Ajay and his wife Reshmi appear independently for two vacancies in the same post. The probability that Ajay is selected is 1/7 and the probability that Reshmi is selected is 1/5. What is the probability that exactly one of them is selected?

Introduction / Context:
This probability question focuses on the idea of exactly one of two independent events occurring. Ajay and Reshmi both apply for two vacancies, and we know their individual selection probabilities. We must find the probability that exactly one of them is selected, meaning either Ajay alone or Reshmi alone, but not both together. This is an example of the event known as the symmetric difference of two independent events.

Given Data / Assumptions:

Probability that Ajay is selected = 1/7.

Probability that Reshmi is selected = 1/5.

Their selection events are independent.

We need the probability that exactly one is selected.

Concept / Approach:
Let A be the event that Ajay is selected and R be the event that Reshmi is selected. The event "exactly one is selected" can be written as (A and not R) or (R and not A). Because A and R are independent, the probabilities of combinations such as A and not R can be found by multiplying the probabilities of the constituent events. We then add the mutually exclusive events A and not R, and R and not A.

Step-by-Step Solution:
Step 1: Let P(A) = 1/7 and P(R) = 1/5.Step 2: Probability that Ajay is selected and Reshmi is not selected = P(A) * P(not R) = (1/7) * (1 - 1/5) = (1/7) * (4/5) = 4/35.Step 3: Probability that Reshmi is selected and Ajay is not selected = P(R) * P(not A) = (1/5) * (1 - 1/7) = (1/5) * (6/7) = 6/35.Step 4: These two events are mutually exclusive because they cannot happen at the same time.Step 5: Probability that exactly one is selected = 4/35 + 6/35 = 10/35.Step 6: Simplify 10/35 by dividing numerator and denominator by 5 to get 2/7.

Verification / Alternative check:
An alternative is to use the formula P(exactly one) = P(A) + P(R) - 2 * P(A and R). Since P(A and R) = (1/7) * (1/5) = 1/35, we obtain P(exactly one) = 1/7 + 1/5 - 2 * (1/35) = 5/35 + 7/35 - 2/35 = 10/35 = 2/7. This confirms the result obtained from direct case analysis.

Why Other Options Are Wrong:
The fraction 2/35 is actually P(A and R), the probability that both are selected, not that exactly one is selected. The value 5/7 is much too large and would imply that in most scenarios exactly one is chosen, which is not supported by the given probabilities. The values 1/5 and 1/7 correspond to the individual selection probabilities and do not capture the combined event correctly.

Common Pitfalls:
A common mistake is to compute P(A) + P(R) and stop there, forgetting to subtract the probability that both are selected, which is counted twice in that sum. Another error is to treat the events as mutually exclusive, which they are not. Always remember that "exactly one" requires including two separate cases and excluding the overlap where both are selected.

Final Answer:
The probability that exactly one of Ajay or Reshmi is selected is 2/7.