<p>Two dices are rolled one after the other. The probability that the number on the first is smaller than the number on the second is</p>
Step-by-Step Solution
Key Concept: Count favorable outcomes where first die < second die systematically for each value on the first die, then divide by total outcomes (36). Recognize the symmetry: P(first < second) = P(first > second), and P(first < second) + P(first = second) + P(first > second) = 1.
<p><strong>Step 1:</strong> Identify total possible outcomes when rolling two dice one after another: 6 × 6 = 36 outcomes.</p><p><strong>Step 2:</strong> Count favorable outcomes where first die < second die:</p><ul><li>If first die = 1: second can be 2,3,4,5,6 → 5 outcomes</li><li>If first die = 2: second can be 3,4,5,6 → 4 outcomes</li><li>If first die = 3: second can be 4,5,6 → 3 outcomes</li><li>If first die = 4: second can be 5,6 → 2 outcomes</li><li>If first die = 5: second can be 6 → 1 outcome</li><li>If first die = 6: second can be nothing → 0 outcomes</li></ul><p><strong>Step 3:</strong> Total favorable outcomes = 5 + 4 + 3 + 2 + 1 + 0 = 15</p><p><strong>Step 4:</strong> Calculate probability = 15/36 = 5/12</p><p><strong>Alternative approach:</strong> By symmetry, P(first < second) = P(first > second). Since outcomes where first = second are 6 (pairs: 1-1, 2-2, 3-3, 4-4, 5-5, 6-6), we have: P(first < second) + P(first > second) + P(first = second) = 1. Therefore: 2·P(first < second) + 6/36 = 1, giving P(first < second) = (36 - 6)/2 ÷ 36 = 30/72 = 5/12</p><p>∴ Answer: <strong>5/12</strong></p>
Correct Answer: D