<p>A father has 3 children with at least one boy. The probability that he has 2 boys and 1 girl is</p>
Step-by-Step Solution
Key Concept: Use conditional probability: P(2 boys, 1 girl | at least 1 boy) = P(2 boys AND 1 girl AND at least 1 boy) / P(at least 1 boy). Since 2 boys and 1 girl automatically satisfies 'at least 1 boy', this simplifies to the ratio of favorable outcomes to all outcomes with at least one boy.
<p><strong>Step 1:</strong> List all possible outcomes for 3 children (B = boy, G = girl).</p><p>Total outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG (8 outcomes)</p><p><strong>Step 2:</strong> Apply the condition 'at least one boy' — exclude GGG.</p><p>Favorable sample space: BBB, BBG, BGB, BGG, GBB, GBG, GGB (7 outcomes)</p><p><strong>Step 3:</strong> Count outcomes with exactly 2 boys and 1 girl: BBG, BGB, GBB (3 outcomes)</p><p><strong>Step 4:</strong> Calculate conditional probability.</p><p>P(2 boys, 1 girl | at least 1 boy) = 3/7</p><p><strong>∴ Answer: B (3/7)</strong></p>
Correct Answer: B