Choose the correct answer from the given four options.If P(A)=2 5 ,P(B)=3 5 and P(A )=1 5 , then P ( A' B' ) ( B' A' ) is equas :

Choose the correct answer from the given four options.If P(A)=2 5…

MCQ

Choose the correct answer from the given four options.If $\text{P}(\text{A})=\frac{2}{5},\text{P}(\text{B})=\frac{3}{5}$ and $\text{P}(\text{A}\cap\text{B})=\frac{1}{5},$ then $\text{P}\Big(\frac{\text{A}'}{\text{B}'}\Big)\cdot\text{P}\Big(\frac{\text{B}'}{\text{A}'}\Big)$ is equas :

✓
$\frac{5}{6}$
B
$\frac{5}{7}$
C
$\frac{25}{42}$
D
$1$

✓

Answer

Correct option: A.

$\frac{5}{6}$

Here, $\text{P}(\text{A})=\frac{2}{5},\text{P}(\text{B})=\frac{3}{5}$ and $\text{P}(\text{A}\cap\text{B})=\frac{1}{5}$
$\text{P}\Big(\frac{\text{A}'}{\text{B}'}\Big)=\frac{\text{P}(\text{A}'\cap\text{B}')}{\text{P}(\text{B}')}+\frac{1-\text{P}(\text{A}\cap\text{B})}{1-\text{P}(\text{B})}$
$=\frac{1-\big[\text{P}(\text{A})+\text{P}(\text{B})-\text{A}(\text{A}\cap\text{B})\big]}{1-\text{P}(\text{B})}$
$=\frac{1-\Big(\frac{2}{5}+\frac{3}{10}-\frac{1}{5}\Big)}{1-\frac{3}{10}}$
$=\frac{1-\Big(\frac{4+3-2}{10}\Big)}{\frac{7}{10}}-\frac{1-\frac{1}{2}}{\frac{7}{10}}$
$=\frac{5}{7}$
And $\text{P}\Big(\frac{\text{B}'}{\text{A}'}\Big)=\frac{\text{P}(\text{B}'\cap\text{A}')}{\text{P}(\text{A}')}$
$=\frac{1-\text{P}(\text{A}\cup\text{B})}{1-\text{P}(\text{A})}$
$=\frac{1-\frac{1}{2}}{1-\frac{2}{5}}$
$\Big[\because\text{P}(\text{A}\cup\text{B})=\frac{1}{2}\Big]$
$=\frac{\frac{1}{2}}{\frac{3}{5}}=\frac{5}{6}$
$\therefore\text{P}\Big(\frac{\text{A}'}{\text{B}'}\Big)\cdot\text{P}\Big(\frac{\text{B}'}{\text{A}'}\Big)=\frac{5}{7}\cdot\frac{5}{6}=\frac{25}{42}$