A and B appear for an interview for two posts. The probability of A's section is `(1//3)` and that of `B's` selection is `(2//5)`. Find the probability that only one of them will be selected.

Options

A. `7//15`
B. `8//15`
C. `2//15`
D. `4//15`

Answer

To solve the problem step by step, we need to find the probability that only one of A or B is selected for the posts.

### Step 1: Determine the probabilities of selection and rejection for A and B.
- The probability of A being selected (P(A)) is given as \( \frac{1}{3} \).
- Therefore, the probability of A being rejected (P(A')) is:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \]

- The probability of B being selected (P(B)) is given as \( \frac{2}{5} \).
- Therefore, the probability of B being rejected (P(B')) is:
\[ P(B') = 1 - P(B) = 1 - \frac{2}{5} = \frac{3}{5} \]

### Step 2: Calculate the probability that only one of them is selected.
To find the probability that only one of them is selected, we consider two scenarios:
1. A is selected and B is rejected.
2. A is rejected and B is selected.

The probability of the first scenario (A selected, B rejected) is:
\[ P(A \text{ and } B') = P(A) \times P(B') = \frac{1}{3} \times \frac{3}{5} \]
Calculating this gives:
\[ P(A \text{ and } B') = \frac{1 \times 3}{3 \times 5} = \frac{3}{15} \]

The probability of the second scenario (A rejected, B selected) is:
\[ P(A' \text{ and } B) = P(A') \times P(B) = \frac{2}{3} \times \frac{2}{5} \]
Calculating this gives:
\[ P(A' \text{ and } B) = \frac{2 \times 2}{3 \times 5} = \frac{4}{15} \]

### Step 3: Add the probabilities of the two scenarios.
Now, we add the probabilities of the two scenarios to find the total probability that only one of them is selected:
\[ P(\text{only one selected}) = P(A \text{ and } B') + P(A' \text{ and } B) = \frac{3}{15} + \frac{4}{15} = \frac{7}{15} \]

### Conclusion
Thus, the probability that only one of A or B is selected is:
\[ \boxed{\frac{7}{15}} \]