Suppose a fair six-sided die is rolled once. If the value on the die is 1, 2, or 3, the die is rolled a second time. What is the probability that the sum total of values that turn up is at least 6? A. 10/21 B. 5/12 C. 2/3 D. 1/6

GATE 2012 · Engineering Mathematics · Conditional Probability · medium

Answer: 5/12

1. Case 1: First roll in {4, 5, 6} — no second roll: Only first roll = 6 gives total >= 6. P(this case contributes) = 1/6.
2. Case 2: First roll in {1, 2, 3} — second roll is made: For first = 1: need second >= 5, so second in {5,6}: 2 values. For first = 2: need second >= 4, so second in {4,5,6}: 3 values. For first = 3: need second >= 3, so second in {3,4,5,6}: 4 values. Total favorable pairs = 2 + 3 + 4 = 9 out of 18. Each first roll has probability 1/6, each second roll 1/6, so contribution = 9 * (1/6)*(1/6) = 9/36 = 1/4.
3. Add both cases: P = 1/6 + 1/4 = 2/12 + 3/12 = 5/12.