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The most likely sum of any roll of n unbiased s-sided dice is given by:

$ P (n, s) = \frac{1}{2} n ( s + 1 )$

This tells us, of course, that for an odd number n, the most likely sum will be fractional; that is, two sums will be equally likely.

Is there a simple, closed formula that would determine the probability of that most likely sum? It strikes me that such a formula should exist, since any particular sum has a definite and predictable number of summands, based upon $n$ and $s$. And since the total number of permutations is $s^n$, the question becomes one of calculating the number of ways the most likely number can be achieved with those dice.

Thank you.

Blueyedaisy
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POD
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    By generating functions, the coefficient of $x^{\frac{n(s+1)}{2}}$ in the expansion of $(x+x^2+x^3+\dots+x^s)^n/s^n$. Simplifying this monstrosity however is easier said than done. – JMoravitz May 05 '20 at 01:14
  • That said, it should be relatively easy to approximate. See here and here for instance. You'll find that as the number of dice grows the curve smooths out and approaches the normal curve. – JMoravitz May 05 '20 at 01:19

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