variance and streaks

Measuring the spread

Variance is the expected squared deviation from the mean; its square root, the standard deviation, is in the same units as the result and is the more useful figure. For a one-unit even-money bet on a single-zero wheel the expectation is −1/37 and the variance is very close to one, so the standard deviation of a single round is almost exactly one unit.

Over many rounds the expectation grows in proportion to the number of rounds while the standard deviation grows only with its square root. That asymmetry is the entire mechanism by which a small edge becomes a certainty, and it is worth seeing numerically.

per round:  E[X]  = -1/37      = -0.0270
            var   = 1 - E^2    ~  0.9993
            sd                 ~  0.9996

over n = 500 independent rounds:
  expected result = 500 x -0.0270      = -13.5 units
  sd of result    = 0.9996 x sqrt(500) =  22.4 units

the noise is larger than the signal: finishing ahead is unremarkable
at n = 50,000 the expectation is -1350 and the sd only 224

How long a streak is normal

Runs of identical results are far more common than intuition suggests. On an even-money proposition close to one half, a run of six is expected roughly once every sixty-four opportunities, and across an evening's play a run of eight or nine is entirely unremarkable. A sequence that looks patterned is what independent trials actually produce; a sequence with no runs in it would be the anomalous one.

This is why sequences of past results, displayed beside a table or a machine, carry no information. They are a record of independent draws, and they contain exactly as much predictive content as a list of previous coin tosses.

The gambler's fallacy

The belief that an independent process compensates for its own history — that red is due after a run of black — is the most durable error in the subject, and it survives because it feels like the law of large numbers. It is the opposite of it. Convergence happens because later results dilute early ones, not because they correct them.

The inverse error is equally common: treating a run as evidence that a mechanism is favourable and will continue. Both readings attribute memory to a memoryless process. The only family of games where history genuinely informs the next trial is the card family, and there the mechanism is dependence, not compensation.

Fair and survivable

Variance also explains why a game with a small edge can be more dangerous than one with a large edge. A low-variance game exposes a bankroll to a great many rounds, and exposure is measured in rounds rather than in sessions. A high-variance game may end a session quickly in either direction. The edge sets the direction; the variance sets how long the journey takes and how convincing the detours look.