<p>Let \(X\) be the random variable which denotes the ₹ gained by a person. Two dice are thrown. The person gets ₹15 if same numbers appear on both dice, ₹12 if the sum is 9, and loses ₹6 otherwise. The expected gain/loss \(E(X)\) is:</p>
Step-by-Step Solution
Key Concept: Expected value is calculated by multiplying each outcome's value by its probability, then summing all contributions. You must ensure all probabilities are mutually exclusive and partition the entire sample space.
<p><strong>Step 1:</strong> Identify all outcomes and their probabilities.</p><p>Total outcomes when throwing two dice = 36</p><p><strong>Step 2:</strong> Find P(same numbers appear).</p><p>Same numbers: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) → 6 outcomes</p><p>P(same) = 6/36 = 1/6, Gain = ₹15</p><p><strong>Step 3:</strong> Find P(sum = 9).</p><p>Sum = 9: (3,6), (4,5), (5,4), (6,3) → 4 outcomes</p><p>P(sum=9) = 4/36 = 1/9, Gain = ₹12</p><p><strong>Step 4:</strong> Find P(otherwise).</p><p>P(otherwise) = 1 - 6/36 - 4/36 = 1 - 10/36 = 26/36 = 13/18, Loss = ₹-6</p><p><strong>Step 5:</strong> Calculate E(X).</p><p>E(X) = 15 × (1/6) + 12 × (1/9) + (-6) × (13/18)</p><p>E(X) = 15/6 + 12/9 - 78/18</p><p>E(X) = 45/18 + 24/18 - 78/18 = -9/18 = -1/2</p><p><strong>∴ Answer: E(X) = -₹0.50 (expected loss of ₹0.50)</strong></p>
Correct Answer: B