1 Probability
$$A$$ and $$B$$ are two independent events. The probability that both $$A$$ and $$B$$ occur is $$1/6$$ and the probability that at least one of them occurs is $$\cfrac{2}{3}$$. The probability of the occurrence of $$A=$$............ if $$P(A)=2P(B)$$.
1) $$\cfrac{2}{9}$$ 2) $$\cfrac{4}{9}$$ 3) $$\cfrac{5}{18}$$ 4) $$\cfrac{5}{9}$$
Here given $$P(A\cap B) = \cfrac{1}{6}, P(A\cup B) = \cfrac{2}{3}, P(A)=2P(B)$$ Thus using $$P(A\cup B) = P(A)+P(B)-P(A\cap B)$$ $$\Rightarrow \cfrac{2}{3}=P(A)+\cfrac{1}{2}P(A)-\cfrac{1}{6}\Rightarrow P(A) = \cfrac{5}{9}$$
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