Probability
Total Probability
Grade None
Question:
<p>There are two urns \(A\) and \(B\). Urn \(A\) contains 5 red, 3 blue, and 2 white balls, urn \(B\) contains 4 red, 3 blue, and 3 white balls. An urn is chosen at random and a ball is drawn. Probability that the ball drawn is red is</p>
<p>(1) 9/10</p>
<p>(2) 1/2</p>
<p>(3) 11/20</p>
<p>(4) 9/20</p>
Step-by-Step Solution
Key Concept: Use the law of total probability: P(red) = P(red|A)·P(A) + P(red|B)·P(B). Since an urn is chosen randomly, P(A) = P(B) = 1/2, and you must find the conditional probability of drawing red from each urn.
<p><strong>Step 1:</strong> Identify the composition of each urn.</p><p>Urn A: 5 red, 3 blue, 2 white → Total = 10 balls</p><p>Urn B: 4 red, 3 blue, 3 white → Total = 10 balls</p><p><strong>Step 2:</strong> Find P(red|A) and P(red|B).</p><p>P(red|A) = 5/10 = 1/2</p><p>P(red|B) = 4/10 = 2/5</p><p><strong>Step 3:</strong> Apply the law of total probability.</p><p>P(urn A chosen) = 1/2 and P(urn B chosen) = 1/2</p><p>P(red) = P(red|A)·P(A) + P(red|B)·P(B)</p><p>P(red) = (1/2)·(1/2) + (2/5)·(1/2)</p><p>P(red) = 1/4 + 2/10 = 1/4 + 1/5</p><p><strong>Step 4:</strong> Find common denominator and add.</p><p>P(red) = 5/20 + 4/20 = 9/20</p><p>∴ Answer: D</p>
Correct Answer: D