7. Conditional Probability

If E1 and E2 are two events, the probability that E2 occurs given that E1 has occurred is denoted by P(E2|E1).

P(E2|E1) is called the conditional probability of E2 given that E1 has occurred.

Calculating Conditional Probability

Let E1 and E2 be any two events defined in a sample space S such that P(E1) > 0.

The conditional probability of E2, assuming E1 has already occurred, is given by

`P(E_2|E_1)=(P(E_2\ "and"\ E_1))/(P(E_1))`

Example 1

Let A denote the event 'student is female' and let B denote the event 'student is French'. In a class of `100` students suppose `60` are French, and suppose that `10` of the French students are females. Find the probability that if I pick a French student, it will be a girl, that is, find P(A|B).

Example 2

What is the probability that the total of two dice will be greater than `8`, given that the first die is a `6`?