If A and B are two events such that $straight P left parenthesis straight A right parenthesis space equals space 1 fourth comma space space straight P left parenthesis straight B right parenthesis space equals space 1 half space and space straight P left parenthesis straight A intersection straight B right parenthesis space equals space 1 over 8 comma$ find P(not A and not B).

Solution

straight P left parenthesis straight A right parenthesis space equals space 1 fourth comma space space straight P left parenthesis straight B right parenthesis space equals space 1 half comma space space straight P left parenthesis straight A intersection straight B right parenthesis space equals space 1 over 8

Now,

straight P left parenthesis straight A space intersection space straight B right parenthesis space equals space 1 over 8 space equals 1 fourth cross times 1 half space equals space straight P left parenthesis straight A right parenthesis cross times straight P left parenthesis straight B right parenthesis comma

therefore

A and B are independent

rightwards double arrow space space space space space space space straight A to the power of straight c space and space straight B to the power of straight c space are space also space independent. rightwards double arrow space space space space space space straight P left parenthesis straight A to the power of straight c space intersection space straight B to the power of straight c right parenthesis space equals space straight P left parenthesis straight A to the power of straight c right parenthesis thin space straight P left parenthesis straight B to the power of straight c right parenthesis rightwards double arrow space space space straight P left parenthesis not space straight A space and space not thin space straight B right parenthesis space equals space open square brackets 1 minus straight P left parenthesis straight A right parenthesis close square brackets space open square brackets 1 minus straight P left parenthesis straight B right parenthesis close square brackets 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 space space space equals space open parentheses 1 minus 1 fourth close parentheses space open parentheses 1 minus 1 half close parentheses space equals 3 over 4 cross times 1 half space equals 3 over 8.

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