Question

{"given":"India's probability of winning each test match is $\\frac{1}{2}$, and matches are independent. We need to find the probability that India's second win occurs at the third test in a 5-match series.","key_observation":"For India's second win to occur at the third test, two conditions must be satisfied: (1) India must win exactly one of the first two matches and lose the other, and (2) India must win the third match. Since matches are independent, we can multiply the probabilities of these events.","option_analysis":[{"label":"(A)","text":"$\\frac{1}{8}$","verdict":"incorrect","explanation":"This would be the probability if we needed three specific outcomes in sequence, but we're looking for one win in two matches plus a third win, which has higher probability."},{"label":"(B)","text":"$\\frac{1}{4}$","verdict":"correct","explanation":"Probability of exactly one win in first two matches is $\\frac{1}{2}$ (either WL or LW), and probability of winning third match is $\\frac{1}{2}$. Total: $\\frac{1}{2} \\times \\frac{1}{2} = \\frac{1}{4}$."},{"label":"(C)","text":"$\\frac{1}{2}$","verdict":"incorrect","explanation":"This is just the probability of winning the third match alone, but doesn't account for the constraint of having exactly one win in the first two matches."},{"label":"(D)","text":"$\\frac{2}{3}$","verdict":"incorrect","explanation":"This probability is too high and doesn't follow from the given constraints. The actual calculation yields a much smaller probability."}],"answer":"(B)","formula_steps":[]}