Problem

A pair of dice is rolled. Find the probability of rolling (a) a sum not more than 8 , (b) a sum not less than 8 , (c) a sum between 5 and 10 (exclusive).

Solution

Step 1 :There are 36 possible outcomes when two dice are rolled.

Step 2 :The outcomes where the sum of the numbers on the two dice is not more than 8 are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (5,1), (5,2), (5,3), (5,4), (6,1), (6,2), (6,3). That's 30 outcomes.

Step 3 :So, the probability of rolling a sum not more than 8 is \(\frac{30}{36} = \frac{5}{6}\).

Step 4 :The outcomes where the sum of the numbers on the two dice is not less than 8 are: (2,6), (3,5), (3,6), (4,4), (4,5), (4,6), (5,3), (5,4), (5,5), (5,6), (6,2), (6,3), (6,4), (6,5), (6,6). That's 15 outcomes.

Step 5 :So, the probability of rolling a sum not less than 8 is \(\frac{15}{36} = \frac{5}{12}\).

Step 6 :The outcomes where the sum of the numbers on the two dice is between 5 and 10 (exclusive) are: (2,3), (2,4), (2,5), (2,6), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (6,1), (6,2), (6,3), (6,4). That's 24 outcomes.

Step 7 :So, the probability of rolling a sum between 5 and 10 (exclusive) is \(\frac{24}{36} = \frac{2}{3}\).

Step 8 :The final answers are: \(\boxed{\frac{5}{6}}\), \(\boxed{\frac{5}{12}}\), and \(\boxed{\frac{2}{3}}\).

From Solvely APP
Source: https://solvelyapp.com/problems/UBYp3OCZYn/

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