Question

For two events $A$ and $B$, let $P\left(A\right)=\frac{1}{2},P(A\cup B)=\frac{2}{3}$ and $P(A\cap B)=\frac{1}{6}$, What is $P(\overline{A}\cap B)$ equal to?

• A. $\frac{1}{6}$
• B. $\frac{1}{4}$
• C. $\frac{1}{3}$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\frac{1}{6}$

$P(A\cap B)=P\left(B\right)-P(\overline{A}\cap B)$

Also, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\frac{2}{3}=\frac{1}{2}+P\left(B\right)-\frac{1}{6}$

$⟹P\left(B\right)=\frac{5}{6}-\frac{3}{6}$

$⟹P\left(B\right)=\frac{1}{3}$

$\Rightarrow P(\overline{A}\cap B)=P\left(B\right)-P(A\cap B)=\frac{1}{3}-\frac{1}{6}$

$\therefore P(\overline{A}\cap B)=\frac{1}{6}$