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All courses Math II · S-CP.3 65 of 73
Understand conditional probability and connect independence to unchanged conditional probabilities.

Find \(P(A \mid B)\) from counts by dividing within the conditioned group

Problem
Among the 30 students who play sports, 12 play soccer. Compute P(soccer | sports) by dividing the overlap count by the conditioned-group count.
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Hint

For \(P(soccer | sports)\), use only the students who play sports as the total group.

Conditional probability is \(\text{part inside the condition} / \text{whole condition group}\).

Solution walkthrough

01

Identify the conditioned group

\[B=\text{sports},~\text{with}~n(B)=30\]

The word among restricts the denominator to students who play sports.

02

Identify the overlap

\[\text{soccer}~\text{and}~\text{sports}~\text{count}=12\]

The 12 soccer players are the favorable outcomes inside the sports group.

03

Form and reduce the conditional ratio

\[P(\text{soccer}~\text{given}~\text{sports})=12/30=2/5\]

Divide favorable members in the conditioned group by the conditioned-group total, then reduce by 6.

04

Interpret and check

\[2/5=0.40,~\text{so}~40\%~\text{of}~\text{sports}~\text{players}~\text{play}~\text{soccer}\]

The probability is between 0 and 1 and answers the within-sports question.

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Another way

  1. Read it as 12 out of 30 in the restricted sports-only sample space.

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Common mistake

Do not use 12/42; the conditioned denominator is the 30 sports players, not a sum of counts.