Which sum shows up most often when you roll 10 dice?

Question

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score 5 · Answer 1 · answered Feb 10 '11 at 16:10

5

The sum that shows up most often when you roll 10 dice is 35.

Are you also interested to see a reason why this is so?

answered Feb 10 '11 at 16:10

Rasmus

18,404

I am interested – ajbeaven Jun 22 '11 at 22:26
@ajbeaven: I think Yuval Filmus suggestion for a proof is very good. See also this question. – Rasmus Jun 23 '11 at 20:52

score 4 · Answer 2 · answered Mar 26 '11 at 17:37

The key to this question is the concept of unimodality (q.v.).

Try to prove by induction that if you take the sum of $n \geq 2$ cubes then the distribution has the following form:

It is supported (has non-zero probability) on the numbers $n, \ldots, 6n$.
It is symmetric around $3.5n$.
It is unimodal: the sequence of probabilities increases monotonically until it reaches its mode (the most probable value), then decreases monotonically.
If $n$ is even then the mode is unique (and so $3.5n$).
If $n$ is odd then there are two modes ($\lfloor 3.5n \rfloor$ and $\lceil 3.5n \rceil$).

If you find the proof difficult, try again with two-sided dice.

Of course, as $n \rightarrow \infty$ the distribution will approach (in some sense) a Gaussian, but I'm not sure this is enough to prove anything specific about the mode (other than its asymptotics).

Which sum shows up most often when you roll 10 dice?

2 Answers2