Given that the events A and B are such that P(A) = Find p if they are independent. - Wired Faculty

Question

Given that the events A and B are such that P(A) = $1 half comma$ $straight P left parenthesis straight A union straight B right parenthesis space equals space 3 over 5 space and space straight P left parenthesis straight B right parenthesis space equals space straight p.$ Find p if they are independent.

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Solution

straight P left parenthesis straight A right parenthesis space equals space 1 half comma space space straight P left parenthesis straight B right parenthesis space equals space straight p comma space space space space straight P left parenthesis straight A union straight B right parenthesis space equals space 3 over 5

Since A and B are independent events

therefore space space space space space space space straight P left parenthesis straight A intersection straight B right parenthesis space equals space straight P left parenthesis straight A right parenthesis space straight P left parenthesis straight B right parenthesis space equals space 1 half cross times straight p therefore space space space space space space space space straight P left parenthesis straight A space intersection space straight B right parenthesis space equals space straight p over 2 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 space space space space space space space space space... left parenthesis 2 right parenthesis

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

therefore space space space space space 3 over 5 space equals 1 half plus straight p minus straight p over 2 space space space space space space space open square brackets because space of space left parenthesis 2 right parenthesis close square brackets therefore space space space space space straight p over 2 space equals 3 over 5 minus 1 half space space space space space space space space space space space rightwards double arrow space space space space straight p over 2 space equals space fraction numerator 6 minus 5 over denominator 10 end fraction rightwards double arrow space space space space space space straight p over 2 space equals space 1 over 10 space space space space space space space space space space space rightwards double arrow space space space straight p space equals space 2 over 10 space equals 1 fifth

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