Question

If A and B are two events such that $P\left(A\right)=\frac{2}{5}$ and $P(A\cap B)=\frac{3}{20}$, then $P\left(A|\right(A^{\prime }\cup B^{\prime }\left)\right)$ is equal to

• A. $\frac{8}{17}$
• B. $\frac{1}{4}$
• C. $\frac{11}{20}$
• D. $\frac{5}{17}$

Answer 1

The correct option is D

$\frac{5}{17}$

$P\left(A\right)=\frac{2}{5}$ and $P(A\cap B)=\frac{3}{20}$

Now, $P(A^{\prime }\cup B^{\prime })=1-P(A\cap B)$

$\therefore P(A^{\prime }\cup B^{\prime })=\frac{17}{20}$ ....(i)

$P\left(\frac{A}{A^{\prime }\cup B^{\prime }}\right)=\frac{P(A\cap (A^{\prime }\cup B^{\prime }\left)\right)}{P(A^{\prime }\cup B^{\prime })}=\frac{P(A\cap B^{\prime })}{P(A^{\prime }\cup B^{\prime })}$

$\Rightarrow \frac{P\left(A\right)-P(A\cap B)}{P(A^{\prime }\cup B^{\prime })}=\frac{\frac{2}{5}-\frac{3}{20}}{\frac{17}{20}}=\frac{5}{17}$