Question

$A$ 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 $\frac{1}{3}$. Find the probability of the occurrence of $A$.

• A. $\frac{4}{5}$
• B. $\frac{2}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{3}$

Answer 1

Answer: (D)

Given $P\left(A\cap B\right)=\frac{1}{6}$
$\Rightarrow P\left(A\right)P\left(B\right)=\frac{1}{6}\dots \left(1\right)$
and $P\left(A^c\cap B^c\right)=\frac{1}{3}$
$\Rightarrow P\left(A^c\right)P\left(B^c\right)=\frac{1}{3}$
$\Rightarrow \left(1-P\left(A\right)\right)\left(1-P\left(B\right)\right)=\frac{1}{3}\dots \left(2\right)$
Eliminating $P\left(B\right)$ between $\left(1\right)$ and $\left(2\right)$, we obtain
$\left(1-P\left(A\right)\right)\left(1-\frac{1}{6P\left(A\right)}\right)=\frac{1}{3}$
$\Rightarrow \left(1-t\right)\left(6t-1\right)=2t$, where $t=P\left(A\right)$
$\Rightarrow 6t^2-5t+1=0$
$\Rightarrow t=\frac{1}{2}$ or $\frac{1}{3}$
$\Rightarrow P\left(A\right)=\frac{1}{2}$ or $\frac{1}{3}$