Two candidates A and B interview for two vacancies. The probabilities that A and B are selected are 1/3 and 1/6 respectively. Assuming independence, what is the probability that neither is selected?

Introduction / Context:
Given marginal selection probabilities for two candidates and assuming independence of their outcomes, the chance that neither is selected is the product of their non-selection probabilities.

Given Data / Assumptions:

• P(A selected) = 1/3 → P(A not selected) = 2/3.
• P(B selected) = 1/6 → P(B not selected) = 5/6.
• Selections are independent (standard assumption unless stated otherwise).

Concept / Approach:
P(neither) = P(A not selected) * P(B not selected).

Step-by-Step Solution:
P(neither) = (2/3) * (5/6) = 10/18 = 5/9.

Verification / Alternative check:
P(at least one selected) = 1 − 5/9 = 4/9. Inclusion–exclusion also yields P(A ∪ B) = 1/3 + 1/6 − (1/3)(1/6) = 1/2 − 1/18 = 8/18 = 4/9.

Why Other Options Are Wrong:
5/12 and smaller values underestimate; independence requires multiplying complements, not adding/subtracting naïvely.

Common Pitfalls:
Assuming mutual exclusivity or that exactly two positions force dependence; the statement provides independent success chances for each candidate.

Final Answer:
5/9