# 7. Conditional Probability

If *E*_{1} and *E*_{2} are two events, the probability that *E*_{2} occurs given that *E*_{1} has occurred is denoted by *P*(*E*_{2}|*E*_{1}).

*P*(*E*_{2}|*E*_{1}) is called the **conditional probability** of *E*_{2} given that *E*_{1} has occurred.

## Calculating Conditional Probability

Let *E*_{1} and *E*_{2} be any two events defined in a sample space *S* such that *P*(*E*_{1}) > 0.

The **conditional probability** of *E*_{2}, assuming *E*_{1} 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`?

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