Two friends, P and Q, appeared in an interview for two vacancies for the same post. The probability of P's selection is $\frac{1}{3}$ and that of Q's selection is $\frac{2}{7}$. What is the probability that at least one of them will be selected?

The probability of P's selection is $\frac{1}{3}$ and the probability of Q's selection is $\frac{2}{7}$.

To find the probability that at least one of them will be selected, we use:

P(At least one selected) = 1 - P(Neither selected)

---

The probability that P is not selected:

$P(\text{P not selected}) = 1 - \frac{1}{3}$

$= \frac{2}{3}$

---

The probability that Q is not selected:

$P(\text{Q not selected}) = 1 - \frac{2}{7}$

$= \frac{5}{7}$

---

Since the selections are independent, the probability that neither gets selected:

$P(\text{Neither selected}) = \frac{2}{3} \times \frac{5}{7}$

$= \frac{10}{21}$

---

The probability that at least one gets selected:

$P(\text{At least one selected}) = 1 - \frac{10}{21}$

$= \frac{21}{21} - \frac{10}{21}$

$= \frac{11}{21}$

Therefore, the probability that at least one of them will be selected is $\frac{11}{21}$.