# Conditional probability

A **conditional probability** is the probability that some event A occurs given that a different event B has already occurred. This is written P(A|B) (often read as "the probability of A given B"). The conditional probability satisfies the equation

where P(AB) is the probability of both events A and B occurring and P(B) is the probability of event B occurring.

### Example

A bag has 5 red balls and 3 green balls inside it. You pick out a ball and do not replace it in the bag afterwards.

Q. What is the probability of removing a red ball?

A. 5/8

Q. What is the probability of removing a red ball, *given that you have already removed a red ball*?

A. There are now 4 red balls and 3 greens, so: 4/7

In the second case, the question requires a conditional probability: P(red on 2nd draw|red on 1st draw).