The Question
Which is more likely: rolling a total of 9 when two dice
are rolled or rolling a total of 9 when three dice are rolled?
My Work
First we have to determine the probability of two die summing to $9$. I began by calculating the sample space. There are $6*6 = 36$ different ways the two die could sum. Now to figure out how many of those ways sum to $9$. This is essentially the same question as how many integer solutions to the equality $x_1 + x_2 = 9$ where $1 \leq x\le6$ ($x$ represents the two dice in this case). The number of ways to do this is $\binom{8}{7} – 2$ (where $2$ represents the case where $x\geq6$. So the probability of rolling two dice which sum to $9$ is $\frac{6}{36}$
Figuring out how many ways $3$ dice sum to $9$ is a similar process. Find all solutions to $x_1 + x_2 + x_3 =9$ where $1\le x\le6$ which is $\binom{8}{6} – 3 = 25$ ways to do this (3 represents all the solutions where $x_i$ is greater than $6$). There are $6^3$ ways $3$ dice can be rolled therefore the probability $\frac{25}{6^3}$.
Computing the probabilities I found $\frac{6}{36} > \frac{25}{6^3}$ Therefore, it is more likely with 2 dice.
The Problem
My book says it is more likely with 3 dice. Can anyone point out where I went wrong in my approach, and how to correct it?
Best Answer
In your computation of how many ways for two dice to sum to 9, you subtracted 2 for the two cases in which $x_1$ is 7 or 8, but forgot to subtract 2 for the two cases in which $x_2$ is 7 or 8.
Using a formula is nice, but especially when what's going on is as simple as two numbers from 1 to 6 summing to 9, double-check it by writing them out!