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Probability

Question

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A can hit a target 4 times with 5 shots. B can hit 3 times with 5 shots and C can hit 3 times with 5 shots. Each fire a volley, what is the probability that 2 shots hit the target?

Solution

Let E₁ , E₂ , E₃ be the events ‘A hits a shot’, ‘B hits a shot’, ‘C hits a shot’ respectively.

therefore space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 4 over 5 comma space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 3 over 5 comma space space straight P left parenthesis straight E subscript 3 right parenthesis space equals 3 over 5

P(two shots hit)

equals space straight P left parenthesis straight E subscript 1 space straight E subscript 2 space straight E with bar on top subscript 3 right parenthesis space plus space straight P left parenthesis straight E subscript 1 space straight E with bar on top subscript 2 space straight E subscript 3 right parenthesis space plus space straight P left parenthesis straight E with bar on top subscript 1 space straight E subscript 2 space straight E subscript 3 right parenthesis space equals space straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight E with bar on top subscript 3 right parenthesis thin space plus thin space straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight E with bar on top subscript 2 right parenthesis thin space straight P left parenthesis straight E subscript 3 right parenthesis space plus space straight P left parenthesis straight E with bar on top subscript 1 right parenthesis thin space straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight E subscript 3 right parenthesis space equals space straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight E subscript 2 right parenthesis space left square bracket 1 minus straight P left parenthesis straight E subscript 3 right parenthesis right square bracket space plus space straight P left parenthesis straight E subscript 1 right parenthesis space left square bracket 1 minus straight P left parenthesis straight E subscript 2 right parenthesis right square bracket space straight P left parenthesis straight E subscript 3 right parenthesis plus open square brackets 1 minus straight P left parenthesis straight E subscript 1 right parenthesis close square brackets space straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight E subscript 3 right parenthesis equals space open parentheses 4 over 5 close parentheses space open parentheses 3 over 5 close parentheses space open parentheses 1 minus 3 over 5 close parentheses space plus 4 over 5 space open parentheses 1 minus 3 over 5 close parentheses space open parentheses 3 over 5 close parentheses space plus space open parentheses 1 minus 4 over 5 close parentheses space open parentheses 3 over 5 close parentheses space open parentheses 3 over 5 close parentheses equals space open parentheses 4 over 5 close parentheses space open parentheses 3 over 5 close parentheses space open parentheses 2 over 5 close parentheses space plus space open parentheses 4 over 5 close parentheses space open parentheses 2 over 5 close parentheses space open parentheses 3 over 5 close parentheses space plus space open parentheses 1 fifth close parentheses space open parentheses 3 over 5 close parentheses space open parentheses 3 over 5 close parentheses space equals 24 over 125 plus 24 over 125 plus 9 over 125 equals 57 over 125

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