The law of large numbers: why the house edge shows up over time

The law of large numbers says that the average result per bet moves towards the game's expected value as the number of bets grows. In casino games that value is negative for the player, so the house edge becomes visible over many rounds. It does not say that a loss is ever 'due' to be repaid.

By Nairobi Ultimate Editorial (prepared with AI tools under our editorial policy) · Last updated 2026-10-29

What the law of large numbers says

Suppose you repeat the same independent bet again and again. The law of large numbers says that the average result per bet gets closer and closer to the bet's expected value. The expected value is the probability-weighted average of every possible outcome, as set out in the guide to expected value.

In a casino game the rules fix that value, and for the player it is below zero. The negative amount, shown as a share of the stake, is the house edge. So the law is the reason a house edge that is invisible in one round becomes visible in thousands of rounds. The wider idea is covered in the guide on how RTP and house edge work.

Under Regulation 20(2) of the Conduct of Gambling Operations Regulations, 2026, a licensee shall display clearly the rules of games, odds, house edge and average return to the player. That published average is the number the law says results drift towards.

Step 1: the expected loss of one bet

Illustrative example

A European roulette wheel has 37 pockets. A straight bet on one number pays 35 to 1, so on a stake of KES 100 you gain KES 3,500 with probability 1/37 and lose KES 100 with probability 36/37.

Expected result = (3,500 x 1 + (-100) x 36) / 37 = (3,500 - 3,600) / 37 = -100/37, which is about -KES 2.70. As a share of the stake that is 1/37, or about 2.70%.

Nothing in this calculation depends on how many times you bet. Each KES 100 stake carries the same expected loss of about KES 2.70, and that is the house edge on this bet.

Step 2: the spread of one bet

Expected loss tells you the centre. You also need the spread, measured by the standard deviation, which the guide to standard deviation in casino games defines.

Illustrative example

Square each result and weigh it by its probability: (3,500 x 3,500 x 1 + 100 x 100 x 36) / 37 = (12,250,000 + 360,000) / 37 = 12,610,000/37, about 340,810.81.

Subtract the square of the mean (100/37 squared, about 7.30). The variance is about 340,803.5, which equals 100 x 100 x 46,656/1,369. Its square root is 100 x 216/37, about KES 583.78.

So one spin has an expected result of about -KES 2.70 and a standard deviation of about KES 583.78. The typical gap from the average is more than 200 times the average itself. That is why a single spin, or a short session, says almost nothing about the house edge.

Step 3: what happens over n bets

For n independent bets of the same kind, expected results add up, so the expected loss is n times (100/37). Variances also add up, so the variance is n times the single-bet variance and the standard deviation is the square root of n times KES 583.78.

That gives two different growth rates. Expected loss grows in proportion to n, while the spread grows only in proportion to the square root of n. Their ratio is (n x 100/37) / (sqrt(n) x 100 x 216/37) = sqrt(n)/216.

Straight bet on one number, European wheel, KES 100 per spin (all figures rounded)
Spins (n)	Expected loss (KES)	Standard deviation (KES)	Expected loss / spread
1	2.70	583.78	0.005
100	270.27	5,837.84	0.046
10,000	27,027.03	58,378.38	0.463
46,656	126,097.30	126,097.30	1.000
1,000,000	2,702,702.70	583,783.78	4.630

At 100 spins the spread is about 22 times the expected loss, so a player finishing ahead is entirely ordinary. At 46,656 spins (216 squared) the two are equal. At a million spins the expected loss is more than four times the spread, so the result lands near the house edge with very little room to escape it.

Average per bet versus running total

The law is about the average per bet. Divide the total result by n and the spread of that average shrinks like 1/sqrt(n), so the average per spin tightens around -KES 2.70. After a million spins the standard deviation of the average is about KES 0.58 per spin.

The running total behaves differently. Its expected value moves further below zero with every spin, and its spread also widens. The law does not pull the total back towards zero. It says only that the total, divided by n, approaches the negative expected value.

Why a loss is never "due" to be corrected

A common misreading is that, because averages settle, a run of losses must be followed by extra wins to balance things. Rounds are independent, so the wheel, the RNG or the crash multiplier has no memory of earlier outcomes. The chance on the next round is the same whatever happened before.

What actually happens is dilution, not correction. A big early loss is not repaid; it is simply spread across a growing number of rounds while new expected losses keep arriving. The same mistake in another form is described in the guide to the gambler's fallacy and session limits.

Reading this for other games

The same reasoning applies to any game with fixed rules and independent rounds: blackjack, baccarat, slots and crash games. Only the numbers change: the expected result per round and the size of the swings. A high-volatility game needs far more rounds before its average looks like its published return, which is the point of the guide on theoretical RTP versus short-session results.

Using the idea to set limits

Since the expected cost grows with every bet, the amount you stake and the time you spend set how much of that cost you take on. Regulation 87 of the same regulations requires a licensee to provide deposit limits (daily, weekly and monthly), loss, session and expenditure limits, reality checks showing duration of play, and an easy to access self-exclusion option for at least twenty-four hours.

Setting these before play fixes the exposure in advance. A limit does not change the house edge; it caps how long the edge works against you. Gambling is addictive: if play stops being a limited pastime, the responsible gambling page lists the GRA toll-free line and the self-exclusion process. This guide is for adults aged 18 and over.

Questions and answers

Does the law of large numbers mean I will win back my losses if I keep playing?

No. The law describes the average result per bet, not the running total. Each round is independent, so past losses are not repaid by later rounds. On a negative-expectation game the total expected loss keeps growing with every extra bet.

Why can one player be ahead after many rounds if the house edge is certain?

The house edge is an average. The spread of a player's total grows with the square root of the number of bets, so for a long while a player can sit above zero. The expected loss grows faster in proportion, and it gradually dominates the spread.

Is the law of large numbers the same as the gambler's fallacy?

No. The law is a true statement about averages over many independent rounds. The gambler's fallacy is the false belief that a short run of results will be 'corrected' by the next ones.

Does the law apply to crash games and slots as well as roulette?

Yes, for any game with fixed rules and independent rounds. The expected result per round is set by the rules, and the average over many rounds approaches it. Only the size of the swings differs between games.

What does the Kenyan regulation say about showing the house edge?

Regulation 20(2) of the Conduct of Gambling Operations Regulations, 2026 says a licensee shall display clearly the rules of games, odds, house edge and average return to the player.