Question

If $A$ and $B$ are two independent events such that $P(A^{\prime }\cap B)=\frac{2}{15}$ and $P(A\cap B^{\prime })=\frac{1}{6},$ then $P\left(A\right)$ is

• A. $\frac{1}{5}$
• B. $\frac{1}{6}$
• C. $\frac{4}{5}$
• D. $\frac{5}{6}$

Answer 1

Answer: (A), (D)

The correct options are
A $\frac{1}{5}$
D $\frac{5}{6}$

$P(A^{\prime }\cap B)=\frac{2}{15},P(A\cap B^{\prime })=\frac{1}{6}$
As $A$ and $B$ are independent events,
$P(A\cap B)=P\left(A\right)\times P\left(B\right)$

and $P(A^{\prime }\cap B)=P\left(A^{\prime }\right)\times P\left(B\right)$
$=(1-P(A\left)\right)\times P\left(B\right)=\frac{2}{15}\cdots \left(1\right)$

$P(A\cap B^{\prime })=P\left(A\right)\times (1-P(B\left)\right)=\frac{1}{6}\cdots \left(2\right)$

Let $P\left(A\right)=p$
From $\left(1\right)$,
$P\left(B\right)=\frac{2}{15}\times \frac{1}{1-p}$
Substituting it in $\left(2\right)$
$p(1-\frac{2}{15}\times \frac{1}{1-p})=\frac{1}{6}$

$\Rightarrow 30p^2-31p+5=0\Rightarrow (5p-1)(6p-5)=0$
$\Rightarrow p=\frac{1}{5}$ or $\frac{5}{6}$