A pair of dice is rolled. Find the probability of rolling a) a sum not more than 8 , b) a sum not less than 6 , c) a sum between 3 and 7 (exclusive).

To find the probabilities for the different scenarios when rolling a pair of dice, we first need to determine the total number of possible outcomes. When rolling two six-sided dice, there are \(6 \times 6 = 36\) possible outcomes. Now, let's analyze each part of the question: ### a) Probability of rolling a sum not more than 8 We need to count the outcomes where the sum of the two dice is 8 or less. The possible sums and their combinations are: - **Sum = 2**: (1,1) → 1 outcome - **Sum = 3**: (1,2), (2,1) → 2 outcomes - **Sum = 4**: (1,3), (2,2), (3,1) → 3 outcomes - **Sum = 5**: (1,4), (2,3), (3,2), (4,1) → 4 outcomes - **Sum = 6**: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes - **Sum = 7**: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes - **Sum = 8**: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes Now, we add these outcomes together: \[ 1 + 2 + 3 + 4 + 5 + 6 + 5 = 26 \] Thus, the probability of rolling a sum not more than 8 is: \[ P(\text{sum} \leq 8) = \frac{26}{36} = \frac{13}{18} \] ### b) Probability of rolling a sum not less than 6 We need to count the outcomes where the sum of the two dice is 6 or more. The possible sums and their combinations are: - **Sum = 6**: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes - **Sum = 7**: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes - **Sum = 8**: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes - **Sum = 9**: (3,6), (4,5), (5,4), (6,3) → 4 outcomes - **Sum = 10**: (4,6), (5,5), (6,4) → 3 outcomes - **Sum = 11**: (5,6), (6,5) → 2 outcomes - **Sum = 12**: (6,6) → 1 outcome Now, we add these outcomes together: \[ 5 + 6 + 5 + 4 + 3 + 2 + 1 = 26 \] Thus, the probability of rolling a sum not less than 6 is: \[ P(\text{sum} \geq 6) = \frac{26}{36} = \frac{13}{18} \] ### c) Probability of rolling a sum between 3 and 7 (exclusive) We need to count the outcomes where the sum is greater than 3 and less than 7. The possible sums and their combinations are: - **Sum = 4**: (1,3), (2,2), (3,1) → 3 outcomes - **Sum = 5**: (1,4), (2,3), (3,2), (4,1) → 4 outcomes - **Sum = 6**: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes Now, we add these outcomes together: \[ 3 + 4 + 5 = 12 \] Thus, the probability of rolling a sum between 3 and 7 (exclusive) is: \[ P(3 < \text{sum} < 7) = \frac{12}{36} = \frac{1}{3} \] ### Summary of Probabilities - a) \(P(\text{sum} \leq 8) = \frac{13}{18}\) - b