# 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.