# I roll a die $N$ times, what is the probability that the sum is exactly $k$? [duplicate]

The die is six-sided. So, this is easy with a two-sided coin showing exactly $$k$$ heads, as that is just a Bernoulli trial being run $$n$$ times (Binomial RV).

You can also approximate this using a normal random variable with $$N(3.5\times n, \frac{35}{12} \times n)$$. But that doesn't yield an exact answer for $$k$$. I can get an answer for the value being $$at\ least\ k$$, that is easy. In fact, as $$n$$ approaches infinity, we know it will in fact be a normal distribution.

I can also run a simulation that will get me to a close value for a given $$n$$ and $$k$$.

But how do I figure out the exact value? This is not a continuous distribution, this is discrete. i.e. what is the generalized formula for this? There are many posts online that show all the previously mentioned methods, I couldn't find anything for a generalized version of the exact solution to this problem, however.

• You can find a thorough explanation and formula in this related post – G Cab Sep 7 '19 at 18:53