<p>The probability of India winning a test match against West Indies is 1/2 assuming independence from match to match. The probability that in a match series India's second win occurs at the third test is:</p>
Step-by-Step Solution
Key Concept: For India's second win to occur at the third test, they must win exactly 1 of the first 2 tests AND win the 3rd test. Use the binomial probability for first two matches multiplied by the probability of winning the third.
<p><strong>Step 1:</strong> For the second win to occur at the 3rd test, we need:</p><ul><li>Exactly 1 win in the first 2 tests</li><li>A win in the 3rd test (mandatory)</li></ul><p><strong>Step 2:</strong> Probability of exactly 1 win in first 2 tests:</p><p>P(exactly 1 win in 2 matches) = C(2,1) × (1/2)¹ × (1/2)¹ = 2 × (1/4) = 1/2</p><p><strong>Step 3:</strong> Probability of winning the 3rd test:</p><p>P(win at 3rd test) = 1/2</p><p><strong>Step 4:</strong> Combined probability (using independence):</p><p>P(second win at 3rd test) = (1/2) × (1/2) = 1/4</p><p><strong>Alternative approach using negative binomial:</strong></p><p>P(rth success at nth trial) = C(n-1, r-1) × p^r × (1-p)^(n-r)</p><p>P(2nd win at 3rd test) = C(2,1) × (1/2)² × (1/2)¹ = 2 × (1/8) = 1/4</p><p>∴ Answer: B (1/4)</p>
Correct Answer: B