Law of large numbers in betting: why the margin wins in the long run
The law of large numbers says the average result per bet moves toward its expected value as the number of bets grows. For a bet priced with a margin that value is negative, so luck matters less and the cost matters more with every extra bet. It does not repay earlier losses.
The short answer
As the number of bets grows, the average result per bet gets closer to the bet's expected value. When a bookmaker has built a margin into the prices, that expected value is below zero, so the average settles on a cost. The expected value of a bet is the number the average heads toward.
The law is about averages, not about totals. Total luck can stay large in shillings while it shrinks as a share of the money staked.
What the law says, and what it does not
Take one bet that you could repeat under identical conditions, where each repeat is independent of the others. Its expected value is the probability-weighted average of the outcomes. The law of large numbers states that the average of the first N results approaches that expected value as N increases.
It does not say that wins and losses "even out" in a way that cancels a bad run. It does not say a loss makes a win more likely. Independent bets have no memory, which is the point developed in the guide on variance and losing streaks.
Two quantities that grow at different speeds
For one bet with stake S, decimal odds D and win probability p, the expected result is S × (p × D − 1). The spread of one result, its standard deviation, is S × D × √(p × (1 − p)). Over N identical independent bets the two combine differently:
The expected result grows in step with N. The spread grows only with the square root of N. Sooner or later the steady cost outgrows the swing, and that crossover is the whole of the "long run".
Illustrative example
A KES 100 bet on an even chance (p = 0.50) at decimal odds 1.90. A win returns KES 190 and a profit of KES 90; a loss costs KES 100.
Expected result: 0.50 × 90 − 0.50 × 100 = −KES 5. Spread: 100 × 1.90 × √(0.50 × 0.50) = 190 × 0.5 = KES 95.
Check: each outcome sits exactly KES 95 from the average (90 is 95 above −5, and −100 is 95 below −5), so the spread is KES 95.
The table: when the cost catches up with the swing
The table applies those two formulas to the example above. The last two columns show the ratio of the spread to the expected loss, and the spread of the average result per bet (the spread of the total divided by N, which equals 95 ÷ √N). Values are rounded to the digits shown.
| Bets N | Total staked | Expected result (−5 × N) | Spread (95 × √N) | Spread ÷ expected loss | Spread of average per bet |
|---|---|---|---|---|---|
| 1 | KES 100 | −KES 5 | KES 95 | 19.00 | KES 95.00 |
| 10 | KES 1,000 | −KES 50 | KES 300.42 | 6.01 | KES 30.04 |
| 100 | KES 10,000 | −KES 500 | KES 950 | 1.90 | KES 9.50 |
| 361 | KES 36,100 | −KES 1,805 | KES 1,805 | 1.00 | KES 5.00 |
| 1,000 | KES 100,000 | −KES 5,000 | KES 3,004.16 | 0.60 | KES 3.00 |
| 10,000 | KES 1,000,000 | −KES 50,000 | KES 9,500 | 0.19 | KES 0.95 |
| 1,000,000 | KES 100,000,000 | −KES 5,000,000 | KES 95,000 | 0.019 | KES 0.095 |
Look at the row for 100 bets. The expected loss is KES 500, yet an ordinary swing is KES 950, so finishing ahead is entirely normal. At 361 bets the two are equal, because 95 × √361 = 95 × 19 = 1,805 and 5 × 361 = 1,805. By 10,000 bets the spread is under a fifth of the expected loss.
Why the margin is the number that wins
The expected result per bet in the example is −KES 5 on a KES 100 stake, which is 5% of the stake. That is the margin at work: the two prices of 1.90 each carry implied probabilities of 1 ÷ 1.90 = 0.5263, which add to 1.0526, a margin of 5.26%. Dividing the margin by 1 plus itself gives 0.0526 ÷ 1.0526 = 0.05, the 5% cost seen above. The guide on calculating the bookmaker's margin shows the first step in full.
The prices are fixed before the bet is placed, so the cost is built in and does not depend on who bets or on a particular result. What the long run removes is the noise that sits on top of it. Regulation 20(2) of the 2026 operations regulations requires a licensee to display clearly the rules of games, odds, house edge and average return to the player, so for casino games the same average is meant to be shown in the open.
Longer odds need a much longer run
The crossover point is where the expected loss equals one spread. For one bet that is when N × |EV| = √N × spread, which gives N = (spread ÷ |EV|)2. The table compares the even-chance bet with a long-odds bet that costs the same on average.
Illustrative example
Bet B is a KES 100 stake with p = 0.10 at decimal odds 9.50. Expected result: 100 × (0.10 × 9.50 − 1) = −KES 5. Spread: 100 × 9.50 × √(0.10 × 0.90) = 950 × 0.3 = KES 285.
Crossover for B: (285 ÷ 5)2 = 572 = 3,249 bets. Crossover for A (the even-chance bet): (95 ÷ 5)2 = 192 = 361 bets.
| Bet | Expected result per bet | Spread per bet | Crossover (N where expected loss equals spread) |
|---|---|---|---|
| A: p = 0.50, odds 1.90 | −KES 5 | KES 95 | 361 bets |
| B: p = 0.10, odds 9.50 | −KES 5 | KES 285 | 3,249 bets |
The spread of B is three times that of A, so it needs nine times as many bets (3,249 ÷ 361 = 9). Long odds therefore hide the same cost for longer. This is one reason a short run on long odds says little about the price.
Early luck is diluted, not repaid
A common reading of the law is that a good start must be followed by a bad stretch. The arithmetic shows otherwise.
Illustrative example
Using bet A, suppose the first 100 bets finish at +KES 400 instead of the expected −KES 500. The next 9,900 bets each have an expected result of −KES 5, so they add an expected −KES 49,500.
Expected total after 10,000 bets: 400 − 49,500 = −KES 49,100. Average per bet: −49,100 ÷ 10,000 = −KES 4.91, close to the −KES 5 cost.
The early surplus was not taken back; it simply became small beside the cost of 9,900 more bets. The same holds for a bad start. The deviation stays in the total, and only its weight in the average falls.
What this means for your spending
- Cost follows the amount staked. The guide on expected loss per KES 1,000 staked converts any margin into a cost for the stake you actually place.
- A good week is not evidence of a better price. At 100 bets the spread is nearly twice the expected loss in the example, so results say little about the margin.
- More bets move you toward the average cost. They do not move you toward break-even.
Treat the expected loss as the price of the activity and decide in advance what you can afford to spend, as set out in the guide on setting a betting budget and limits. If a run of results creates urgency or a wish to recover money, stop and read the responsible gambling page, which lists the national helpline and the self-exclusion process. 18+. Gambling is addictive. Play responsibly.
Questions and answers
What is the law of large numbers in simple terms?
It says that the average of many repeated, independent results gets closer to the expected value as the number of results grows. For one bet the outcome is win or lose, but for thousands of identical bets the average result per bet settles near the expected value.
Does the law of large numbers mean I will win back my losses?
No. It describes the average per bet, not a debt that the future repays. Earlier luck, good or bad, is not reversed. It is only diluted by the bets that follow, each of which carries the same negative expected value.
How many bets does it take for the margin to show?
There is no fixed number. In the worked example with an even chance at odds of 1.90, the expected loss equals one standard deviation after 361 bets. A bet with a smaller chance of winning and longer odds needs far more, 3,249 bets in the second example.
Is the margin the same as the amount I lose?
Not exactly. The average loss is the stake times margin divided by 1 plus the margin. A bookmaker margin of 5.26% costs 5% of the amount staked on average, and the actual result of any period can be better or worse.
Why does the law apply more strongly to the bookmaker than to one bettor?
The bookmaker handles a very large number of bets at the same prices, so its result per shilling staked is close to the margin. A single bettor places far fewer bets, so the spread of results is wide relative to the average cost.
Sources
- Gambling Control (Conduct of Gambling Operations) Regulations, 2026, reg. 20(2), accessed 2026-11-02