Question 6
2 pts
A gambler is monitoring a craps game in a casino and keeping track of the results of the dice he rolls. In this game, there are 3 key groups of sums that can be rolled. If the sum is 2,3 , or 12 on the first roll, the player loses. If the sum is 7 or 11 on the first roll, the player wins. If the sum on the first roll is any other total, the game continues. After observing 185 rolls, he has observed the following frequencies for the listed groups of sums:
\begin{tabular}{|l|l|l|l|}
\hline Sum: & $2,3,12$ & 7,11 & $4,5,6,8,9,10$ \\
\hline Frequency: & 24 & 51 & 110 \\
\hline
\end{tabular}
The gambler wants to use the data to test whether or not the dice are producing the 3 groups of sums in the proper proportions. What is the expected frequency for the group $\{7,11\}$ ? Round your answer using 2 decimal places. Hint: Make a chart listing all possible sums for the two dice.

Step 1 ：The gambler is observing a craps game and keeping track of the results of the dice he rolls. In this game, there are 3 key groups of sums that can be rolled. If the sum is 2,3 , or 12 on the first roll, the player loses. If the sum is 7 or 11 on the first roll, the player wins. If the sum on the first roll is any other total, the game continues. After observing 185 rolls, he has observed the following frequencies for the listed groups of sums: \begin{tabular}{|l|l|l|l|} \hline Sum: & $2,3,12$ & 7,11 & $4,5,6,8,9,10$ \ \hline Frequency: & 24 & 51 & 110 \ \hline \end{tabular}

Step 2 ：The gambler wants to use the data to test whether or not the dice are producing the 3 groups of sums in the proper proportions. The expected frequency for a group is calculated by multiplying the total number of observations by the probability of the event. In this case, the total number of observations is 185.

Step 3 ：The probability of rolling a 7 or 11 with two dice can be calculated by listing all possible outcomes and counting the ones that sum to 7 or 11. There are 36 possible outcomes when rolling two dice (6 outcomes for the first die times 6 outcomes for the second die).

Step 4 ：The sums that result in 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - a total of 6 outcomes. The sums that result in 11 are (5,6), (6,5) - a total of 2 outcomes. So, the probability of rolling a 7 or 11 is (6+2)/36 = 8/36 = 2/9.

Step 5 ：Therefore, the expected frequency for the group {7,11} is (2/9)*185 = 41.11.

Step 6 ：Final Answer: The expected frequency for the group \{7,11\} is approximately \(\boxed{41.11}\).