A husband and wife appear in an interview for two vacancies in the same post. The probability of husband's selection is 1/5 and that of wife's selection is 1/3. What is the probability that only one of them will be selected ?

Options

A. `1//5`
B. `2//5`
C. `3//5`
D. `4//5`

Explanation

To solve the problem, we need to find the probability that only one of the husband or wife is selected for the vacancies.

Let's denote:
- Event A: Husband is selected
- Event B: Wife is selected

Given:
- Probability of husband being selected, P(A) = 1/5
- Probability of wife being selected, P(B) = 1/3

We need to find the probability that only one of them is selected. This can happen in two scenarios:
1. The husband is selected and the wife is not selected.
2. The wife is selected and the husband is not selected.

### Step 1: Calculate the probability that the husband is selected and the wife is not selected.
The probability that the wife is not selected is given by:
\[ P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \]

Now, the probability that the husband is selected and the wife is not selected is:
\[ P(A \cap B') = P(A) \times P(B') = \frac{1}{5} \times \frac{2}{3} = \frac{2}{15} \]

### Step 2: Calculate the probability that the wife is selected and the husband is not selected.
The probability that the husband is not selected is given by:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{5} = \frac{4}{5} \]

Now, the probability that the wife is selected and the husband is not selected is:
\[ P(A' \cap B) = P(A') \times P(B) = \frac{4}{5} \times \frac{1}{3} = \frac{4}{15} \]

### Step 3: Add the probabilities of the two scenarios.
Now, we can find the total probability that only one of them is selected by adding the probabilities from Step 1 and Step 2:
\[ P(\text{Only one selected}) = P(A \cap B') + P(A' \cap B) = \frac{2}{15} + \frac{4}{15} = \frac{6}{15} \]

### Step 4: Simplify the result.
Finally, we simplify the fraction:
\[ \frac{6}{15} = \frac{2}{5} \]

Thus, the probability that only one of them will be selected is:
\[ \boxed{\frac{2}{5}} \]