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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.

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