Law of large numbers explained for betting

In one sentence

The law of large numbers says that as you place more bets, your average profit per bet settles closer and closer to its true expected value.

How it works

Toss a fair coin ten times and seven heads is common. Toss it ten thousand times and you will be very close to half heads. The coin has no memory; the extra tosses simply swamp the early noise.

Betting works the same way. A bet with a 5% edge can lose five times in a row, and a losing bet can win. Only over a long run does the average result drift towards the true edge.

The catch is how long "long" is. The spread of your average shrinks with the square root of the number of bets, so four times as many bets only halves the noise. That is why short-term results say so little about skill.

The maths

\bar{X}_n = \frac{X_1 + X_2 + \dots + X_n}{n} \;\longrightarrow\; \mu \quad \text{as } n \to \infty

\text{SD}(\bar{X}_n) = \frac{\sigma}{\sqrt{n}}

X with a bar: your average profit per bet after n bets.
X1, X2 and so on: the profit or loss on each individual bet.
n: the number of bets.
μ: the true expected profit per bet (the EV).
σ: the standard deviation of a single bet's profit, a measure of how much one result swings.

In plain English: the average of your results heads for the true EV, and the wobble around it shrinks with the square root of the number of bets.

Worked betting example

Football, Match Odds. You back away teams at 3.00 with £10 stakes, and each one truly wins 35% of the time. The market implies 33.3%, so you have an edge.

EV per bet = 0.35 × £20 − 0.65 × £10 = £7.00 − £6.50 = £0.50, a 5% return on stake.
Swing of a single bet: σ = √(0.35 × 0.65) × £30 = £14.31.
After 100 bets: expected profit £50, but the average per bet has an SD of £14.31 ÷ √100 = £1.43. The exact binomial chance of being in loss is 38.0%.
After 1,000 bets: expected profit £500, SD of the average £0.45, chance of being in loss 13.7%.
After 10,000 bets: expected profit £5,000, SD of the average £0.14, chance of being in loss about 0.02%.

So a bettor with a real 5% edge has well over a one-in-three chance of showing a loss after 100 bets. Commission would push these loss chances higher.

Where it's good

Setting realistic expectations before starting a new strategy.
Deciding how many bets to wait before judging a system or tipster.
Explaining why short-term profit or loss is weak evidence either way.
Understanding why high-odds strategies need far more bets than short-priced ones.
Planning bankroll size so you survive long enough for the edge to appear.

Limitations and pitfalls

It only guarantees convergence to your true EV. If that EV is negative, more bets just make the loss more certain.
It assumes the edge stays the same. In real markets edges shrink as others find them, so the long run may never arrive.
Bets must be roughly independent. Ten correlated bets on one match are closer to one bet than ten.
It is not the gambler's fallacy. A losing run does not make a win "due"; the early losses are diluted, not reversed.
Longshots converge slowly. At odds of 20.0 the swings are huge and thousands of bets may be needed.
Big stakes early can ruin you before the long run arrives, so pair this with sensible staking.

How to build it

numpy for simulation, scipy.stats.binom for exact loss probabilities.
Data: your typical odds, stake size and a realistic edge estimate (not your hoped-for one).
Practical tip: simulate 10,000 seasons of your strategy and plot the spread of outcomes at 100, 500 and 1,000 bets. It is the fastest cure for over-reading a hot streak.

Expected value: the number your average converges to.
Central limit theorem: describes the shape of the wobble around that number.
Risk of ruin: the chance you go bust before the long run arrives.
Hypothesis testing: judging whether a record shows real skill.
Power analysis: working out how many bets you need up front.