Count a union of selection events

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Problem

Two people are chosen uniformly from five people, including Ana and Ben. Let $A$ be the event that Ana is chosen and $B$ the event that Ben is chosen. Compute $P(A)$ , $P(B)$ , $P(A\cap B)$ , $P(A\cup B)$ , and $P(A^c\cap B^c)$ .

Reveal answer or reference solution

There are $\binom52=10$ equally likely pairs. Four include Ana and four include Ben, while one includes both. Thus

P(A)=P(B)=\frac4{10}=\frac25,\qquad P(A\cap B)=\frac1{10}

Inclusion-exclusion gives

P(A\cup B)=\frac25+\frac25-\frac1{10}=\frac7{10}

By complementation, $P(A^c\cap B^c)=1-7/10=3/10$ .

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Count a union of selection events

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