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Probability

Question
CBSEENMA12033734

A can hit a target 3 times in 5 shots, B 2 times in 5 shots, C 3 times in 4 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                straight P space left parenthesis straight E subscript 1 right parenthesis space space equals space 3 over 5 comma space space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 2 over 5 comma space space straight P left parenthesis straight F subscript 3 right parenthesis space equals space 3 over 4
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
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 space plus 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 space straight P left parenthesis straight E subscript 3 right parenthesis
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 space minus space 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 space 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 space plus space left square bracket 1 minus straight P left parenthesis straight E subscript 1 right parenthesis right square bracket 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 3 over 5 cross times 2 over 5 cross times open parentheses 1 minus 3 over 4 close parentheses plus 3 over 5 cross times open parentheses 1 minus 2 over 5 close parentheses space cross times 3 over 4 plus open parentheses 1 minus 3 over 5 close parentheses cross times 2 over 5 cross times 3 over 4
equals space 3 over 5 cross times 2 over 5 cross times 1 fourth plus 3 over 5 cross times 3 over 5 cross times 3 over 4 plus 2 over 5 cross times 2 over 5 cross times 3 over 4 equals 6 over 100 plus 27 over 100 plus 12 over 100
equals space fraction numerator 6 plus 27 plus 12 over denominator 100 end fraction space equals 45 over 100 space equals 9 over 20

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