Law of large numbers (casino example)
The law of large numbers says averages settle toward the expected value as rounds pile up. Why a 97 percent RTP shows over 10,000 rounds, not one session.
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The law of large numbers says that the average result of many independent rounds moves toward the true expected value as the number of rounds grows. It is why a game's 97 percent RTP describes thousands of rounds but tells you little about the next ten.
Crucially, it concerns the average per round, not the running total of wins and losses. The total can drift further from zero even as the average settles.
Worked example: 100-coin bets on Dice at 50 percent
Each bet returns 194 coins half the time and nothing half the time. The average return is 97, so the expected result is minus 3 coins per round. One round's standard deviation is 97 coins. For n rounds the expected total is minus 3n, and the standard deviation of the total is 97 times the square root of n.
| Rounds | Expected total | Typical swing (1 standard deviation) | Average per round |
|---|---|---|---|
| 100 | minus 300 | plus or minus 970 | minus 3, give or take 9.7 |
| 10,000 | minus 30,000 | plus or minus 9,700 | minus 3, give or take 0.97 |
| 1,000,000 | minus 3,000,000 | plus or minus 97,000 | minus 3, give or take 0.097 |
After 100 rounds the swing (970) is over three times the expected loss (300), so being ahead is common (roughly 4 times in 10). After 10,000 rounds the expected loss is three times the swing, so being ahead is very unlikely (about 1 time in 1,000). After a million rounds the average per round sits almost exactly at minus 3. That is the edge showing through.
What the law does not say
It does not say that early luck gets corrected. A player who is 500 coins ahead after 100 rounds is not "due" to lose it back; they simply carry on from where they are, and the larger number of later rounds dilutes the early result. The mistaken belief that it does correct is the Gambler's fallacy (definition and example).
It also does not make any single session predictable. The law deals with averages across large samples, and the size of the sample matters, as in House Edge Explained: How Casinos Make Their Money.
Common misconceptions
- "Over time I will break even." Over time you converge on the game's return, which is below 100 percent.
- "A million-round simulation tells me my next session." It tells you the average, not the spread of one session.
- "The law needs a small bet size." Bet size scales results; it does not change convergence.
Coins in the app are virtual and have no cash value. Chances for each game are on the Odds page.