Fraction owning both a car and a bicycle

Question

In a town survey, 65\% of households own a car, 40\% own a bicycle, and 80\% own at least one of the two. What fraction of households own both a car and a bicycle?

Accepted Answer

Here we know the union (80\% own at least one) and want the overlap. Inclusion-exclusion for two events says P(	ext{car or bike}) = P(	ext{car}) + P(	ext{bike}) - P(	ext{both}). Rearrange to isolate the overlap: P(	ext{both}) = P(	ext{car}) + P(	ext{bike}) - P(	ext{car or bike}) = 0.65 + 0.40 - 0.80 = 0.25. So 25\% own both. The reason it works: 0.65 + 0.40 = 1.05 overcounts the union by exactly the both-owners, and that excess 1.05 - 0.80 = 0.25 is the overlap. Disjoint check: car only 0.40, bike only 0.15, both 0.25, neither 0.20, summing to 1.00. Takeaway. When the union is given i…

Fraction owning both a car and a bicycle

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