Question

A problem is given to three students, whose chances of solving it are $1 third comma space 1 fifth space and space 1 over 6$ respectively. What is 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 space straight P left parenthesis straight A right parenthesis space equals space 1 third comma space space space straight P left parenthesis straight B right parenthesis space equals space 1 fifth comma space space straight P left parenthesis straight C right parenthesis space equals space 1 over 6

straight P left parenthesis straight A with bar on top right parenthesis space equals space 1 minus space straight P left parenthesis straight A right parenthesis space equals space 1 minus 1 third space equals space 2 over 3 comma space space straight P left parenthesis straight B with bar on top right parenthesis space equals space 1 minus 1 fifth space equals 4 over 5 comma space space straight P left parenthesis straight C with bar on top right parenthesis space equals space 1 minus 1 over 6 space equals 5 over 6

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 space space or space space space straight A with bar on top 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 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 straight A with bar on top space straight B space straight C with bar on top right parenthesis space plus space straight P left parenthesis straight A with bar on top space straight B with bar on top space straight C right parenthesis space equals space straight P left parenthesis straight A right parenthesis space straight P left parenthesis straight B with bar on top right parenthesis space straight P left parenthesis straight C with bar on top right parenthesis space plus space straight P open parentheses straight A with bar on top close parentheses space straight P left parenthesis straight B right parenthesis space straight P open parentheses top enclose straight C close parentheses space plus space straight P open parentheses straight A with bar on top close parentheses space straight P open parentheses straight B with bar on top close parentheses space straight P left parenthesis straight C right parenthesis equals space 1 third cross times 4 over 5 cross times 5 over 6 plus 2 over 3 cross times 1 fifth cross times 5 over 6 plus 2 over 3 cross times 4 over 5 cross times 1 over 6 equals space 20 over 90 plus 10 over 90 plus 8 over 90 space equals fraction numerator 20 plus 10 plus 8 over denominator 90 end fraction space equals 38 over 90 space equals 19 over 45

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