Choose the correct answer from the given four options. If A and B… - Vidyadip

MCQ

Choose the correct answer from the given four options.
If A and B are two events such that $\text{P}(\text{A})=\frac{1}{2},\text{P}(\text{B})=\frac{1}{3},$ $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{1}{4},$ then $\text{P}(\text{A}'\cap\text{B}')$ equals:

A
$\frac{1}{12}$
B
$\frac{3}{4}$
✓
$\frac{1}{4}$
D
$\frac{3}{16}$

✓

Answer

Correct option: C.

$\frac{1}{4}$

We have, $\text{P}(\text{A})=\frac{1}{2},\text{P}(\text{B})=\frac{1}{3}$ and $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{1}{4}$

$\Rightarrow\text{P}(\text{A}\cap\text{B})=\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)\cdot\text{P}(\text{B})$

$=\frac{1}{4}\cdot\frac{1}{3}=\frac{1}{12}$

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

$=1-\big[\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A}\cap\text{B})\big]$

$=1-\Big[\frac{1}{2}+\frac{1}{3}-\frac{1}{12}\Big]=1-\frac{9}{12}$

$=\frac{3}{12}=\frac{1}{4}$

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