Hi. Is there a way to solve the following without actually listing out all the possibilities?
What is the probability of rolling a total sum of 8 using three dice?
Thank you!
Yes, there is a way.
There are 3 dice and you want to find all the sums of 8.
$\displaystyle \left(\sum_{k=1}^{6}x^{k}\right)^{3}$
Expand this out and check the coefficient of $\displaystyle x^{8}$
That is the number of ways to sum to 8.
The coefficent is $\displaystyle 21x^{8}$
Therefore, there are 21 ways.
There are $\displaystyle 6^{3}=216$ possible rolls with 3 dice.
Probability is $\displaystyle \frac{21}{216}$
This is a more direct approach for those not familiar with generating functions.
Let Z be the sum of three dice. Let Y be the sum of the first two dice.
$\displaystyle P(Z=8)=\sum_{k=2}^{12}P(Z=8|Y=k)P(Y=k)$
Now
$\displaystyle P(Z = 8|Y = k) = 1/6$, for k = 2, 3,...,7, and 0 elsewhere. Also it's easy to show that
$\displaystyle P(Y=k)=\frac{k-1}{6^{2}}$, for k = 2, 3,...,7 (we won't consider higher k's)
Hence we end up with
$\displaystyle P(Z=8)=\sum_{k=2}^{7}\frac{1}{6}\frac{k-1}{6^{2}}=\frac{21}{6^{3}}=\frac{21}{216}$