Hiring scenario with two candidates: A brother and sister apply for two vacant posts. P(brother selected) = 1/5 and P(sister selected) = 1/3, independently. What is the probability that exactly one of them is selected?

Introduction / Context:
We calculate the probability that exactly one of two independent events occurs. This is a common inclusion–exclusion style computation for Bernoulli outcomes (selected vs not selected).

Given Data / Assumptions:

• P(Brother selected) = 1/5 ⇒ P(B̄) = 4/5.
• P(Sister selected) = 1/3 ⇒ P(S̄) = 2/3.
• Selections are independent (as stated).

Concept / Approach:
“Exactly one” happens if the brother is selected and the sister is not, OR the sister is selected and the brother is not. Add these disjoint probabilities.

Step-by-Step Solution:

Verification / Alternative check:
Use complement: 1 − P(both) − P(neither) = 1 − (1/5)*(1/3) − (4/5)*(2/3) = 1 − 1/15 − 8/15 = 1 − 9/15 = 6/15 = 2/5.

Why Other Options Are Wrong:
1/5 or 1/3 capture only one piece; 2/3 is the complement of “at most one,” not “exactly one.”

Common Pitfalls:
Confusing “exactly one” with “at least one,” or forgetting independence while multiplying.

Final Answer:
2/5