<p>A bag contains 3 white, 3 black and 2 red balls. One by one, three balls are drawn without replacing them. Find the probability that the third ball is red.</p>
Step-by-Step Solution
Key Concept: By symmetry principle, each ball has equal probability of being drawn in any specific position. The probability that the third ball is red equals the proportion of red balls in the bag, regardless of what happens in draws 1 and 2.
<p><strong>Step 1:</strong> Identify total balls: 3 white + 3 black + 2 red = 8 balls total</p><p><strong>Step 2:</strong> Apply symmetry principle: When drawing without replacement, by symmetry each ball has equal probability of appearing in any specific position (1st, 2nd, 3rd, etc.)</p><p><strong>Step 3:</strong> Therefore, P(3rd ball is red) = (Number of red balls)/(Total balls) = 2/8 = 1/4</p><p><strong>Verification:</strong> This can be verified by considering all cases:<br/>P(3rd red) = P(RR in first 2) × P(red|2 drawn) + P(exactly 1R in first 2) × P(red|1R drawn) + P(no R in first 2) × P(red|0R drawn)<br/>= (2/8 × 1/7 × 2/6) + (2 × 2/8 × 6/7 × 2/6) + (6/8 × 5/7 × 2/6) = 1/4</p><p>∴ <strong>Answer: 1/4</strong></p>
Correct Answer: 1