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Question

Evaluate $straight P left parenthesis straight A union straight B right parenthesis comma space space if space 2 space straight P left parenthesis straight A right parenthesis space equals space straight P left parenthesis straight B right parenthesis space equals 5 over 13 space and space straight P left parenthesis straight A space left enclose straight B right parenthesis space equals 2 over 5.$

Solution

2 P(A) = P(B) = $5 over 13$ (given)
$therefore space space space space space space straight P left parenthesis straight A right parenthesis space equals space 5 over 26 comma space space space straight P left parenthesis straight B right parenthesis space equals space 5 over 13$
Also, $straight P left parenthesis straight A space left enclose straight B right parenthesis space equals space 2 over 5 space space space space space rightwards double arrow space space space space fraction numerator straight P left parenthesis straight A intersection straight B right parenthesis over denominator straight P left parenthesis straight B right parenthesis end fraction space equals space 2 over 5 space space space rightwards double arrow space space space space space fraction numerator straight P left parenthesis straight A intersection straight B right parenthesis over denominator begin display style 5 over 13 end style end fraction space equals 2 over 5$
$therefore space space space space space space space straight P left parenthesis straight A intersection straight B right parenthesis space equals space 2 over 5 cross times 5 over 13 space equals space 2 over 13$
Now, $straight P left parenthesis straight A union straight B right parenthesis space equals space straight P left parenthesis straight A right parenthesis plus straight P left parenthesis straight B right parenthesis minus straight P left parenthesis straight A intersection straight B right parenthesis$
$equals space 5 over 26 plus 5 over 13 minus 2 over 13 space equals fraction numerator 5 plus 10 minus 4 over denominator 26 end fraction space equals 11 over 26$

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