Fraction subscribing to both magazines

Question

Among readers, 55\% subscribe to magazine X, 35\% subscribe to magazine Y, and 75\% subscribe to at least one of the two. What fraction subscribe to both magazines?

Accepted Answer

We know the union and want the overlap. Rearranging inclusion-exclusion for two events, P(	ext{both}) = P(X) + P(Y) - P(X 	ext{ or } Y) = 0.55 + 0.35 - 0.75 = 0.15. So 15\% subscribe to both. The intuition: 0.55 + 0.35 = 0.90 overshoots the 0.75 union by the amount double counted, namely the both-subscribers. Disjoint check: X only 0.40, Y only 0.20, both 0.15, neither 0.25, which sum to 1.00. Takeaway. Overlap equals the sum of the two group sizes minus the size of their union.

Fraction subscribing to both magazines

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