# 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).

Since 10 out of 100 students are both French and female, then

P(A\ "and"\ B) = 10/100

Also, 60 out of the 100 students are French, so

P(B) = 60/100

So the required probability is:

P(A|B)=frac{P(A\ "and"\ B)}{P(B)}=frac{10/100}{60/100}=1/6

### 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?

Let E1 = first die is 6;

Let E2 = total of two dice is  > 8

Then "E1 and E2" will be given by (6, 3),\ (6, 4),\ (6, 5),\ (6, 6).

There are 36 possible outcomes when we throw 2 dice.

So

P(E_2\ "and"\ E_1)=4/36=1/9

Therefore

P(E_2|E_1) = frac{P(E_2\ "and"\ E_1)}{P(E_1)} =frac{1/9}{1/6}=6/9 =2/3