A man and his wife apply independently for two vacancies on the same post. The probability that the man is selected is 1/7, and the probability that the wife is selected is 1/5. What is the probability that exactly one of them is selected?

Introduction / Context:
This question is about independent events and the probability that exactly one of two candidates is selected. It appears frequently in aptitude tests and recruitment exam practice, where candidates must understand how to combine probabilities of mutually exclusive scenarios and use complement events correctly. The key is to recognize that "exactly one" means one selected and the other rejected, which can happen in two distinct ways.

Given Data / Assumptions:

Probability that the man is selected = 1/7.

Probability that the wife is selected = 1/5.

Their selections are assumed independent.

We are interested in the probability that exactly one of them is selected.

Concept / Approach:
There are two mutually exclusive ways for exactly one of them to be selected: (1) the man is selected and the wife is not; (2) the wife is selected and the man is not. Because the events are independent, the probability of a specific combination is the product of the individual selection or rejection probabilities. The total probability for exactly one selection is the sum of the probabilities of these two mutually exclusive cases.

Step-by-Step Solution:
Let M be the event that the man is selected and W be the event that the wife is selected.Given: P(M) = 1/7, so P(not M) = 1 - 1/7 = 6/7.Given: P(W) = 1/5, so P(not W) = 1 - 1/5 = 4/5.Exactly one of them is selected means either (M and not W) or (not M and W).Because the events are independent, we can multiply probabilities within each case.P(M and not W) = P(M) * P(not W) = (1/7) * (4/5) = 4/35.P(not M and W) = P(not M) * P(W) = (6/7) * (1/5) = 6/35.Total probability for exactly one selection = 4/35 + 6/35 = 10/35.Simplify 10/35 by dividing numerator and denominator by 5 to get 2/7.
Verification / Alternative check:
We can check consistency by also considering all possible outcomes.Possible outcomes: both selected, neither selected, exactly one selected.If we compute these separately and sum them, the total must be 1.However, for the purposes of this problem we only need the exactly one case, and the calculation gives 2/7, which is between 0 and 1 and seems reasonable given the relatively small individual probabilities.
Why Other Options Are Wrong:
1/7 corresponds to one portion of the exact one case, but omits the scenario where the wife is selected and the man is not. 3/4 and 4/5 are too large to be plausible given the small individual selection probabilities. 1/5 is simply the wife's selection probability and does not reflect the condition about the man's non selection.

Common Pitfalls:
Students sometimes confuse "exactly one" with "at least one" and compute 1 minus the probability that neither is selected. That approach would produce a different value. Another common mistake is forgetting that both ways of having exactly one selected must be counted. Always list the mutually exclusive cases explicitly when dealing with "exactly one" type problems.

Final Answer:
The probability that exactly one of them is selected is 2/7.