For two events A and B,if P(A) = Pleft( frac{ | vedclass

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(D) Given $P(A) = \frac{1}{4}$ and $P\left( \frac{A}{B} \right) = \frac{1}{4}$.Since $P(A) = P\left( \frac{A}{B} \right)$,the events $A$ and $B$ are i

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(D) Given $P(A) = \frac{1}{4}$ and $P\left( \frac{A}{B} \right) = \frac{1}{4}$.Since $P(A) = P\left( \frac{A}{B} \right)$,the events $A$ and $B$ are independent.Therefore,option $A$ is correct.For independent events $A$ and $B$,$A'$ and $B$ are also independent.Thus,$P\left( \frac{A'}{B} \right) = P(A') = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4}$.Therefore,option $B$ is correct.Similarly,for independent events $A$ and $B$,$A'$ and $B'$ are also independent.Thus,$P\left( \frac{B'}{A'} \right) = P(B')$.We know $P\left( \frac{B}{A} \right) = P(B) = \frac{1}{2}$ (since $A$ and $B$ are independent).So,$P(B') = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2}$.Therefore,option $C$ is correct.Since all options $A, B,$ and $C$ are correct,the correct answer is $D$.

For two events $A$ and $B$,if $P(A) = P\left( \frac{A}{B} \right) = \frac{1}{4}$ and $P\left( \frac{B}{A} \right) = \frac{1}{2},$ then:

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