Law of Large Numbers Law of Large Numbers
A probability principle stating that as the number of trials grows, the average of actual results gets closer to the theoretical expected value. A slot's RTP only lines up after hundreds of thousands to millions of spins.
| English | Law of Large Numbers |
|---|---|
| Classification | Math & Probability |
| English | Law of Large Numbers |
| Category | Math / Probability |
| First proved by | Jacob Bernoulli (published 1713) |
| Core idea | More trials → average approaches expected value |
| Slot application | RTP only holds over a very large number of spins |
| Common misconception | The belief that results "return" to average in the short term (gambler's fallacy) |
Overview
The law of large numbers is a basic law of probability: the more times you repeat the same trial, the closer the average of the actual results gets to the theoretical expected value. Swiss mathematician Jacob Bernoulli first proved it, and his book was published in 1713.
If you flip a coin 10 times, you might get 7 heads, but if you flip it 10,000 times, the share of heads gets very close to 50%. The RTP of a slot works on the same principle: it is a value that holds true “over a very large number of spins.”
How many spins are enough for slots
We simulated two hypothetical games with 96% RTP. For each number of spins, we repeated the run thousands of times and calculated the range over which the actual return rate was spread (bottom 5% to top 5%).
| Spin count | Low volatility (2x with 48% probability) | High volatility (100x with 0.96% probability) |
|---|---|---|
| 100 spins | 80% ~ 112% | 0% ~ 300% |
| 1,000 spins | 90.8% ~ 101.2% | 50% ~ 150% |
| 10,000 spins | 94.3% ~ 97.7% | 80% ~ 110% |
| 100,000 spins | 95.5% ~ 96.5% | 90% ~ 100% |
A low-volatility game clusters near the RTP (within ±2%) after about 10,000 spins, but a high-volatility game still spreads about ±5% even after 100,000 spins. Real slots have max wins of thousands to tens of thousands of times the bet, so they need even more spins than this. That’s why providers and certification labs run millions to billions of simulated spins when verifying RTP (Monte Carlo simulation).
Why does it take so long
The fluctuation of the average shrinks by standard deviation ÷ √(number of spins). You need to increase the number of spins 100 times to cut the fluctuation to one tenth. The larger a game’s standard deviation (variance and standard deviation), the more spins you need to reach the same accuracy.
What the law of large numbers does not say
The law of large numbers says “the average gets closer to the expected value,” not “you’ll soon win back what you’ve lost.” Confusing the two leads to the gambler’s fallacy.
| The correct understanding | The misconception |
|---|---|
| The overall average gets closer to the RTP as spins increase | It hasn’t hit for a while, so it’s due |
| Past results do not affect future probabilities | Results are adjusted so losses get “compensated” |
| The longer you play, the closer your total loss gets to house edge × total bets | If you play long enough, you’ll eventually break even |
In reality, the average gets closer to the RTP not because “past losses are made up,” but because as countless future results pile up, the weight of past results gets smaller and smaller.
Why the casino always wins
For one person it’s a few hundred spins, but from the casino’s point of view it’s billions of spins from countless players added together. According to the law of large numbers, the casino’s overall results land very close to the house edge. So while an individual can win or lose a lot depending on luck, the casino’s profit is predicted very reliably.