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#

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.
• Focus on process, not single outcomes. A good decision can lose and a bad decision can win. See Hindsight and Outcome Bias.
• Size positions to survive until the law of large numbers can work. See Risk of Ruin.
• Backtests need many trades to be meaningful. See 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.

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.

When the law does not help#

• The edge changes: markets evolve; a past edge may fade. See 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.
• 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.