• # question_answer If A and B are two events such that$P(A\cup B)=\frac{5}{6},P(A\cap B)=\frac{1}{3},P(\bar{B})=\frac{1}{2},$ then the events A and B are A)  dependent B)  independent C)  mutually exclusive D)  None of the above

Solution :

We have $P(A\cup B)=\frac{5}{6},(A\cap B)=\frac{1}{3},P(B)=\frac{1}{2}$ Now,     $P(B)=1-P(\bar{B})$                 $=1-\frac{1}{2}=\frac{1}{2}$ Also by addition theorem,                 $P(A\cup B)=P(A)+P(B)-P(A\cap B)$                 $\Rightarrow$$P(A)=P(A\cup B)+P(A\cap B)-P(B)$                 $\Rightarrow$$P(A)=\frac{5}{6}+\frac{1}{3}-\frac{1}{2}$                 $\Rightarrow$$P(A)=\frac{2}{3}$                 $\therefore$  $P(A)P(B)=\frac{2}{3}.\frac{1}{2}=\frac{1}{3}$ Thus, $P(A\cap B)=P(A)P(B)$ Hence, A and B are independent events.

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