<p>A person throws two fair dice. He wins ₹15 for throwing a doublet (same numbers on the two dice), wins ₹12 when the throw results in the sum of 9, and loses ₹6 for any other outcome on the throw. Then the expected gain/loss (in ₹) of the person is</p>
<p>\(\dfrac{1}{2}\) gain</p>
<p>\(\dfrac{1}{4}\) loss</p>
<p>\(\dfrac{1}{2}\) loss</p>
<p>2 gain</p>
Step-by-Step Solution
Key Concept: Expected value is calculated as the sum of (probability × outcome) for all mutually exclusive events. Ensure events don't overlap when counting favorable outcomes.
<p><strong>Step 1: Identify all mutually exclusive events</strong></p><p>Total outcomes when throwing two dice = 36</p><p><strong>Doublets (same numbers):</strong> (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) = 6 outcomes<br>Win ₹15 per doublet</p><p><strong>Sum equals 9 (excluding doublets):</strong> (3,6), (6,3), (4,5), (5,4) = 4 outcomes<br>Win ₹12 per sum of 9</p><p><strong>Other outcomes:</strong> 36 - 6 - 4 = 26 outcomes<br>Lose ₹6 per other outcome</p><p><strong>Step 2: Calculate expected value</strong></p><p>E(X) = P(doublet)×15 + P(sum=9)×12 + P(other)×(-6)</p><p>E(X) = (6/36)×15 + (4/36)×12 + (26/36)×(-6)</p><p>E(X) = 90/36 + 48/36 - 156/36</p><p>E(X) = (90 + 48 - 156)/36</p><p>E(X) = -18/36 = -1/2</p><p><strong>∴ Answer: Expected loss of ₹0.50 (or -₹0.5)</strong></p>
Correct Answer: C