Expected Sum of Several Dice

One fair die is a uniform distribution over its faces, so its average is the midpoint of the first and last: $(1 + d)/2$, which is 3.5 for a six-sided die and 10.5 for a twenty-sided one. Roll $n$ of them and add, and the average is $n$ times that. The step that makes this trivial is worth naming, because it is one of the most useful facts in probability: expectation is linear. The average of a sum is the sum of the averages, always, whether or not the things being added are independent of one another.

That is why this page has no distribution in it. To know the average total of three dice you do not need to know how often three dice add to ten; you need only know the average of one die, three times over. The full distribution of the sum is a much harder object — it is the triangular hump everyone recognises for two dice and a bell-shaped one for several — and none of it is required here.

The averages land on halves whenever the number of dice is odd, and that is a real feature rather than an artefact. A three-die total averages 10.5, and 10.5 is not a possible roll. An expected value need not be a value the thing can actually take. It is the long-run average, and long-run averages sit between the outcomes rather than among them.

Now the thing that costs people more than any arithmetic slip: the dice have no memory. A die that has not shown its highest face in forty rolls is not "due" for one. Its next roll has exactly the chance it always had, because nothing about the die has changed and nothing about it records what it did before. That belief is the gambler's fallacy, and the reason it is so persistent is that the law of large numbers really is true — the running average really does converge on 3.5 — so it feels as though something must be pulling it there.

Nothing is. The convergence happens by DILUTION, not by correction. If your first ten rolls average 2.5, you are one full point of average down over ten rolls; after another ten thousand rolls averaging 3.5, that deficit is still exactly the same ten points of shortfall, but it is now spread across ten thousand and ten rolls instead of ten, so it barely shows. The average arrives because the early results are swamped, never because they are repaid. A run of bad luck is not compensated. It is simply outvoted.