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If we roll a six face die $k$ times and sum all the results, which is the probability to get $n$ in the sum? ($k ≤ n ≤ 6k$). Which is the most probable $n$?

I think that the most probable one is $\frac{7k}{2}$ but I am not sure why.

Andarrkor
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    If $k$ is odd, a sum of $7k/2$ is impossible, since the sum must be a whole number. $7k/2$ is the expected value, but as we see that's not the same as the most likely value. – saulspatz Oct 30 '19 at 14:42
  • @saulspatz well in that case the most probable $n$s would be $\frac{7k}{2} +- \frac{1}{2}$. The ones next to the $n$ "in the middle". The question is that I do not know how to justify that. – Andarrkor Oct 30 '19 at 14:48

2 Answers2

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This is not solution but a lead

(x1+x2+x3+x4+x5+x6)^k

6 class 1,2,3,4,5,6

The maximum coefficient occurs if K is distributed equally in 1 to 6 as so each one have k/6

So sum will ( 21×k/6)=7k/2

This is perfect if K is divisible by 6 but for rest need little bit more work

Naresh Teli
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Sketch for a solution

Let $X_k$ a random variable that represent the throw of a six-sided fair dice, and set $Y:=X_1+X_2+\ldots +X_n$. Then $Y$ is the random variable that represent the sum of a throw of $n$ fair six-sided dice.

Suppose that $(d_1,d_2,\ldots ,d_n)$ are the values of a throw that add up to $s$, then necessarily $(7-d_1,7-d_2,\ldots ,7-d_n)$ add up to $7n-s$, and viceversa. Hence $P[Y=s]=P[Y=7n-s]$. This shows that the PMF of the random variable that represent the sum of $n$ fair six-sided dice is symmetric respect to the mean.

Then to show that the modes of this distribution are $\lfloor 7n/2 \rfloor$ and $\lceil 7n/2 \rceil$ you need to show that $P[Y=s]$ increases from it minimum value ($n$) up to $\lfloor 7n/2 \rfloor$. However I dont see a simple way to show this even when I can write explicitly the PMF as $$ P[Y=s]=\frac1{6^n}\sum_{j\geqslant 0}(-1)^j\binom{n}{j}\binom{s-6j-1}{n-1} $$ The explanation for this formula is here.

Probably with a good bound for the error term from an approximation to the distribution of $Y$ using the central limit theorem we can show this, however I dont know a good bound for the error.

Masacroso
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