Conditional Probability

How to find the conditional probability P(B|A): formula, 2 examples, and their solutions.

Formula

Formula

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

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

[B|A] is read as
[B bar A] or [B given A].

Example 1

Example

Solution

The probability of Sam oversleeping
is 4%.

Set A as
Sam is oversleeping.

Then P(A) = 0.04.

The probability of
Sam oversleeping and getting late for school
is 3%.

Set B as
Sam is getting late for school.

Then P(A and B) = 0.03.

Sam woke up and realized that he overslept (A).

This means
the event A has already happened.

In this case,
the probability of
Sam getting late for school (B)
is P(B|A).

So it says to find P(B|A).

P(A) = 0.04
P(A and B) = 0.03

Then
P(B|A) = 0.03/0.04.

0.03/0.04 = 3/4

So 3/4 is the answer.

This means
if Sam overslept,
then the probability of
Sam getting late for school is
3/4 = 75%.

Example 2

Example

Solution

80% of students saw Mary.

Set A as
choosing a student who saw Mary.

Then P(A) = 0.8.

50% of students saw both Mary and John.

Set B as
choosing a student who saw John.

Then P(A and B) = 0.5.

If you choose a student who saw Mary, ...

This means
you already chose a student who saw Mary (A).

So the event A has already happened.

In this case,
the probability of
the chosen student also saw John (B)
is P(B|A).

So it says to find P(B|A).

P(A) = 0.8
P(A and B) = 0.5

Then
P(B|A) = 0.5/0.8.

0.5/0.8 = 5/8

So 5/8 is the answer.