A problem is given to three students, whose chances of solving it are respectively. Find the probability that exactly one of them may solve it. - Wired Faculty

Question

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A problem is given to three students, whose chances of solving it are $1 fourth comma space 1 third space and space 1 half$ respectively. Find the probability that exactly one of them may solve it.

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Solution

Let A, B, C be the events that three students solve the problem.

therefore space space space straight P left parenthesis straight A right parenthesis space equals space 1 fourth comma space space space straight P left parenthesis straight B right parenthesis space equals space 1 third comma space space straight P left parenthesis straight C right parenthesis space equals 1 half

straight P open parentheses straight A with bar on top close parentheses space equals space 1 space minus space straight P left parenthesis straight A right parenthesis space equals space 1 minus 1 fourth space equals 3 over 4 comma space space straight P left parenthesis straight B with bar on top right parenthesis space space equals space 1 minus 1 third space equals 2 over 3 comma space space straight P open parentheses straight C with bar on top close parentheses space equals space 1 minus 1 half space equals 1 half

P(exactly one student solves the problem)

equals space straight P left parenthesis straight A space straight B with bar on top space straight C with bar on top space or space straight A with bar on top space straight B space straight C with bar on top space or space straight A with bar on top space straight B with bar on top space straight C right parenthesis equals space straight P left parenthesis straight A space straight B with bar on top space straight C with bar on top right parenthesis space plus space straight P left parenthesis top enclose straight A space straight B thin space top enclose straight C right parenthesis space plus space straight P left parenthesis top enclose straight A space top enclose straight B space straight C right parenthesis space equals space straight P left parenthesis straight A right parenthesis space straight P left parenthesis top enclose straight B right parenthesis space straight P left parenthesis top enclose straight C right parenthesis space plus space straight P left parenthesis top enclose straight A right parenthesis thin space straight P left parenthesis straight B right parenthesis space straight P left parenthesis top enclose straight C right parenthesis space plus space straight P left parenthesis top enclose straight A right parenthesis thin space straight P left parenthesis top enclose straight B right parenthesis thin space straight P left parenthesis straight C right parenthesis equals space 1 fourth cross times 2 over 3 cross times 1 half plus 3 over 4 cross times 1 third cross times 1 half plus 3 over 4 cross times 2 over 3 cross times 1 half space equals 2 over 24 plus 3 over 24 plus 6 over 24 space equals fraction numerator 2 plus 3 plus 6 over denominator 24 end fraction equals 11 over 24

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Probability

A problem is given to three students, whose chances of solving it are $1 fourth comma space 1 third space and space 1 half$ respectively. Find the probability that exactly one of them may solve it.

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