Fraction who play both guitar and piano

Question

At a music school, 30\% of students play guitar, 25\% play piano, and 45\% play at least one of these two instruments. What fraction play both guitar and piano?

Accepted Answer

The union is given and the overlap is unknown. Rearranging inclusion-exclusion for two events, P(	ext{both}) = P(	ext{guitar}) + P(	ext{piano}) - P(	ext{either}) = 0.30 + 0.25 - 0.45 = 0.10. So 10\% play both. Disjoint check: guitar only 0.20, piano only 0.15, both 0.10, neither 0.55, which sum to 1.00. Note that "neither" here is large because fewer than half play either instrument. Takeaway. Sum of parts minus union gives the overlap, regardless of how large or small the "neither" group turns out to be.

Fraction who play both guitar and piano

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