Randomness and variance

Two events are independent when the outcome of one leaves the probabilities of the other exactly as they were. A die, a wheel and a properly seeded random number generator all produce independent trials. This is a strong condition, and it forbids a great deal that intuition wants to believe. It forbids a number being due. It forbids a machine being warm or cold. It forbids the past of a sequence carrying any information whatsoever about its future.

The mechanism has no memory because there is nowhere for a memory to be stored. A die is a piece of geometry; it is not connected to the record of its previous throws. The belief that it is—that a long run of one result makes the opposite more likely next—is the gambler's fallacy, and it survives because it borrows the shape of a true statement. The long-run proportions really do settle down. They settle by accumulating a growing mass of ordinary trials that dilutes the early imbalance, not by producing compensating results.

A genuinely random sequence looks far streakier than most people expect. In a few hundred throws of a fair coin, runs of six or seven identical results are ordinary rather than remarkable. When people are asked to write down an invented random sequence they alternate far too often, because they are producing what randomness feels like rather than what it is. Statisticians can usually distinguish an invented sequence from a real one on exactly this basis.

This has a direct consequence for anyone reading a record of results from a game of chance. A visible pattern in a short record is not evidence of a pattern in the mechanism, because short records of random processes are full of visible patterns. Establishing that a mechanism is biased requires a sample large enough that the observed deviation would be improbable under fairness, and for a small bias that sample is very large indeed.

Variance measures how widely results scatter around their average. It is a separate quantity from expected value, and confusing the two accounts for a large share of the mistaken reasoning about games of chance. A high-variance game scatters widely: long stretches of nothing punctuated by large returns. A low-variance game returns small amounts often. Two games can have identical expected values and completely different experiences, and two games with identical experiences can have very different expected values.

Variance is also what makes a negative expectation survivable in the short run. Over a small number of decisions the scatter dominates the drift, so results above the average are common and feel like skill or luck depending on temperament. As the number of decisions grows the scatter grows with the square root of the count while the drift grows in proportion to it, and the drift wins. That relationship is the subject of the next section.

The practical question behind all of this is how anyone establishes that a physical or computational mechanism is actually producing independent, uniformly distributed outcomes. For physical devices the answer is measurement: dimensional tolerance, balance, wear inspection and long observed sequences tested against the distribution fairness predicts. Historic cases of biased wheels and shaved dice were found precisely this way, by people willing to record tens of thousands of results.

For computational generators the answer is different in kind. A deterministic algorithm is not random at all; it is a sequence that passes statistical tests for randomness while being entirely reproducible from its seed. Assessment therefore covers the algorithm's period and distribution properties, the entropy source that seeds it, and the mapping from raw output to game outcome, where an unevenly sized mapping can introduce bias into a perfectly good generator.