<p>Susmit met his fast friend Ricky. Susmit asked Ricky, How many children do you have? Ricky replied 2, further Ricky added that one of his son born on Sunday then what is the probability that he has two sons?</p>
Step-by-Step Solution
Key Concept: This is a conditional probability problem where we must use Bayes' theorem with the constraint that AT LEAST one child is a son born on Sunday. We count favorable outcomes (both sons with at least one born on Sunday) divided by all possible outcomes matching the given condition.
<p><strong>Step 1:</strong> Define the sample space. Each child can be either Boy (B) or Girl (G), and born on any of 7 days. This gives 14 possibilities per child.</p><p><strong>Step 2:</strong> Given condition: "one son born on Sunday." This means at least one child is a boy born on Sunday.</p><p><strong>Step 3:</strong> Find outcomes where at least one child is a boy born on Sunday:</p><p>• Both sons with first born on Sunday: (B_Sun, B_any) = 1 × 14 = 14 outcomes</p><p>• Both sons with second born on Sunday: (B_any, B_Sun) = 14 × 1 = 14 outcomes</p><p>• Subtract overlap (both born on Sunday): 1 × 1 = 1 outcome</p><p>• Total with at least one son born on Sunday: 14 + 14 - 1 = 27 outcomes</p><p><strong>Step 4:</strong> Find outcomes with TWO SONS and at least one born on Sunday:</p><p>• (B_Sun, B_any): 1 × 14 = 14 outcomes</p><p>• (B_any, B_Sun): 14 × 1 = 14 outcomes</p><p>• Subtract overlap: 1 outcome</p><p>• Total: 14 + 14 - 1 = 27 outcomes (but we want BOTH boys)</p><p>Actually: Both sons where at least one born on Sunday = 14 + 14 - 1 = 27 outcomes</p><p><strong>Step 5:</strong> Probability = (Favorable outcomes with 2 sons AND at least 1 son on Sunday) / (All outcomes with at least 1 son on Sunday)</p><p>= 27/27 × (probability both are boys) = Wait, recalculate:</p><p>Outcomes with at least 1 B_Sun: 27 total outcomes across all gender combinations</p><p>Outcomes with 2 sons AND at least 1 B_Sun: Both children must be boys, at least one on Sunday = 27 outcomes</p><p>But total matching condition includes girls: (B_Sun, G_any) = 14; (G_any, B_Sun) = 14; (B_Sun, B_Sun) already counted</p><p>Total with at least one B_Sun = 14 + 14 + 27 - 1 = ... recalculating systematically:</p><p><strong>Correct:</strong> P(2 sons | at least 1 son on Sunday) = 27/(27 + 14) = 27/41</p><p>∴ Answer: <strong>27/41</strong> (Option D)</p>
Correct Answer: D