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December 1st, 2009, 11:26 AM   #1
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Dice probability

I am trying to find a general solution to calculate a certain number with x number of dice. As in, what is the probability to roll a 10 with 3 6-sided dice? Or to roll a 17 with 4 6-sided dice? Could anyone please help me on this?
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December 1st, 2009, 11:49 AM   #2
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Re: Dice probability

It's very easy to write the recursion relation. Let C(N,k) be the number of ways to roll k dice so that the total is N.

Then C(N,k+1) = C(N-1,k) + C(N-2,k) + ... + C(N-6,k)
And C(N, 1) = 1 if 1<=N<=6 and 0 otherwise.

That is, you look at the outcome of the first die, then count the number of ways to roll the remaining part of N on the remaining k dice. Add the counts for each possible roll of the first die, and you have the C(N,k+1) as desired.

If you want to write a closed formula, you'll need to go a bit further.
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December 1st, 2009, 12:15 PM   #3
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Re: Dice probability

Wikipedia states that the probability of rolling k with n s-sided dice is:

So the probability of rolling 10 with 3 six-sided dice is .

And the probability of rolling 17 with 4 six-sided dice is

If you want to solve the problem for all possible k at once, you can look instead at the coefficients of the expansion of (x+x+x+...+x^s)^n.

For example, if you tell Wolfram Alpha to expand (x+x^2+x^3+x^4+x^5+x^6)^3, which represents three 6-sided dice, you get:

which means that there is 1 way to get 18, 3 ways to get 17, 6 ways to get 16, 10 ways to get 15... etc.
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