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Q&A

How to visualize multiplication in the Odds form of Bayes's Theorem?

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I don't understand

1. How do I "visually" multiply Circle 1 (representing $P(D)$) by Circle 2 ($P(+|D)$) into Circle 3 ($P(+ \cap D)$)?

2. The red shaded areas in Circles 1 and 2 disappear in Circle 3, but how does this pictorialize multiplication?

3. Pictorially, how do the $\color{red}{9 ; +s}$ reduce to $\color{red}{3 ; +s}$?

4. Why does the shaded gray area increase in Circle 3?

I first precis the problem statement and percentages. Abbreviate Disease to D, positive test result to +. 1. The website postulates P(D) = 20%, P(+|D) = 90%, P(+|D^C) = 30%. What fraction of patients who tested positive are diseased?

3/7 or 43%, quickly obtainable as follows: In the screened population, there's 1 sick patient for 4 healthy patients. Sick patients are 3 times more likely to turn the tongue depressor black than healthy patients. $(1:4)⋅(3:1)=(3:4)$ or 3 sick patients to 4 healthy patients among those that turn the tongue depressor black, corresponding to a probability of 3/7=43% that the patient is sick.

Using red for sick, blue for healthy, grey for a mix of sick and healthy patients, and + signs for positive test results, the proof above can be visualized as follows:

"the proof above" refers to the Odds form of Bayes's Rule. For clarity, I replace the website's $H_j$ with $D$, $H_k$ with $D^C$ and $e_0$ with $+$.

$\dfrac{P(D)}{P(D^C)} \times \dfrac{\color{red}P(+|D)} {\color{deepskyblue}{P(+|D^C)}} = \dfrac{P(+ \cap D)}{P(+ \cap D^C)} = \dfrac{P(+ \cap D)/P(+)}{P(+ \cap D^C)/P(+)} = \dfrac{P(D|+)}{P(D^C|+)}$

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