Chance of points after multiple rolls on 2 die

Question

I was wondering what the probability is of rolling at least a certain amount of points $x$, after a certain amount of rolls $y$. For example, what is the probability of rolling at least 15 in 3 rolls on 2 die. I know that you could draw out a tree of all possible cases and count how many end cases are greater than $x$, but is there an easier, less tedious way to figure it out?

Answer 1

There is an easy way to do this in an excel spreadsheet.

Start at least 6 rows down from the top. Label column A with 1-36.

In column B, place 1 in the 6 rows next to [1,6] this is the number of ways to roll a each number on 1 die.

Column C, starting in with the row labeled 2, make a formula that sums the 6 numbers in column B, starting with 5 empty cells and ending with the 1 next to the 1 label. Copy this formula down. You should have a 1 in the in the row labeled 12, and a 6 in the row labeled 7.

If you have the correct results, copy and paste this formula into the next 4 columns.

Each column will then show the number of ways to row each number for the number of dice rolled. Divide by the sum of the column to convert to probabilities.

If you want to learn the combinatorics, this is a "stars and bars" or a "multi-choose probability. I am not going to detail how you do multi-choose.

But, the number of ways to roll exactly $15$ is ${15-1\choose 5} - {6\choose 1}{15-7\choose 5}$

If you want a more detailed rundown on multi-choose, leave a comment.