Two random subsets that overlap

Let AA and BB be independent, uniformly random subsets of {1,2,3,4,5}\{1, 2, 3, 4, 5\} (each element included in each set independently with probability 12\tfrac12). Find the probability that AA and BB are not disjoint (i.e. they share at least one element).

Show hints (2)+
  1. Complement: disjoint means no element is in both sets. Treat each of the 55 elements independently.
  2. Each element avoids "both" with probability 34\tfrac34, so P[disjoint]=(3/4)5\mathbb{P}[\text{disjoint}]=(3/4)^5.

Answer

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0.7627 (± 0.002)

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Asked at: Citadel, Akuna Capital

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