Variance, expected value and what 'due' really means

Expected value describes the long-run average; variance describes how far a short session can sit from it. Most arguments about luck are really arguments about the second quantity while using the vocabulary of the first.

Expected value

Expected value is the average result per decision, computed by weighting each possible outcome by its probability. It is a single number and it applies to the process rather than to any instance of it. No individual session is obliged to resemble it, and a great many sessions will not.

Variance

Variance measures spread. A game with a small edge and wide payouts can produce dramatic short-run results in both directions, while a game with the same edge and narrow payouts will grind along close to its average. Two games can be identically priced and feel completely different to play for exactly this reason.

The word 'due'

Calling an outcome due imports an idea of correction that independent trials do not have. What is true is that the average of many results tends toward the expected value. What is not true is that any particular future result is more likely because of what came before it.

Why the distinction matters in practice

Confusing the two leads people to read a good or bad session as evidence about the game's pricing, when in most cases the sample is far too small to be evidence about anything. Holding the two ideas apart is most of what separates a clear reading of a result from a story told about it.