Probability
Bayes Theorem
Grade 12
Question:
<p>There are 3 bags which are known to contain 2 white and 3 black, 4 white and 1 black, and 3 white and 7 black balls, respectively. A ball is drawn at random from one of the bags and found to be a black ball. Then the probability that it was drawn from the bag containing the most black balls is</p>
<p>(1) 7/15</p>
<p>(2) 5/19</p>
<p>(3) 3/4</p>
<p>(4) None of these</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find the probability of drawing from a specific bag given that a black ball was observed. The denominator requires calculating the total probability of drawing a black ball from any bag.
<p><strong>Step 1:</strong> Identify the bags and their compositions.</p><ul><li>Bag 1: 2 white, 3 black (total 5 balls)</li><li>Bag 2: 4 white, 1 black (total 5 balls)</li><li>Bag 3: 3 white, 7 black (total 10 balls) — most black balls</li></ul><p><strong>Step 2:</strong> Calculate P(black ball | each bag).</p><ul><li>P(B|Bag₁) = 3/5</li><li>P(B|Bag₂) = 1/5</li><li>P(B|Bag₃) = 7/10</li></ul><p><strong>Step 3:</strong> Each bag is selected with probability 1/3. Calculate total probability of drawing a black ball.</p><p>P(B) = (1/3)(3/5) + (1/3)(1/5) + (1/3)(7/10)</p><p>P(B) = (1/3)[3/5 + 1/5 + 7/10] = (1/3)[6/10 + 2/10 + 7/10] = (1/3)(15/10) = 1/2</p><p><strong>Step 4:</strong> Apply Bayes' theorem to find P(Bag₃ | B).</p><p>P(Bag₃|B) = P(B|Bag₃) × P(Bag₃) / P(B)</p><p>P(Bag₃|B) = (7/10 × 1/3) / (1/2) = (7/30) × 2 = 7/15</p><p>∴ <strong>Answer: 7/15</strong></p>
Correct Answer: 4