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A survey show that $63 \% $ of the Amrican people like cheese whereas $76 \%$ like apples. What can you say about the percentage of the American people that like both cheese and apples?

I have worked out on that problem and deduced that the number of people who like both apples and cheese is within $39 \%$ and $63 \%$.

Is it correct?

RFZ
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  • Yes, that is correct. – Sarvesh Ravichandran Iyer Nov 01 '17 at 10:47
  • @RFZ By inclusion-exclusion, $P(A) + P(B) - P(A \cup B) < 1$. This means that $P(A \cup B)$ must equal at least $0.63+0.76-1$, or $0.39$. On the other hand, since $0.63$ like cheese, then if $P(A \cup B)$ were greater than $0.63$ then it would imply more than $0.63$ of people like cheese. – Toby Mak Nov 01 '17 at 10:50

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You are correct, it is easier when you say that there are 100 people in America. 63 like cheese and 76 like apples. The maximum amount of people that will like both cheese and apples is equal to: min(63,76)=63. The minimum amount of people that will like both cheese and apples is equal to: 76-(100-63) = 63-(100-76) = 39. So indeed the number of people who like both apples and cheese is within 39% and 63%.

T C Molenaar
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