Probability, variance & luck in dice

Almost every “the dice hate me” feeling comes from a handful of misunderstandings about randomness. None of them are stupid — our brains are built to find patterns, and dice are built to defeat that instinct. Here’s what’s really going on.

Every roll is independent

A die has no memory. The result of one roll has zero influence on the next. A fair d20 that just rolled three 1s is still exactly 1-in-20 to roll a 1 again, and exactly 1-in-20 to roll a 20. The dice can’t “know” they owe you a good result, because there is no mechanism — no memory, no ledger — connecting one throw to the next. This single fact dissolves most dice superstition.

The gambler’s fallacy

The gambler’s fallacy is the belief that a run of one outcome makes the opposite “due”. After five low rolls, a high roll feels overdue — but the die isn’t balancing a cosmic account. Each roll starts fresh at the same odds. The flip side is just as wrong: a die on a hot streak isn’t “on fire” and more likely to keep winning. Streaks are simply what randomness looks like — over enough rolls, clusters of highs and lows are guaranteed to appear.

Why 2d6 bells toward 7

Roll one d6 and every number is equally likely — a flat line. Roll two and something changes: totals near the middle become far more common, because there are more ways to make them. There’s only one way to roll a 2 (1+1) but six ways to roll a 7 (1+6, 2+5, 3+4, and their mirrors). That’s the famous bell curve, and it’s why 7 dominates games like craps and Catan.

Notice the counts add to 36 — every possible pair of dice — and mirror perfectly around 7. A 7 is six times more likely than a 12.

Expected value vs variance

Two ideas do most of the work in understanding luck:

• Expected value is the long-run average of a roll. A single d20 averages 10.5; 2d6 averages 7. It’s what you’d get per roll if you rolled forever.
• Variance (and its friendlier cousin, standard deviation) measures how far results typically stray from that average. A flat single die is swingy — high variance. Adding dice together, like 2d6 or 3d6, pulls results toward the middle and shrinks the variance, even though the average stays put.

Standard deviation is just “the usual size of a surprise”. A high-variance roll like one d20 will regularly hand you results far from 10.5; a low-variance roll like 3d6 clusters tightly around 10.5, rarely producing extremes. Same average, very different feel — which is exactly why designers choose one over the other. It’s also why advantage feels so different from a flat bonus: it reshapes the distribution rather than shifting it.

“Hot dice” is memoryless

Because rolls are independent, there is no such thing as a die that’s “running hot” in any predictive sense. You can absolutely have a lucky night — you just can’t know you’re in one until it’s over, and it tells you nothing about the next roll. Randomness is memoryless: the future doesn’t depend on the past. The streak you notice is real; the momentum you infer from it is not.

The law of large numbers (the honest version)

Over thousands of rolls, a fair die’s results do converge on the expected average — that’s the law of large numbers. But it works by swamping early flukes with sheer volume, not by correcting them. A run of bad luck is never “paid back”; it’s simply diluted as the count grows. Understanding that keeps you from chasing losses or reading destiny into a single evening’s dice.

Keep reading: dice superstitions & rituals — the fun folklore that grows in the gaps this maths leaves behind.