(i) A husband and wife appear in an interview for two vacancies in the same post. The probability of husband's selection is `(1)/(7)` and that of wife's selection is `(1)/(5)`. Find the probability that both of them will be selected.
(ii) Amit and Nisha appear for an interview for two vacancies in a company. The probability of Amit's selection is `(1)/(5)` and that of Nisha's selection is `(1)/(6)`. What is the probability that both of them are selected ?

Answer

To solve the given problems step by step, we will calculate the probability of both individuals being selected in each scenario.

### Part (i): Husband and Wife Selection

1. **Identify the probabilities**:
- Probability of husband's selection, \( P(H) = \frac{1}{7} \)
- Probability of wife's selection, \( P(W) = \frac{1}{5} \)

2. **Calculate the probability that both are selected**:
The events of selection are independent, so we multiply the probabilities:
\[ P(\text{Both selected}) = P(H) \times P(W) = \frac{1}{7} \times \frac{1}{5} \]

3. **Perform the multiplication**:
\[ P(\text{Both selected}) = \frac{1 \times 1}{7 \times 5} = \frac{1}{35} \]

### Final Answer for Part (i):
The probability that both the husband and wife will be selected is \( \frac{1}{35} \).

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### Part (ii): Amit and Nisha Selection

1. **Identify the probabilities**:
- Probability of Amit's selection, \( P(A) = \frac{1}{5} \)
- Probability of Nisha's selection, \( P(N) = \frac{1}{6} \)

2. **Calculate the probability that both are selected**:
Again, since the events are independent, we multiply the probabilities:
\[ P(\text{Both selected}) = P(A) \times P(N) = \frac{1}{5} \times \frac{1}{6} \]

3. **Perform the multiplication**:
\[ P(\text{Both selected}) = \frac{1 \times 1}{5 \times 6} = \frac{1}{30} \]

### Final Answer for Part (ii):
The probability that both Amit and Nisha will be selected is \( \frac{1}{30} \).

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