location=(-0.7818314824680299, 0.6234898018587334)) Actions: - [04:46.856](https://academa.ai/lectures/the-birthday-paradox?t=286.8556666666666): star moves to a new place on the board. - [04:46.856](https://academa.ai/lectures/the-birthday-paradox?t=286.8556666666666): left\_label is shown on the screen, written out. - [04:50.304](https://academa.ai/lectures/the-birthday-paradox?t=290.3036666666666): star\_lines is shown on the screen, written out. - [04:50.424](https://academa.ai/lectures/the-birthday-paradox?t=290.4236666666666): star\_lines\_2 is shown on the screen, written out. - [04:50.544](https://academa.ai/lectures/the-birthday-paradox?t=290.5436666666666): star\_lines\_3 is shown on the screen, written out. - [04:50.664](https://academa.ai/lectures/the-birthday-paradox?t=290.6636666666666): star\_lines\_4 is shown on the screen, written out. - [04:50.784](https://academa.ai/lectures/the-birthday-paradox?t=290.7836666666666): star\_lines\_5 is shown on the screen, written out. - [04:50.904](https://academa.ai/lectures/the-birthday-paradox?t=290.9036666666666): star\_lines\_6 is shown on the screen, written out. ##### [04:57.881](https://academa.ai/lectures/the-birthday-paradox?t=297.8811666666666) Narration: For the original birthday problem, every pair matters. The same seven people create twenty one comparisons, because every person is checked against every other person. Board: left\_label — a Tex \[text\] that says "Everyone compared with me"; star — a Figure (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), aspect=(1.0, 1.0)); heading\_compare — a Heading that says "Two Questions That Sound Similar"; narrator — a Point \[red\] labelled "upright("me")" drawn in star (location=(6.123233995736766e-17, 1.0)); star\_others — a Point \[blue\] drawn in star (location=(0.7818314824680298, 0.6234898018587335)); star\_others\_2 — a Point \[blue\] drawn in star (location=(0.9749279121818236, -0.2225209339563144)); star\_others\_3 — a Point \[blue\] drawn in star (location=(0.4338837391175582, -0.9009688679024191)); star\_others\_4 — a Point \[blue\] drawn in star (location=(-0.43388373911755806, -0.9009688679024191)); star\_others\_5 — a Point \[blue\] drawn in star (location=(-0.9749279121818236, -0.2225209339563145)); star\_others\_6 — a Point \[blue\] drawn in star (location=(-0.7818314824680299, 0.6234898018587334)); star\_lines — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.7818314824680298, 0.6234898018587335)); star\_lines\_2 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.9749279121818236, -0.2225209339563144)); star\_lines\_3 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.4338837391175582, -0.9009688679024191)); star\_lines\_4 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.43388373911755806, -0.9009688679024191)); star\_lines\_5 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.9749279121818236, -0.2225209339563145)); star\_lines\_6 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.7818314824680299, 0.6234898018587334)) Actions: - [04:57.881](https://academa.ai/lectures/the-birthday-paradox?t=297.8811666666666): mesh is shown on the screen, written out. - [05:0.006](https://academa.ai/lectures/the-birthday-paradox?t=300.00566666666657): right\_label is shown on the screen, written out. - [05:2.235](https://academa.ai/lectures/the-birthday-paradox?t=302.2346666666666): mesh\_dots is shown on the screen, written out. - [05:2.315](https://academa.ai/lectures/the-birthday-paradox?t=302.3146666666666): mesh\_dots\_2 is shown on the screen, written out. - [05:2.395](https://academa.ai/lectures/the-birthday-paradox?t=302.3946666666666): mesh\_dots\_3 is shown on the screen, written out. - [05:2.475](https://academa.ai/lectures/the-birthday-paradox?t=302.4746666666666): mesh\_dots\_4 is shown on the screen, written out. - [05:2.555](https://academa.ai/lectures/the-birthday-paradox?t=302.5546666666666): mesh\_dots\_5 is shown on the screen, written out. - [05:2.635](https://academa.ai/lectures/the-birthday-paradox?t=302.6346666666666): mesh\_dots\_6 is shown on the screen, written out. - [05:2.715](https://academa.ai/lectures/the-birthday-paradox?t=302.7146666666666): mesh\_dots\_7 is shown on the screen, written out. - [05:3.187](https://academa.ai/lectures/the-birthday-paradox?t=303.1866666666666): mesh\_lines is shown on the screen, written out. - [05:3.227](https://academa.ai/lectures/the-birthday-paradox?t=303.2266666666666): mesh\_lines\_2 is shown on the screen, written out. - [05:3.267](https://academa.ai/lectures/the-birthday-paradox?t=303.2666666666666): mesh\_lines\_3 is shown on the screen, written out. - [05:3.307](https://academa.ai/lectures/the-birthday-paradox?t=303.3066666666666): mesh\_lines\_4 is shown on the screen, written out. - [05:3.347](https://academa.ai/lectures/the-birthday-paradox?t=303.3466666666666): mesh\_lines\_5 is shown on the screen, written out. - [05:3.387](https://academa.ai/lectures/the-birthday-paradox?t=303.3866666666666): mesh\_lines\_6 is shown on the screen, written out. - [05:3.427](https://academa.ai/lectures/the-birthday-paradox?t=303.4266666666666): mesh\_lines\_7 is shown on the screen, written out. - [05:3.467](https://academa.ai/lectures/the-birthday-paradox?t=303.4666666666666): mesh\_lines\_8 is shown on the screen, written out. - [05:3.507](https://academa.ai/lectures/the-birthday-paradox?t=303.5066666666666): mesh\_lines\_9 is shown on the screen, written out. - [05:3.547](https://academa.ai/lectures/the-birthday-paradox?t=303.5466666666666): mesh\_lines\_10 is shown on the screen, written out. - [05:3.587](https://academa.ai/lectures/the-birthday-paradox?t=303.5866666666666): mesh\_lines\_11 is shown on the screen, written out. - [05:3.627](https://academa.ai/lectures/the-birthday-paradox?t=303.6266666666666): mesh\_lines\_12 is shown on the screen, written out. - [05:3.667](https://academa.ai/lectures/the-birthday-paradox?t=303.6666666666666): mesh\_lines\_13 is shown on the screen, written out. - [05:3.707](https://academa.ai/lectures/the-birthday-paradox?t=303.7066666666666): mesh\_lines\_14 is shown on the screen, written out. - [05:3.747](https://academa.ai/lectures/the-birthday-paradox?t=303.7466666666666): mesh\_lines\_15 is shown on the screen, written out. - [05:3.787](https://academa.ai/lectures/the-birthday-paradox?t=303.7866666666666): mesh\_lines\_16 is shown on the screen, written out. - [05:3.827](https://academa.ai/lectures/the-birthday-paradox?t=303.8266666666666): mesh\_lines\_17 is shown on the screen, written out. - [05:3.867](https://academa.ai/lectures/the-birthday-paradox?t=303.8666666666666): mesh\_lines\_18 is shown on the screen, written out. - [05:3.907](https://academa.ai/lectures/the-birthday-paradox?t=303.9066666666666): mesh\_lines\_19 is shown on the screen, written out. - [05:3.947](https://academa.ai/lectures/the-birthday-paradox?t=303.9466666666666): mesh\_lines\_20 is shown on the screen, written out. - [05:3.987](https://academa.ai/lectures/the-birthday-paradox?t=303.9866666666666): mesh\_lines\_21 is shown on the screen, written out. ##### [05:8.663](https://academa.ai/lectures/the-birthday-paradox?t=308.6631666666666) Narration: The left count grows one at a time as people arrive. The right count grows by the whole number already in the room. That is why the original any-pair question reaches one half so much sooner. Board: left\_label — a Tex \[text\] that says "Everyone compared with me"; star — a Figure (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), aspect=(1.0, 1.0)); right\_label — a Tex \[text\] that says "Everyone compared with everyone"; mesh — a Figure (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), aspect=(1.0, 1.0)); heading\_compare — a Heading that says "Two Questions That Sound Similar"; narrator — a Point \[red\] labelled "upright("me")" drawn in star (location=(6.123233995736766e-17, 1.0)); star\_others — a Point \[blue\] drawn in star (location=(0.7818314824680298, 0.6234898018587335)); star\_others\_2 — a Point \[blue\] drawn in star (location=(0.9749279121818236, -0.2225209339563144)); star\_others\_3 — a Point \[blue\] drawn in star (location=(0.4338837391175582, -0.9009688679024191)); star\_others\_4 — a Point \[blue\] drawn in star (location=(-0.43388373911755806, -0.9009688679024191)); star\_others\_5 — a Point \[blue\] drawn in star (location=(-0.9749279121818236, -0.2225209339563145)); star\_others\_6 — a Point \[blue\] drawn in star (location=(-0.7818314824680299, 0.6234898018587334)); star\_lines — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.7818314824680298, 0.6234898018587335)); star\_lines\_2 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.9749279121818236, -0.2225209339563144)); star\_lines\_3 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(0.4338837391175582, -0.9009688679024191)); star\_lines\_4 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.43388373911755806, -0.9009688679024191)); star\_lines\_5 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.9749279121818236, -0.2225209339563145)); star\_lines\_6 — a Line \[yellow\] drawn in star (start=(6.123233995736766e-17, 1.0), end=(-0.7818314824680299, 0.6234898018587334)); mesh\_dots — a Point \[blue\] drawn in mesh (location=(6.123233995736766e-17, 1.0)); mesh\_dots\_2 — a Point \[blue\] drawn in mesh (location=(0.7818314824680298, 0.6234898018587335)); mesh\_dots\_3 — a Point \[blue\] drawn in mesh (location=(0.9749279121818236, -0.2225209339563144)); mesh\_dots\_4 — a Point \[blue\] drawn in mesh (location=(0.4338837391175582, -0.9009688679024191)); mesh\_dots\_5 — a Point \[blue\] drawn in mesh (location=(-0.43388373911755806, -0.9009688679024191)); mesh\_dots\_6 — a Point \[blue\] drawn in mesh (location=(-0.9749279121818236, -0.2225209339563145)); mesh\_dots\_7 — a Point \[blue\] drawn in mesh (location=(-0.7818314824680299, 0.6234898018587334)); mesh\_lines — a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(0.7818314824680298, 0.6234898018587335)); mesh\_lines\_2 — a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(0.9749279121818236, -0.2225209339563144)); mesh\_lines\_3 — a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(0.4338837391175582, -0.9009688679024191)); mesh\_lines\_4 — a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(-0.43388373911755806, -0.9009688679024191)); mesh\_lines\_5 — a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(-0.9749279121818236, -0.2225209339563145)); mesh\_lines\_6 — a Line \[yellow\] drawn in mesh (start=(6.123233995736766e-17, 1.0), end=(-0.7818314824680299, 0.6234898018587334)); mesh\_lines\_7 — a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(0.9749279121818236, -0.2225209339563144)); mesh\_lines\_8 — a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(0.4338837391175582, -0.9009688679024191)); mesh\_lines\_9 — a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(-0.43388373911755806, -0.9009688679024191)); mesh\_lines\_10 — a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(-0.9749279121818236, -0.2225209339563145)); mesh\_lines\_11 — a Line \[yellow\] drawn in mesh (start=(0.7818314824680298, 0.6234898018587335), end=(-0.7818314824680299, 0.6234898018587334)); mesh\_lines\_12 — a Line \[yellow\] drawn in mesh (start=(0.9749279121818236, -0.2225209339563144), end=(0.4338837391175582, -0.9009688679024191)); mesh\_lines\_13 — a Line \[yellow\] drawn in mesh (start=(0.9749279121818236, -0.2225209339563144), end=(-0.43388373911755806, -0.9009688679024191)); mesh\_lines\_14 — a Line \[yellow\] drawn in mesh (start=(0.9749279121818236, -0.2225209339563144), end=(-0.9749279121818236, -0.2225209339563145)); mesh\_lines\_15 — a Line \[yellow\] drawn in mesh (start=(0.9749279121818236, -0.2225209339563144), end=(-0.7818314824680299, 0.6234898018587334)); mesh\_lines\_16 — a Line \[yellow\] drawn in mesh (start=(0.4338837391175582, -0.9009688679024191), end=(-0.43388373911755806, -0.9009688679024191)); mesh\_lines\_17 — a Line \[yellow\] drawn in mesh (start=(0.4338837391175582, -0.9009688679024191), end=(-0.9749279121818236, -0.2225209339563145)); mesh\_lines\_18 — a Line \[yellow\] drawn in mesh (start=(0.4338837391175582, -0.9009688679024191), end=(-0.7818314824680299, 0.6234898018587334)); mesh\_lines\_19 — a Line \[yellow\] drawn in mesh (start=(-0.43388373911755806, -0.9009688679024191), end=(-0.9749279121818236, -0.2225209339563145)); mesh\_lines\_20 — a Line \[yellow\] drawn in mesh (start=(-0.43388373911755806, -0.9009688679024191), end=(-0.7818314824680299, 0.6234898018587334)); mesh\_lines\_21 — a Line \[yellow\] drawn in mesh (start=(-0.9749279121818236, -0.2225209339563145), end=(-0.7818314824680299, 0.6234898018587334)) Actions: - [05:10.01](https://academa.ai/lectures/the-birthday-paradox?t=310.0096666666666): left\_label is indicated — a transient flash. - [05:13.493](https://academa.ai/lectures/the-birthday-paradox?t=313.4926666666666): right\_label is indicated — a transient flash. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): heading\_compare is hidden from the screen — left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): left\_label is hidden from the screen — left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh is hidden from the screen — left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_dots is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_dots\_2 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_dots\_3 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_dots\_4 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_dots\_5 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_dots\_6 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_dots\_7 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_2 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_3 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_4 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_5 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_6 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_7 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_8 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_9 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_10 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_11 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_12 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_13 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_14 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_15 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_16 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_17 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_18 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_19 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_20 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): mesh\_lines\_21 is hidden from the screen — mesh left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): right\_label is hidden from the screen — left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star is hidden from the screen — left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): narrator is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_others is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_others\_2 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_others\_3 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_others\_4 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_others\_5 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_others\_6 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_lines is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_lines\_2 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_lines\_3 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_lines\_4 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_lines\_5 is hidden from the screen — star left the board. - [05:20.006](https://academa.ai/lectures/the-birthday-paradox?t=320.00566666666657): star\_lines\_6 is hidden from the screen — star left the board. ##### [05:21.206](https://academa.ai/lectures/the-birthday-paradox?t=321.2056666666666) Narration: Let n mean the number of other people, not counting me. One person fails to match my birthday with probability three hundred sixty four over three hundred sixty five. For nobody to match me, all n people must fail. Board: Empty. Actions: - [05:21.206](https://academa.ai/lectures/the-birthday-paradox?t=321.2056666666666): heading\_calc is shown on the screen, written out. - [05:33.478](https://academa.ai/lectures/the-birthday-paradox?t=333.4776666666666): calculation is shown on the screen, written out. ##### [05:35.889](https://academa.ai/lectures/the-birthday-paradox?t=335.8886666666666) Narration: A match becomes more likely than not when the no-match probability falls below one half. So we set the shrinking power below one half. Board: heading\_calc — a Heading that says "How Many People Match One Fixed Birthday" Actions: - [05:40.034](https://academa.ai/lectures/the-birthday-paradox?t=340.0336666666666): calculation is shown on the screen, written out. ##### [05:45.069](https://academa.ai/lectures/the-birthday-paradox?t=345.06866666666656) Narration: Take logarithms to bring n down from the exponent. Because the logarithm of three hundred sixty four over three hundred sixty five is negative, dividing by it reverses the inequality. Board: Unchanged from the preceding beat in this scene. Actions: - [05:45.719](https://academa.ai/lectures/the-birthday-paradox?t=345.7186666666666): calculation is shown on the screen, written out. - [05:53.73](https://academa.ai/lectures/the-birthday-paradox?t=353.7296666666666): calculation is shown on the screen, written out. ##### [05:56.803](https://academa.ai/lectures/the-birthday-paradox?t=356.8026666666666) Narration: The result is n greater than two hundred fifty two point six. Since n must be a whole number, we need two hundred fifty three other people. Including me, that is two hundred fifty four people in the room. Board: Unchanged from the preceding beat in this scene. Actions: - [05:57.291](https://academa.ai/lectures/the-birthday-paradox?t=357.2906666666666): calculation is shown on the screen, written out. - [06:3.34](https://academa.ai/lectures/the-birthday-paradox?t=363.3396666666666): answer\_others is shown on the screen, written out. - [06:5.778](https://academa.ai/lectures/the-birthday-paradox?t=365.77766666666656): answer\_total is shown on the screen, written out. - [06:9.103](https://academa.ai/lectures/the-birthday-paradox?t=369.1031666666666): answer\_others is hidden from the screen — left the board. - [06:9.103](https://academa.ai/lectures/the-birthday-paradox?t=369.1031666666666): answer\_total is hidden from the screen — left the board. - [06:9.103](https://academa.ai/lectures/the-birthday-paradox?t=369.1031666666666): calculation is hidden from the screen — left the board. - [06:9.103](https://academa.ai/lectures/the-birthday-paradox?t=369.1031666666666): heading\_calc is hidden from the screen — left the board. - [06:9.103](https://academa.ai/lectures/the-birthday-paradox?t=369.1031666666666): both is shown on the screen, written out. - [06:9.103](https://academa.ai/lectures/the-birthday-paradox?t=369.1031666666666): half is shown on the screen, written out. - [06:9.103](https://academa.ai/lectures/the-birthday-paradox?t=369.1031666666666): A box is drawn around answer\_total. ##### [06:10.303](https://academa.ai/lectures/the-birthday-paradox?t=370.3031666666666) Narration: Put both questions on one graph. The horizontal axis now counts everyone in the room, including me. The dashed line is again a probability of one half. Board: both — an Axes (x\_range=(1.0, 300.0), x\_ticks\_every=50.0, y\_ticks\_every=0.25); half — a Line \[gray\] drawn in both (start=(1.0, 0.5), end=(300.0, 0.5), dashed=True) Actions: - [06:10.303](https://academa.ai/lectures/the-birthday-paradox?t=370.3031666666666): heading\_curves is shown on the screen, written out. ##### [06:21.01](https://academa.ai/lectures/the-birthday-paradox?t=381.0101666666666) Narration: The blue curve asks whether any pair matches. It crosses one half at twenty three people because those people create two hundred fifty three pairs. Board: both — an Axes (x\_range=(1.0, 300.0), x\_ticks\_every=50.0, y\_ticks\_every=0.25); heading\_curves — a Heading that says "The Two Questions on One Graph"; half — a Line \[gray\] drawn in both (start=(1.0, 0.5), end=(300.0, 0.5), dashed=True) Actions: - [06:21.521](https://academa.ai/lectures/the-birthday-paradox?t=381.52066666666656): curve\_any is shown on the screen, drawn. - [06:25.353](https://academa.ai/lectures/the-birthday-paradox?t=385.35266666666655): mark\_any is shown on the screen, written out. - [06:28.661](https://academa.ai/lectures/the-birthday-paradox?t=388.66066666666654): note\_any is shown on the screen, written out. ##### [06:30.074](https://academa.ai/lectures/the-birthday-paradox?t=390.07366666666655) Narration: The green curve asks whether someone matches one fixed birthday. It rises far more slowly and crosses one half at two hundred fifty four people, meaning two hundred fifty three comparisons with me. Board: note\_any — a Tex \[text\] that says "Any pair: above one half at 23 people."; both — an Axes (x\_range=(1.0, 300.0), x\_ticks\_every=50.0, y\_ticks\_every=0.25); heading\_curves — a Heading that says "The Two Questions on One Graph"; half — a Line \[gray\] drawn in both (start=(1.0, 0.5), end=(300.0, 0.5), dashed=True); curve\_any — a FunctionPlot \[blue\] labelled "upright("any pair")" drawn in both (function=\, x\_range=(1.0, 300.0)); mark\_any — a Point \[blue\] labelled "23" drawn in both (location=(23.0, 0.5073)) Actions: - [06:30.597](https://academa.ai/lectures/the-birthday-paradox?t=390.5966666666666): curve\_mine is shown on the screen, drawn. - [06:36.773](https://academa.ai/lectures/the-birthday-paradox?t=396.77266666666657): mark\_mine is shown on the screen, written out. - [06:39.653](https://academa.ai/lectures/the-birthday-paradox?t=399.6526666666666): note\_mine is shown on the screen, written out. ##### [06:41.89](https://academa.ai/lectures/the-birthday-paradox?t=401.8896666666666) Narration: So the intuition that twenty three feels too small was not foolish. It was quietly answering the green question while the birthday paradox asks the blue one. Always ask which pairs actually count. Board: note\_any — a Tex \[text\] that says "Any pair: above one half at 23 people."; note\_mine — a Tex \[text\] that says "My birthday: above one half at 254 people."; both — an Axes (x\_range=(1.0, 300.0), x\_ticks\_every=50.0, y\_ticks\_every=0.25); heading\_curves — a Heading that says "The Two Questions on One Graph"; half — a Line \[gray\] drawn in both (start=(1.0, 0.5), end=(300.0, 0.5), dashed=True); curve\_any — a FunctionPlot \[blue\] labelled "upright("any pair")" drawn in both (function=\, x\_range=(1.0, 300.0)); mark\_any — a Point \[blue\] labelled "23" drawn in both (location=(23.0, 0.5073)); curve\_mine — a FunctionPlot \[green\] labelled "upright("matches me")" drawn in both (function=\, x\_range=(1.0, 300.0)); mark\_mine — a Point \[green\] labelled "254" drawn in both (location=(254.0, 0.5005)) Actions: - [06:47.706](https://academa.ai/lectures/the-birthday-paradox?t=407.7056666666666): curve\_mine is indicated — a transient flash. - [06:50.156](https://academa.ai/lectures/the-birthday-paradox?t=410.15566666666655): curve\_any is indicated — a transient flash. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): both is hidden from the screen — left the board. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): half is hidden from the screen — both left the board. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): curve\_any is hidden from the screen — both left the board. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): mark\_any is hidden from the screen — both left the board. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): curve\_mine is hidden from the screen — both left the board. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): mark\_mine is hidden from the screen — both left the board. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): heading\_curves is hidden from the screen — left the board. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): note\_any is hidden from the screen — left the board. - [06:53.862](https://academa.ai/lectures/the-birthday-paradox?t=413.8617499999999): note\_mine is hidden from the screen — left the board. ### Scene 4: [The Same Collision in a Hash Function](https://academa.ai/lectures/the-birthday-paradox?t=414.90341666666654) Span: 06:54.903–09:51.727 (414.90341666666654s–591.7273333333333s). #### Objects - bound: a Math \[text\] that says "$k approx 1.18 sqrt(N) thin upright("at 50%")$" - bound\_work: a Derivation \[text\] that says "$upright("pairs") &= frac(k(k - 1), 2) \\ P(upright("no collision")) &approx exp(-frac(k(k - 1), 2 N)) \\ P(upright("collision")) &approx 1 - exp(-frac(k(k - 1), 2 N))$" - buckets: a Figure (x\_range=(0.0, 16.0), y\_range=(-0.7, 1.5), aspect=(16.0, 2.2)) - cells: a Polygon \[gray\] drawn in buckets (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)), filled=False) - cells\_10: a Polygon \[gray\] drawn in buckets (vertices=((9.0, 0.0), (10.0, 0.0), (10.0, 1.0), (9.0, 1.0)), filled=False) - cells\_11: a Polygon \[gray\] drawn in buckets (vertices=((10.0, 0.0), (11.0, 0.0), (11.0, 1.0), (10.0, 1.0)), filled=False) - cells\_12: a Polygon \[gray\] drawn in buckets (vertices=((11.0, 0.0), (12.0, 0.0), (12.0, 1.0), (11.0, 1.0)), filled=False) - cells\_13: a Polygon \[gray\] drawn in buckets (vertices=((12.0, 0.0), (13.0, 0.0), (13.0, 1.0), (12.0, 1.0)), filled=False) - cells\_14: a Polygon \[gray\] drawn in buckets (vertices=((13.0, 0.0), (14.0, 0.0), (14.0, 1.0), (13.0, 1.0)), filled=False) - cells\_15: a Polygon \[gray\] drawn in buckets (vertices=((14.0, 0.0), (15.0, 0.0), (15.0, 1.0), (14.0, 1.0)), filled=False) - cells\_16: a Polygon \[gray\] drawn in buckets (vertices=((15.0, 0.0), (16.0, 0.0), (16.0, 1.0), (15.0, 1.0)), filled=False) - cells\_2: a Polygon \[gray\] drawn in buckets (vertices=((1.0, 0.0), (2.0, 0.0), (2.0, 1.0), (1.0, 1.0)), filled=False) - cells\_3: a Polygon \[gray\] drawn in buckets (vertices=((2.0, 0.0), (3.0, 0.0), (3.0, 1.0), (2.0, 1.0)), filled=False) - cells\_4: a Polygon \[gray\] drawn in buckets (vertices=((3.0, 0.0), (4.0, 0.0), (4.0, 1.0), (3.0, 1.0)), filled=False) - cells\_5: a Polygon \[gray\] drawn in buckets (vertices=((4.0, 0.0), (5.0, 0.0), (5.0, 1.0), (4.0, 1.0)), filled=False) - cells\_6: a Polygon \[gray\] drawn in buckets (vertices=((5.0, 0.0), (6.0, 0.0), (6.0, 1.0), (5.0, 1.0)), filled=False) - cells\_7: a Polygon \[gray\] drawn in buckets (vertices=((6.0, 0.0), (7.0, 0.0), (7.0, 1.0), (6.0, 1.0)), filled=False) - cells\_8: a Polygon \[gray\] drawn in buckets (vertices=((7.0, 0.0), (8.0, 0.0), (8.0, 1.0), (7.0, 1.0)), filled=False) - cells\_9: a Polygon \[gray\] drawn in buckets (vertices=((8.0, 0.0), (9.0, 0.0), (9.0, 1.0), (8.0, 1.0)), filled=False) - collision\_cell: a Polygon \[red\] drawn in buckets (vertices=((9.0, 0.0), (10.0, 0.0), (10.0, 1.0), (9.0, 1.0))) - hash\_definition: a Panel that says "A hash function maps an input to one of a fixed number of outputs. Different inputs with the same output form a collision." - heading\_bound: a Heading that says "Why Collisions Arrive Near a Square Root" - heading\_hash: a Heading that says "Different Inputs, the Same Hash" - heading\_recap: a Heading that says "What to Carry Away" - heading\_scale: a Heading that says "What Hash Lengths Really Buy" - items: a Point \[blue\] drawn in buckets (location=(2.5, 0.5)) - items\_2: a Point \[blue\] drawn in buckets (location=(9.35, 0.5)) - items\_3: a Point \[blue\] drawn in buckets (location=(13.5, 0.5)) - items\_4: a Point \[blue\] drawn in buckets (location=(5.5, 0.5)) - items\_5: a Point \[blue\] drawn in buckets (location=(9.65, 0.5)) - scale: a Table \[text\] that says "Outputs Items near 50% 365 23 $2^64$ $2^32$ $2^128$ $2^64$ $2^256$ $2^128$" (rows=(('Outputs', 'Items near 50%'), ('365', '23'), ('$2^64$', '$2^3…, header=True) - takeaways: a Block \[text\] that says "Count comparisons, not objects. Use the chance that every outcome differs. Check whether every pair counts or only pairs involving one target. With $N$ possible outputs, collisions become likely near $sqrt(N)$ trials." #### Beats ##### [06:54.903](https://academa.ai/lectures/the-birthday-paradox?t=414.90341666666654) Narration: The birthday idea appears anywhere many objects are assigned to a limited set of labels. A hash function turns a file, password, or message into one fixed-size output. Two different inputs that receive the same output create a collision. Board: Empty. Actions: - [06:54.903](https://academa.ai/lectures/the-birthday-paradox?t=414.90341666666654): heading\_hash is shown on the screen, written out. - [06:54.903](https://academa.ai/lectures/the-birthday-paradox?t=414.90341666666654): buckets is shown on the screen, written out. - [07:5.979](https://academa.ai/lectures/the-birthday-paradox?t=425.97941666666657): cells is shown on the screen, written out. - [07:6.009](https://academa.ai/lectures/the-birthday-paradox?t=426.00941666666654): cells\_2 is shown on the screen, written out. - [07:6.039](https://academa.ai/lectures/the-birthday-paradox?t=426.03941666666657): cells\_3 is shown on the screen, written out. - [07:6.069](https://academa.ai/lectures/the-birthday-paradox?t=426.06941666666654): cells\_4 is shown on the screen, written out. - [07:6.099](https://academa.ai/lectures/the-birthday-paradox?t=426.09941666666657): cells\_5 is shown on the screen, written out. - [07:6.129](https://academa.ai/lectures/the-birthday-paradox?t=426.12941666666654): cells\_6 is shown on the screen, written out. - [07:6.159](https://academa.ai/lectures/the-birthday-paradox?t=426.1594166666665): cells\_7 is shown on the screen, written out. - [07:6.189](https://academa.ai/lectures/the-birthday-paradox?t=426.18941666666655): cells\_8 is shown on the screen, written out. - [07:6.219](https://academa.ai/lectures/the-birthday-paradox?t=426.2194166666665): cells\_9 is shown on the screen, written out. - [07:6.249](https://academa.ai/lectures/the-birthday-paradox?t=426.24941666666655): cells\_10 is shown on the screen, written out. - [07:6.279](https://academa.ai/lectures/the-birthday-paradox?t=426.2794166666665): cells\_11 is shown on the screen, written out. - [07:6.309](https://academa.ai/lectures/the-birthday-paradox?t=426.30941666666655): cells\_12 is shown on the screen, written out. - [07:6.339](https://academa.ai/lectures/the-birthday-paradox?t=426.3394166666665): cells\_13 is shown on the screen, written out. - [07:6.369](https://academa.ai/lectures/the-birthday-paradox?t=426.36941666666655): cells\_14 is shown on the screen, written out. - [07:6.399](https://academa.ai/lectures/the-birthday-paradox?t=426.3994166666665): cells\_15 is shown on the screen, written out. - [07:6.429](https://academa.ai/lectures/the-birthday-paradox?t=426.42941666666655): cells\_16 is shown on the screen, written out. - [07:10.181](https://academa.ai/lectures/the-birthday-paradox?t=430.18141666666656): buckets moves to a new place on the board. - [07:10.181](https://academa.ai/lectures/the-birthday-paradox?t=430.18141666666656): hash\_definition is shown on the screen, written out. ##### [07:11.687](https://academa.ai/lectures/the-birthday-paradox?t=431.68741666666654) Narration: Here are sixteen possible outputs. Send in one item, then a second, a third, and a fourth. So far every item has landed in a different slot. Board: hash\_definition — a Panel that says "A hash function maps an input to one of a fixed number of outputs. Different inputs with the same output form a collision."; buckets — a Figure (x\_range=(0.0, 16.0), y\_range=(-0.7, 1.5), aspect=(16.0, 2.2)); heading\_hash — a Heading that says "Different Inputs, the Same Hash"; cells — a Polygon \[gray\] drawn in buckets (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)), filled=False); cells\_2 — a Polygon \[gray\] drawn in buckets (vertices=((1.0, 0.0), (2.0, 0.0), (2.0, 1.0), (1.0, 1.0)), filled=False); cells\_3 — a Polygon \[gray\] drawn in buckets (vertices=((2.0, 0.0), (3.0, 0.0), (3.0, 1.0), (2.0, 1.0)), filled=False); cells\_4 — a Polygon \[gray\] drawn in buckets (vertices=((3.0, 0.0), (4.0, 0.0), (4.0, 1.0), (3.0, 1.0)), filled=False); cells\_5 — a Polygon \[gray\] drawn in buckets (vertices=((4.0, 0.0), (5.0, 0.0), (5.0, 1.0), (4.0, 1.0)), filled=False); cells\_6 — a Polygon \[gray\] drawn in buckets (vertices=((5.0, 0.0), (6.0, 0.0), (6.0, 1.0), (5.0, 1.0)), filled=False); cells\_7 — a Polygon \[gray\] drawn in buckets (vertices=((6.0, 0.0), (7.0, 0.0), (7.0, 1.0), (6.0, 1.0)), filled=False); cells\_8 — a Polygon \[gray\] drawn in buckets (vertices=((7.0, 0.0), (8.0, 0.0), (8.0, 1.0), (7.0, 1.0)), filled=False); cells\_9 — a Polygon \[gray\] drawn in buckets (vertices=((8.0, 0.0), (9.0, 0.0), (9.0, 1.0), (8.0, 1.0)), filled=False); cells\_10 — a Polygon \[gray\] drawn in buckets (vertices=((9.0, 0.0), (10.0, 0.0), (10.0, 1.0), (9.0, 1.0)), filled=False); cells\_11 — a Polygon \[gray\] drawn in buckets (vertices=((10.0, 0.0), (11.0, 0.0), (11.0, 1.0), (10.0, 1.0)), filled=False); cells\_12 — a Polygon \[gray\] drawn in buckets (vertices=((11.0, 0.0), (12.0, 0.0), (12.0, 1.0), (11.0, 1.0)), filled=False); cells\_13 — a Polygon \[gray\] drawn in buckets (vertices=((12.0, 0.0), (13.0, 0.0), (13.0, 1.0), (12.0, 1.0)), filled=False); cells\_14 — a Polygon \[gray\] drawn in buckets (vertices=((13.0, 0.0), (14.0, 0.0), (14.0, 1.0), (13.0, 1.0)), filled=False); cells\_15 — a Polygon \[gray\] drawn in buckets (vertices=((14.0, 0.0), (15.0, 0.0), (15.0, 1.0), (14.0, 1.0)), filled=False); cells\_16 — a Polygon \[gray\] drawn in buckets (vertices=((15.0, 0.0), (16.0, 0.0), (16.0, 1.0), (15.0, 1.0)), filled=False) Actions: - [07:14.891](https://academa.ai/lectures/the-birthday-paradox?t=434.89141666666654): items is shown on the screen, written out. - [07:15.82](https://academa.ai/lectures/the-birthday-paradox?t=435.8204166666666): items\_2 is shown on the screen, written out. - [07:16.772](https://academa.ai/lectures/the-birthday-paradox?t=436.7724166666666): items\_3 is shown on the screen, written out. - [07:17.655](https://academa.ai/lectures/the-birthday-paradox?t=437.65541666666655): items\_4 is shown on the screen, written out. ##### [07:22.736](https://academa.ai/lectures/the-birthday-paradox?t=442.73641666666657) Narration: Now send in the fifth. It lands in the same slot as the second item. Sixteen slots sounded generous for five items, but five items already create ten pairs that might collide. Board: hash\_definition — a Panel that says "A hash function maps an input to one of a fixed number of outputs. Different inputs with the same output form a collision."; buckets — a Figure (x\_range=(0.0, 16.0), y\_range=(-0.7, 1.5), aspect=(16.0, 2.2)); heading\_hash — a Heading that says "Different Inputs, the Same Hash"; cells — a Polygon \[gray\] drawn in buckets (vertices=((0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)), filled=False); cells\_2 — a Polygon \[gray\] drawn in buckets (vertices=((1.0, 0.0), (2.0, 0.0), (2.0, 1.0), (1.0, 1.0)), filled=False); cells\_3 — a Polygon \[gray\] drawn in buckets (vertices=((2.0, 0.0), (3.0, 0.0), (3.0, 1.0), (2.0, 1.0)), filled=False); cells\_4 — a Polygon \[gray\] drawn in buckets (vertices=((3.0, 0.0), (4.0, 0.0), (4.0, 1.0), (3.0, 1.0)), filled=False); cells\_5 — a Polygon \[gray\] drawn in buckets (vertices=((4.0, 0.0), (5.0, 0.0), (5.0, 1.0), (4.0, 1.0)), filled=False); cells\_6 — a Polygon \[gray\] drawn in buckets (vertices=((5.0, 0.0), (6.0, 0.0), (6.0, 1.0), (5.0, 1.0)), filled=False); cells\_7 — a Polygon \[gray\] drawn in buckets (vertices=((6.0, 0.0), (7.0, 0.0), (7.0, 1.0), (6.0, 1.0)), filled=False); cells\_8 — a Polygon \[gray\] drawn in buckets (vertices=((7.0, 0.0), (8.0, 0.0), (8.0, 1.0), (7.0, 1.0)), filled=False); cells\_9 — a Polygon \[gray\] drawn in buckets (vertices=((8.0, 0.0), (9.0, 0.0), (9.0, 1.0), (8.0, 1.0)), filled=False); cells\_10 — a Polygon \[gray\] drawn in buckets (vertices=((9.0, 0.0), (10.0, 0.0), (10.0, 1.0), (9.0, 1.0)), filled=False); cells\_11 — a Polygon \[gray\] drawn in buckets (vertices=((10.0, 0.0), (11.0, 0.0), (11.0, 1.0), (10.0, 1.0)), filled=False); cells\_12 — a Polygon \[gray\] drawn in buckets (vertices=((11.0, 0.0), (12.0, 0.0), (12.0, 1.0), (11.0, 1.0)), filled=False); cells\_13 — a Polygon \[gray\] drawn in buckets (vertices=((12.0, 0.0), (13.0, 0.0), (13.0, 1.0), (12.0, 1.0)), filled=False); cells\_14 — a Polygon \[gray\] drawn in buckets (vertices=((13.0, 0.0), (14.0, 0.0), (14.0, 1.0), (13.0, 1.0)), filled=False); cells\_15 — a Polygon \[gray\] drawn in buckets (vertices=((14.0, 0.0), (15.0, 0.0), (15.0, 1.0), (14.0, 1.0)), filled=False); cells\_16 — a Polygon \[gray\] drawn in buckets (vertices=((15.0, 0.0), (16.0, 0.0), (16.0, 1.0), (15.0, 1.0)), filled=False); items — a Point \[blue\] drawn in buckets (location=(2.5, 0.5)); items\_2 — a Point \[blue\] drawn in buckets (location=(9.35, 0.5)); items\_3 — a Point \[blue\] drawn in buckets (location=(13.5, 0.5)); items\_4 — a Point \[blue\] drawn in buckets (location=(5.5, 0.5)) Actions: - [07:23.851](https://academa.ai/lectures/the-birthday-paradox?t=443.8514166666665): items\_5 is shown on the screen, written out. - [07:25.592](https://academa.ai/lectures/the-birthday-paradox?t=445.59241666666657): collision\_cell is shown on the screen, written out. - [07:26.521](https://academa.ai/lectures/the-birthday-paradox?t=446.52141666666654): items\_2 is indicated — a transient flash. - [07:32.418](https://academa.ai/lectures/the-birthday-paradox?t=452.41841666666653): items\_5 is indicated — a transient flash. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): buckets is hidden from the screen — left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_2 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_3 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_4 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_5 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_6 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_7 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_8 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_9 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_10 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_11 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_12 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_13 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_14 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_15 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): cells\_16 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): items is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): items\_2 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): items\_3 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): items\_4 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): items\_5 is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): collision\_cell is hidden from the screen — buckets left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): hash\_definition is hidden from the screen — left the board. - [07:34.601](https://academa.ai/lectures/the-birthday-paradox?t=454.6014166666665): heading\_hash is hidden from the screen — left the board. ##### [07:35.201](https://academa.ai/lectures/the-birthday-paradox?t=455.20141666666655) Narration: Suppose a hash has N equally likely outputs and we try k inputs. The number of input pairs is k times k minus one over two, exactly the same pair count as before. Board: Empty. Actions: - [07:35.201](https://academa.ai/lectures/the-birthday-paradox?t=455.20141666666655): heading\_bound is shown on the screen, written out. - [07:41.087](https://academa.ai/lectures/the-birthday-paradox?t=461.0874166666665): bound\_work is shown on the screen, written out. ##### [07:48.259](https://academa.ai/lectures/the-birthday-paradox?t=468.25891666666655) Narration: Each pair has about one chance in N of sharing an output. Combining all those small risks gives this standard approximation for no collision. Subtracting from one gives the chance of at least one collision. Board: heading\_bound — a Heading that says "Why Collisions Arrive Near a Square Root" Actions: - [07:55.573](https://academa.ai/lectures/the-birthday-paradox?t=475.57341666666656): bound\_work is shown on the screen, written out. - [07:57.999](https://academa.ai/lectures/the-birthday-paradox?t=477.99941666666655): bound\_work is shown on the screen, written out. ##### [08:2.35](https://academa.ai/lectures/the-birthday-paradox?t=482.34991666666656) Narration: Set that collision chance to one half and solve. The result is about one point one eight times the square root of N. The constant matters for an exact estimate, but the memorable fact is the square root. Board: Unchanged from the preceding beat in this scene. Actions: - [08:5.902](https://academa.ai/lectures/the-birthday-paradox?t=485.90241666666657): bound is shown on the screen, written out. - [08:8.979](https://academa.ai/lectures/the-birthday-paradox?t=488.97941666666657): bound (the "sqrt(N)" part) is emphasized. - [08:15.324](https://academa.ai/lectures/the-birthday-paradox?t=495.32391666666655): bound moves to a new place on the board. - [08:15.324](https://academa.ai/lectures/the-birthday-paradox?t=495.32391666666655): bound\_work is hidden from the screen — left the board. - [08:15.324](https://academa.ai/lectures/the-birthday-paradox?t=495.32391666666655): heading\_bound is hidden from the screen — left the board. - [08:15.324](https://academa.ai/lectures/the-birthday-paradox?t=495.32391666666655): A box is drawn around bound. - [08:15.324](https://academa.ai/lectures/the-birthday-paradox?t=495.32391666666655): bound (the "sqrt(N)" part) is no longer emphasized. ##### [08:16.524](https://academa.ai/lectures/the-birthday-paradox?t=496.52391666666654) Narration: The birthday example checks the scale. Three hundred sixty five possible days have a square root near nineteen, and the more accurate crossing is twenty three. Board: bound — a Math \[text\] that says "$k approx 1.18 sqrt(N) thin upright("at 50%")$" Actions: - [08:16.524](https://academa.ai/lectures/the-birthday-paradox?t=496.52391666666654): heading\_scale is shown on the screen, written out. - [08:18.155](https://academa.ai/lectures/the-birthday-paradox?t=498.15541666666655): scale is shown on the screen, written out. - [08:25.04](https://academa.ai/lectures/the-birthday-paradox?t=505.04041666666654): scale is shown on the screen, written out. ##### [08:26.766](https://academa.ai/lectures/the-birthday-paradox?t=506.76641666666654) Narration: A sixty four bit hash has two to the sixty fourth possible outputs, but collisions become likely after roughly two to the thirty second inputs. That is about four billion, far below the number of available outputs. Board: bound — a Math \[text\] that says "$k approx 1.18 sqrt(N) thin upright("at 50%")$"; heading\_scale — a Heading that says "What Hash Lengths Really Buy" Actions: - [08:27.312](https://academa.ai/lectures/the-birthday-paradox?t=507.31241666666654): scale is shown on the screen, written out. - [08:36.332](https://academa.ai/lectures/the-birthday-paradox?t=516.3324166666665): scale (the "row=3" part) is emphasized. - [08:40.001](https://academa.ai/lectures/the-birthday-paradox?t=520.0014166666665): scale (the "row=3" part) is no longer emphasized. ##### [08:40.601](https://academa.ai/lectures/the-birthday-paradox?t=520.6014166666665) Narration: For a one hundred twenty eight bit hash, the birthday scale is about two to the sixty fourth trials. For a two hundred fifty six bit hash, it is about two to the one hundred twenty eighth. Collision resistance therefore provides roughly half as many security bits as the output length. Board: Unchanged from the preceding beat in this scene. Actions: - [08:41.274](https://academa.ai/lectures/the-birthday-paradox?t=521.2744166666665): scale is shown on the screen, written out. - [08:47.115](https://academa.ai/lectures/the-birthday-paradox?t=527.1154166666665): scale is shown on the screen, written out. - [08:54.382](https://academa.ai/lectures/the-birthday-paradox?t=534.3824166666666): scale (the "column=2" part) is emphasized. - [08:57.366](https://academa.ai/lectures/the-birthday-paradox?t=537.3664166666665): scale (the "column=2" part) is no longer emphasized. ##### [08:57.966](https://academa.ai/lectures/the-birthday-paradox?t=537.9664166666665) Narration: That matters when hashes identify files, support digital signatures, or detect duplicates. A real design either chooses enough output bits to make accidental collisions fantastically unlikely, or includes a clear plan for handling collisions when they occur. Board: Unchanged from the preceding beat in this scene. Actions: - [09:9.901](https://academa.ai/lectures/the-birthday-paradox?t=549.9014166666666): bound is indicated — a transient flash. - [09:14.278](https://academa.ai/lectures/the-birthday-paradox?t=554.2779166666666): bound is hidden from the screen — left the board. - [09:14.278](https://academa.ai/lectures/the-birthday-paradox?t=554.2779166666666): heading\_scale is hidden from the screen — left the board. - [09:14.278](https://academa.ai/lectures/the-birthday-paradox?t=554.2779166666666): scale is hidden from the screen — left the board. ##### [09:15.478](https://academa.ai/lectures/the-birthday-paradox?t=555.4779166666665) Narration: Four ideas now hold the whole lecture together. Count comparisons rather than objects. Compute the easier no-collision probability. Check whether every pair counts or only comparisons with one fixed target. And expect collisions near the square root of the number of possible outcomes. Board: Empty. Actions: - [09:15.478](https://academa.ai/lectures/the-birthday-paradox?t=555.4779166666665): heading\_recap is shown on the screen, written out. - [09:15.826](https://academa.ai/lectures/the-birthday-paradox?t=555.8264166666665): takeaways is shown on the screen, written out. - [09:19.088](https://academa.ai/lectures/the-birthday-paradox?t=559.0884166666665): takeaways (the "Count comparisons, not objects." part) is emphasized. - [09:23.279](https://academa.ai/lectures/the-birthday-paradox?t=563.2794166666665): takeaways (the "Count comparisons, not objects." part) is no longer emphasized. - [09:23.279](https://academa.ai/lectures/the-birthday-paradox?t=563.2794166666665): takeaways (the "Use the chance that every outcome differs." part) is emphasized. - [09:28.992](https://academa.ai/lectures/the-birthday-paradox?t=568.9924166666665): takeaways (the "Check whether every pair counts or only pairs involving one target." part) is emphasized. - [09:28.992](https://academa.ai/lectures/the-birthday-paradox?t=568.9924166666665): takeaways (the "Use the chance that every outcome differs." part) is no longer emphasized. - [09:31.813](https://academa.ai/lectures/the-birthday-paradox?t=571.8134166666665): takeaways (the "Check whether every pair counts or only pairs involving one target." part) is no longer emphasized. - [09:31.813](https://academa.ai/lectures/the-birthday-paradox?t=571.8134166666665): takeaways (the "With $N$ possible outputs, collisions become likely near $sqrt(N)$ trials." part) is emphasized. ##### [09:35.257](https://academa.ai/lectures/the-birthday-paradox?t=575.2574166666666) Narration: Twenty three people are enough because they do not create twenty three chances to match. They create two hundred fifty three pairs. Once that hidden web is visible, the birthday result stops being a trick and becomes an ordinary collision problem. Board: takeaways — a Block \[text\] that says "Count comparisons, not objects. Use the chance that every outcome differs. Check whether every pair counts or only pairs involving one target. With $N$ possible outputs, collisions become likely near $sqrt(N)$ trials."; heading\_recap — a Heading that says "What to Carry Away" Actions: - [09:35.257](https://academa.ai/lectures/the-birthday-paradox?t=575.2574166666666): takeaways (the "With $N$ possible outputs, collisions become likely near $sqrt(N)$ trials." part) is no longer emphasized. - [09:42.955](https://academa.ai/lectures/the-birthday-paradox?t=582.9554166666666): takeaways (the "Count comparisons, not objects." part) is indicated — a transient flash. - [09:50.686](https://academa.ai/lectures/the-birthday-paradox?t=590.6856666666665): heading\_recap is hidden from the screen — left the board. - [09:50.686](https://academa.ai/lectures/the-birthday-paradox?t=590.6856666666665): takeaways is hidden from the screen — left the board.