Skill: Calculate conditional probabilities

A conditional probability is the probability that event $A$ occurs when we know that event $B$ has occurred.

Use the formula

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)},$$

where $(A\cap B)$ means that both $A$ and $B$ occur.

Think of $B$ as the new, restricted sample space. Once $B$ is known, only the outcomes in $B$ matter.

Example

A school surveys 100 students about club membership and whether they play a sport.

	Plays a sport	Does not play a sport	Total
Club member	18	12	30
Not a club member	42	28	70
Total	60	40	100

Find the probability that a student plays a sport, given that the student is a club member.

Step 1: Identify the events

Let

• $A$: the student plays a sport
• $B$: the student is a club member

We want $(P(A\mid B))$, which means “the probability of playing a sport given club membership.”

Step 2: Restrict the sample space

Because we know the student is a club member, consider only the 30 club members. Thus,

$$P(B)=\frac{30}{100}.$$

Step 3: Count students in both events

There are 18 students who are both club members and play a sport. Therefore,

$$P(A\cap B)=\frac{18}{100}.$$

Step 4: Apply the formula

$$P(A\mid B) =\frac{P(A\cap B)}{P(B)} =\frac{\frac{18}{100}}{\frac{30}{100}} =\frac{18}{30} =\frac{3}{5} =0.6.$$

Therefore, the probability is

$$\boxed{P(A\mid B)=0.6}$$

or $(60\%)$.

The denominator is $(30)$, not $(100)$, because the information that the student is a club member restricts the sample space to the 30 club members.

Be careful not to reverse the events. For example,

$$P(B\mid A)=\frac{18}{60}=0.3,$$

which is different from $(P(A\mid B)=0.6)$. In general,

$$P(A\mid B)\ne P(B\mid A).$$