Let's pick an arbitrary person from this study of 100 people. Let's also assume that you are told that their test result was positive. What is the probability of them actually having cancer? So, we are told that event B has already taken place, and that their test came back positive. The question now is: what is the probability that they have cancer, that is P(A)? This is called a conditional probability of A given B or P(A|B). Effectively, it is asking you to calculate the probability of an event given that another event has already happened.

You can think of conditional probability as changing the relevant universe. P(A|B) (called the probability of A given B) is a way of saying, given that my entire universe is now B, what is the probability of A? This is also known as transforming the sample space.

Zooming in on our previous diagram, our universe is now B, and we are concerned with AB (A and B) inside B.

The formula can be given as follows:

P(A|B) = P(A and B) / P(B) = (20/100) / (30/100) = 20/30 = .66 = 66%

There is a 66% chance that if a test result came back positive, that person had cancer. In reality, this is the main probability that the experimenters want. They want to know how good the test is at predicting cancer.