Sample averages converge to the true mean as the number of observations grows. It says nothing about how fast, which is the part traders need.
The law guarantees that if your edge is real, enough trades will reveal it. It does not guarantee this happens inside your account's lifetime or your patience, and it offers no protection against ruin on the way.
How fast is governed by the ratio of edge to noise. A system with a 0.05% edge and 1.5% trade volatility has a signal-to-noise ratio of 0.033 per trade. To get the standard error down to a quarter of the edge you need n such that 1.5/sqrt(n) = 0.0125, which is n = 14,400 trades. At ten trades a week that is 28 years.
This is the honest reason most traders cannot prove their edge. It also explains why professionals diversify across many instruments and signals: twenty weakly correlated bets per day accumulate sample far faster than one.
Original diagrams for the ideas on this page. Illustrative, not real market data.
Expectancy: the average trade. Forty trades sorted by outcome: 24 small losses and 16 larger wins. Weighting each side by how often it happens gives the average result per trade, marked here by the dashed line at +$120.Risk of ruin. The chance of losing the whole account, plotted against the share of it staked on each trade, for a method that wins 52% of the time at even money. The edge is the same all along the curve; only the bet size changes.
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