Bayes' theorem allows us to reverse conditional probabilities, determining the probability of an event based on prior knowledge of another event. If we have events A and B, Bayes' theorem states:

In other words, Bayes' theorem lets us update our beliefs or predictions after observing new evidence. It's particularly useful when dealing with sequential information or adjusting probabilities based on new data.

Example

Consider a medical test for a rare disease that affects 1 in 1000 people. The test correctly identifies the disease in 99% of patients who have it, and has a false positivity rate of 1%. Find the probability that a person who tests positive has the disease.

Let B be the event "a person has the disease" with P(B)=0.001.

Let A be the event "a person tests positive" with an accuracy of P(A∣B)=0.99, and a false positivity rate of P(A∣B′)=0.01.

The probability of A given B is

Or an approximately 9% chance.