A man and his wife appeared for an interview for the two vacancies. If the probability of husband's selection is 1 3 and the probability of wife's…

The probability of husband's selection is $\frac{1}{3}$31​ and the probability of wife's selection is $\frac{1}{5}$51​.

To find the probability that none of them will be selected, we need the probability that both get rejected.

Probability that husband is not selected:

$$P(\text{husband not selected}) = 1 - \frac{1}{3}$$P(husband not selected)=1−31​

$$= \frac{2}{3}$$=32​

Probability that wife is not selected:

$$P(\text{wife not selected}) = 1 - \frac{1}{5}$$P(wife not selected)=1−51​

$$= \frac{4}{5}$$=54​

---

Since the selections are independent events, the probability that both are not selected is the product of their individual probabilities.

Probability that none of them will be selected:

$$P(\text{none selected}) = P(\text{husband not selected}) \times P(\text{wife not selected})$$P(none selected)=P(husband not selected)×P(wife not selected)

$$= \frac{2}{3} \times \frac{4}{5}$$=32​×54​

$$= \frac{8}{15}$$=158​

---

Therefore, the probability that none of them will be selected is $\frac{8}{15}$158​.