Two different questions
“What happens on average?” and “what happens in the next ten plays?” are different questions with different answers. The average describes the long run. The next ten plays belong to the short run, and in the short run chance has a lot of room to move.
Mixing the two up is behind many wrong ideas about luck, such as expecting a game to “even out” within a single evening.
Ten coin flips
A fair coin has a long-run proportion of heads of 50%. Yet in just 10 flips the exact split of 5 heads and 5 tails is not the most likely thing to happen overall; it is simply the most likely single count.
The number of ways to get each count of heads in 10 flips, out of 1,024 equally likely sequences, includes 252 ways for exactly 5 heads, 210 ways each for 4 or for 6 heads, and 120 ways each for 3 or for 7 heads.
- Exactly 5 heads: 252 / 1,024, about 25%.
- Between 4 and 6 heads: (210 + 252 + 210) / 1,024 = 672 / 1,024, about 66%.
- 7 or more heads: (120 + 45 + 10 + 1) / 1,024 = 176 / 1,024, about 17%.
So in a typical 10-flip experiment, roughly a third of the time the result falls outside 4 to 6 heads. Seeing 70% heads is not a sign of a rigged coin. It is ordinary short-run behaviour.
A hundred flips
With 100 flips the standard deviation of the count of heads is 5 (the square root of 100 x 0.25). A common rule of thumb says most results land within two standard deviations of 50, which is roughly 40 to 60 heads. The probability of landing in that band is about 96%.
Now compare proportions. A result of 60 heads is 60% heads, still a visible departure from 50%, but 60 is much more ordinary in 100 flips than 6 heads out of 10 might feel. Longer runs make the same raw gap look smaller in percentage terms.
Counts diverge while proportions converge
This is the key point, and it can look contradictory. As the number of flips grows:
- The proportion of heads gets closer to 50% on average, because the standard deviation of the proportion shrinks.
- The raw difference between heads and tails does not shrink. Its typical size grows, roughly with the square root of the number of flips.
For 100 flips the standard deviation of the count is 5. For 10,000 flips it is 50. A result of 5,050 heads is 50.5%, very close to even, yet it is 50 heads above the midpoint. The gap in the count got bigger while the percentage got tighter. That is the idea behind the law of large numbers.
Nothing is “owed”
It is tempting to think the long run needs to repair early swings. It does not. If a fair coin shows 7 heads in the first 10 flips, the next flips are still 50/50. The early excess is not corrected; it is just diluted by the many flips that follow. After another 990 flips the expected total is 7 + 495 = 502 heads out of 1,000, or 50.2%. This is the difference between dilution and compensation, and the belief in compensation is the gambler’s fallacy, sometimes called the law of averages.
What it means for a game with a house edge
Consider an even-money bet on a single-zero roulette wheel. Each bet has an expected result of about -2.7% of the stake. Over a few bets that edge is tiny compared with the swing from a single win or loss, so many people finish a short session ahead. Over thousands of bets, the swing grows more slowly than the accumulated expected loss, and the edge dominates.
In other words, the short run is dominated by variance and the long run is dominated by the edge. A short run of wins is therefore not evidence that the game is fair or favourable, and it does not predict the next result. This is also why a handful of results gives such weak information about a true rate: small sample sizes are noisy.
Practical takeaways
- A few plays tell you very little about the underlying odds.
- Streaks and swings in a short run are normal and expected from chance alone.
- Average results become reliable only across many independent repetitions.
- The long-run average is what the rules of the game dictate, not what luck “owes” you.
None of this makes gambling a safe or sensible way to earn money. Outcomes are uncertain, a house edge works against the player, and spending can quickly exceed what one intended. If you find it hard to stop, a local support service can help.