<p>An unbiased cubic die marked with 1, 2, 2, 3, 3, 3 is rolled 3 times. The probability of getting a total score of 4 or 6 is</p>
Step-by-Step Solution
Key Concept: List all possible outcomes across 3 rolls that sum to 4 or 6, then count favorable outcomes considering the non-standard die has repeated faces (1 appears once, 2 twice, 3 thrice). The probability of each face is: P(1)=1/6, P(2)=2/6, P(3)=3/6.
<p><strong>Step 1:</strong> Identify the probability of each face on the die: P(1) = 1/6, P(2) = 2/6 = 1/3, P(3) = 3/6 = 1/2</p><p><strong>Step 2:</strong> Find all ways to get sum = 4 in 3 rolls:</p><p>• (1,1,2): coefficient = 3!/(2!·1!) = 3 arrangements, probability = 3 × (1/6)² × (1/3) = 3 × (1/108) = 1/36</p><p>• (1,3): impossible (need 3 rolls)</p><p>∴ Sum = 4: P = 1/36</p><p><strong>Step 3:</strong> Find all ways to get sum = 6 in 3 rolls:</p><p>• (1,2,3): coefficient = 3! = 6 arrangements, probability = 6 × (1/6) × (1/3) × (1/2) = 6/36 = 1/6</p><p>• (2,2,2): coefficient = 1 arrangement, probability = (1/3)³ = 1/27 = 4/108</p><p>∴ Sum = 6: P = 1/6 + 1/27 = 9/54 + 2/54 = 11/54</p><p><strong>Step 4:</strong> Total probability = P(sum=4) + P(sum=6) = 1/36 + 11/54 = 3/108 + 22/108 = 25/108</p><p>∴ Answer: B (25/108)</p>
Correct Answer: B