<p>Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is:</p>
Step-by-Step Solution
Key Concept: Use the definition of conditional probability P(A|B) = P(A∩B)/P(B). The event 'all girls' is a subset of 'at least two girls', so we need to count favorable outcomes carefully from the sample space of two families with two children each.
<p><strong>Step 1:</strong> Define the sample space. Each child is B or G with equal probability. Two families, each with 2 children = 4 children total. Total outcomes = 2⁴ = 16.</p><p><strong>Step 2:</strong> Find P(at least 2 girls). Count outcomes with 2, 3, or 4 girls:</p><ul><li>Exactly 2 girls: C(4,2) = 6 outcomes</li><li>Exactly 3 girls: C(4,3) = 4 outcomes</li><li>Exactly 4 girls: C(4,4) = 1 outcome</li></ul><p>Total: 6 + 4 + 1 = 11 outcomes</p><p><strong>Step 3:</strong> Find P(all 4 are girls AND at least 2 are girls). All 4 girls = 1 outcome, which satisfies 'at least 2 girls'.</p><p><strong>Step 4:</strong> Apply conditional probability formula:<br/>P(all girls | at least 2 girls) = P(all girls ∩ at least 2 girls) / P(at least 2 girls)<br/>= (1/16) / (11/16) = 1/11</p><p>∴ Answer: <strong>1/11</strong></p>
Correct Answer: A