<p>A bag contains some white and some black balls, all combinations of balls being equally likely. The total number of balls in the bag is 10. If three balls are drawn at random without replacement and all of them are found to be black, the probability that the bag contains 1 white and 9 black balls is</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find the posterior probability of a specific composition given the observed event (three black balls drawn). We need P(1W, 9B | 3 black balls drawn) = P(3B | 1W, 9B) × P(1W, 9B) / P(3B).
<p><strong>Step 1: Define the problem</strong><br/>Let B_k denote the event that the bag contains k black balls and (10-k) white balls, where k ∈ {0,1,2,...,10}. Since all combinations are equally likely, P(B_k) = 1/11 for each k.</p><p><strong>Step 2: Find P(3 black balls drawn | B_k)</strong><br/>If the bag has k black balls, the probability of drawing 3 black balls without replacement is:<br/>P(3B | B_k) = C(k,3)/C(10,3) = [k(k-1)(k-2)]/[10×9×8] = [k(k-1)(k-2)]/720</p><p><strong>Step 3: Calculate P(3B | specific compositions)</strong><br/>• P(3B | B_9) = [9×8×7]/720 = 504/720 = 7/10<br/>• P(3B | B_8) = [8×7×6]/720 = 336/720 = 7/15<br/>• P(3B | B_k) = 0 for k < 3</p><p><strong>Step 4: Find total probability P(3B)</strong><br/>P(3B) = Σ P(3B | B_k) × P(B_k)<br/>Since only k ≥ 3 contribute:<br/>P(3B) = (1/11) × Σ_{k=3}^{10} [k(k-1)(k-2)]/720</p><p>Computing the sum:<br/>• k=3: 3×2×1 = 6<br/>• k=4: 4×3×2 = 24<br/>• k=5: 5×4×3 = 60<br/>• k=6: 6×5×4 = 120<br/>• k=7: 7×6×5 = 210<br/>• k=8: 8×7×6 = 336<br/>• k=9: 9×8×7 = 504<br/>• k=10: 10×9×8 = 720<br/>Sum = 1980<br/>P(3B) = (1/11) × (1980/720) = 1980/7920 = 11/44</p><p><strong>Step 5: Apply Bayes' theorem for P(B_9 | 3B)</strong><br/>P(B_9 | 3B) = [P(3B | B_9) × P(B_9)] / P(3B)<br/>= [(7/10) × (1/11)] / (11/44)<br/>= [7/110] / [11/44]<br/>= (7/110) × (44/11)<br/>= (7 × 44)/(110 × 11)<br/>= 308/1210<br/>= 14/55</p><p><strong>∴ Answer:</strong> The probability that the bag contains 1 white and 9 black balls is 14/55, which corresponds to option (1).</p>
Correct Answer: 1