Problem
A pair of dice is rolled. Find the probability of rolling (a) a sum not more than 10 , (b) a sum not less than 4 , (c) a sum between 3 and 8 (exclusive) (Type an integer or a simplified fraction.) (b) How many ways can two dice be rolled such that their sum is not less than 4 ? $33^{\top}$ (Simplify your answer.) The probability of rolling a sum not less than 4 is $\frac{11}{12}$. (Type an integer or a simplified fraction.) (c) How many ways can two dice be rolled such that their sum is between 3 and 8 (exclusive)? 18 (Simplify your answer.) The probability of rolling a sum between 3 and 8 (exclusive) is $\square$. (Type an integer or a simplified fraction.)
Solution
Step 1 ：The total number of outcomes when two dice are rolled is 36.
Step 2 ：The only sums that are more than 10 when two dice are rolled are 11 and 12.
Step 3 ：There are 2 outcomes that result in a sum of 11 and 1 outcome that results in a sum of 12.
Step 4 ：Therefore, the number of outcomes that result in a sum not more than 10 is \(36 - 2 - 1 = 33\).
Step 5 ：So, the probability of rolling a sum not more than 10 is \(\frac{33}{36}\).
Step 6 ：Final Answer: The probability of rolling a sum not more than 10 is \(\boxed{\frac{33}{36}}\) or \(\boxed{0.9167}\) when rounded to four decimal places.
