Question

A man and a woman appear in an interview for two vacancies in the same post. The probability of man's selection is $\frac{1}{4}$ and that of the woman's selection is $\frac{1}{3}$. What is the probability that none of them will be selected.

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{8}$

Answer 1

The correct option is A $\frac{1}{2}$

Let $E_1$ be the event that man will be selected and $E_2$ be the event that woman will be selected.
$\Rightarrow$ $P\left(E_1\right)=\frac{1}{4}$ and $P\left(E_2\right)=\frac{1}{3}$
$\Rightarrow$ $P\left({\overline{E}}_1\right)=1-\frac{1}{4}=\frac{3}{4}$
and $P\left({\overline{E}}_2\right)=\frac{2}{3}$
Clearly, $E_1$ and $E_2$ are independent events.
So, $P(\overline{E}\cap {\overline{E}}_2)=P\left({\overline{E}}_1\right)\times P\left({\overline{E}}_2\right)=\frac{3}{4}\times \frac{2}{3}=\frac{1}{2}$