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.
Expand this out and check the coefficient of
That is the number of ways to sum to 8.
The coefficent is
Therefore, there are 21 ways.
There are possible rolls with 3 dice.
Probability is
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.
Now
, for k = 2, 3,...,7, and 0 elsewhere. Also it's easy to show that
, for k = 2, 3,...,7 (we won't consider higher k's)
Hence we end up with