Conditional Probability

Conditional Probability

How to solve conditional probability problems: formula, examples, and their solutions.

Formula

P(B|A) = P(A and B)/P(A)

P(B | A) means the probability of (A and B)
when A has already happened.

So P(B | A) = P(B | A) / P(A).

This is why it is called the 'conditional' probability.

Example 1

A bag is filled with 15 colored and shaped cards. Given that a card selected is a gray card. Find the probability that it is a gray square card.

Selecting a gray card (A) has already happened.
So this is a conditional probability problem.

So find P(A) and P(A and B).
P(A) = 8/15
P(A and B) = 3/15

P(B | A) = (3/15) / (8/15) = 3/8.

Example 2

The probability of Sam oversleeping is 4%. And the probability of Sam getting late for school (because of oversleeping) is 3%. If Sam woke up and realized that he overslept, find the probability that he will be late for school.

Sam oversleeping (A) has already happened.
So this is a conditional probability problem.

So find P(A) and P(A and B).
P(A) = 0.04
P(A and B) = 0.03

P(B | A) = 0.03/0.04 = 3/4.