Final answer:
The events of rolling a sum of 2, 10, or 12 on two dice are mutually exclusive (non-overlapping).
The combined probability of getting either of these sums in a single roll is 5/36.
Step-by-step explanation:
The student's question involves determining whether certain dice rolls are overlapping or non-overlapping events, and calculating the probability of a combined event.
When rolling two dice, the sum of the numbers on the two dice can be 2, 10, or 12 in a non-overlapping manner since none of these sums can be achieved simultaneously on a single roll.
To calculate the probability of getting a sum of either 2, 10, or 12 on a roll of two dice, we examine each possible combination.
There is only one way to get a sum of 2: rolling a 1 on both dice. There are three ways to get a sum of 10: (4, 6), (5, 5), and (6, 4).
There is only one way to get a sum of 12: rolling a 6 on both dice.
Since a standard die has 6 sides, the total number of outcomes when rolling two dice is 66 = 36.
Therefore, the probability of rolling a sum of 2, 10, or 12 is:
P(sum of 2, 10, or 12) = (Number of favorable outcomes for sum of 2 + Number of favorable outcomes for sum of 10 + Number of favorable outcomes for sum of 12) / Total number of outcomes
P(sum of 2, 10, or 12) = (1 + 3 + 1) / 36
P(sum of 2, 10, or 12) = 5 / 36
The events of rolling a sum of 2, 10, or 12 are mutually exclusive because they cannot occur at the same time with a single pair of dice.
The combined probability of these events happening in one roll is 5/36.