A man and his wife appear in an interview for two vacancies in the same post. The probability two of husband’s selection is 1/7 and the probability of wife’s selection is 1/5 . What is the probability that only one of them is selected?

Answer: Option B

Let $$E_1$$ = event that the husband is selected and $$E_2$$ = event that the wife is selected.
Then, $$P(E_1)=\frac{1}{7}$$   and $$P(E_2)=\frac{1}{5}$$
∴ $$P({\overline{E}}_1)=\left(1-\frac{1}{7}\right)$$   $$\text{ = }\frac{6}{7}$$
and
$$P({\overline{E}}_2)=$$   $$\left(1-\frac{1}{5}\right)=\frac{4}{5}$$
∴ Required probability = P [( A and not B) or (B and not A)]
= P $$\left[\left(E_1\cap {\bar{E}}_2\right)\text{or}\left(E_2\cap {\bar{E}}_1\right)\right]$$
= P $$\left(E_1\cap {\overline{E}}_2\right)+\text{ P}\left(E_2\cap {\overline{E}}_1\right)$$
= $$\text{P}\left(E_1\right).\text{ P}\left({\overline{E}}_2\right)+P\left(E_2\right).P\left({\overline{E}}_1\right)$$
$$\begin{array}{rl}& =\left(\frac{1}{7}\times \frac{4}{5}\right)+\left(\frac{1}{5}\times \frac{6}{7}\right)\\ & =\frac{10}{35}\\ & =\frac{2}{7}\end{array}$$