<p>India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points 0, 1 and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least 7 points is __________ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: India plays 4 matches total (2 with each team) and needs at least 7 points. Since max points per match is 2, the only way to get ≥7 points is to score exactly 7 or 8 points. Identify all favorable outcomes: getting 2 points in exactly 3 matches and 1 point in 1 match (sum=7), or getting 2 points in all 4 matches (sum=8).
<p><strong>Step 1:</strong> Identify the constraint. India plays 4 matches total. Points per match: 0 (prob 0.45), 1 (prob 0.05), 2 (prob 0.50). Need P(Total ≥ 7 points).</p><p><strong>Step 2:</strong> Find favorable cases. Maximum possible = 8 points. For ≥7 points, we need either exactly 7 or exactly 8 points.</p><p><strong>Case 1 - Exactly 8 points:</strong> All 4 matches yield 2 points each.<br>P(8) = (0.50)^4 = 0.0625</p><p><strong>Case 2 - Exactly 7 points:</strong> Three matches yield 2 points, one match yields 1 point. This can happen in C(4,3) = 4 ways.<br>P(7) = C(4,3) × (0.50)^3 × (0.05)^1 = 4 × 0.125 × 0.05 = 0.025</p><p><strong>Step 3:</strong> Apply addition rule for mutually exclusive events.<br>P(≥7) = P(7) + P(8) = 0.025 + 0.0625 = 0.0875</p><p>∴ Answer: <strong>0.0875</strong></p>
Correct Answer: 0.0875