# probability of rolling at least $n$ on $k$ 6-sided dice

Is there a simple form for the probability of rolling at least $n$ on $k$ 6-sided dice? Of course you can do it by recursion (see here). But is there a way to do it with just a few binomial coefficients, without the recursion?

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If you're willing to sacrifice accuracy, you can use the normal approximation, if $k$ is large. There isn't much alternative apart form doing the calculation. –  Calvin Lin Jan 27 '13 at 19:18

What you want to know is here.

The formula for a homogeneous and fair dice from 1 to D values is this

$$P(S,n,D)=\frac{1}{D^n}\sum_{k=0}^{\lfloor(S-n)/D\rfloor}(-1)^k\binom{n}{k}\binom{S-Dk-1}{n-1}$$

where S is the sum that you want, n is the number of dice you roll and D is the number of sides of the dice (D6, D8, etc.)

To know the sum of at least S just need sum what you want over S.

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