Fraction who like both tea and coffee

Question

In a taste survey, 45\% of people like tea, 60\% like coffee, and 85\% like at least one of the two. What fraction like both tea and coffee?

Accepted Answer

We are given the union and want the overlap. Inclusion-exclusion rearranged gives P(	ext{both}) = P(	ext{tea}) + P(	ext{coffee}) - P(	ext{either}) = 0.45 + 0.60 - 0.85 = 0.20. So 20\% like both. Disjoint check: tea only 0.25, coffee only 0.40, both 0.20, neither 0.15, which sum to 1.00. Takeaway. The both-group is the amount by which the two shares together exceed the share that likes at least one.

Fraction who like both tea and coffee

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