Bayesian probability
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Bayesian probability is a likelihood of an event A happening given event B happens where event B is an independent event. (Throwing 6 on dice given it is thursday is an example of independent. Getting wet given it is raining would be dependent.)
Since it is very often used for this purposes, think about a test (cancer test for example)
You get a positive cancer test (event B). What is the probability, given positive test, you have cancer (event A)? This question can be expressed as probability of A being true given event B or P(A|B)
We know that approximately 2% of the people have cancer that is a general probability of event A happening or P(A) = 0.02
We also know that if you have cancer the likelihood of getting positive test is about 95% this is expressed as a probability of event B happening given A being true P(B|A) = 0.95.
The last number we sould use is P(B) or a general likelyhood of geting a positive cancer test. Both correct and incorrect positive test and out historical data claim 10% are positive Or P(B)=0.1
We can now answer the original question
What is the probability, given positive test, you have cancer?
The bayesian theorem supplies that P(A|B) = P(B|A) * P(A) / P(B)
And it will come down to about 20%
It is a very elegant way to show, that just because something is 95% accurate, it doesn’t mean it is necessarily very likely.
It may seem a very contrary to intuition so let me add some numbers to show where this discrepancy comes from.
You are one of 100 000 people. Only 2000 P(A) are expected to have cancer.
You test all of them
Given the likelihood of 10% for positive test P(B) = 0.1, 10 000 people are marked positive.
Out of 2000 actually ill, 100 are incorrectly misidentified as healthy (due to P(B|A)= 0.95) and 1900 are correctly identified as ill.
That makes 8100 people who are diagnosed positive despite being healthy.
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