Probability
Total Probability Theorem
Grade 12
Question:
<p>An urn contains 5 red and 5 black balls. A ball is drawn at random, its color is noted and is returned to the urn. Moreover, 2 additional balls of the color drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?</p>
Step-by-Step Solution
Key Concept: Use the law of total probability by conditioning on the color of the first ball drawn. The composition of the urn changes after the first draw, affecting the probability of the second draw.
<p><strong>Step 1:</strong> Set up using the law of total probability. Let R₁ = first ball is red, B₁ = first ball is black.</p><p>P(2nd red) = P(2nd red|R₁)·P(R₁) + P(2nd red|B₁)·P(B₁)</p><p><strong>Step 2:</strong> Calculate initial probabilities: P(R₁) = 5/10 = 1/2 and P(B₁) = 5/10 = 1/2</p><p><strong>Step 3:</strong> Find conditional probabilities after composition changes:</p><p>• If 1st ball is red: ball is returned, 2 red balls added → urn has 7 red, 5 black (12 total)</p><p> P(2nd red|R₁) = 7/12</p><p>• If 1st ball is black: ball is returned, 2 black balls added → urn has 5 red, 7 black (12 total)</p><p> P(2nd red|B₁) = 5/12</p><p><strong>Step 4:</strong> Apply the law of total probability:</p><p>P(2nd red) = (7/12)·(1/2) + (5/12)·(1/2) = 7/24 + 5/24 = 12/24 = 1/2</p><p>∴ Answer: <strong>1/2</strong></p>
Correct Answer: 1