Foundations
Why the arithmetic wins in the end
The edge is small and the swings are large, which is exactly why the edge is reliable. Both facts come from the same square root.
| Rounds | Expected loss (units) | Typical swing (units) | Swing as a multiple of the expected loss |
|---|---|---|---|
| 100 | 2.70 | 10.00 | 3.70 |
| 1,000 | 27.03 | 31.62 | 1.17 |
| 10,000 | 270.27 | 100.00 | 0.37 |
| 100,000 | 2,702.70 | 316.23 | 0.12 |
| 1,000,000 | 27,027.03 | 1,000.00 | 0.04 |
A 2.70 per cent edge sounds negligible, and over one round it is. Over one round it does not exist at all: the result is a win or a loss, and no fraction of a per cent appears anywhere. The percentage is a statement about averages, and averages only assert themselves over volume. The question this page answers is how much volume, and why the answer is so much smaller than most people expect.
Two quantities that grow at different speeds
Consider one unit staked per round on an even-money proposition carrying a 2.70 per cent edge. Two things accumulate. The expected loss accumulates in direct proportion to the number of rounds: play ten times as long and expect to lose ten times as much. The typical size of the random swing accumulates far more slowly, in proportion to the square root of the number of rounds. Play a hundred times as long and the swing is only ten times as large.
Working, the two growth rates
expected loss after n rounds = 0.0270 x ntypical swing after n rounds is about 1.00 x square root of nn = 100: loss 2.70, swing 10.00n = 10,000: loss 270.27, swing 100.00n = 1,000,000: loss 27,027.03, swing 1,000.00linear growth overtakes square-root growth, and never gives it backThe crossover is the important moment. Early on, the swing dominates: the expected loss is buried inside the noise, and a session can end well ahead without anything unusual having happened. As rounds accumulate, the linear term overtakes the square-root term permanently. After a few thousand rounds the expected loss is comparable to a typical swing; after a hundred thousand it is several times larger; after a million it is so much larger that finishing ahead requires an outcome that will not occur.
This is the law of large numbers, and it is a theorem rather than a tendency. The average result per round converges on the expected value as rounds accumulate, and nothing about the sequence of results can prevent it.
The same law from the other side of the table
The identical mathematics explains why a game with a small edge is a dependable business. A venue does not need to win any particular round, and does not care about any particular result. It needs volume, and it obtains volume by aggregating across many players, many rounds and many days. The convergence that takes an individual a lifetime to experience takes a busy floor an afternoon, because the relevant count is total rounds played, not rounds played by any one person.
That is why the edge does not need to be large. A percentage that is imperceptible in one round is entirely reliable across a million, and reliability is what a business requires. The same reasoning underlies insurance and bookmaking, both of which sell propositions they cannot predict individually and can predict in aggregate.
The gambler's fallacy
The commonest misreading of the law of large numbers is that outcomes must even out, so a long run of one result makes the opposite result more likely. For independent trials this is simply false. A wheel does not know it has produced red eight times, and the ninth spin has the same probabilities as the first.
What the law actually says is subtler and less comforting. Deviations are not corrected. They are diluted. A surplus of eight reds does not attract eight compensating blacks; it is swamped by the growing count of later spins until it becomes an insignificant fraction of the total. The absolute discrepancy can grow without limit while the proportional discrepancy shrinks towards zero, and both things happen at once.
Working, dilution rather than correction
after 100 spins: 58 red, 42 black, difference 16, that is 16% of 100after 10,000 spins the same absolute difference of 16is 0.16% of the totalthe difference was not repaid; the denominator grewRuin
One further consequence deserves stating plainly. A player has a finite bankroll and a negative expectation; the game has a much larger bankroll and a positive one. Under those conditions, continued play tends towards the exhaustion of the smaller bankroll, and the probability of that outcome rises with the edge and with the number of rounds played. Betting systems that increase stakes after losses do not alter this: they redistribute the loss into a smaller chance of a much larger one, which is a change in the shape of the distribution and not in its mean.
This is the honest summary of everything else on this desk. The percentages derived on the other pages are small, fixed and known. The only variable that a player controls is how many rounds those percentages are applied to, and the arithmetic runs in one direction as that number grows.