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Probability

Question
CBSEENMA12033725

The probability of A hitting a target is 4 over 5 and that of B hitting it is 2 over 3. They both fire at the target. Find the probability that 
(i) at least one of them will hit the target.
(ii) only one of them will hit the target.

Solution
Let E denote the event ‘A hits the target’ and F denote the event ‘B hits the target’.
therefore                            straight P left parenthesis straight E right parenthesis space equals 4 over 5 comma space space space straight P left parenthesis straight F right parenthesis space equals space 2 over 3
therefore space space space space space straight P left parenthesis straight E with bar on top right parenthesis space equals space 1 minus space straight P left parenthesis straight E right parenthesis space equals space 1 minus 4 over 5 space equals 1 fifth comma
            straight P left parenthesis straight F with bar on top right parenthesis space equals space 1 minus straight P left parenthesis straight F right parenthesis space equals space 1 minus 2 over 3 space equals 1 third
(i) P(none of them hits the target) = straight P left parenthesis straight E with bar on top space and space straight F with bar on top right parenthesis
                          equals space straight P open parentheses straight E with bar on top close parentheses space straight P open parentheses straight F with bar on top close parentheses space equals space 1 fifth cross times 1 third space equals 1 over 15
therefore space space space straight P left parenthesis at space least space one space of space them space hits space the space target right parenthesis space equals space 1 space minus space straight P left parenthesis none space of space them space hits space the space target right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 space minus space 1 over 15 space equals space 14 over 15
(ii) P(only one of them hits the target) = straight P left parenthesis apostrophe straight E space and space straight F with bar on top apostrophe space space or space space straight E with bar on top space and space straight F apostrophe right parenthesis
                                  equals space straight P open parentheses straight E space and space straight F with bar on top close parentheses space plus space straight P open parentheses straight E with bar on top space and space straight F close parentheses
                                   equals space 4 over 5 cross times 1 third plus 2 over 3 cross times 2 over 5 space equals space 4 over 15 plus 4 over 15 space equals space 8 over 15

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