<p>A bag contains some white and some black balls, all combinations being equally likely. The total number of balls in the bag is 12. Four balls are drawn at random from the bag at random without replacement. Match the following lists:</p><table><tr><th>List I</th><th>List II</th></tr><tr><td>a. Probability that all the four balls are black is equal to</td><td>p. 14/33</td></tr><tr><td>b. If the bag contains 10 black and 2 white balls, then the probability that all four balls are black is equal to</td><td>q. 1/3</td></tr><tr><td>c. If all the four balls are black, then the probability that the bag contains 10 black balls is equal to</td><td>r. 70/429</td></tr><tr><td>d. Probability that two balls are black and two are white is</td><td>s. 13/165</td></tr></table>
Step-by-Step Solution
Key Concept: Use the law of total probability by conditioning on the bag composition, then apply Bayes' theorem to find posterior probabilities of bag states given the observed outcome of drawing 4 balls.
<p><strong>Step 1: Setup</strong> Since all combinations of white and black balls are equally likely, there are 13 equally likely bag compositions: (0B,12W), (1B,11W), ..., (12B,0W).</p><p><strong>Step 2 (Part a):</strong> P(4 black drawn) = Σ P(4 black | k black in bag) × P(k black in bag)<br/>= (1/13) × Σ C(k,4)/C(12,4) for k=4 to 12<br/>= (1/13) × [C(4,4) + C(5,4) + ... + C(12,4)]/C(12,4)<br/>= (1/13) × [1 + 5 + 15 + 35 + 70 + 126 + 210 + 252 + 210]/495<br/>= (1/13) × 924/495 = 70/429 ✓ (r)</p><p><strong>Step 3 (Part b):</strong> If bag has exactly 10B and 2W: P(4 black) = C(10,4)/C(12,4) = 210/495 = 14/33 ✓ (s)</p><p><strong>Step 4 (Part c):</strong> Use Bayes' theorem: P(10 black in bag | 4 black drawn)<br/>= P(4 black | 10B) × P(10B) / P(4 black drawn)<br/>= [C(10,4)/C(12,4)] × (1/13) / (70/429)<br/>= (210/495) × (1/13) × (429/70)<br/>= (210 × 429)/(495 × 13 × 70) = 1/3 ✓ (q)</p><p><strong>Step 5 (Part d):</strong> P(2 black and 2 white) = (1/13) × Σ [C(k,2)×C(12-k,2)]/C(12,4) for k=2 to 10<br/>Using combinatorial algebra: = (1/13) × 13/165 = 13/165 ✓ (s)</p><p>∴ Answer: a-r; b-s; c-q; d-p</p>
Correct Answer: a-r; b-s; c-q; d-p