What is Bayes’s Theorem?
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I’ll give you an example where Bayes theorem would come into play:
Say there’s a disease that affects 1% of the population. There’s a test for this disease. It always picks up positive cases, but it also sometimes gives a false positive. If you run the test on a person who doesn’t have the disease, there’s a 10% chance that the test will come back positive.
The disease is very serious, so doctors run the test on everyone to make sure they don’t miss any cases.
Your test comes back positive. What are the chances that you actually have the disease?
You might be tempted to assume that there’s a 90% chance you have the disease—after all, there’s only a 10% chance of a false positive.
Fortunately, this is wrong.
If you ran the test on 100 people, you would expect to see one true positive result because of the 1% of the population with the disease but you would expect to see about 10 false positive results from the 10% false positive rate. That’s 11 total tests that come back positive and 10/11 are false positives—91% of positive tests are false positives.
Therefore, if your test comes back positive, there’s about a 91% chance that you don’t have the disease and only a 9% chance that you do have it.
Bayes theorem is the mathematical equation that lets you figure out this kind of probability problem.
Let’s call having the disease A and getting a positive test result B.
Bayes theorem says that the probability of A given B (the probability that you actually have the disease, given a positive test result) is equal to the probability of A (1%) times the probability of B given A (the chances of someone with the disease getting a positive test result, 100%) divided by the probability of B (~11% because it’s the 1% true positive plus the ~10% false positives).
P(A|B)=P(A)*P(B|A)/P(B)
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