# Law of Large Numbers > The law of large numbers says averages converge to the true value as samples grow. Learn what it means for judging strategies and how many trades you need. Source: https://learn.tradelabsai.com/math/law-of-large-numbers/ Track: Math and Statistics · Level: Intermediate · Updated: 2026-10-03 Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice. Cite as: TradeLabs Learn, "Law of Large Numbers", https://learn.tradelabsai.com/math/law-of-large-numbers/ The law of large numbers says that as you repeat a random process more times, the average of the results gets closer to the true expected value. Flip a fair coin 10 times and you might see 7 heads; flip it 10,000 times and the share of heads will be very close to 50%. For traders, this law explains why a strategy's real edge only shows up over many trades, why short term results are mostly noise and why judging a strategy on a handful of trades leads to bad decisions. ## The idea ``` sample average → expected value, as the number of trials → ∞ ``` Each individual outcome stays random. What converges is the average. ## Small samples are noisy **Example: A 55% strategy over different sample sizes** A strategy truly wins 55% of the time. How much can its observed win rate vary? | Number of trades | Standard deviation of win rate | Typical range (about 95%) | |---|---|---| | 10 | 15.7% | About 24% to 86% | | 50 | 7.0% | About 41% to 69% | | 100 | 5.0% | About 45% to 65% | | 500 | 2.2% | About 51% to 59% | | 1,000 | 1.6% | About 52% to 58% | The standard deviation is √(0.55 × 0.45 / n). After 10 trades, a genuinely good strategy can easily look terrible, and a bad one can look brilliant. See [Sampling and Standard Error](https://learn.tradelabsai.com/math/sampling-and-standard-error/). ## What this means for traders - **Do not judge a strategy on a few trades.** Ten or twenty trades say very little. - **Expect streaks.** Clusters of wins and losses are normal. See [Losing and Winning Streaks](https://learn.tradelabsai.com/risk/losing-and-winning-streaks/). - **Focus on process, not single outcomes.** A good decision can lose and a bad decision can win. See [Hindsight and Outcome Bias](https://learn.tradelabsai.com/psychology/hindsight-and-outcome-bias/). - **Size positions to survive** until the law of large numbers can work. See [Risk of Ruin](https://learn.tradelabsai.com/risk/risk-of-ruin/). - **Backtests need many trades** to be meaningful. See [Backtesting Methodology](https://learn.tradelabsai.com/research/backtesting-methodology/). ## The law does not "correct" past results A common misunderstanding is that after a run of losses, wins must come to "balance out". The law of large numbers does not work by reversing past results; it works by diluting them with many future results. Past losses stay; their effect on the average shrinks as the sample grows. Expecting a correction is the gambler's fallacy. See [Gambler's Fallacy](https://learn.tradelabsai.com/psychology/gamblers-fallacy/). ## Low win rate strategies need more trades Strategies with low win rates and large winners, such as trend following, have higher variance in results. They need even more trades for their edge to show, and they can go through long flat periods. See [Trend Following](https://learn.tradelabsai.com/strategies/trend-following/). **Example: Trend following variance** A trend strategy wins 35% of the time, with winners averaging 3 times losers (EV per trade = 0.35 × 3 minus 0.65 × 1 = 0.40 units). Over 30 trades, the chance of being down overall is about 12% if every win and loss is exactly that size, and higher when sizes vary, even though the strategy has a real edge. Over 300 trades, that chance falls to a small fraction of a percent. ## When the law does not help - **The edge changes:** markets evolve; a past edge may fade. See [Signal and Alpha Decay](https://learn.tradelabsai.com/research/signal-and-alpha-decay/). - **Dependence:** if trades are correlated, the effective sample is smaller than the number of trades. - **Fat tails:** with extremely heavy tailed outcomes, averages converge slowly. See [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/). - **Ruin first:** if you run out of capital, you never reach the long run. ## Frequently asked questions ### What is the law of large numbers? The principle that the average of results from a random process gets closer to its true expected value as the number of trials increases. ### How many trades do I need to judge a strategy? It depends on the win rate and payoff spread, but dozens give a rough idea and hundreds give much more reliable estimates. ### Does the law of large numbers mean losses will be made up? No. Past results are not reversed; they are diluted by many future results. Next, learn how new information changes probabilities in [Conditional Probability](https://learn.tradelabsai.com/math/conditional-probability/). ## Continue learning - Next lesson: [Conditional Probability](https://learn.tradelabsai.com/math/conditional-probability/) - Previous lesson: [Expected Value](https://learn.tradelabsai.com/math/expected-value/) - Related: [Expected Value](https://learn.tradelabsai.com/math/expected-value/): Expected value is the average result of a bet over many repetitions. Learn the formula, trading and prediction market examples, and why EV alone is not enough. - Related: [Sampling and Standard Error](https://learn.tradelabsai.com/math/sampling-and-standard-error/): Standard error measures how much an estimate like a win rate or average return varies between samples. Learn the formulas and what they mean for backtests. - Related: [Statistical Significance in Trading](https://learn.tradelabsai.com/math/statistical-significance/): Statistical significance helps judge whether trading results reflect a real edge or luck. Learn the t statistic rule of thumb, sample size and multiple testing. - Related: [Recency Bias](https://learn.tradelabsai.com/psychology/recency-bias/): Recency bias makes recent events feel more important than they are. Learn how it distorts strategy judgement, risk and market views, and how to counter it. - Related: [Central Limit Theorem](https://learn.tradelabsai.com/math/central-limit-theorem/): The central limit theorem says averages of many independent values tend toward a normal distribution. Learn what it means for trading statistics and when it fails.