Independence and the gambler's fallacy

Why a run of results leaves the next trial exactly as it was, what the law of large numbers actually claims, and why staking patterns do not move an average.

What independence means

Two trials are independent when the outcome of one leaves the probabilities of the other exactly as they were. A wheel, a die and a certified electronic selection are independent between trials, because nothing in the mechanism carries information from one trial to the next. There is no register of recent results inside a wheel, and no property of a die is altered by having been thrown.

The consequence is stated most usefully as a conditional probability. After any sequence of prior results, however long and however lopsided, the probability of the next result is what it always was.

P(group of 18 on the next trial)                        = 18/37 = 0.486486

P(group of 18 on the next trial | previous 10 were all
  from the other group)                                 = 18/37 = 0.486486

The fallacy

The gambler's fallacy is the belief that the second figure should be higher than the first: that a run of one result makes the opposite result "due". It rests on a real theorem applied to the wrong quantity. The law of large numbers says the observed proportion of a result approaches its probability as trials accumulate. It does not say the observed count approaches the expected count, and in fact the absolute gap between them tends to widen.

Suppose one group runs 10 behind after 100 trials.

  proportion so far  =  (48.65 - 10) / 100        = 38.65 %
  gap                =  10

After a further 9,900 trials in which the two groups
split exactly as expected, the gap is unchanged at 10,
but the proportion has moved:

  proportion         =  (4,864.86 - 10) / 10,000  = 48.55 %

The proportion converged. Nothing repaid the gap.

Convergence in proportion is achieved by dilution, not by correction. Later trials do not compensate for earlier ones; they merely outnumber them. No force exists that could push results back towards an average, because a force of that kind would require the mechanism to know its own history.

Regression to the mean is a different thing

Regression to the mean is sometimes offered as the correcting mechanism, and it is not one. It says only that a result selected for being extreme is likely to be followed by a result nearer the average, because extreme results are rare and average results are common. It describes what typically follows an unusual observation. It does not describe a tendency to cancel it out, and it says nothing at all about running totals.

Staking patterns and the same arithmetic

The fallacy has a practical descendant in staking patterns that raise the stake after losses, of which the doubling pattern is the best known. Their appeal is that they convert many small wins into an apparent reliability. Their arithmetic is unchanged, because expected values add regardless of stake size: a sequence of bets each carrying −2.70 % per unit staked carries −2.70 % per unit staked in total, whatever the stakes are.

Doubling after each loss, on an even-money bet at 18/37,
starting at 1 unit, with a limit of 8 units:

  P(at least one win in 4 trials) = 1 - (19/37)^4
                                  = 1 - 0.069415
                                  = 0.930585

  outcome if a win occurs           +1 unit
  outcome if all four lose         -15 units

  EV = (0.930585 x +1) + (0.069415 x -15)
     = 0.930585 - 1.041225
     = -0.110640 units over the sequence

  total staked across the sequence in the losing case
  is 15 units, and the loss rate remains -2.70 % of
  every unit staked.

The pattern has redistributed the outcomes, making a small gain very likely and a large loss unlikely but far larger. It has not moved the average, and it cannot: the average is a weighted sum over the same independent trials whatever order the stakes are placed in.