{"version":1,"lectureId":"01M14V09JNY5H01VZKTHC75907","attempt":2,"publication":{"slug":"the-birthday-paradox","title":"Why 23 People Make a Shared Birthday Likely","subject":"statistics","summary":"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.","metaDescription":"A visual explanation of why 23 people cross 50%, why one fixed birthday needs 253 others, and how the same logic predicts hash collisions.","transcript":"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. 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. 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. 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.","watch":{"version":1,"scenes":[{"title":"Count the Pairs, Not the People","start":0,"end":109.17004166666666,"objects":{"card":"a Title that says \"Probability for Everyone — Why 23 People Make a Shared Birthday Likely\"","count_23":"a Math [text] that says \"$frac(23 dot.op 22, 2) = 253$\"","five_count":"a Math [text] that says \"$5 thin upright(\"people\") arrow.r 10 thin upright(\"pairs\")$\"","full":"a Figure (x_range=(-1.35, 1.35), y_range=(-1.35, 1.35), aspect=(1.0, 1.0))","full_chords":"a Line [yellow] drawn in full (start=(6.123233995736766e-17, 1.0), end=(0.2697967711570242, 0.9629172873477994))","full_chords_10":"a Line [yellow] drawn in full (start=(6.123233995736766e-17, 1.0), end=(0.39840108984624134, -0.917211301505453))","full_chords_100":"a Line [yellow] drawn in full (start=(0.8878852184023752, 0.46006503773115215), end=(-0.26979677115702516, 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That is two hundred fifty three. 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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.","live":["pair_formula","count_23","full","heading_full","full_dots","full_dots_2","full_dots_3","full_dots_4","full_dots_5","full_dots_6","full_dots_7","full_dots_8","full_dots_9","full_dots_10","full_dots_11","full_dots_12","full_dots_13","full_dots_14","full_dots_15","full_dots_16","full_dots_17","full_dots_18","full_dots_19","full_dots_20","full_dots_21","full_dots_22","full_dots_23","full_chords","full_chords_2","full_chords_3","full_chords_4","full_chords_5","full_chords_6","full_chords_7","full_chords_8","full_chords_9","full_chords_10","full_chords_11","full_chords_12","full_chords_13","full_chords_14","full_chords_15","full_chords_16","full_chords_17","full_chords_18","full_chords_19","full_chords_20","full_chords_21","full_chords_22","full_chords_23","full_chords_24","full_chords_25","full_chords_26","full_chords_27","full_chords_28","full_chords_29","full_chords_30","full_chords_31","full_chords_32","full_chords_33","full_chords_34","full_chords_35","full_chords_36","full_chords_37","full_chords_38","full_chords_39","full_chords_40","full_chords_41","full_chords_42","full_chords_43","full_chords_44","full_chords_45","full_chords_46","full_chords_47","full_chords_48","full_chords_49","full_chords_50","full_chords_51","full_chords_52","full_chords_53","full_chords_54","full_chords_55","full_chords_56","full_chords_57","full_chords_58","full_chords_59","full_chords_60","full_chords_61","full_chords_62","full_chords_63","full_chords_64","full_chords_65","full_chords_66","full_chords_67","full_chords_68","full_chords_69","full_chords_70","full_chords_71","full_chords_72","full_chords_73","full_chords_74","full_chords_75","full_chords_76","full_chords_77","full_chords_78","full_chords_79","full_chords_80","full_chords_81","full_chords_82","full_chords_83","full_chords_84","full_chords_85","full_chords_86","full_chords_87","full_chords_88","full_chords_89","full_chords_90","full_chords_91","full_chords_92","full_chords_93","full_chords_94","full_chords_95","full_chords_96","full_chords_97","full_chords_98","full_chords_99","full_chords_100","full_chords_101","full_chords_102","full_chords_103","full_chords_104","full_chords_105","full_chords_106","full_chords_107","full_chords_108","full_chords_109","full_chords_110","full_chords_111","full_chords_112","full_chords_113","full_chords_114","full_chords_115","full_chords_116","full_chords_117","full_chords_118","full_chords_119","full_chords_120","full_chords_121","full_chords_122","full_chords_123","full_chords_124","full_chords_125","full_chords_126","full_chords_127","full_chords_128","full_chords_129","full_chords_130","full_chords_131","full_chords_132","full_chords_133","full_chords_134","full_chords_135","full_chords_136","full_chords_137","full_chords_138","full_chords_139","full_chords_140","full_chords_141","full_chords_142","full_chords_143","full_chords_144","full_chords_145","full_chords_146","full_chords_147","full_chords_148","full_chords_149","full_chords_150","full_chords_151","full_chords_152","full_chords_153","full_chords_154","full_chords_155","full_chords_156","full_chords_157","full_chords_158","full_chords_159","full_chords_160","full_chords_161","full_chords_162","full_chords_163","full_chords_164","full_chords_165","full_chords_166","full_chords_167","full_chords_168","full_chords_169","full_chords_170","full_chords_171","full_chords_172","full_chords_173","full_chords_174","full_chords_175","full_chords_176","full_chords_177","full_chords_178","full_chords_179","full_chords_180","full_chords_181","full_chords_182","full_chords_183","full_chords_184","full_chords_185","full_chords_186","full_chords_187","full_chords_188","full_chords_189","full_chords_190","full_chords_191","full_chords_192","full_chords_193","full_chords_194","full_chords_195","full_chords_196","full_chords_197","full_chords_198","full_chords_199","full_chords_200","full_chords_201","full_chords_202","full_chords_203","full_chords_204","full_chords_205","full_chords_206","full_chords_207","full_chords_208","full_chords_209","full_chords_210","full_chords_211","full_chords_212","full_chords_213","full_chords_214","full_chords_215","full_chords_216","full_chords_217","full_chords_218","full_chords_219","full_chords_220","full_chords_221","full_chords_222","full_chords_223","full_chords_224","full_chords_225","full_chords_226","full_chords_227","full_chords_228","full_chords_229","full_chords_230","full_chords_231","full_chords_232","full_chords_233","full_chords_234","full_chords_235","full_chords_236","full_chords_237","full_chords_238","full_chords_239","full_chords_240","full_chords_241","full_chords_242","full_chords_243","full_chords_244","full_chords_245","full_chords_246","full_chords_247","full_chords_248","full_chords_249","full_chords_250","full_chords_251","full_chords_252","full_chords_253"],"does":[[99.19699999999997,"punch is shown on the screen, written out."],[99.71899999999998,"count_23 is indicated — a transient flash."],[108.12837499999999,"count_23 is hidden from the screen — left the board."],[108.12837499999999,"full is hidden from the screen — left the board."],[108.12837499999999,"full_dots is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_2 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_3 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_4 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_5 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_6 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_7 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_8 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_9 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_10 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_11 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_12 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_13 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_14 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_15 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_16 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_17 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_18 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_19 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_20 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_21 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_22 is hidden from the screen — full left the board."],[108.12837499999999,"full_dots_23 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_2 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_3 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_4 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_5 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_6 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_7 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_8 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_9 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_10 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_11 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_12 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_13 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_14 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_15 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_16 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_17 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_18 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_19 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_20 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_21 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_22 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_23 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_24 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_25 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_26 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_27 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_28 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_29 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_30 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_31 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_32 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_33 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_34 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_35 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_36 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_37 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_38 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_39 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_40 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_41 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_42 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_43 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_44 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_45 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_46 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_47 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_48 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_49 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_50 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_51 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_52 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_53 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_54 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_55 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_56 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_57 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_58 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_59 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_60 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_61 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_62 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_63 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_64 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_65 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_66 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_67 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_68 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_69 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_70 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_71 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_72 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_73 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_74 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_75 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_76 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_77 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_78 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_79 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_80 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_81 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_82 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_83 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_84 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_85 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_86 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_87 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_88 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_89 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_90 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_91 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_92 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_93 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_94 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_95 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_96 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_97 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_98 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_99 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_100 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_101 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_102 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_103 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_104 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_105 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_106 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_107 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_108 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_109 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_110 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_111 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_112 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_113 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_114 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_115 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_116 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_117 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_118 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_119 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_120 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_121 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_122 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_123 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_124 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_125 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_126 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_127 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_128 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_129 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_130 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_131 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_132 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_133 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_134 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_135 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_136 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_137 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_138 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_139 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_140 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_141 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_142 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_143 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_144 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_145 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_146 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_147 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_148 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_149 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_150 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_151 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_152 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_153 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_154 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_155 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_156 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_157 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_158 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_159 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_160 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_161 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_162 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_163 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_164 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_165 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_166 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_167 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_168 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_169 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_170 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_171 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_172 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_173 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_174 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_175 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_176 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_177 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_178 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_179 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_180 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_181 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_182 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_183 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_184 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_185 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_186 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_187 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_188 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_189 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_190 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_191 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_192 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_193 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_194 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_195 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_196 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_197 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_198 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_199 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_200 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_201 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_202 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_203 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_204 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_205 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_206 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_207 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_208 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_209 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_210 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_211 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_212 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_213 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_214 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_215 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_216 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_217 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_218 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_219 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_220 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_221 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_222 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_223 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_224 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_225 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_226 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_227 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_228 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_229 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_230 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_231 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_232 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_233 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_234 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_235 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_236 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_237 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_238 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_239 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_240 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_241 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_242 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_243 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_244 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_245 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_246 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_247 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_248 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_249 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_250 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_251 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_252 is hidden from the screen — full left the board."],[108.12837499999999,"full_chords_253 is hidden from the screen — full left the board."],[108.12837499999999,"heading_full is hidden from the screen — left the board."],[108.12837499999999,"pair_formula is hidden from the screen — left the board."],[108.12837499999999,"punch is hidden from the screen — left the board."]]}]},{"title":"Multiply the Shrinking Fractions","start":109.17004166666666,"end":272.9036666666666,"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=<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=<VariableNumber group = 50.0>)","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":[{"start":109.17004166666666,"say":"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.","live":[],"does":[[109.17004166666666,"heading_model is shown on the screen, written out."],[122.22004166666666,"assumption is shown on the screen, written out."],[125.09854166666666,"assumption is hidden from the screen — left the board."],[125.09854166666666,"heading_model is hidden from the screen — left the board."]]},{"start":126.29854166666667,"say":"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?","live":null,"does":[[126.29854166666667,"heading_product is shown on the screen, written out."],[136.78204166666666,"work is shown on the screen, written out."]]},{"start":139.54154166666666,"say":"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.","live":["heading_product"],"does":[[142.87404166666664,"work is shown on the screen, written out."]]},{"start":151.93754166666668,"say":"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.","live":null,"does":[[152.98204166666665,"work is shown on the screen, written out."],[157.75404166666667,"work (the \"frac(363, 365)\" part) is emphasized."],[164.45304166666665,"work (the \"frac(363, 365)\" part) is no longer emphasized."]]},{"start":165.05304166666667,"say":"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.","live":null,"does":[[170.78804166666666,"work is shown on the screen, written out."]]},{"start":180.24654166666664,"say":"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.","live":null,"does":[[182.18504166666665,"work is shown on the screen, written out."],[188.93104166666666,"work is shown on the screen, written out."],[195.92004166666663,"A box is drawn around work."],[197.15104166666666,"heading_product is hidden from the screen — left the board."],[197.15104166666666,"work is hidden from the screen — left the board."],[197.15104166666666,"plot is shown on the screen, written out."]]},{"start":198.35104166666667,"say":"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.","live":["plot"],"does":[[198.35104166666667,"plot moves to a new place on the board."],[198.35104166666667,"heading_curve is shown on the screen, written out."],[200.74204166666667,"curve is shown on the screen, drawn."],[203.42404166666665,"readout is shown on the screen, written out."],[210.62304166666667,"half is shown on the screen, written out."]]},{"start":212.03554166666666,"say":"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.","live":["plot","heading_curve","curve","half"],"does":[[212.62704166666666,"probe is shown on the screen, written out."],[214.54304166666665,"readout is shown on the screen, written out."]]},{"start":221.43554166666667,"say":"Move to twenty two people. The chance has climbed to about forty seven point six percent, still below the dashed half line.","live":["plot","heading_curve","curve","half","probe"],"does":[[222.20204166666667,"probe is redrawn as the numbers it depends on change."],[222.20204166666667,"group ticks to 22.0."],[225.11604166666666,"readout is shown on the screen, written out."]]},{"start":229.65154166666667,"say":"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.","live":null,"does":[[230.54504166666666,"probe is redrawn as the numbers it depends on change."],[230.54504166666666,"group ticks to 23.0."],[239.86804166666664,"readout is shown on the screen, written out."],[241.28504166666667,"crossing is shown on the screen, written out."],[241.80704166666666,"crossing is indicated — a transient flash."]]},{"start":243.02254166666665,"say":"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.","live":["plot","heading_curve","curve","half","probe","crossing"],"does":[[245.37904166666667,"probe is redrawn as the numbers it depends on change."],[245.37904166666667,"group ticks to 30.0."],[246.83004166666666,"readout is shown on the screen, written out."],[248.65304166666667,"probe is redrawn as the numbers it depends on change."],[248.65304166666667,"group ticks to 50.0."],[249.84904166666666,"readout is shown on the screen, written out."]]},{"start":257.13604166666664,"say":"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.","live":null,"does":[[258.08804166666664,"crossing is indicated — a transient flash."],[270.2090416666666,"readout is indicated — a transient flash."],[271.86199999999997,"heading_curve is hidden from the screen — left the board."],[271.86199999999997,"plot is hidden from the screen — left the board."],[271.86199999999997,"curve is hidden from the screen — plot left the board."],[271.86199999999997,"half is hidden from the screen — plot left the board."],[271.86199999999997,"probe is hidden from the screen — plot left the board."],[271.86199999999997,"crossing is hidden from the screen — plot left the board."],[271.86199999999997,"readout is hidden from the screen — left the board."]]}]},{"title":"Someone Who Matches Your Birthday","start":272.9036666666666,"end":414.90341666666654,"objects":{"answer_others":"a Math [text] that says \"$n = 253 thin upright(\"other 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The dashed line is again a probability of one half.","live":["both","half"],"does":[[370.3031666666666,"heading_curves is shown on the screen, written out."]]},{"start":381.0101666666666,"say":"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.","live":["both","heading_curves","half"],"does":[[381.52066666666656,"curve_any is shown on the screen, drawn."],[385.35266666666655,"mark_any is shown on the screen, written out."],[388.66066666666654,"note_any is shown on the screen, written out."]]},{"start":390.07366666666655,"say":"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.","live":["note_any","both","heading_curves","half","curve_any","mark_any"],"does":[[390.5966666666666,"curve_mine is shown on the screen, drawn."],[396.77266666666657,"mark_mine is shown on the screen, written out."],[399.6526666666666,"note_mine is shown on the screen, written out."]]},{"start":401.8896666666666,"say":"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.","live":["note_any","note_mine","both","heading_curves","half","curve_any","mark_any","curve_mine","mark_mine"],"does":[[407.7056666666666,"curve_mine is indicated — a transient flash."],[410.15566666666655,"curve_any is indicated — a transient flash."],[413.8617499999999,"both is hidden from the screen — left the board."],[413.8617499999999,"half is hidden from the screen — both left the board."],[413.8617499999999,"curve_any is hidden from the screen — both left the board."],[413.8617499999999,"mark_any is hidden from the screen — both left the board."],[413.8617499999999,"curve_mine is hidden from the screen — both left the board."],[413.8617499999999,"mark_mine is hidden from the screen — both left the board."],[413.8617499999999,"heading_curves is hidden from the screen — left the board."],[413.8617499999999,"note_any is hidden from the screen — left the board."],[413.8617499999999,"note_mine is hidden from the screen — left the board."]]}]},{"title":"The Same Collision in a Hash Function","start":414.90341666666654,"end":591.7273333333333,"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":[{"start":414.90341666666654,"say":"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.","live":[],"does":[[414.90341666666654,"heading_hash is shown on the screen, written out."],[414.90341666666654,"buckets is shown on the screen, written out."],[425.97941666666657,"cells is shown on the screen, written out."],[426.00941666666654,"cells_2 is shown on the screen, written out."],[426.03941666666657,"cells_3 is shown on the screen, written out."],[426.06941666666654,"cells_4 is shown on the screen, written out."],[426.09941666666657,"cells_5 is shown on the screen, written out."],[426.12941666666654,"cells_6 is shown on the screen, written out."],[426.1594166666665,"cells_7 is shown on the screen, written out."],[426.18941666666655,"cells_8 is shown on the screen, written out."],[426.2194166666665,"cells_9 is shown on the screen, written out."],[426.24941666666655,"cells_10 is shown on the screen, written out."],[426.2794166666665,"cells_11 is shown on the screen, written out."],[426.30941666666655,"cells_12 is shown on the screen, written out."],[426.3394166666665,"cells_13 is shown on the screen, written out."],[426.36941666666655,"cells_14 is shown on the screen, written out."],[426.3994166666665,"cells_15 is shown on the screen, written out."],[426.42941666666655,"cells_16 is shown on the screen, written out."],[430.18141666666656,"buckets moves to a new place on the board."],[430.18141666666656,"hash_definition is shown on the screen, written out."]]},{"start":431.68741666666654,"say":"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.","live":["hash_definition","buckets","heading_hash","cells","cells_2","cells_3","cells_4","cells_5","cells_6","cells_7","cells_8","cells_9","cells_10","cells_11","cells_12","cells_13","cells_14","cells_15","cells_16"],"does":[[434.89141666666654,"items is shown on the screen, written out."],[435.8204166666666,"items_2 is shown on the screen, written out."],[436.7724166666666,"items_3 is shown on the screen, written out."],[437.65541666666655,"items_4 is shown on the screen, written out."]]},{"start":442.73641666666657,"say":"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.","live":["hash_definition","buckets","heading_hash","cells","cells_2","cells_3","cells_4","cells_5","cells_6","cells_7","cells_8","cells_9","cells_10","cells_11","cells_12","cells_13","cells_14","cells_15","cells_16","items","items_2","items_3","items_4"],"does":[[443.8514166666665,"items_5 is shown on the screen, written out."],[445.59241666666657,"collision_cell is shown on the screen, written out."],[446.52141666666654,"items_2 is indicated — a transient flash."],[452.41841666666653,"items_5 is indicated — a transient flash."],[454.6014166666665,"buckets is hidden from the screen — left the board."],[454.6014166666665,"cells is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_2 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_3 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_4 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_5 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_6 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_7 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_8 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_9 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_10 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_11 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_12 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_13 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_14 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_15 is hidden from the screen — buckets left the board."],[454.6014166666665,"cells_16 is hidden from the screen — buckets left the board."],[454.6014166666665,"items is hidden from the screen — buckets left the board."],[454.6014166666665,"items_2 is hidden from the screen — buckets left the board."],[454.6014166666665,"items_3 is hidden from the screen — buckets left the board."],[454.6014166666665,"items_4 is hidden from the screen — buckets left the board."],[454.6014166666665,"items_5 is hidden from the screen — buckets left the board."],[454.6014166666665,"collision_cell is hidden from the screen — buckets left the board."],[454.6014166666665,"hash_definition is hidden from the screen — left the board."],[454.6014166666665,"heading_hash is hidden from the screen — left the board."]]},{"start":455.20141666666655,"say":"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.","live":[],"does":[[455.20141666666655,"heading_bound is shown on the screen, written out."],[461.0874166666665,"bound_work is shown on the screen, written out."]]},{"start":468.25891666666655,"say":"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.","live":["heading_bound"],"does":[[475.57341666666656,"bound_work is shown on the screen, written out."],[477.99941666666655,"bound_work is shown on the screen, written out."]]},{"start":482.34991666666656,"say":"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.","live":null,"does":[[485.90241666666657,"bound is shown on the screen, written out."],[488.97941666666657,"bound (the \"sqrt(N)\" part) is emphasized."],[495.32391666666655,"bound moves to a new place on the board."],[495.32391666666655,"bound_work is hidden from the screen — left the board."],[495.32391666666655,"heading_bound is hidden from the screen — left the board."],[495.32391666666655,"A box is drawn around bound."],[495.32391666666655,"bound (the \"sqrt(N)\" part) is no longer emphasized."]]},{"start":496.52391666666654,"say":"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.","live":["bound"],"does":[[496.52391666666654,"heading_scale is shown on the screen, written out."],[498.15541666666655,"scale is shown on the screen, written out."],[505.04041666666654,"scale is shown on the screen, written out."]]},{"start":506.76641666666654,"say":"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.","live":["bound","heading_scale"],"does":[[507.31241666666654,"scale is shown on the screen, written out."],[516.3324166666665,"scale (the \"row=3\" part) is emphasized."],[520.0014166666665,"scale (the \"row=3\" part) is no longer emphasized."]]},{"start":520.6014166666665,"say":"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.","live":null,"does":[[521.2744166666665,"scale is shown on the screen, written out."],[527.1154166666665,"scale is shown on the screen, written out."],[534.3824166666666,"scale (the \"column=2\" part) is emphasized."],[537.3664166666665,"scale (the \"column=2\" part) is no longer emphasized."]]},{"start":537.9664166666665,"say":"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.","live":null,"does":[[549.9014166666666,"bound is indicated — a transient flash."],[554.2779166666666,"bound is hidden from the screen — left the board."],[554.2779166666666,"heading_scale is hidden from the screen — left the board."],[554.2779166666666,"scale is hidden from the screen — left the board."]]},{"start":555.4779166666665,"say":"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.","live":[],"does":[[555.4779166666665,"heading_recap is shown on the screen, written out."],[555.8264166666665,"takeaways is shown on the screen, written out."],[559.0884166666665,"takeaways (the \"Count comparisons, not objects.\" part) is emphasized."],[563.2794166666665,"takeaways (the \"Count comparisons, not objects.\" part) is no longer emphasized."],[563.2794166666665,"takeaways (the \"Use the chance that every outcome differs.\" part) is emphasized."],[568.9924166666665,"takeaways (the \"Check whether every pair counts or only pairs involving one target.\" part) is emphasized."],[568.9924166666665,"takeaways (the \"Use the chance that every outcome differs.\" part) is no longer emphasized."],[571.8134166666665,"takeaways (the \"Check whether every pair counts or only pairs involving one target.\" part) is no longer emphasized."],[571.8134166666665,"takeaways (the \"With $N$ possible outputs, collisions become likely near $sqrt(N)$ trials.\" part) is emphasized."]]},{"start":575.2574166666666,"say":"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.","live":["takeaways","heading_recap"],"does":[[575.2574166666666,"takeaways (the \"With $N$ possible outputs, collisions become likely near $sqrt(N)$ trials.\" part) is no longer emphasized."],[582.9554166666666,"takeaways (the \"Count comparisons, not objects.\" part) is indicated — a transient flash."],[590.6856666666665,"heading_recap is hidden from the screen — left the board."],[590.6856666666665,"takeaways is hidden from the screen — left the board."]]}]}]},"durationSeconds":592,"chapters":[{"title":"Count the Pairs, Not the People","startSeconds":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. 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."},{"title":"Multiply the Shrinking Fractions","startSeconds":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. 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."},{"title":"Someone Who Matches Your Birthday","startSeconds":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. 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."},{"title":"The Same Collision in a Hash Function","startSeconds":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. 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."}]}}