Question

$P(A\cap B)=\frac{1}{2},P(\overline{A}\cap \overline{B})=\frac{1}{2}$ and $2P\left(A\right)=P\left(B\right)=p$, then the value of $p$ is equal to

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

Answer 1

Answer: (B)

$\frac{2}{3}$

$P(A\cap B)=\frac{1}{2},P(\overline{A}\cap \overline{B})=\frac{1}{2}$ and $2P\left(A\right)=P\left(B\right)=p$
Now, $P(\overline{A}\cap \overline{B})=\frac{1}{2}$
$\Rightarrow P(\overline{A}\cup \overline{B})=\frac{1}{2}$
$\Rightarrow 1-P(A\cup B)=\frac{1}{2}$
$\Rightarrow P(A\cup B)=\frac{1}{2}$
$\Rightarrow P\left(A\right)+P\left(B\right)-P(A\cap B)=\frac{1}{2}$
$\Rightarrow \frac{p}{2}+p-\frac{1}{2}=\frac{1}{2}\Rightarrow \frac{3p}{2}=1\Rightarrow p=\frac{2}{3}$