A and B are two independent events. probability that both A and B occur is frac{1}{6} and probabilit

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14.Probability

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$A$ and $B$ are two independent events. The probability that both $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$. Then the probability of the two events are respectively

A

$\frac{1}{2}$ and $\frac{1}{3}$

B

$\frac{1}{5}$ and $\frac{1}{6}$

C

$\frac{1}{2}$ and $\frac{1}{6}$

D

$\frac{2}{3}$ and $\frac{1}{4}$

Solution

(a) $P(A \cap B) = P(A).P(B) = \frac{1}{6}$

$P(\bar A \cap \bar B) = \frac{1}{3} = 1 – P(A \cup B)$

$ \Rightarrow \frac{1}{3} = 1 – [P(A) + P(B)] + \frac{1}{6} $

$\Rightarrow P(A) + P(B) = \frac{5}{6}.$

Hence $P(A)$ and $P(B)$ are $\frac{1}{2}$ and $\frac{1}{3}.$

Standard 11

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