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Probability

Question
CBSEENMA12033719

The probability of hitting a target by three marksmen are 1 half comma space 1 third space and space 1 fourth respectively. Find the probability that one and only one of them will hit the target when they fire simultaneously.

Solution
Let A denote the event ‘first marksman hits the target’, B denote the event ‘second marksman hits the target’, and C denote the event ‘third marksman hits the target’.
therefore space space space space straight P left parenthesis straight A right parenthesis space equals space 1 half 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 space 1 fourth              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 half space equals space 1 half comma space space straight P left parenthesis straight B with bar on top right parenthesis space equals space 1 minus 1 third space equals space 2 over 3 comma space space straight P left parenthesis straight C with bar on top right parenthesis space equals space 1 minus 1 fourth space equals 3 over 4
Required probability
                               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 left parenthesis straight A with bar on top right parenthesis space straight P left parenthesis straight B right parenthesis space straight P left parenthesis straight C with bar on top right parenthesis space plus space straight P left parenthesis straight A with bar on top right parenthesis space straight P left parenthesis straight B with bar on top right parenthesis space straight P left parenthesis straight C right parenthesis
equals space open parentheses 1 half close parentheses space open parentheses 2 over 3 close parentheses space open parentheses 3 over 4 close parentheses space plus space open parentheses 1 half close parentheses space open parentheses 1 third close parentheses space open parentheses 3 over 4 close parentheses space plus space open parentheses 1 half close parentheses space open parentheses 2 over 3 close parentheses space open parentheses 1 fourth close parentheses
space equals space 1 fourth plus 1 over 8 plus 1 over 12 space equals fraction numerator 6 plus 3 plus 2 over denominator 24 end fraction space equals space 11 over 24

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