Two random subsets that overlap

HardProbability Combinatorics~6m

Let $A$ and $B$ be independent, uniformly random subsets of $\{1, 2, 3, 4, 5\}$ (each element included in each set independently with probability $\tfrac12$ ). Find the probability that $A$ and $B$ are not disjoint (i.e. they share at least one element).

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Complement: disjoint means no element is in both sets. Treat each of the $5$ elements independently.
Each element avoids "both" with probability $\tfrac34$ , so $\mathbb{P}[\text{disjoint}]=(3/4)^5$ .

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0.7627 (± 0.002)

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Two random subsets that overlap

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