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

In summary: The probability of the patient being sick given a positive test result can be calculated as 3/7 or 43%, which is visually represented by the ratio of the red area to the total area in Circle 3.
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Here I'm asking solely about the circle pictograms. Please eschew referring to, or using, numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT asking about how to divide numbers.

I don't understand
NN9Arf1jAT3gbr7pVT1cUUUy.jpg


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

2. I see that the bottom gray areas in Circles 1 and 3 disappear, but how does this pictorialize division?

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:

HsMfASb.jpg


"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{P(+|D)} {{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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The above equation can be visually represented by the following circles: Circle 1 (representing ##P(+ \cap D)##) divided by Circle 2 (##P(+)##) into Circle 3 (##P(+)##). The bottom gray areas in Circles 1 and 3 represent the mix of healthy and sick patients. The top red and blue areas represent the fractions of healthy and sick patients, respectively, among those that test positive for the disease.
 

1. What is the Odds form of Bayes's Theorem?

The Odds form of Bayes's Theorem is a mathematical formula used to calculate the probability of an event occurring based on prior knowledge or evidence. It is often used in Bayesian statistics to update the probability of a hypothesis as new information becomes available.

2. How do you calculate the odds in the Odds form of Bayes's Theorem?

To calculate the odds in the Odds form of Bayes's Theorem, you need to divide the probability of the event occurring by the probability of the event not occurring. This will give you the odds in favor of the event occurring. For example, if the probability of an event occurring is 0.6, the odds would be 0.6/0.4 = 1.5.

3. Can you give an example of visualizing division in the Odds form of Bayes's Theorem?

Yes, for example, if we have a bag of 100 marbles, 60 of which are red and 40 are blue. The probability of picking a red marble is 0.6 (60/100) and the probability of picking a blue marble is 0.4 (40/100). The odds in favor of picking a red marble would be 0.6/0.4 = 1.5.

4. How can visualizing division in the Odds form of Bayes's Theorem help in understanding probabilities?

Visualizing division in the Odds form of Bayes's Theorem can help in understanding probabilities by providing a clearer representation of the relationship between the probability of an event occurring and the probability of the event not occurring. It can also help in understanding how prior knowledge or evidence can affect the probability of an event.

5. Are there any limitations to visualizing division in the Odds form of Bayes's Theorem?

Yes, there are some limitations to visualizing division in the Odds form of Bayes's Theorem. It may not always be easy to visualize the division, especially when dealing with complex probabilities or multiple events. It is also important to note that the Odds form of Bayes's Theorem is just one way of representing probabilities and may not always be the most appropriate or accurate method for a given scenario.

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