From 13 applicants (5 women, 8 men), two persons are selected at random. What is the probability that at least one selected person is a woman?

Introduction / Context:
“At least one” probability questions are usually simpler via complement: subtract the probability that no one satisfies the property from 1. Here, “no woman” means both selected are men.

Given Data / Assumptions:

• Total applicants = 13 (5 women, 8 men).
• Two are selected uniformly without order.

Concept / Approach:
P(≥1 woman) = 1 − P(both men) = 1 − C(8,2)/C(13,2).

Step-by-Step Solution:
C(8,2) = 28; C(13,2) = 78.P(≥1 woman) = 1 − 28/78 = 50/78 = 25/39.

Verification / Alternative check:
Direct count of favourable outcomes: total pairs (78) minus all-men pairs (28) = 50 favourable → 50/78 = 25/39.

Why Other Options Are Wrong:
10/13 and 5/13 arise from incorrect linear approximations; 14/35 is unrelated.

Common Pitfalls:
Forgetting to use combinations (unordered selection) or to apply the complement method.

Final Answer:
25/39