<p>The probability of India winning a test match against West Indies is 1/2. Assuming independence from match to match, find the probability that in a match series India's second win occurs at the third test.</p>
Step-by-Step Solution
Key Concept: For the second win to occur at the third test, India must win exactly 1 of the first 2 tests and then win the third test. This requires careful sequencing: one loss and one win in first two matches, followed by a mandatory win.
<p><strong>Step 1:</strong> For India's second win to occur at the third test, we need:</p><ul><li>Exactly 1 win in the first 2 tests</li><li>A win on the third test (mandatory)</li></ul><p><strong>Step 2:</strong> Probability of exactly 1 win in first 2 tests:</p><p>Number of ways = C(2,1) = 2 (win-loss or loss-win)</p><p>P(exactly 1 win in 2 tests) = C(2,1) × (1/2)¹ × (1/2)¹ = 2 × 1/4 = 1/2</p><p><strong>Step 3:</strong> Probability of win on third test = 1/2</p><p><strong>Step 4:</strong> By independence:</p><p>P(second win at third test) = P(exactly 1 win in first 2) × P(win on test 3)</p><p>= 1/2 × 1/2 = <strong>1/4</strong></p>
Correct Answer: 1/4