Question

$A$ and $B$ are two independent events such that $P\left(A\right)=\frac{1}{2}$ and $P\left(B\right)=\frac{1}{3}$. Then $P$ (neither $A$ nor $B$) is equal to

• A. $\frac{2}{3}$
• B. $\frac{1}{6}$
• C. $\frac{5}{6}$
• D. $\frac{1}{3}$

Answer 1

Answer: (D)

$\frac{1}{3}$

P(neither $A$ nor $B$)=$=p\left(\overline{A}\cap \overline{B}\right)$

$\begin{array}{c}=1-p\left(A\cap B\right)\\ =1-p\left(A\right)-p\left(B\right)+p\left(A\cap B\right)\\ =1-\frac{1}{2}-\frac{1}{3}+\frac{1}{6}\\ =\frac{6-3-2+1}{6}\\ =\frac{2}{6}\\ =\frac{1}{3}\end{array}$

Option $D$ is correct answer.