A brother and a sister appear for an interview for two vacant posts in an office. The probability that the brother is selected is 1/5 and the probability that the sister is selected is 1/3. Assuming independence, what is the probability that exactly one of them is selected?

Introduction / Context:
This problem illustrates probability with independent events and focuses on the situation where exactly one of two candidates is selected. We know the individual probabilities of selection for the brother and the sister and want the probability that either the brother or the sister, but not both, get selected.

Given Data / Assumptions:

• Probability that the brother is selected, P(B) = 1/5.
• Probability that the sister is selected, P(S) = 1/3.
• The selection events for brother and sister are assumed to be independent.
• We seek P(exactly one is selected) = P(B and not S) + P(S and not B).

Concept / Approach:
We compute the probability that the brother is selected while the sister is not, plus the probability that the sister is selected while the brother is not. Because of independence, these can be expressed as products of the individual probabilities and their complements. This is a standard pattern for exactly one of two independent events occurring.

Step-by-Step Solution:
Complement probabilities: P(not B) = 1 - 1/5 = 4/5 and P(not S) = 1 - 1/3 = 2/3.Probability that the brother is selected but the sister is not: P(B and not S) = P(B) * P(not S) = (1/5) * (2/3) = 2 / 15.Probability that the sister is selected but the brother is not: P(S and not B) = P(S) * P(not B) = (1/3) * (4/5) = 4 / 15.Probability that exactly one of them is selected = 2/15 + 4/15 = 6/15.Simplify 6/15 by dividing numerator and denominator by 3 to get 2/5.

Verification / Alternative check:
We can also compute the probability that neither is selected and that both are selected, and ensure that all four basic possibilities sum to 1. P(neither) = P(not B) * P(not S) = (4/5) * (2/3) = 8/15. P(both) = P(B) * P(S) = (1/5) * (1/3) = 1/15. So P(exactly one) = 1 - P(neither) - P(both) = 1 - 8/15 - 1/15 = 6/15 = 2/5, confirming the result.

Why Other Options Are Wrong:
The value 1/5 is just the probability of the brother being selected alone and ignores the sister. The value 3/5 would imply 9/15 for exactly one selection, which does not fit with the other probabilities. The probability 3/4 is too high given that both individual selection probabilities are significantly below 1. Only 2/5 correctly sums the two disjoint cases where exactly one of them is selected.

Common Pitfalls:
Some students add P(B) and P(S) directly to get 1/5 + 1/3 without subtracting the probability that both are selected, which overcounts the overlapping case. Others forget to use the complements P(not B) and P(not S) correctly. Drawing a small table of outcomes (B selected or not, S selected or not) and filling in probabilities can help visualise and avoid such mistakes.

Final Answer:
The probability that exactly one of the brother and sister is selected is 2/5.