7
$\begingroup$

Is there a general formula to determine the probability of unbounded, cumulative dice rolls hitting a specified number?

For Example, with a D6 and 14:

5 + 2 + 3 + 4 = 14 : success

1 + 1 + 1 + 6 + 5 + 4 = 17 : failure

  • 0
    Ahh, ok, thanks. Unfortunately that only defines an upper limit, not a good approximation.2010-11-15

1 Answers 1

8

Assuming the order matters (i,e 1+2 is a different outcome from 2+1)

The probability of getting the sum $n$ with dice numbered $1,2,\dots,6$ is the coefficient of $x^n$ in

$\sum_{j=0}^{\infty}(\frac{x+x^2+x^3+x^4+x^5+x^6}{6})^j = \frac{6}{6-x-x^2-x^3-x^4-x^5-x^6}$

Writing it as partial fractions (using roots of $6-x-x^2-x^3-x^4-x^5-x^6=0$) or using Cauchy's integral formula to find the coefficient of $x^n$, Taylor series, etc should work.

  • 0
    @Qia: Yes, I know that method (one of my favourites to compute fibonacci in O(logn) time:-)). If you write the above as a power series and multiply that by the denominator we got above, and equate coefficients, we get a linear recurrence which can be read off from the powers of a 7$x$7 matrix. (I realized this immediately after writing my previous comment). So the above is useful too :-)2010-11-07