The question
"We've had eight unders in a row in this league. The over has to be due, doesn't it?"
Is a result due in betting? No. This lesson explains why, and why the thing people are half-remembering, the law of large numbers, is real but works in a completely different way from how most people think.
The idea in one sentence
Over thousands of bets your win rate settles near its true value, not because luck evens out, but because new results swamp the old ones.
The picture
Toss a fair coin 10 times and 7 heads is nothing unusual. Toss it 10,000 times and 7,000 heads would be astonishing. The proportion of heads closes in on 50% as you keep going. That's the law of large numbers.
But look at how it closes in. Say you're 20 heads ahead after 100 tosses. The coin doesn't then start throwing extra tails to pay you back. Your 20-toss lead just gets buried under the next 10,000 tosses, where it becomes a rounding error.
The chart below plays this out. The top line is your running win rate, which wobbles wildly at first and then flattens toward the true chance. Underneath, the strip of results shows every streak, and the panel counts what happened straight after each long run.
Two things are always true in the simulation, however many times you run it:
- After a run of losers, the next result wins at about the true rate. The coin, and the football, has no memory.
- Long runs are common in long samples. They're not a sign that anything has changed.
Worked Betfair example
You back Over 2.5 in matches where your model says 53%, and you get 2.00 on the exchange. (Illustrative figures, and a genuinely good edge.)
- Your edge per bet. EV per £1 after 2% commission = 0.53 × £1 × 0.98 − 0.47 × £1 = +4.94p. A strong, real edge.
- You hit seven losers in a row. The chance of any particular seven bets all losing is 0.47⁷ = 0.51%, about 1 in 200. It feels like something must be wrong.
- But how often does it happen at all? Over 500 bets there are hundreds of places a losing run could start. The chance of at least one run of 7 or more losers is 74%. Over 1,000 bets it's 93%.
- How long is a typical worst run? Over 1,000 bets at a 47% loss rate, the typical longest losing run is about 8. Even 10 in a row turns up in 24% of 1,000-bet samples.
- What's the next bet worth? Still 53%, still +4.94p per £1. The seven losers haven't made a winner more likely and they haven't made it less likely.
- What if you doubled the stake to "win it back"? You'd still earn +4.94p per £1, just on bigger stakes at a worse moment for your bank. The edge per £1 never changes (Lesson 6.5).
Verdict: a run of seven losers is expected, not a warning sign, for a real 53% edge. Judge whether your edge is still alive with a proper test (Lesson 7.3), not with how the last week felt.
How the law of large numbers actually works
| Bets | Range your win rate lands in, 95% of the time | Typical gap in number of winners |
|---|---|---|
| 100 | 43.2% to 62.8% | ±5 |
| 1,000 | 49.9% to 56.1% | ±16 |
| 10,000 | 52.0% to 54.0% | ±50 |
The percentage tightens. The gap measured in winners gets bigger. Nothing is being paid back; the gap is just shrinking as a share of an ever-larger total.
The league version
The same maths answers the eight-unders question. If every match in a 380-game season were a genuine 50-50 on over 2.5, a run of 8 or more unders in a row somewhere in the season happens 52% of the time. It's more likely than not.
The formula
Law of large numbers
- p̂ₙ ("p-hat") is your win rate after n bets.
- p is the true chance of winning each bet.
In plain English: the more bets you place, the closer your win rate gets to the true chance. It says nothing about the next bet.
How wide the wobble is
- SE is the standard error of your win rate.
In plain English: the wobble in your win rate shrinks with the square root of the number of bets. The 95% range is about p ± 1.96 × SE.
The gambler's fallacy, written down
- k is the length of the losing run.
In plain English: for independent bets, the chance of winning next is the same whatever just happened. That's what independent means (Lesson 3.1).
Chance of a run somewhere
- q is the chance of losing one bet, 1 − p.
In plain English: a particular run is rare, but a long sample has many starting points. The chance of at least one run is much larger and is worked out by counting step by step (the tool does it exactly).
Try it
Set win chance to 53%, 500 bets and a run length of 7, and check the exact chance of at least one run is about 74%. Then set 10,000 bets, run the simulation a few times and look at the "next result after 3 or more losers" panel: it lands within a couple of points of 53% every time.
Common mistakes
- Thinking a result is "due". Coins, dice and independent football matches have no memory. Eight unders don't make an over more likely.
- Reading a streak as a change. Long runs are normal in long samples. A run is only evidence of change when it's longer than your edge would produce, tested properly.
- The opposite mistake: the "hot hand". Backing a team because it's won five in a row is the same error in reverse, unless you have a reason the chances really changed (Lesson 5.2).
- Chasing losses with bigger stakes. Martingale and similar systems don't change the edge per £1. They just raise the stakes when your bank is weakest.
- Expecting the law of large numbers to work quickly. After 100 bets a true 53% bettor can easily show anywhere from 43% to 63%.
Mistaking normal runs for a broken model, or a broken model for a normal run, is at the heart of why good models still lose money.