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

• A

  $\frac{11}{20}$
• Correct answer
  B

  $\frac{5}{17}$
• C

  $\frac{8}{17}$
• D

  $\frac{1}{4}$

Solution

Given $P\left(A\right)=\frac{2}{5}=\frac{8}{20}$ & $P\left(A\cap B\right)=\frac{3}{20}$

$\therefore P\left(A^{\text{'}}\cup B^{\text{'}}\right)=P{\left(A\cap B\right)}^{\text{'}}$

$=1-P\left(A\cap B\right)$

$=1-\frac{3}{20}=\frac{17}{20}$

$P\left(A\cap \left(A^{\text{'}}\cup B^{\text{'}}\right)\right)=P\left(A\cap B^{\text{'}}\right)$

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

$=\frac{8}{20}-\frac{3}{20}=\frac{5}{20}$

$\therefore P\left(A|\left(A^{\text{'}}\cup B^{\text{'}}\right)\right)=\frac{P\left(A\cap \left(A^{\text{'}}\cup B^{\text{'}}\right)\right)}{P\left(A^{\text{'}}\cup B^{\text{'}}\right)}$

$=\frac{\left(\frac{5}{20}\right)}{\left(\frac{17}{20}\right)}=\frac{5}{17}$

Hence, the correct answer is option: 'B'.