Question

A man and his wife appear in an interview for two vacancies in the same post. The probability of husband's selection is (1/7) and the probability of wife's selection is (1/5). What is the probability that only one of them is selected ?

• A. 2/7
• B. 1/7
• C. 3/4
• D. 4/5

Hint:

For any two independent events with probabilities \(p_1\) and \(p_2\):
The probability that exactly one occurs is:
\[ P = p_1(1-p_2) + p_2(1-p_1) \]
Here: \(\frac{1}{7}\left(\frac{4}{5}\right) + \frac{1}{5}\left(\frac{6}{7}\right) = \frac{4+6}{35} = \frac{10}{35} = \frac{2}{7}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a probability problem involving independent events.
The selection of the husband and the selection of the wife are two independent events.
The event "only one of them is selected" can happen in two mutually exclusive ways:
1. The husband is selected, and the wife is not selected.
2. The wife is selected, and the husband is not selected.
Key Formula or Approach:
Let \(H\) be the event that the husband is selected, and \(W\) be the event that the wife is selected.
Since they are independent:
\[ P(\text{only one}) = P(H \cap W^c) + P(H^c \cap W) \]
\[ P(\text{only one}) = P(H)P(W^c) + P(H^c)P(W) \]
Where \(P(A^c) = 1 - P(A)\) represents the probability of the complement of event \(A\).

Step 2: Detailed Explanation:
Let us identify the given probabilities:
- Probability of husband's selection, \(P(H) = \frac{1}{7}\)
- Probability of husband's non-selection, \(P(H^c) = 1 - \frac{1}{7} = \frac{6}{7}\)
- Probability of wife's selection, \(P(W) = \frac{1}{5}\)
- Probability of wife's non-selection, \(P(W^c) = 1 - \frac{1}{5} = \frac{4}{5}\)
Substitute these values into the formula:
\[ P(\text{only one}) = \left(\frac{1}{7} \times \frac{4}{5}\right) + \left(\frac{6}{7} \times \frac{1}{5}\right) \]
Calculate each term:
\[ \text{First term} = \frac{4}{35} \]
\[ \text{Second term} = \frac{6}{35} \]
Sum the two terms:
\[ P(\text{only one}) = \frac{4}{35} + \frac{6}{35} = \frac{10}{35} \]
Reduce the fraction to its simplest form by dividing the numerator and denominator by 5:
\[ P(\text{only one}) = \frac{2}{7} \]
This matches Option (A).

Step 3: Final Answer:
The correct option is (A).