Probability
Conditional Probability
Grade 12
Question:
<p>A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is</p>
<p>3/4</p>
<p>3/10</p>
<p>2/5</p>
<p>1/5</p>
Step-by-Step Solution
Key Concept: Use the law of total probability by conditioning on the color of the first draw. The first draw changes the composition of the bag, affecting the second draw's probability.
<p><strong>Step 1:</strong> Initial composition: 4 red, 6 black balls (total 10)</p><p><strong>Step 2:</strong> Let R₁ = first ball is red, B₁ = first ball is black</p><p>P(R₁) = 4/10 = 2/5, P(B₁) = 6/10 = 3/5</p><p><strong>Step 3:</strong> Case 1 - If first ball is RED: Ball is returned + 2 red balls added → Bag has 6 red, 6 black (total 12)</p><p>P(R₂|R₁) = 6/12 = 1/2</p><p><strong>Step 4:</strong> Case 2 - If first ball is BLACK: Ball is returned + 2 black balls added → Bag has 4 red, 8 black (total 12)</p><p>P(R₂|B₁) = 4/12 = 1/3</p><p><strong>Step 5:</strong> By law of total probability:</p><p>P(R₂) = P(R₂|R₁)·P(R₁) + P(R₂|B₁)·P(B₁)</p><p>P(R₂) = (1/2)·(2/5) + (1/3)·(3/5)</p><p>P(R₂) = 2/10 + 3/15 = 2/10 + 1/5 = 2/10 + 2/10 = 4/10 = <strong>2/5</strong></p><p>∴ Answer: C</p>
Correct Answer: C