Question

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

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

Answer 1

Answer: (B)

$P\left(A\right)=\frac{2}{5}=\frac{8}{20};P\left(A\cap B\right)=\frac{3}{20}$
$P\left(\overline{A\cup B}\right)=1-\frac{3}{20}$
$\Rightarrow P\left(\bar{A}\cup \bar{B}\right)=\frac{17}{20}$
$A\cap \left(A^{\prime }\cup B^{\prime }\right)$
$=A-\left(A\cap B\right)$
$\therefore P\left(A-\left(A\cap B\right)\right)=\frac{5}{20}$
$\therefore P\left(A/\left(A^{\prime }\cap B^{\prime }\right)\right)=\frac{P\left(A-\left(A\cap B\right)\right)}{P\left(\bar{A}\cup \bar{B}\right)}=\frac{5}{17}$