Final answer:
The probability that a sum of 3 occurs first when rolling two dice is 1/3, since there are two ways to get a sum of 3 and four ways to get a sum of 5.
Step-by-step explanation:
To find the probability that a sum of 3 occurs first when rolling two six-sided dice, we must consider the possible outcomes for the sum to be either 3 or 5. The sum of 3 can be obtained in two ways: (1,2) or (2,1). The sum of 5 can be obtained in four ways: (1,4), (4,1), (2,3), or (3,2).
The total number of possible outcomes when rolling two dice is 6 * 6 = 36. Therefore, the probability of rolling a sum of 3 is P(3) = 2/36, and the probability of rolling a sum of 5 is P(5) = 4/36. To find the probability that a sum of 3 occurs first, we only consider the first occurrence, so we compare the probabilities of obtaining a sum of 3 against obtaining a sum of 5 on a single roll.
The probability that a sum of 3 occurs first is simply the probability of rolling a 3, which is P(3)/(P(3) + P(5)) = 2/36 / (2/36 + 4/36) = 2/6 = 1/3. Thus, the probability that a sum of 3 occurs before a sum of 5 is 1/3.