Question

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

• A. $\frac{1}{2}$
• B. $\frac{1}{12}$
• C. $\frac{1}{4}$
• D. None of these

Answer 1

Answer: (A)

Let $E_{1}$ be the events that man will be selected and $E_{2}$ the events that woman will be selected.
Then, $P\left(E_{1}\right)=\frac{1}{4}$
So, $P\left(\bar{E}_{1}\right)=1-\frac{1}{4}=\frac{3}{4}$
and $P\left(E_{2}\right)=\frac{1}{3}$
$\Rightarrow P\left(\bar{E}_{2}\right)=\frac{2}{3}$
$\therefore E_{1}$ and $E_{2}$ are independent events.
$\therefore P\left(\bar{E}_{1} \cap \bar{E}_{2}\right)=P\left(\bar{E}_{1}\right) \times P\left(\bar{E}_{2}\right)$
$=\frac{3}{4} \times \frac{2}{3}=\frac{1}{2}$