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Let A and B be two events such that ,P(A cap B)=(1)/(4)` and `Poverset(-)(A)=(1)/(4)`, where `overset(-)A` stands for the complement of the event A. Then, the events A and B are A. independent but not equally likely B. independent nad equally likely C. mutually exclusive and independent D. equally likely but not independent Select the correct answer from above options

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Correct Answer - A Given , `P(bar(A cup B)) =(1)/(6) , P(A cap B) =(1)/(4) ,P(bar(A)) =(1)/(4)` `:. P(A cup B )=1-P(bar(A cup B)) =1-(1)/(6)=(5)/(6)` ` and P(A) =1-P(bar(A)) =1-(1)/(4)=(3)/(4)` `:. P(A cup B ) =P(A) + P(B) -P( cap B ) ` `(5)/(6) =(3)/(4) +P(B)-(1)/(4)` `implies P(B)=(1)/(3) implies ` A and are not equally likely `P( A cap B ) =P(A) * P(B) =(1)/(4)` So , events are independent .

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