Calculate conditional probabilities

Conditional probability describes the likelihood of event $A$ when event $B$ is known to have occurred, calculated as $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ for $P(B)>0$ . The learner interprets $B$ as a restricted sample space, calculates conditional probabilities from tables, tree diagrams, or stated probabilities, distinguishes $P(A\mid B)$ from $P(B\mid A)$ , and identifies independence when conditioning does not change probability; this supports the multiplication rule for dependent events. Continuous distributions and measure-theoretic generalizations are outside this scope.

Detailed Explanation: 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).

Learn by doing: Calculate conditional probabilities

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