<p>From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one woman. Then the probability for these committees to have more women than men is</p>
Step-by-Step Solution
Key Concept: Split the favorable outcomes into mutually exclusive cases based on gender composition (3W1M and 4W0M), then divide by total committees with at least one woman to find the conditional probability.
<p><strong>Step 1:</strong> Find total 4-member committees with at least one woman.</p><p>Total committees with at least one woman = C(15,4) - C(10,4) = 1365 - 210 = 1155</p><p><strong>Step 2:</strong> Find committees with more women than men (i.e., 3W1M or 4W0M).</p><p>Committees with 3 women and 1 man = C(5,3) × C(10,1) = 10 × 10 = 100</p><p>Committees with 4 women and 0 men = C(5,4) × C(10,0) = 5 × 1 = 5</p><p>Total favorable outcomes = 100 + 5 = 105</p><p><strong>Step 3:</strong> Calculate conditional probability.</p><p>Probability = 105/1155 = 105/1155 = 7/77 = 1/11</p><p>∴ Answer: B</p>
Correct Answer: B