If we need to find the probability of an event occurring given that another event has already occurred, then we are dealing with conditional probability.

If A and B are two events, then the conditional probability that A occurs given that B already has is written as P(A | B) where:

or:

Example:

A frog climbing out of a well is affected by the weather. When it rains, he falls back down the well with a probability of 1/10. In dry weather, he only falls back down with probability of 1/25. The probability of rain is 1/5.

Find the probability that given he falls it was a rainy day.

Let's start by drawing the tree diagram of these events:

Another way to think about conditional probability is:

How many ways there are of obtaining 'A and B' out of a total possible number of ways of getting the 'B':

If we rearrange this formula we obtain another useful result:

If the two events A and B are independent (i.e. one doesn't affect the other) then quite clearly: