ou roll two 12-sided dice. What is the probability that roll a double or a sum of 10?

To solve this problem, we need to understand the total number of outcomes and the number of favorable outcomes.

Step 1: Determine the total number of outcomes When you roll two 12-sided dice, the total number of outcomes is 12*12 = 144. This is because each die has 12 sides, and you are rolling two dice.

Step 2: Determine the number of outcomes that are doubles A double is when both dice show the same number. Since each die has 12 sides, there are 12 possible doubles (1-1, 2-2, 3-3, ..., 12-12).

Step 3: Determine the number of outcomes that sum to 10 The pairs that sum to 10 are (4,6), (5,5), (6,4), (2,8), (3,7), (7,3), (8,2). So there are 7 pairs. But each pair can occur in two ways (for example, the first die could show a 4 and the second die a 6, or the first die could show a 6 and the second die a 4). So there are 7*2 = 14 outcomes that sum to 10.

Step 4: Determine the number of favorable outcomes The favorable outcomes are those that are either doubles or sum to 10. But we have to be careful not to double-count the doubles that also sum to 10. The doubles that sum to 10 are (5,5). So the total number of favorable outcomes is 12 (from the doubles) + 14 (from the sums to 10) - 2 (because we counted (5,5) twice) = 24.

Step 5: Calculate the probability The probability is the number of favorable outcomes divided by the total number of outcomes. So the probability is 24/144 = 1/6.