If two dice are rolled to obtain a sum or 5 or 7. What is the probability of…

Answer: 2/5

Key probabilities per single roll:

• P(sum = 5) = 4/36 = 1/9

• P(sum = 7) = 6/36 = 1/6

• P(neither 5 nor 7) = 1 - (1/9 + 1/6) = 13/18

Reasoning (short method): When rolling until either a 5 or 7 appears, the probability that the first such outcome is a 5 equals the conditional probability P(sum = 5) divided by P(sum = 5 or sum = 7).

Compute:

• P(5 before 7) = P(sum = 5) / (P(sum = 5) + P(sum = 7)) = (1/9) / (1/9 + 1/6) = 2/5

• Equivalent geometric-series view: P = 1/9 + (13/18)(1/9) + (13/18)^2(1/9) + ... = (1/9) / (1 - 13/18) = 2/5

Therefore the probability of getting a sum of 5 before a sum of 7 is 2/5.