# Why a 99% Accurate Test Can Still Be Wrong

> A visual introduction to base rates and Bayes' theorem for viewers without statistics training. Starting with 10,000 people and a rare disease, the lecture separates true positives from false positives, reads the probability directly from those populations, and only then introduces Bayes' formula. It shows how prevalence and symmetric test accuracy change the meaning of a positive result, then follows the same population through a second conditionally independent test.

- Canonical watch page: [Why a 99% Accurate Test Can Still Be Wrong](https://academa.ai/lectures/bayes-through-the-medical-test-paradox)
- Publisher: [Academa, Inc.](https://academa.ai)
- Subject: Statistics
- Published: 2026-08-28T22:51:56.000Z
- Updated: 2026-08-28T22:51:56.000Z
- Duration: PT695S (11 minutes 35 seconds)
- Chapters: 5
- Views: 0
- Language: en-US
- Access: Free
- Video stream: [HLS content](https://academa.ai/media/l/01M14V03BNSC4HNC1D114614TX/0/dark/master.m3u8)
- Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14V03BNSC4HNC1D114614TX/0/semantic.json)
- Thumbnail: [Image](https://academa.ai/media/l/01M14V03BNSC4HNC1D114614TX/0/dark/poster.jpg)

## Description

See how a 99% sensitive and 99% specific test can be wrong for a rare disease, then watch prevalence, accuracy, and retesting change the odds.

## Chapters

- [00:00–02:11.741 · Ten Thousand People](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=0)
- [02:11.741–04:11.752 · Read the Positive Pile](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=131.74064583333333)
- [04:11.752–06:25.53 · Bayes Names the Count](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=251.75231250000002)
- [06:25.53–08:55.657 · Watch the Answer Swing](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=385.5296875)
- [08:55.657–11:35 · Test Again](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=535.6572708333333)

## Transcript

### [00:00 · Ten Thousand People](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=0)

Suppose a medical test is described as ninety-nine percent accurate, and it comes back positive. That sounds almost conclusive. But if the disease is rare, the positive result can still be more likely wrong than right. We are going to see why by counting people before writing any probability formula. Here is the question in its most personal form. A positive result says you have a rare disease. Does ninety-nine percent accurate mean there is a ninety-nine percent chance you have it? No. That number describes how the test behaves inside known groups. It does not yet answer what group a positive person probably came from. Take ten thousand people. The large gray field represents the people in this cohort who do not have the disease. I have magnified the affected people so we can actually see them. Let the disease affect one person in a thousand. That is a prevalence of zero point one percent. In ten thousand people, only ten actually have the disease. The remaining nine thousand nine hundred ninety are healthy. Now we must say exactly what ninety-nine percent accurate means. For this lecture, it means two things. Sensitivity is ninety-nine percent, so among people who truly have the disease, the test is positive ninety-nine percent of the time. Specificity is also ninety-nine percent, so among healthy people, the test is negative ninety-nine percent of the time. Apply sensitivity to the ten affected people. Ninety-nine percent of ten is nine point nine. So across many cohorts like this one, we expect about nine point nine true positive results. Now turn to the healthy majority. Ninety-nine percent specificity leaves a one percent false-positive rate. One percent sounds tiny, but it acts on nine thousand nine hundred ninety people. One percent of that enormous healthy group is ninety-nine point nine false positives. The decimal counts are expected counts, averages over many equally sized cohorts. In one real cohort we would see whole people, very close to these values.

### [02:11.741 · Read the Positive Pile](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=131.74064583333333)

The test has now run on all ten thousand people. But after a positive result, most of that original cohort is no longer relevant. We need a new reference group: everyone whose result was positive. The green pile contains the positive results from people who truly have the disease. Its expected size is nine point nine. These are the true positives. The yellow pile contains positive results from healthy people. Its expected size is ninety-nine point nine. These are false positives, contributed by the enormous healthy majority. Pause on the picture. The test is excellent inside either group. Yet the false-positive pile is about ten times taller, because the healthy group supplying it began nine hundred ninety-nine times larger than the disease group. Now gather the two piles. Write nine point nine true positives above ninety-nine point nine false positives. Rule beneath them and add. The positive-test group contains about one hundred nine point eight people. To answer our question, ask what fraction of that positive group came from the green pile. The numerator is nine point nine true positives. The denominator is every positive result, one hundred nine point eight. That fraction is about nine percent. So after one positive test, the chance of actually having the disease is only about nine percent under our assumptions. The complementary probability is about ninety-one percent. In other words, this positive result is probably wrong, even though the test has ninety-nine percent sensitivity and ninety-nine percent specificity. Nothing paradoxical happened. The test made errors on only one percent of healthy people. There were simply so many healthy people that their small error rate produced far more positive results than the rare disease did.

### [04:11.752 · Bayes Names the Count](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=251.75231250000002)

Now that the populations are visible, we can compress the same reasoning into Bayes' theorem. Start with the rule we already used: true positives divided by all positive results. The vertical bar means given. P of D given positive asks: among people known to have a positive result, what fraction have disease? The phrase after the bar names the reference group. Before building the formula, name its ingredients. P of D is prevalence, the fraction who have the disease before testing. Here it is zero point zero zero one. P of positive given D is sensitivity, the positive rate inside the disease group. Here it is zero point nine nine. P of D complement is the healthy share, zero point nine nine nine. And P of positive given D complement is the false-positive rate, zero point zero one. Now replace each pile by the probability that creates it. Sensitivity times prevalence creates the true-positive share. False-positive rate times healthy share creates the false-positive share. The numerator keeps the true-positive route. The denominator adds both routes into the positive group. This is exactly what the two visible piles did, with the common population size canceled out. Substitute our values. The true-positive route is zero point nine nine times zero point zero zero one. The false-positive route is zero point zero one times zero point nine nine nine. The result is about zero point zero nine zero, or nine percent. Bayes' theorem has not introduced a new argument. It has merely named the count in a form that works for any cohort size. The common mistake is to reverse the condition. Ninety-nine percent sensitivity describes positive results among people already known to have disease. We wanted disease among people already known to have a positive result. Those are different questions.

### [06:25.53 · Watch the Answer Swing](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=385.5296875)

Bayes' formula lets us change one ingredient at a time. First keep the test fixed at ninety-nine percent sensitivity and specificity, and vary only the disease prevalence. The horizontal coordinate is prevalence as a percentage. The vertical coordinate is the chance of disease after a positive result. At zero point one percent prevalence, our yellow point reads about nine percent. Raise prevalence to one percent. Now one person in a hundred has the disease before testing. The point climbs to fifty percent, because the expected true-positive and false-positive piles are equal. Raise prevalence to five percent. The test has not improved at all, but the positive result now means about eighty-three point nine percent. At ten percent prevalence, the posterior reaches about ninety-one point seven percent. The same test result means something very different in a high-risk population than in a low-risk population. Prevalence is the starting information, sometimes called the prior probability. A positive test updates that starting point. It does not erase it. Now restore the very rare prevalence of zero point one percent and change the test itself. To keep the phrase accuracy unambiguous, a will mean both sensitivity and specificity. At ninety-nine percent accuracy, the point again sits near nine percent. Its label gives accuracy first and the posterior probability second. Drop accuracy to ninety-five percent. The posterior falls below two percent. A five percent false-positive rate applied to nearly ten thousand healthy people overwhelms the true-positive pile. Return to ninety-nine percent, and we recover about nine percent. Now push the accuracy to ninety-nine point nine percent. The false-positive rate falls from one percent to one tenth of one percent. That extra nine in the accuracy raises the posterior to about fifty percent. For an extremely rare disease, tiny changes in the false-positive rate can matter enormously because that rate acts on the healthy majority. So the phrase ninety-nine percent accurate is incomplete on its own. We need sensitivity, specificity, and prevalence. Change any one of them and the meaning of a positive result can swing dramatically.

### [08:55.657 · Test Again](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=535.6572708333333)

Suppose the same person is tested again and the second result is also positive. Begin with the group that survived the first test: about nine point nine true positives and ninety-nine point nine false positives. Assume the second test is conditionally independent of the first. Among the people who truly have disease, it again detects ninety-nine percent. Ninety-nine percent of nine point nine is nine point eight zero one. Among the healthy people who produced the first false positive, only one percent produce another false positive independently. One percent of ninety-nine point nine is zero point nine nine nine. Now read the two surviving piles. About nine point eight people are true positives twice, while about one person is falsely positive twice. The green pile is finally much larger than the yellow pile. The probability of disease after two positive results is the green count divided by the two surviving counts together. That is about ninety point eight percent. One independent repeat test has moved the answer from about nine percent to about ninety-one percent by filtering both piles again. There is a compact way to understand that jump. Start with disease odds of ten to nine thousand nine hundred ninety, which reduce to one to nine hundred ninety-nine. A positive result is ninety-nine times more likely when disease is present than when it is absent. That factor, ninety-nine, is called the positive likelihood ratio. The first positive result multiplies the prior odds by ninety-nine. That produces the same roughly nine percent probability we found from the first two piles. Under conditional independence, the second positive multiplies by the same factor again. Two positives contribute ninety-nine times ninety-nine, changing the odds by a factor of nine thousand eight hundred one. Convert those final odds back to a probability and we recover ninety point eight percent. The count method and the odds method are two views of the same update. The independence assumption matters. If both tests use the same sample, the same instrument, or the same biological signal, their errors may be correlated. A repeated error can then be more likely than this calculation assumes, so the second positive may add less evidence. The lesson is not to distrust accurate tests. It is to ask the complete question. How rare is the disease? What are the sensitivity and specificity? And is new evidence genuinely independent? With those facts, a surprising positive result becomes a count we can understand.

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## Complete audiovisual record

Immutable source: [semantic.json](https://academa.ai/media/l/01M14V03BNSC4HNC1D114614TX/0/semantic.json)

Record version: 1. Render attempt: 0.

### 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. Action times are absolute positions in the published video.

### Scene 1: [Ten Thousand People](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=0)

Span: 00:00–02:11.741 (0s–131.74064583333333s).

#### Objects

- assumption: a Text \[text\] that says "Assumption: 99% sensitivity and 99% specificity."
- card: a Title that says "Probability for Everyday Decisions — Why a 99% Accurate Test Can Still Be Wrong"
- crowd: a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 6.0), aspect=(5.0, 3.0))
- disease\_dots: a Point \[red\] drawn in crowd (location=(0.75, 4.9))
- disease\_dots\_10: a Point \[red\] drawn in crowd (location=(2.4299999999999997, 5.32))
- disease\_dots\_2: a Point \[red\] drawn in crowd (location=(1.17, 4.9))
- disease\_dots\_3: a Point \[red\] drawn in crowd (location=(1.5899999999999999, 4.9))
- disease\_dots\_4: a Point \[red\] drawn in crowd (location=(2.01, 4.9))
- disease\_dots\_5: a Point \[red\] drawn in crowd (location=(2.4299999999999997, 4.9))
- disease\_dots\_6: a Point \[red\] drawn in crowd (location=(0.75, 5.32))
- disease\_dots\_7: a Point \[red\] drawn in crowd (location=(1.17, 5.32))
- disease\_dots\_8: a Point \[red\] drawn in crowd (location=(1.5899999999999999, 5.32))
- disease\_dots\_9: a Point \[red\] drawn in crowd (location=(2.01, 5.32))
- disease\_label: a Math \[red\] that says "$10 thin upright("people with disease")$" drawn in crowd
- false\_positive\_count: a Math \[text\] that says "$upright("false positives") = 0.01 dot.op 9990 = 99.9$"
- false\_positive\_rate: a Math \[text\] that says "$upright("false-positive rate") = 100% - 99% = 1%$"
- healthy\_block: a Polygon \[gray\] drawn in crowd (vertices=((0.4, 0.4), (9.6, 0.4), (9.6, 4.4), (0.4, 4.4)), fill\_opacity=0.18)
- healthy\_count: a Math \[text\] that says "$upright("without disease") = 9990$"
- healthy\_label: a Math \[gray\] that says "$9990 thin upright("people without disease")$" drawn in crowd
- population: a Math \[text\] that says "$upright("population") = 10000$"
- prevalence: a Math \[text\] that says "$P(D) = 0.1% = frac(10, 10000)$"
- question: a Panel that says "A test described as 99% accurate says you have a rare disease. How likely is it that the positive result is correct?"
- true\_positive\_count: a Math \[text\] that says "$upright("true positives") = 0.99 dot.op 10 = 9.9$"

#### Beats

##### [00:00](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=0)

Narration: Suppose a medical test is described as ninety-nine percent accurate, and it comes back positive. That sounds almost conclusive. But if the disease is rare, the positive result can still be more likely wrong than right. We are going to see why by counting people before writing any probability formula.

Board: Empty.

Actions:
- [00:00](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=0): card is shown on the screen, written out.
- [00:1.5](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=1.5): card: enter:write-left-to-right.
- [00:17.96](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=17.9605): card is hidden from the screen — left the board.

##### [00:19.16](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=19.1605)

Narration: Here is the question in its most personal form. A positive result says you have a rare disease. Does ninety-nine percent accurate mean there is a ninety-nine percent chance you have it? No. That number describes how the test behaves inside known groups. It does not yet answer what group a positive person probably came from.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [00:19.16](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=19.1605): question is shown on the screen, written out.
- [00:38.793](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=38.792500000000004): question moves to a new place on the board.

##### [00:39.392](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=39.3925)

Narration: Take ten thousand people. The large gray field represents the people in this cohort who do not have the disease. I have magnified the affected people so we can actually see them.

Board: question — a Panel that says "A test described as 99% accurate says you have a rare disease. How likely is it that the positive result is correct?"

Actions:
- [00:39.392](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=39.3925): crowd is shown on the screen, written out.
- [00:40.066](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=40.066): population is shown on the screen, written out.
- [00:42.365](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=42.364999999999995): healthy\_block is shown on the screen, written out.
- [00:45.024](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=45.024): healthy\_label is shown on the screen, written out.

##### [00:51.103](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=51.1035)

Narration: Let the disease affect one person in a thousand. That is a prevalence of zero point one percent. In ten thousand people, only ten actually have the disease. The remaining nine thousand nine hundred ninety are healthy.

Board: question — a Panel that says "A test described as 99% accurate says you have a rare disease. How likely is it that the positive result is correct?"; population — a Math \[text\] that says "$upright("population") = 10000$"; crowd — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 6.0), aspect=(5.0, 3.0)); healthy\_block — a Polygon \[gray\] drawn in crowd (vertices=((0.4, 0.4), (9.6, 0.4), (9.6, 4.4), (0.4, 4.4)), fill\_opacity=0.18); healthy\_label — a Math \[gray\] that says "$9990 thin upright("people without disease")$" drawn in crowd

Actions:
- [00:55.132](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=55.132000000000005): prevalence is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_2 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_3 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_4 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_5 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_6 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_7 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_8 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_9 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_dots\_10 is shown on the screen, written out.
- [00:58.081](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=58.080999999999996): disease\_label is shown on the screen, written out.
- [01:2.296](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=62.296): healthy\_count is shown on the screen, written out.
- [01:4.374](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=64.374): healthy\_block is indicated — a transient flash.
- [01:5.233](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=65.233): crowd moves to a new place on the board.
- [01:5.233](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=65.233): healthy\_count is hidden from the screen — left the board.
- [01:5.233](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=65.233): population is hidden from the screen — left the board.
- [01:5.233](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=65.233): prevalence is hidden from the screen — left the board.

##### [01:5.833](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=65.833)

Narration: Now we must say exactly what ninety-nine percent accurate means. For this lecture, it means two things. Sensitivity is ninety-nine percent, so among people who truly have the disease, the test is positive ninety-nine percent of the time. Specificity is also ninety-nine percent, so among healthy people, the test is negative ninety-nine percent of the time.

Board: question — a Panel that says "A test described as 99% accurate says you have a rare disease. How likely is it that the positive result is correct?"; crowd — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 6.0), aspect=(5.0, 3.0)); healthy\_block — a Polygon \[gray\] drawn in crowd (vertices=((0.4, 0.4), (9.6, 0.4), (9.6, 4.4), (0.4, 4.4)), fill\_opacity=0.18); healthy\_label — a Math \[gray\] that says "$9990 thin upright("people without disease")$" drawn in crowd; disease\_dots — a Point \[red\] drawn in crowd (location=(0.75, 4.9)); disease\_dots\_2 — a Point \[red\] drawn in crowd (location=(1.17, 4.9)); disease\_dots\_3 — a Point \[red\] drawn in crowd (location=(1.5899999999999999, 4.9)); disease\_dots\_4 — a Point \[red\] drawn in crowd (location=(2.01, 4.9)); disease\_dots\_5 — a Point \[red\] drawn in crowd (location=(2.4299999999999997, 4.9)); disease\_dots\_6 — a Point \[red\] drawn in crowd (location=(0.75, 5.32)); disease\_dots\_7 — a Point \[red\] drawn in crowd (location=(1.17, 5.32)); disease\_dots\_8 — a Point \[red\] drawn in crowd (location=(1.5899999999999999, 5.32)); disease\_dots\_9 — a Point \[red\] drawn in crowd (location=(2.01, 5.32)); disease\_dots\_10 — a Point \[red\] drawn in crowd (location=(2.4299999999999997, 5.32)); disease\_label — a Math \[red\] that says "$10 thin upright("people with disease")$" drawn in crowd

Actions:
- [01:5.833](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=65.833): assumption is shown on the screen, written out.
- [01:13.032](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=73.032): disease\_label is indicated — a transient flash.
- [01:21.031](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=81.03099999999999): healthy\_label is indicated — a transient flash.

##### [01:28.875](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=88.875)

Narration: Apply sensitivity to the ten affected people. Ninety-nine percent of ten is nine point nine. So across many cohorts like this one, we expect about nine point nine true positive results.

Board: question — a Panel that says "A test described as 99% accurate says you have a rare disease. How likely is it that the positive result is correct?"; crowd — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 6.0), aspect=(5.0, 3.0)); healthy\_block — a Polygon \[gray\] drawn in crowd (vertices=((0.4, 0.4), (9.6, 0.4), (9.6, 4.4), (0.4, 4.4)), fill\_opacity=0.18); healthy\_label — a Math \[gray\] that says "$9990 thin upright("people without disease")$" drawn in crowd; disease\_dots — a Point \[red\] drawn in crowd (location=(0.75, 4.9)); disease\_dots\_2 — a Point \[red\] drawn in crowd (location=(1.17, 4.9)); disease\_dots\_3 — a Point \[red\] drawn in crowd (location=(1.5899999999999999, 4.9)); disease\_dots\_4 — a Point \[red\] drawn in crowd (location=(2.01, 4.9)); disease\_dots\_5 — a Point \[red\] drawn in crowd (location=(2.4299999999999997, 4.9)); disease\_dots\_6 — a Point \[red\] drawn in crowd (location=(0.75, 5.32)); disease\_dots\_7 — a Point \[red\] drawn in crowd (location=(1.17, 5.32)); disease\_dots\_8 — a Point \[red\] drawn in crowd (location=(1.5899999999999999, 5.32)); disease\_dots\_9 — a Point \[red\] drawn in crowd (location=(2.01, 5.32)); disease\_dots\_10 — a Point \[red\] drawn in crowd (location=(2.4299999999999997, 5.32)); disease\_label — a Math \[red\] that says "$10 thin upright("people with disease")$" drawn in crowd; assumption — a Text \[text\] that says "Assumption: 99% sensitivity and 99% specificity."

Actions:
- [01:32.706](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=92.70599999999999): true\_positive\_count is shown on the screen, written out.
- [01:38.546](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=98.54599999999998): true\_positive\_count (the "9.9" part) is emphasized.
- [01:41.448](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=101.4485): true\_positive\_count (the "9.9" part) is no longer emphasized.

##### [01:42.049](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=102.0485)

Narration: Now turn to the healthy majority. Ninety-nine percent specificity leaves a one percent false-positive rate. One percent sounds tiny, but it acts on nine thousand nine hundred ninety people.

Board: question — a Panel that says "A test described as 99% accurate says you have a rare disease. How likely is it that the positive result is correct?"; crowd — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 6.0), aspect=(5.0, 3.0)); healthy\_block — a Polygon \[gray\] drawn in crowd (vertices=((0.4, 0.4), (9.6, 0.4), (9.6, 4.4), (0.4, 4.4)), fill\_opacity=0.18); healthy\_label — a Math \[gray\] that says "$9990 thin upright("people without disease")$" drawn in crowd; disease\_dots — a Point \[red\] drawn in crowd (location=(0.75, 4.9)); disease\_dots\_2 — a Point \[red\] drawn in crowd (location=(1.17, 4.9)); disease\_dots\_3 — a Point \[red\] drawn in crowd (location=(1.5899999999999999, 4.9)); disease\_dots\_4 — a Point \[red\] drawn in crowd (location=(2.01, 4.9)); disease\_dots\_5 — a Point \[red\] drawn in crowd (location=(2.4299999999999997, 4.9)); disease\_dots\_6 — a Point \[red\] drawn in crowd (location=(0.75, 5.32)); disease\_dots\_7 — a Point \[red\] drawn in crowd (location=(1.17, 5.32)); disease\_dots\_8 — a Point \[red\] drawn in crowd (location=(1.5899999999999999, 5.32)); disease\_dots\_9 — a Point \[red\] drawn in crowd (location=(2.01, 5.32)); disease\_dots\_10 — a Point \[red\] drawn in crowd (location=(2.4299999999999997, 5.32)); disease\_label — a Math \[red\] that says "$10 thin upright("people with disease")$" drawn in crowd; assumption — a Text \[text\] that says "Assumption: 99% sensitivity and 99% specificity."; true\_positive\_count — a Math \[text\] that says "$upright("true positives") = 0.99 dot.op 10 = 9.9$"

Actions:
- [01:43.465](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=103.46499999999999): healthy\_block is indicated — a transient flash.
- [01:47.204](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=107.204): false\_positive\_rate is shown on the screen, written out.

##### [01:55.014](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=115.01350000000001)

Narration: One percent of that enormous healthy group is ninety-nine point nine false positives. The decimal counts are expected counts, averages over many equally sized cohorts. In one real cohort we would see whole people, very close to these values.

Board: question — a Panel that says "A test described as 99% accurate says you have a rare disease. How likely is it that the positive result is correct?"; crowd — a Figure (x\_range=(0.0, 10.0), y\_range=(0.0, 6.0), aspect=(5.0, 3.0)); healthy\_block — a Polygon \[gray\] drawn in crowd (vertices=((0.4, 0.4), (9.6, 0.4), (9.6, 4.4), (0.4, 4.4)), fill\_opacity=0.18); healthy\_label — a Math \[gray\] that says "$9990 thin upright("people without disease")$" drawn in crowd; disease\_dots — a Point \[red\] drawn in crowd (location=(0.75, 4.9)); disease\_dots\_2 — a Point \[red\] drawn in crowd (location=(1.17, 4.9)); disease\_dots\_3 — a Point \[red\] drawn in crowd (location=(1.5899999999999999, 4.9)); disease\_dots\_4 — a Point \[red\] drawn in crowd (location=(2.01, 4.9)); disease\_dots\_5 — a Point \[red\] drawn in crowd (location=(2.4299999999999997, 4.9)); disease\_dots\_6 — a Point \[red\] drawn in crowd (location=(0.75, 5.32)); disease\_dots\_7 — a Point \[red\] drawn in crowd (location=(1.17, 5.32)); disease\_dots\_8 — a Point \[red\] drawn in crowd (location=(1.5899999999999999, 5.32)); disease\_dots\_9 — a Point \[red\] drawn in crowd (location=(2.01, 5.32)); disease\_dots\_10 — a Point \[red\] drawn in crowd (location=(2.4299999999999997, 5.32)); disease\_label — a Math \[red\] that says "$10 thin upright("people with disease")$" drawn in crowd; assumption — a Text \[text\] that says "Assumption: 99% sensitivity and 99% specificity."; true\_positive\_count — a Math \[text\] that says "$upright("true positives") = 0.99 dot.op 10 = 9.9$"; false\_positive\_rate — a Math \[text\] that says "$upright("false-positive rate") = 100% - 99% = 1%$"

Actions:
- [01:55.42](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=115.42): false\_positive\_count is shown on the screen, written out.
- [01:58.171](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=118.17099999999999): false\_positive\_count (the "99.9" part) is emphasized.
- [02:10.449](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.44897916666667): false\_positive\_count (the "99.9" part) is no longer emphasized.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): assumption is hidden from the screen — left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): crowd is hidden from the screen — left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): healthy\_block is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): healthy\_label is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_2 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_3 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_4 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_5 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_6 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_7 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_8 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_9 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_dots\_10 is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): disease\_label is hidden from the screen — crowd left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): false\_positive\_count is hidden from the screen — left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): false\_positive\_rate is hidden from the screen — left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): question is hidden from the screen — left the board.
- [02:10.699](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=130.69897916666667): true\_positive\_count is hidden from the screen — left the board.

### Scene 2: [Read the Positive Pile](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=131.74064583333333)

Span: 02:11.741–04:11.752 (131.74064583333333s–251.75231250000002s).

#### Objects

- baseline: a Line \[gray\] drawn in piles (start=(0.25, 0.0), end=(4.75, 0.0))
- false\_bar: a Polygon \[yellow\] drawn in piles (vertices=((3.15, 0.0), (4.45, 0.0), (4.45, 99.9), (3.15, 99.9)), fill\_opacity=0.55)
- false\_name: a Math \[yellow\] that says "$upright("false positives")$" drawn in piles
- false\_value: a Math \[yellow\] that says "$99.9$" drawn in piles
- piles: a Figure (x\_range=(0.0, 5.0), y\_range=(-18.0, 112.0))
- positive\_sum: an Arithmetic \[text\] that says "$9.9 99.9 109.8$" (operator='+', operands=('9.9', '99.9'), result='109.8')
- posterior: a Math \[text\] that says "$P(D \| +) = frac(9.9, 109.8)$"
- prompt: a Tex \[text\] that says "Gather everyone whose test is positive."
- true\_bar: a Polygon \[green\] drawn in piles (vertices=((0.55, 0.0), (1.85, 0.0), (1.85, 9.9), (0.55, 9.9)), fill\_opacity=0.75)
- true\_name: a Math \[green\] that says "$upright("true positives")$" drawn in piles
- true\_value: a Math \[green\] that says "$9.9$" drawn in piles
- wrong: a Math \[text\] that says "$P(D^c \| +) approx 91.0%$"

#### Beats

##### [02:11.741](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=131.74064583333333)

Narration: The test has now run on all ten thousand people. But after a positive result, most of that original cohort is no longer relevant. We need a new reference group: everyone whose result was positive.

Board: Empty.

Actions:
- [02:11.741](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=131.74064583333333): piles is shown on the screen, written out.
- [02:11.741](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=131.74064583333333): baseline is shown on the screen, written out.
- [02:15.619](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=135.61864583333335): piles moves to a new place on the board.
- [02:15.619](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=135.61864583333335): prompt is shown on the screen, written out.

##### [02:25.077](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=145.07664583333334)

Narration: The green pile contains the positive results from people who truly have the disease. Its expected size is nine point nine. These are the true positives.

Board: piles — a Figure (x\_range=(0.0, 5.0), y\_range=(-18.0, 112.0)); prompt — a Tex \[text\] that says "Gather everyone whose test is positive."; baseline — a Line \[gray\] drawn in piles (start=(0.25, 0.0), end=(4.75, 0.0))

Actions:
- [02:25.588](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=145.58764583333334): true\_bar is shown on the screen, written out.
- [02:31.869](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=151.86864583333335): true\_value is shown on the screen, written out.
- [02:33.645](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=153.64464583333333): true\_name is shown on the screen, written out.

##### [02:35.534](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=155.53364583333334)

Narration: The yellow pile contains positive results from healthy people. Its expected size is ninety-nine point nine. These are false positives, contributed by the enormous healthy majority.

Board: piles — a Figure (x\_range=(0.0, 5.0), y\_range=(-18.0, 112.0)); prompt — a Tex \[text\] that says "Gather everyone whose test is positive."; baseline — a Line \[gray\] drawn in piles (start=(0.25, 0.0), end=(4.75, 0.0)); true\_bar — a Polygon \[green\] drawn in piles (vertices=((0.55, 0.0), (1.85, 0.0), (1.85, 9.9), (0.55, 9.9)), fill\_opacity=0.75); true\_value — a Math \[green\] that says "$9.9$" drawn in piles; true\_name — a Math \[green\] that says "$upright("true positives")$" drawn in piles

Actions:
- [02:36.115](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=156.11464583333333): false\_bar is shown on the screen, written out.
- [02:40.851](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=160.85064583333332): false\_value is shown on the screen, written out.
- [02:43](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=162.99964583333332): false\_name is shown on the screen, written out.

##### [02:47.616](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=167.61614583333335)

Narration: Pause on the picture. The test is excellent inside either group. Yet the false-positive pile is about ten times taller, because the healthy group supplying it began nine hundred ninety-nine times larger than the disease group.

Board: piles — a Figure (x\_range=(0.0, 5.0), y\_range=(-18.0, 112.0)); prompt — a Tex \[text\] that says "Gather everyone whose test is positive."; baseline — a Line \[gray\] drawn in piles (start=(0.25, 0.0), end=(4.75, 0.0)); true\_bar — a Polygon \[green\] drawn in piles (vertices=((0.55, 0.0), (1.85, 0.0), (1.85, 9.9), (0.55, 9.9)), fill\_opacity=0.75); true\_value — a Math \[green\] that says "$9.9$" drawn in piles; true\_name — a Math \[green\] that says "$upright("true positives")$" drawn in piles; false\_bar — a Polygon \[yellow\] drawn in piles (vertices=((3.15, 0.0), (4.45, 0.0), (4.45, 99.9), (3.15, 99.9)), fill\_opacity=0.55); false\_value — a Math \[yellow\] that says "$99.9$" drawn in piles; false\_name — a Math \[yellow\] that says "$upright("false positives")$" drawn in piles

Actions:
- [02:50.38](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=170.37964583333334): true\_bar is indicated — a transient flash.
- [02:54.501](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=174.50064583333335): false\_bar is indicated — a transient flash.
- [02:56.115](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=176.11464583333333): false\_name is indicated — a transient flash.

##### [03:2.044](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=182.04414583333335)

Narration: Now gather the two piles. Write nine point nine true positives above ninety-nine point nine false positives. Rule beneath them and add. The positive-test group contains about one hundred nine point eight people.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [03:5.133](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=185.13264583333333): positive\_sum is shown on the screen, written out.
- [03:7.432](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=187.43164583333333): positive\_sum is shown on the screen, written out.
- [03:10.334](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=190.33364583333332): positive\_sum is shown on the screen, drawn.
- [03:11.134](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=191.13364583333333): positive\_sum is shown on the screen, drawn.
- [03:13.109](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=193.10864583333333): positive\_sum is indicated — a transient flash.
- [03:15.129](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=195.12864583333334): positive\_sum is shown on the screen, written out.

##### [03:17.877](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=197.87664583333333)

Narration: To answer our question, ask what fraction of that positive group came from the green pile. The numerator is nine point nine true positives. The denominator is every positive result, one hundred nine point eight.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [03:20.361](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=200.36064583333334): posterior is shown on the screen, written out.
- [03:24.042](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=204.04164583333332): posterior (the "9.9" part) is emphasized.
- [03:27.397](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=207.39664583333334): posterior (the "109.8" part) is emphasized.
- [03:27.397](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=207.39664583333334): posterior (the "9.9" part) is no longer emphasized.
- [03:31.542](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=211.54164583333332): posterior (the "109.8" part) is no longer emphasized.

##### [03:32.142](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=212.1416458333333)

Narration: That fraction is about nine percent. So after one positive test, the chance of actually having the disease is only about nine percent under our assumptions.

Board: posterior — a Math \[text\] that says "$P(D \| +) = frac(9.9, 109.8)$"; piles — a Figure (x\_range=(0.0, 5.0), y\_range=(-18.0, 112.0)); prompt — a Tex \[text\] that says "Gather everyone whose test is positive."; baseline — a Line \[gray\] drawn in piles (start=(0.25, 0.0), end=(4.75, 0.0)); true\_bar — a Polygon \[green\] drawn in piles (vertices=((0.55, 0.0), (1.85, 0.0), (1.85, 9.9), (0.55, 9.9)), fill\_opacity=0.75); true\_value — a Math \[green\] that says "$9.9$" drawn in piles; true\_name — a Math \[green\] that says "$upright("true positives")$" drawn in piles; false\_bar — a Polygon \[yellow\] drawn in piles (vertices=((3.15, 0.0), (4.45, 0.0), (4.45, 99.9), (3.15, 99.9)), fill\_opacity=0.55); false\_value — a Math \[yellow\] that says "$99.9$" drawn in piles; false\_name — a Math \[yellow\] that says "$upright("false positives")$" drawn in piles

Actions:
- [03:33.802](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=213.80164583333334): posterior becomes "$P(D \| +) = frac(9.9, 109.8) approx 9.0%$".
- [03:42.382](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=222.38164583333332): A box is drawn around posterior.

##### [03:42.982](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=222.98164583333335)

Narration: The complementary probability is about ninety-one percent. In other words, this positive result is probably wrong, even though the test has ninety-nine percent sensitivity and ninety-nine percent specificity.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [03:45.594](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=225.5936458333333): wrong is shown on the screen, written out.
- [03:45.594](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=225.5936458333333): wrong (the "91.0%" part) is emphasized.
- [03:56.425](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=236.42514583333332): wrong (the "91.0%" part) is no longer emphasized.

##### [03:57.025](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=237.02514583333334)

Narration: Nothing paradoxical happened. The test made errors on only one percent of healthy people. There were simply so many healthy people that their small error rate produced far more positive results than the rare disease did.

Board: posterior — a Math \[text\] that says "$P(D \| +) = frac(9.9, 109.8)$"; wrong — a Math \[text\] that says "$P(D^c \| +) approx 91.0%$"; piles — a Figure (x\_range=(0.0, 5.0), y\_range=(-18.0, 112.0)); prompt — a Tex \[text\] that says "Gather everyone whose test is positive."; baseline — a Line \[gray\] drawn in piles (start=(0.25, 0.0), end=(4.75, 0.0)); true\_bar — a Polygon \[green\] drawn in piles (vertices=((0.55, 0.0), (1.85, 0.0), (1.85, 9.9), (0.55, 9.9)), fill\_opacity=0.75); true\_value — a Math \[green\] that says "$9.9$" drawn in piles; true\_name — a Math \[green\] that says "$upright("true positives")$" drawn in piles; false\_bar — a Polygon \[yellow\] drawn in piles (vertices=((3.15, 0.0), (4.45, 0.0), (4.45, 99.9), (3.15, 99.9)), fill\_opacity=0.55); false\_value — a Math \[yellow\] that says "$99.9$" drawn in piles; false\_name — a Math \[yellow\] that says "$upright("false positives")$" drawn in piles

Actions:
- [04:4.387](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=244.38664583333332): false\_bar is indicated — a transient flash.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): piles is hidden from the screen — left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): baseline is hidden from the screen — piles left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): true\_bar is hidden from the screen — piles left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): true\_value is hidden from the screen — piles left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): true\_name is hidden from the screen — piles left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): false\_bar is hidden from the screen — piles left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): false\_value is hidden from the screen — piles left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): false\_name is hidden from the screen — piles left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): positive\_sum is hidden from the screen — left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): posterior is hidden from the screen — left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): prompt is hidden from the screen — left the board.
- [04:10.711](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=250.71064583333333): wrong is hidden from the screen — left the board.

### Scene 3: [Bayes Names the Count](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=251.75231250000002)

Span: 04:11.752–06:25.53 (251.75231250000002s–385.5296875s).

#### Objects

- bayes\_work: a Derivation \[text\] that says "$P(D \| +) &= frac(upright("true positives"), upright("true positives") + upright("false positives")) \\ &= frac(P(+ \| D) P(D), P(+ \| D) P(D) + P(+ \| D^c) P(D^c)) \\ &= frac(0.99 dot.op 0.001, 0.99 dot.op 0.001 + 0.01 dot.op 0.999) \\ &approx 0…$"
- distinction: a Text \[text\] that says "$P(+\|D)$ asks about test results among diseased people. $P(D\|+)$ asks about disease among positive results."
- heading: a Heading that says "The Same Count, Written Generally"
- terms: a Table \[text\] that says "Term Meaning Here $P(D)$ prevalence $0.001$ $P(+\|D)$ sensitivity $0.99$ $P(D^c)$ healthy share $0.999$ $P(+\|D^c)$ false-positive rate $0.01$" (rows=(('Term', 'Meaning', 'Here'), ('$P(D)$', 'prevalence', '$0.001$…, header=True)

#### Beats

##### [04:11.752](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=251.75231250000002)

Narration: Now that the populations are visible, we can compress the same reasoning into Bayes' theorem. Start with the rule we already used: true positives divided by all positive results.

Board: Empty.

Actions:
- [04:11.752](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=251.75231250000002): heading is shown on the screen, written out.
- [04:20.018](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=260.01831250000004): bayes\_work is shown on the screen, written out.

##### [04:24.044](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=264.0438125)

Narration: The vertical bar means given. P of D given positive asks: among people known to have a positive result, what fraction have disease? The phrase after the bar names the reference group.

Board: heading — a Heading that says "The Same Count, Written Generally"

Actions:
- [04:25.727](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=265.72731250000004): bayes\_work (the "D \| +" part) is emphasized.
- [04:36.431](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=276.4313125): bayes\_work (the "D \| +" part) is no longer emphasized.

##### [04:38.122](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=278.1223125)

Narration: Before building the formula, name its ingredients. P of D is prevalence, the fraction who have the disease before testing. Here it is zero point zero zero one.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [04:40.816](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=280.81631250000004): terms is shown on the screen, written out.
- [04:42.894](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=282.8943125): terms is shown on the screen, written out.
- [04:42.894](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=282.8943125): terms (the "row=2" part) is emphasized.
- [04:50.139](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=290.13881250000003): terms (the "row=2" part) is no longer emphasized.

##### [04:50.739](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=290.7388125)

Narration: P of positive given D is sensitivity, the positive rate inside the disease group. Here it is zero point nine nine.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [04:53.269](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=293.2693125): terms is shown on the screen, written out.
- [04:53.269](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=293.2693125): terms (the "row=3" part) is emphasized.
- [05:0.142](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=300.1423125): terms (the "row=3" part) is no longer emphasized.

##### [05:0.742](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=300.7423125)

Narration: P of D complement is the healthy share, zero point nine nine nine. And P of positive given D complement is the false-positive rate, zero point zero one.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:2.6](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=302.60031250000003): terms is shown on the screen, written out.
- [05:2.6](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=302.60031250000003): terms (the "row=4" part) is emphasized.
- [05:9.264](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=309.2643125): terms is shown on the screen, written out.
- [05:9.264](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=309.2643125): terms (the "row=4" part) is no longer emphasized.
- [05:9.264](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=309.2643125): terms (the "row=5" part) is emphasized.
- [05:12.597](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=312.5968125): terms (the "row=5" part) is no longer emphasized.

##### [05:13.197](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=313.1968125)

Narration: Now replace each pile by the probability that creates it. Sensitivity times prevalence creates the true-positive share. False-positive rate times healthy share creates the false-positive share.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:13.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=313.90531250000004): bayes\_work is shown on the screen, written out.
- [05:17.585](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=317.5853125): bayes\_work (the "P(+ \| D) P(D)" part) is emphasized.
- [05:21.695](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=321.6953125): bayes\_work (the "P(+ \| D) P(D)" part) is no longer emphasized.
- [05:21.695](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=321.6953125): bayes\_work (the "P(+ \| D^c) P(D^c)" part) is emphasized.
- [05:27.152](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=327.1518125): bayes\_work (the "P(+ \| D^c) P(D^c)" part) is no longer emphasized.

##### [05:27.752](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=327.7518125)

Narration: The numerator keeps the true-positive route. The denominator adds both routes into the positive group. This is exactly what the two visible piles did, with the common population size canceled out.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:28.135](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=328.1353125): bayes\_work (the "P(+ \| D) P(D)" part) is emphasized.
- [05:30.794](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=330.7943125): bayes\_work (the "P(+ \| D) P(D)" part) is no longer emphasized.
- [05:30.794](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=330.7943125): bayes\_work (the "P(+ \| D) P(D) + P(+ \| D^c) P(D^c)" part) is emphasized.
- [05:40.361](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=340.3608125): bayes\_work (the "P(+ \| D) P(D) + P(+ \| D^c) P(D^c)" part) is no longer emphasized.

##### [05:40.961](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=340.9608125)

Narration: Substitute our values. The true-positive route is zero point nine nine times zero point zero zero one. The false-positive route is zero point zero one times zero point nine nine nine.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:41.367](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=341.3673125): bayes\_work is shown on the screen, written out.
- [05:42.377](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=342.3773125): terms (the "column=3" part) is emphasized.
- [05:54.777](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=354.7773125): terms (the "column=3" part) is no longer emphasized.

##### [05:55.377](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=355.3773125)

Narration: The result is about zero point zero nine zero, or nine percent. Bayes' theorem has not introduced a new argument. It has merely named the count in a form that works for any cohort size.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:55.969](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=355.9693125): bayes\_work is shown on the screen, written out.
- [05:59.173](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=359.1733125): A box is drawn around bayes\_work.

##### [06:8.435](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=368.4348125)

Narration: The common mistake is to reverse the condition. Ninety-nine percent sensitivity describes positive results among people already known to have disease. We wanted disease among people already known to have a positive result. Those are different questions.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [06:9.921](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=369.9213125): distinction is shown on the screen, written out.
- [06:12.788](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=372.7883125): distinction (the "$P(+\|D)$" part) is emphasized.
- [06:18.744](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=378.7443125): distinction (the "$P(D\|+)$" part) is emphasized.
- [06:24.238](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=384.23802083333334): distinction (the "$P(+\|D)$" part) is no longer emphasized.
- [06:24.238](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=384.23802083333334): distinction (the "$P(D\|+)$" part) is no longer emphasized.
- [06:24.488](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=384.48802083333334): bayes\_work is hidden from the screen — left the board.
- [06:24.488](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=384.48802083333334): distinction is hidden from the screen — left the board.
- [06:24.488](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=384.48802083333334): heading is hidden from the screen — left the board.
- [06:24.488](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=384.48802083333334): terms is hidden from the screen — left the board.

### Scene 4: [Watch the Answer Swing](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=385.5296875)

Span: 06:25.53–08:55.657 (385.5296875s–535.6572708333333s).

#### Objects

- accuracy: a VariableNumber (initial\_value=99.0, format\_spec='.1f')
- accuracy\_axes: an Axes (x\_range=(90.0, 99.95), y\_range=(0.0, 70.0), x\_ticks\_every=2.0)
- accuracy\_curve: a FunctionPlot \[blue\] drawn in accuracy\_axes (function=\<function\>, x\_range=(90.0, 99.95))
- accuracy\_formula: a Math \[text\] that says "$P(D \| +) = frac(a p, a p + (1-a)(1-p))$"
- accuracy\_heading: a Heading that says "Change the Accuracy"
- accuracy\_note: a Text \[text\] that says "Here a is both sensitivity and specificity, while prevalence stays at 0.1%."
- accuracy\_point: a PlotPoint \[yellow\] labelled "(99.0, 9.0)" drawn in accuracy\_axes (target='accuracy\_curve', x=\<VariableNumber accuracy = 99.9\>)
- p: a VariableNumber (initial\_value=0.1, format\_spec='.1f')
- point: a Point \[yellow\] drawn in prevalence\_axes (location=(1.0, 50.0))
- posterior\_a: a VariableNumber (initial\_value=9.0, format\_spec='.1f')
- posterior\_p: a VariableNumber (initial\_value=9.0, format\_spec='.1f')
- prevalence\_axes: an Axes (x\_range=(0.0, 10.0), y\_range=(0.0, 100.0), x\_ticks\_every=1.0)
- prevalence\_curve: a FunctionPlot \[blue\] drawn in prevalence\_axes (function=\<function\>, x\_range=(0.05, 10.0))
- prevalence\_formula: a Math \[text\] that says "$P(D \| +) = frac(0.99 p, 0.99 p + 0.01 (1-p))$"
- prevalence\_heading: a Heading that says "Change the Prevalence"
- prevalence\_note: a Text \[text\] that says "The test stays fixed. Only the fraction of people with disease changes."
- prevalence\_point: a PlotPoint \[yellow\] labelled "(0.1, 9.0)" drawn in prevalence\_axes (target='prevalence\_curve', x=\<VariableNumber p = 10.0\>)

#### Beats

##### [06:25.53](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=385.5296875)

Narration: Bayes' formula lets us change one ingredient at a time. First keep the test fixed at ninety-nine percent sensitivity and specificity, and vary only the disease prevalence.

Board: Empty.

Actions:
- [06:25.53](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=385.5296875): prevalence\_heading is shown on the screen, written out.
- [06:25.948](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=385.94768750000003): prevalence\_formula is shown on the screen, written out.
- [06:30.406](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=390.4056875): prevalence\_note is shown on the screen, written out.
- [06:34.586](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=394.5856875): prevalence\_axes is shown on the screen, written out.
- [06:34.586](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=394.5856875): prevalence\_curve is shown on the screen, drawn.

##### [06:37.531](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=397.5306875)

Narration: The horizontal coordinate is prevalence as a percentage. The vertical coordinate is the chance of disease after a positive result. At zero point one percent prevalence, our yellow point reads about nine percent.

Board: prevalence\_formula — a Math \[text\] that says "$P(D \| +) = frac(0.99 p, 0.99 p + 0.01 (1-p))$"; prevalence\_note — a Text \[text\] that says "The test stays fixed. Only the fraction of people with disease changes."; prevalence\_axes — an Axes (x\_range=(0.0, 10.0), y\_range=(0.0, 100.0), x\_ticks\_every=1.0); prevalence\_heading — a Heading that says "Change the Prevalence"; prevalence\_curve — a FunctionPlot \[blue\] drawn in prevalence\_axes (function=\<function\>, x\_range=(0.05, 10.0))

Actions:
- [06:48.595](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=408.5946875): prevalence\_point is shown on the screen, written out.
- [06:49.687](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=409.6866875): prevalence\_point is indicated — a transient flash.

##### [06:51.436](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=411.43568750000003)

Narration: Raise prevalence to one percent. Now one person in a hundred has the disease before testing. The point climbs to fifty percent, because the expected true-positive and false-positive piles are equal.

Board: prevalence\_formula — a Math \[text\] that says "$P(D \| +) = frac(0.99 p, 0.99 p + 0.01 (1-p))$"; prevalence\_note — a Text \[text\] that says "The test stays fixed. Only the fraction of people with disease changes."; prevalence\_axes — an Axes (x\_range=(0.0, 10.0), y\_range=(0.0, 100.0), x\_ticks\_every=1.0); prevalence\_heading — a Heading that says "Change the Prevalence"; prevalence\_curve — a FunctionPlot \[blue\] drawn in prevalence\_axes (function=\<function\>, x\_range=(0.05, 10.0)); prevalence\_point — a PlotPoint \[yellow\] labelled "(0.1, 9.0)" drawn in prevalence\_axes (target='prevalence\_curve', x=\<VariableNumber p = 10.0\>)

Actions:
- [06:52.876](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=412.8756875): prevalence\_point is redrawn as the numbers it depends on change.
- [06:52.876](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=412.8756875): p ticks to 1.0.
- [06:52.876](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=412.8756875): posterior\_p ticks to 50.0.
- [06:58.947](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=418.94668750000005): point is shown on the screen, grown.
- [07:0.947](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=420.94668750000005): point is hidden from the screen.

##### [07:4.644](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=424.6436875)

Narration: Raise prevalence to five percent. The test has not improved at all, but the positive result now means about eighty-three point nine percent.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [07:6.084](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=426.0836875): prevalence\_point is redrawn as the numbers it depends on change.
- [07:6.084](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=426.0836875): p ticks to 5.0.
- [07:6.084](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=426.0836875): posterior\_p ticks to 83.9.

##### [07:13.87](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=433.8696875)

Narration: At ten percent prevalence, the posterior reaches about ninety-one point seven percent. The same test result means something very different in a high-risk population than in a low-risk population.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [07:14.427](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=434.4266875): prevalence\_point is redrawn as the numbers it depends on change.
- [07:14.427](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=434.4266875): p ticks to 10.0.
- [07:14.427](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=434.4266875): posterior\_p ticks to 91.7.
- [07:21.567](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=441.5666875): prevalence\_point is indicated — a transient flash.

##### [07:26.289](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=446.28868750000004)

Narration: Prevalence is the starting information, sometimes called the prior probability. A positive test updates that starting point. It does not erase it.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [07:26.637](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=446.6366875): prevalence\_formula (the "p" part) is emphasized.
- [07:36.308](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=456.30818750000003): prevalence\_axes is hidden from the screen — left the board.
- [07:36.308](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=456.30818750000003): prevalence\_curve is hidden from the screen — prevalence\_axes left the board.
- [07:36.308](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=456.30818750000003): prevalence\_point is hidden from the screen — prevalence\_axes left the board.
- [07:36.308](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=456.30818750000003): prevalence\_formula is hidden from the screen — left the board.
- [07:36.308](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=456.30818750000003): prevalence\_heading is hidden from the screen — left the board.
- [07:36.308](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=456.30818750000003): prevalence\_note is hidden from the screen — left the board.
- [07:36.308](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=456.30818750000003): prevalence\_formula (the "p" part) is no longer emphasized.

##### [07:37.508](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=457.5081875)

Narration: Now restore the very rare prevalence of zero point one percent and change the test itself. To keep the phrase accuracy unambiguous, a will mean both sensitivity and specificity.

Board: Empty.

Actions:
- [07:37.508](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=457.5081875): accuracy\_heading is shown on the screen, written out.
- [07:41.897](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=461.89668750000004): accuracy\_axes is shown on the screen, written out.
- [07:41.897](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=461.89668750000004): accuracy\_curve is shown on the screen, drawn.
- [07:46.75](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=466.74968750000005): accuracy\_axes moves to a new place on the board.
- [07:46.75](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=466.74968750000005): accuracy\_formula is shown on the screen, written out.
- [07:47.319](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=467.3186875): accuracy\_note is shown on the screen, written out.

##### [07:50.264](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=470.26418750000005)

Narration: At ninety-nine percent accuracy, the point again sits near nine percent. Its label gives accuracy first and the posterior probability second.

Board: accuracy\_formula — a Math \[text\] that says "$P(D \| +) = frac(a p, a p + (1-a)(1-p))$"; accuracy\_note — a Text \[text\] that says "Here a is both sensitivity and specificity, while prevalence stays at 0.1%."; accuracy\_axes — an Axes (x\_range=(90.0, 99.95), y\_range=(0.0, 70.0), x\_ticks\_every=2.0); accuracy\_heading — a Heading that says "Change the Accuracy"; accuracy\_curve — a FunctionPlot \[blue\] drawn in accuracy\_axes (function=\<function\>, x\_range=(90.0, 99.95))

Actions:
- [07:52.9](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=472.8996875): accuracy\_point is shown on the screen, written out.
- [07:54.084](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=474.0836875): accuracy\_point is indicated — a transient flash.

##### [08:0.686](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=480.68568750000003)

Narration: Drop accuracy to ninety-five percent. The posterior falls below two percent. A five percent false-positive rate applied to nearly ten thousand healthy people overwhelms the true-positive pile.

Board: accuracy\_formula — a Math \[text\] that says "$P(D \| +) = frac(a p, a p + (1-a)(1-p))$"; accuracy\_note — a Text \[text\] that says "Here a is both sensitivity and specificity, while prevalence stays at 0.1%."; accuracy\_axes — an Axes (x\_range=(90.0, 99.95), y\_range=(0.0, 70.0), x\_ticks\_every=2.0); accuracy\_heading — a Heading that says "Change the Accuracy"; accuracy\_curve — a FunctionPlot \[blue\] drawn in accuracy\_axes (function=\<function\>, x\_range=(90.0, 99.95)); accuracy\_point — a PlotPoint \[yellow\] labelled "(99.0, 9.0)" drawn in accuracy\_axes (target='accuracy\_curve', x=\<VariableNumber accuracy = 99.9\>)

Actions:
- [08:2.161](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=482.1606875): accuracy\_point is redrawn as the numbers it depends on change.
- [08:2.161](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=482.1606875): accuracy ticks to 95.0.
- [08:2.161](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=482.1606875): posterior\_a ticks to 1.9.

##### [08:13.918](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=493.9176875)

Narration: Return to ninety-nine percent, and we recover about nine percent. Now push the accuracy to ninety-nine point nine percent. The false-positive rate falls from one percent to one tenth of one percent.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [08:14.812](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=494.8116875): accuracy\_point is redrawn as the numbers it depends on change.
- [08:14.812](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=494.8116875): accuracy ticks to 99.0.
- [08:14.812](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=494.8116875): posterior\_a ticks to 9.0.
- [08:19.816](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=499.81568749999997): accuracy\_point is redrawn as the numbers it depends on change.
- [08:19.816](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=499.81568749999997): accuracy ticks to 99.9.
- [08:19.816](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=499.81568749999997): posterior\_a ticks to 50.0.

##### [08:27.335](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=507.3351875)

Narration: That extra nine in the accuracy raises the posterior to about fifty percent. For an extremely rare disease, tiny changes in the false-positive rate can matter enormously because that rate acts on the healthy majority.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [08:31.098](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=511.0976875): accuracy\_point is indicated — a transient flash.
- [08:39.596](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=519.5956874999999): accuracy\_formula (the "(1-a)(1-p)" part) is emphasized.
- [08:41.105](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=521.1051875): accuracy\_formula (the "(1-a)(1-p)" part) is no longer emphasized.

##### [08:41.705](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=521.7051875)

Narration: So the phrase ninety-nine percent accurate is incomplete on its own. We need sensitivity, specificity, and prevalence. Change any one of them and the meaning of a positive result can swing dramatically.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [08:54.616](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=534.6156041666667): accuracy\_axes is hidden from the screen — left the board.
- [08:54.616](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=534.6156041666667): accuracy\_curve is hidden from the screen — accuracy\_axes left the board.
- [08:54.616](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=534.6156041666667): accuracy\_point is hidden from the screen — accuracy\_axes left the board.
- [08:54.616](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=534.6156041666667): accuracy\_formula is hidden from the screen — left the board.
- [08:54.616](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=534.6156041666667): accuracy\_heading is hidden from the screen — left the board.
- [08:54.616](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=534.6156041666667): accuracy\_note is hidden from the screen — left the board.

### Scene 5: [Test Again](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=535.6572708333333)

Span: 08:55.657–11:35.267 (535.6572708333333s–695.2673125s).

#### Objects

- after\_false: a Polygon \[yellow\] drawn in filter\_picture (vertices=((6.75, 0.0), (7.85, 0.0), (7.85, 0.999), (6.75, 0.999)), fill\_opacity=0.65)
- after\_false\_value: a Math \[yellow\] that says "$0.999$" drawn in filter\_picture
- after\_label: a Math \[text\] that says "$upright("positive twice")$" drawn in filter\_picture
- after\_true: a Polygon \[green\] drawn in filter\_picture (vertices=((5.15, 0.0), (6.25, 0.0), (6.25, 9.801), (5.15, 9.801)), fill\_opacity=0.78)
- after\_true\_value: a Math \[green\] that says "$9.801$" drawn in filter\_picture
- baseline: a Line \[gray\] drawn in filter\_picture (start=(0.25, 0.0), end=(8.75, 0.0))
- before\_false: a Polygon \[yellow\] drawn in filter\_picture (vertices=((2.15, 0.0), (3.25, 0.0), (3.25, 99.9), (2.15, 99.9)), fill\_opacity=0.24)
- before\_false\_value: a Math \[yellow\] that says "$99.9$" drawn in filter\_picture
- before\_label: a Math \[gray\] that says "$upright("after one positive")$" drawn in filter\_picture
- before\_line: a Math \[text\] that says "$upright("one positive"): thin 9.9 thin upright("true"), thin 99.9 thin upright("false")$"
- before\_true: a Polygon \[green\] drawn in filter\_picture (vertices=((0.55, 0.0), (1.65, 0.0), (1.65, 9.9), (0.55, 9.9)), fill\_opacity=0.28)
- before\_true\_value: a Math \[green\] that says "$9.9$" drawn in filter\_picture
- filter\_picture: a Figure (x\_range=(0.0, 9.0), y\_range=(-18.0, 112.0), aspect=(9.0, 4.5))
- heading: a Heading that says "A Second Independent Positive Test"
- independence: a Text \[text\] that says "Assumption: given disease status, the two test errors are independent."
- odds\_heading: a Heading that says "Why Two Positives Are So Much Stronger"
- odds\_result: a Math \[text\] that says "$frac(9801, 9801 + 999) approx 90.8%$"
- odds\_work: a Derivation \[text\] that says "$upright("prior odds") &= frac(10,9990) = frac(1,999) \\ upright("positive likelihood ratio") &= frac(0.99,0.01) = 99 \\ upright("after one positive") &= frac(1,999) dot.op 99 \\ upright("after two positives") &= frac(1,999) dot.op 99 dot.op 99$"
- second\_false: a Math \[text\] that says "$99.9 dot.op 0.01 = 0.999$"
- second\_fraction: a Math \[text\] that says "$P(D \| +,+) = frac(9.801, 9.801 + 0.999)$"
- second\_posterior: a Math \[text\] that says "$P(D \| +,+) approx 90.8%$"
- second\_true: a Math \[text\] that says "$9.9 dot.op 0.99 = 9.801$"

#### Beats

##### [08:55.657](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=535.6572708333333)

Narration: Suppose the same person is tested again and the second result is also positive. Begin with the group that survived the first test: about nine point nine true positives and ninety-nine point nine false positives.

Board: Empty.

Actions:
- [08:55.657](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=535.6572708333333): heading is shown on the screen, written out.
- [08:55.657](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=535.6572708333333): filter\_picture is shown on the screen, written out.
- [08:55.657](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=535.6572708333333): baseline is shown on the screen, written out.
- [09:0.591](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=540.5912708333333): filter\_picture moves to a new place on the board.
- [09:0.591](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=540.5912708333333): before\_line is shown on the screen, written out.
- [09:2.275](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=542.2752708333334): before\_label is shown on the screen, written out.
- [09:3.714](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=543.7142708333333): before\_true is shown on the screen, written out.
- [09:3.714](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=543.7142708333333): before\_true\_value is shown on the screen, written out.
- [09:5.839](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=545.8392708333333): before\_false is shown on the screen, written out.
- [09:5.839](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=545.8392708333333): before\_false\_value is shown on the screen, written out.

##### [09:8.779](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=548.7787708333333)

Narration: Assume the second test is conditionally independent of the first. Among the people who truly have disease, it again detects ninety-nine percent. Ninety-nine percent of nine point nine is nine point eight zero one.

Board: before\_line — a Math \[text\] that says "$upright("one positive"): thin 9.9 thin upright("true"), thin 99.9 thin upright("false")$"; filter\_picture — a Figure (x\_range=(0.0, 9.0), y\_range=(-18.0, 112.0), aspect=(9.0, 4.5)); heading — a Heading that says "A Second Independent Positive Test"; baseline — a Line \[gray\] drawn in filter\_picture (start=(0.25, 0.0), end=(8.75, 0.0)); before\_true — a Polygon \[green\] drawn in filter\_picture (vertices=((0.55, 0.0), (1.65, 0.0), (1.65, 9.9), (0.55, 9.9)), fill\_opacity=0.28); before\_true\_value — a Math \[green\] that says "$9.9$" drawn in filter\_picture; before\_false — a Polygon \[yellow\] drawn in filter\_picture (vertices=((2.15, 0.0), (3.25, 0.0), (3.25, 99.9), (2.15, 99.9)), fill\_opacity=0.24); before\_false\_value — a Math \[yellow\] that says "$99.9$" drawn in filter\_picture; before\_label — a Math \[gray\] that says "$upright("after one positive")$" drawn in filter\_picture

Actions:
- [09:9.423](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=549.4232708333333): after\_label is shown on the screen, written out.
- [09:15.634](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=555.6342708333333): second\_true is shown on the screen, written out.
- [09:19.628](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=559.6282708333333): after\_true is shown on the screen, written out.
- [09:19.628](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=559.6282708333333): after\_true\_value is shown on the screen, written out.

##### [09:22.062](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=562.0622708333333)

Narration: Among the healthy people who produced the first false positive, only one percent produce another false positive independently. One percent of ninety-nine point nine is zero point nine nine nine.

Board: before\_line — a Math \[text\] that says "$upright("one positive"): thin 9.9 thin upright("true"), thin 99.9 thin upright("false")$"; second\_true — a Math \[text\] that says "$9.9 dot.op 0.99 = 9.801$"; filter\_picture — a Figure (x\_range=(0.0, 9.0), y\_range=(-18.0, 112.0), aspect=(9.0, 4.5)); heading — a Heading that says "A Second Independent Positive Test"; baseline — a Line \[gray\] drawn in filter\_picture (start=(0.25, 0.0), end=(8.75, 0.0)); before\_true — a Polygon \[green\] drawn in filter\_picture (vertices=((0.55, 0.0), (1.65, 0.0), (1.65, 9.9), (0.55, 9.9)), fill\_opacity=0.28); before\_true\_value — a Math \[green\] that says "$9.9$" drawn in filter\_picture; before\_false — a Polygon \[yellow\] drawn in filter\_picture (vertices=((2.15, 0.0), (3.25, 0.0), (3.25, 99.9), (2.15, 99.9)), fill\_opacity=0.24); before\_false\_value — a Math \[yellow\] that says "$99.9$" drawn in filter\_picture; before\_label — a Math \[gray\] that says "$upright("after one positive")$" drawn in filter\_picture; after\_true — a Polygon \[green\] drawn in filter\_picture (vertices=((5.15, 0.0), (6.25, 0.0), (6.25, 9.801), (5.15, 9.801)), fill\_opacity=0.78); after\_true\_value — a Math \[green\] that says "$9.801$" drawn in filter\_picture; after\_label — a Math \[text\] that says "$upright("positive twice")$" drawn in filter\_picture

Actions:
- [09:25.847](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=565.8472708333334): second\_false is shown on the screen, written out.
- [09:30.967](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=570.9672708333333): after\_false is shown on the screen, written out.
- [09:30.967](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=570.9672708333333): after\_false\_value is shown on the screen, written out.

##### [09:33.285](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=573.2852708333334)

Narration: Now read the two surviving piles. About nine point eight people are true positives twice, while about one person is falsely positive twice. The green pile is finally much larger than the yellow pile.

Board: before\_line — a Math \[text\] that says "$upright("one positive"): thin 9.9 thin upright("true"), thin 99.9 thin upright("false")$"; second\_true — a Math \[text\] that says "$9.9 dot.op 0.99 = 9.801$"; second\_false — a Math \[text\] that says "$99.9 dot.op 0.01 = 0.999$"; filter\_picture — a Figure (x\_range=(0.0, 9.0), y\_range=(-18.0, 112.0), aspect=(9.0, 4.5)); heading — a Heading that says "A Second Independent Positive Test"; baseline — a Line \[gray\] drawn in filter\_picture (start=(0.25, 0.0), end=(8.75, 0.0)); before\_true — a Polygon \[green\] drawn in filter\_picture (vertices=((0.55, 0.0), (1.65, 0.0), (1.65, 9.9), (0.55, 9.9)), fill\_opacity=0.28); before\_true\_value — a Math \[green\] that says "$9.9$" drawn in filter\_picture; before\_false — a Polygon \[yellow\] drawn in filter\_picture (vertices=((2.15, 0.0), (3.25, 0.0), (3.25, 99.9), (2.15, 99.9)), fill\_opacity=0.24); before\_false\_value — a Math \[yellow\] that says "$99.9$" drawn in filter\_picture; before\_label — a Math \[gray\] that says "$upright("after one positive")$" drawn in filter\_picture; after\_true — a Polygon \[green\] drawn in filter\_picture (vertices=((5.15, 0.0), (6.25, 0.0), (6.25, 9.801), (5.15, 9.801)), fill\_opacity=0.78); after\_true\_value — a Math \[green\] that says "$9.801$" drawn in filter\_picture; after\_label — a Math \[text\] that says "$upright("positive twice")$" drawn in filter\_picture; after\_false — a Polygon \[yellow\] drawn in filter\_picture (vertices=((6.75, 0.0), (7.85, 0.0), (7.85, 0.999), (6.75, 0.999)), fill\_opacity=0.65); after\_false\_value — a Math \[yellow\] that says "$0.999$" drawn in filter\_picture

Actions:
- [09:42.689](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=582.6892708333334): after\_true is indicated — a transient flash.
- [09:44.569](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=584.5692708333333): after\_false is indicated — a transient flash.

##### [09:46.261](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=586.2607708333333)

Narration: The probability of disease after two positive results is the green count divided by the two surviving counts together.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [09:46.83](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=586.8302708333333): second\_fraction is shown on the screen, written out.
- [09:49.964](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=589.9642708333333): second\_fraction (the "9.801" part) is emphasized.
- [09:52.112](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=592.1122708333334): second\_fraction (the "9.801" part) is no longer emphasized.
- [09:52.112](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=592.1122708333334): second\_fraction (the "9.801 + 0.999" part) is emphasized.
- [09:52.948](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=592.9482708333333): second\_fraction (the "9.801 + 0.999" part) is no longer emphasized.

##### [09:53.548](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=593.5482708333333)

Narration: That is about ninety point eight percent. One independent repeat test has moved the answer from about nine percent to about ninety-one percent by filtering both piles again.

Board: before\_line — a Math \[text\] that says "$upright("one positive"): thin 9.9 thin upright("true"), thin 99.9 thin upright("false")$"; second\_true — a Math \[text\] that says "$9.9 dot.op 0.99 = 9.801$"; second\_false — a Math \[text\] that says "$99.9 dot.op 0.01 = 0.999$"; second\_fraction — a Math \[text\] that says "$P(D \| +,+) = frac(9.801, 9.801 + 0.999)$"; filter\_picture — a Figure (x\_range=(0.0, 9.0), y\_range=(-18.0, 112.0), aspect=(9.0, 4.5)); heading — a Heading that says "A Second Independent Positive Test"; baseline — a Line \[gray\] drawn in filter\_picture (start=(0.25, 0.0), end=(8.75, 0.0)); before\_true — a Polygon \[green\] drawn in filter\_picture (vertices=((0.55, 0.0), (1.65, 0.0), (1.65, 9.9), (0.55, 9.9)), fill\_opacity=0.28); before\_true\_value — a Math \[green\] that says "$9.9$" drawn in filter\_picture; before\_false — a Polygon \[yellow\] drawn in filter\_picture (vertices=((2.15, 0.0), (3.25, 0.0), (3.25, 99.9), (2.15, 99.9)), fill\_opacity=0.24); before\_false\_value — a Math \[yellow\] that says "$99.9$" drawn in filter\_picture; before\_label — a Math \[gray\] that says "$upright("after one positive")$" drawn in filter\_picture; after\_true — a Polygon \[green\] drawn in filter\_picture (vertices=((5.15, 0.0), (6.25, 0.0), (6.25, 9.801), (5.15, 9.801)), fill\_opacity=0.78); after\_true\_value — a Math \[green\] that says "$9.801$" drawn in filter\_picture; after\_label — a Math \[text\] that says "$upright("positive twice")$" drawn in filter\_picture; after\_false — a Polygon \[yellow\] drawn in filter\_picture (vertices=((6.75, 0.0), (7.85, 0.0), (7.85, 0.999), (6.75, 0.999)), fill\_opacity=0.65); after\_false\_value — a Math \[yellow\] that says "$0.999$" drawn in filter\_picture

Actions:
- [09:54.466](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=594.4662708333333): second\_posterior is shown on the screen, written out.
- [10:2.453](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=602.4532708333334): after\_true is indicated — a transient flash.
- [10:2.72](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=602.7202708333333): after\_false is indicated — a transient flash.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): before\_line is hidden from the screen — left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): filter\_picture is hidden from the screen — left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): baseline is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): before\_true is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): before\_true\_value is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): before\_false is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): before\_false\_value is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): before\_label is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): after\_true is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): after\_true\_value is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): after\_label is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): after\_false is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): after\_false\_value is hidden from the screen — filter\_picture left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): heading is hidden from the screen — left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): second\_false is hidden from the screen — left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): second\_fraction is hidden from the screen — left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): second\_posterior is hidden from the screen — left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): second\_true is hidden from the screen — left the board.
- [10:3.905](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=603.9047708333333): A box is drawn around second\_posterior.

##### [10:4.505](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=604.5047708333333)

Narration: There is a compact way to understand that jump. Start with disease odds of ten to nine thousand nine hundred ninety, which reduce to one to nine hundred ninety-nine.

Board: Empty.

Actions:
- [10:4.505](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=604.5047708333333): odds\_heading is shown on the screen, written out.
- [10:6.826](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=606.8262708333333): independence is shown on the screen, written out.
- [10:8.707](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=608.7072708333333): odds\_work is shown on the screen, written out.

##### [10:14.868](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=614.8682708333333)

Narration: A positive result is ninety-nine times more likely when disease is present than when it is absent. That factor, ninety-nine, is called the positive likelihood ratio.

Board: independence — a Text \[text\] that says "Assumption: given disease status, the two test errors are independent."; odds\_heading — a Heading that says "Why Two Positives Are So Much Stronger"

Actions:
- [10:16.459](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=616.4592708333333): odds\_work is shown on the screen, written out.
- [10:21.137](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=621.1372708333333): odds\_work (the "99" part) is emphasized.
- [10:25.398](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=625.3982708333333): odds\_work (the "99" part) is no longer emphasized.

##### [10:25.998](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=625.9982708333333)

Narration: The first positive result multiplies the prior odds by ninety-nine. That produces the same roughly nine percent probability we found from the first two piles.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [10:27.658](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=627.6582708333333): odds\_work is shown on the screen, written out.
- [10:29.226](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=629.2262708333333): odds\_work (the "99" part) is emphasized.
- [10:35.286](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=635.2862708333333): odds\_work (the "99" part) is no longer emphasized.

##### [10:35.886](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=635.8862708333334)

Narration: Under conditional independence, the second positive multiplies by the same factor again. Two positives contribute ninety-nine times ninety-nine, changing the odds by a factor of nine thousand eight hundred one.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [10:37.767](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=637.7672708333333): odds\_work is shown on the screen, written out.
- [10:42.573](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=642.5732708333333): odds\_work (the "99 dot.op 99" part) is emphasized.
- [10:47.774](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=647.7742708333333): odds\_work (the "99 dot.op 99" part) is no longer emphasized.

##### [10:48.374](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=648.3742708333333)

Narration: Convert those final odds back to a probability and we recover ninety point eight percent. The count method and the odds method are two views of the same update.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [10:50.476](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=650.4762708333333): odds\_result is shown on the screen, written out.
- [10:52.101](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=652.1012708333333): A box is drawn around odds\_result.

##### [10:59.029](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=659.0287708333333)

Narration: The independence assumption matters. If both tests use the same sample, the same instrument, or the same biological signal, their errors may be correlated. A repeated error can then be more likely than this calculation assumes, so the second positive may add less evidence.

Board: independence — a Text \[text\] that says "Assumption: given disease status, the two test errors are independent."; odds\_result — a Math \[text\] that says "$frac(9801, 9801 + 999) approx 90.8%$"; odds\_heading — a Heading that says "Why Two Positives Are So Much Stronger"

Actions:
- [10:59.563](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=659.5632708333333): independence (the "independent" part) is emphasized.
- [11:15.794](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=675.7942708333333): independence (the "independent" part) is no longer emphasized.

##### [11:16.394](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=676.3942708333333)

Narration: The lesson is not to distrust accurate tests. It is to ask the complete question. How rare is the disease? What are the sensitivity and specificity? And is new evidence genuinely independent? With those facts, a surprising positive result becomes a count we can understand.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [11:20.353](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=680.3532708333332): odds\_result is indicated — a transient flash.
- [11:34.226](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=694.2256458333334): independence is hidden from the screen — left the board.
- [11:34.226](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=694.2256458333334): odds\_heading is hidden from the screen — left the board.
- [11:34.226](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=694.2256458333334): odds\_result is hidden from the screen — left the board.
- [11:34.226](https://academa.ai/lectures/bayes-through-the-medical-test-paradox?t=694.2256458333334): odds\_work is hidden from the screen — left the board.