Points: 0 of 1 An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space provided and, assuming each simple event is as likely as any other, find the probability that the sum of the dots is greater than 8. \begin{tabular}{|c|ccccccc} \multicolumn{10}{c}{} & \multicolumn{7}{c}{ Second Die } \\ \cline { 3 - 8 } First Die & 1 & 2 & 3 & 4 & 5 & 6 \\ \cline { 2 - 8 } & 1 & $(1,1)$ & $(1,2)$ & $(1,3)$ & $(1,4)$ & $(1,5)$ & $(1,6)$ \\ & 2 & $(2,1)$ & $(2,2)$ & $(2,3)$ & $(2,4)$ & $(2,5)$ & $(2,6)$ \\ & 3 & $(3,1)$ & $(3,2)$ & $(3,3)$ & $(3,4)$ & $(3,5)$ & $(3,6)$ \\ & 4 & $(4,1)$ & $(4,2)$ & $(4,3)$ & $(4,4)$ & $(4,5)$ & $(4,6)$ \\ & 5 & $(5,1)$ & $(5,2)$ & $(5,3)$ & $(5,4)$ & $(5,5)$ & $(5,6)$ \\ & 6 & $(6,1)$ & $(6,2)$ & $(6,3)$ & $(6,4)$ & $(6,5)$ & $(6,6)$ \end{tabular} The probability that the sum of the dots is greater than 8 is $\square$. (Type an integer or a simplified fraction.)
Solution
Step 1 ：An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. We are asked to find the probability that the sum of the dots is greater than 8.
Step 2 ：First, we need to find the number of outcomes where the sum of the dots is greater than 8. From the sample space provided, we can see that there are 10 such outcomes.
Step 3 ：Next, we need to find the total number of outcomes. Since each die has 6 faces, and we are rolling two dice, the total number of outcomes is \(6 \times 6 = 36\).
Step 4 ：Now, we can calculate the probability. The probability of an event is the number of favorable outcomes divided by the total number of outcomes. So, the probability that the sum of the dots is greater than 8 is \(\frac{10}{36}\).
Step 5 ：We can simplify this fraction to \(\frac{5}{18}\).
Step 6 ：Final Answer: The probability that the sum of the dots is greater than 8 is \(\boxed{\frac{5}{18}}\).
