Search IntMath
Close

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

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))

Continues below

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