Question 11 \$ Two fair number cubes have faces labeled with the numbers 1 through 6 . The two cubes will be rolled simultaneously. What is the probability that at least one of the cubes will have the face labeled 5 showing and that the sum of the showing faces will be greater than 8 ?

To find the probability that at least one of the cubes shows a 5 and that the sum of the showing faces is greater than 8, we can break this problem down into steps. 1. **Total Outcomes**: When rolling two fair number cubes, each cube has 6 faces. Therefore, the total number of outcomes when rolling two cubes is: \[ 6 \times 6 = 36 \] 2. **Favorable Outcomes**: We need to find the outcomes where at least one cube shows a 5 and the sum of the two cubes is greater than 8. Let's consider the cases where at least one cube shows a 5: - **Case 1**: The first cube shows a 5. - Possible outcomes for the second cube: 1, 2, 3, 4, 5, 6. - The sums for each outcome: - (5,1) → 6 - (5,2) → 7 - (5,3) → 8 - (5,4) → 9 - (5,5) → 10 - (5,6) → 11 - Only the outcomes (5,4), (5,5), and (5,6) have sums greater than 8. This gives us 3 favorable outcomes. - **Case 2**: The second cube shows a 5 (and the first cube does not show a 5). - Possible outcomes for the first cube: 1, 2, 3, 4, 6. - The sums for each outcome: - (1,5) → 6 - (2,5) → 7 - (3,5) → 8 - (4,5) → 9 - (6,5) → 11 - Only the outcomes (4,5) and (6,5) have sums greater than 8. This gives us 2 favorable outcomes. 3. **Total Favorable Outcomes**: Adding the favorable outcomes from both cases: - From Case 1: 3 outcomes - From Case 2: 2 outcomes - Total favorable outcomes = 3 + 2 = 5 4. **Probability Calculation**: The probability of the event is given by the ratio of favorable outcomes to total outcomes: \[ P(\text{at least one 5 and sum > 8}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{36} \] Thus, the probability that at least one of the cubes will have the face labeled 5 showing and that the sum of the showing faces will be greater than 8 is: \[ \boxed{\frac{5}{36}} \]