Conditional probability

Objective. Calculate and interpret P(A|B).

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Learn: the key idea

Conditional probability measures the chance of an event occurring given that another event is already known to have happened, calculated with the formula P(A|B) = P(A∩B)/P(B), restricting the sample space to only outcomes where B occurs. This formula can be rearranged to find the intersection: P(A∩B) = P(A|B)×P(B). For example, if P(A∩B) = 0.12 and P(B) = 0.3, then P(A|B) = 0.12/0.3 = 0.4, meaning given B has occurred, A occurs 40% of the time. Conditional probability is essential for interpreting dependent events, medical testing results, and tree diagrams with sequential outcomes.

Worked example

In a class, P(studies French) = 0.5 and P(studies French and German) = 0.2. Find the conditional probability that a student studies German given they study French.

1 $Recall the formula: P(German|French) = P(French\capGerman)/P(French).$
2 $Substitute the values: P(German|French) = 0.2/0.5.$
3 $Simplify the fraction: 0.2/0.5 = 0.4.$
4 $State the conditional probability as 0.4.$

Answer: 0.4

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Common mistake. Students often confuse P(A|B) with P(B|A), treating them as equal when they generally differ. Always divide by the probability of the event that is given (the condition), not the event being asked about.

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