$A$ and $B$ are the independent events. The probability that both occur simultaneously is $\frac{1}{6}$ and the probability that neither occur is $\frac{1}{3}$. The probability of occurrence of the events $A$ and $B$ is

Question

Dr. Abubacker · Accepted Answer

(b)
Given, $P(A \cap B) = \frac{1}{6}$
$\Rightarrow P(A)P(B) = \frac{1}{6}$ ...(i)
and $P\left( \overline{A} \cap \overline{B} \right) = \frac{1}{3}$
$\Rightarrow P\left( \overline{A} \right)P\left( \overline{B} \right) = \frac{1}{3}$
$\Rightarrow \left\{ 1 - P(A) \right\}\left\{ 1 - P(B) \right\} = \frac{1}{3}$
$\Rightarrow 1 - \frac{1}{3} + P(A)P(B) = P(A) + P(B)$
$\Rightarrow \frac{2}{3} + \frac{1}{6} = P(A) + P(B)$ [from Eq.(i)]
$\Rightarrow P(A) + P(B) = \frac{5}{6}$ ...(ii)
On solving Eqs. (i) and (ii), we get
$P(A) = \frac{1}{2},\ P(B) = \frac{1}{3}$
or $P(A) = \frac{1}{3},\ P(B) = \frac{1}{2}$

Q. $A$ and $B$ are the independent events. The probability that both occur simultaneously is $\frac{1}{6}$ and the probability that neither occur is $\frac{1}{3}$. The probability of occurrence of the events $A$ and $B$ is

Explanation

Reviewed by Dr. Abubacker

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