Question

If $A$ and $B$ are any two events such that $P\left(A\right)=\frac{2}{5}$ and $P(A\cap B)=\frac{3}{20}$, then the conditional probability, $P\left(A|\right(A^{\prime }\cup B^{\prime }\left)\right)$, where A' denotes the complement of $A$, is equal to:

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

Answer 1

The correct option is A $\frac{5}{17}$

$P\left(A\right)=\frac{2}{5}$ and $P(A\cap B)=\frac{3}{20}$, then
$P(A|A^{\prime }\cup B^{\prime })=\frac{P(A\cap (A^{\prime }\cup B^{\prime }\left)\right)}{1-P(A\cap B)}$,
$P(A|A^{\prime }\cup B^{\prime })=\frac{P\left(A\right)-P(A\cap B)}{1-P(A\cap B)}$
$P(A|A^{\prime }\cup B^{\prime })=\frac{\frac{2}{5}-\frac{3}{20}}{1-\frac{3}{20}}=\frac{5}{17}$