The question
"My records say +3% ROI over 2,000 bets. What could my real ROI actually be?"
A confidence interval for betting ROI answers exactly that. Your +3% is one noisy measurement, and the interval shows the range of true edges that could sensibly have produced it.
The idea in one sentence
A confidence interval is your measured ROI plus or minus about two standard errors, and it's the range your true long-run ROI most plausibly sits in.
The picture
In Lesson 1.1 we asked one question: how often would a bettor with no edge match your result? A confidence interval turns that round. It asks: which true edges would comfortably produce the result you got?
Picture your ROI as a dot on a line, with a bar either side of it. The bar is the interval. With 200 bets the bar is huge and your dot tells you little. With 10,000 bets the bar has shrunk to a sliver and the dot means something.
The question to ask of every interval is simple: does the bar cross zero?
- Bar entirely above zero. The data rules out "no edge" at that level of confidence.
- Bar crosses zero. You might have an edge, you might not. The data hasn't decided.
- Bar entirely below zero. You're very likely losing, whatever the recent run says.
| Bets | Same odds and ROI, 95% range |
|---|---|
| 200 | -12.1% to 18.1% |
| 500 | -6.5% to 12.5% |
| 2,000 | -1.8% to 7.8% |
| 5,000 | -0.0% to 6.0% |
| 10,000 | 0.9% to 5.1% |
Worked Betfair example
You've placed 2,000 bets at average odds of 2.20, mostly Match Odds and Over/Under 2.5, and your ROI after Betfair's 2% commission is +3%. (Illustrative figures.)
- Net odds after commission. A winner at 2.20 pays £1.20 per £1, less 2%, so £1.176. Your net odds are 2.176.
- Implied win rate. A +3% ROI at net odds of 2.176 means you've won 1.03 ÷ 2.176 = 47.3% of your bets.
- Swing of one bet. σ = 2.176 × √(0.473 × 0.527) ≈ 1.086 per £1 staked (the formula from Lesson 1.2).
- Standard error of your ROI. 1.086 ÷ √2,000 = 1.086 ÷ 44.7 ≈ 0.0243, or 2.43%.
- 95% interval. 3% ± 1.96 × 2.43% = 3% ± 4.76%. That gives −1.8% to +7.8%.
Verdict: your true ROI could be a small loss or a very healthy edge. The data can't yet tell those apart. You would need about 5,040 bets at the same +3% for the lower end to reach zero.
Other confidence levels for the same record
| Level | Multiplier | Interval |
|---|---|---|
| 90% | 1.645 | −1.0% to +7.0% |
| 95% | 1.960 | −1.8% to +7.8% |
| 99% | 2.576 | −3.3% to +9.3% |
A higher level makes a safer claim, so the range gets wider.
How the interval narrows as the bets pile up
Same odds (2.20) and the same +3% ROI after commission:
| Bets | Standard error | 95% interval | Width |
|---|---|---|---|
| 200 | 7.7% | −12.1% to +18.1% | 30.1 points |
| 500 | 4.9% | −6.5% to +12.5% | 19.0 points |
| 2,000 | 2.4% | −1.8% to +7.8% | 9.5 points |
| 5,000 | 1.5% | 0.0% to +6.0% | 6.0 points |
| 10,000 | 1.1% | +0.9% to +5.1% | 4.3 points |
At 200 bets the true ROI could be anywhere from −12% to +18%. That's the honest size of a "great first month".
The formula
The confidence interval
- ROI is your measured return: profit ÷ total staked, after commission.
- z is the multiplier for your chosen confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.
- SE is the standard error of your ROI.
In plain English: start at your measured ROI and step about two standard errors each way. Anything inside that range is a true edge your results can't rule out.
The standard error
- σ is the standard deviation of one bet per £1 staked.
- n is the number of bets.
- O_net is your average odds after commission: 1 + (odds − 1) × 0.98.
- p is the win rate your ROI implies.
In plain English: the interval's width depends on how lumpy each bet is and how many you've placed. Four times the bets halves the width.
Bets needed for the interval to clear zero
- z, σ and ROI are as above.
In plain English: this is how many bets it takes, at your current ROI, for the bottom of the interval to reach zero. For the example: (1.96 × 1.086 ÷ 0.03)² ≈ 5,040.
If you have your own P&L data
If your stakes vary, skip the formula for σ. Take the profit of each bet divided by its stake, and work out the standard deviation of that column in a spreadsheet (STDEV). That's your σ. The bootstrap is another route: resample your own bets thousands of times and read the range off the results.
Try it
Enter 2,000 bets, odds 2.20, ROI 3% and 95%, and check you get −1.8% to +7.8%. Then find the number of bets where the bar first sits wholly above zero, and see how much longer it takes at odds of 5.0.
Common mistakes
- Quoting the headline ROI without the range. "+8% ROI" over 300 bets sounds like an edge. With its interval attached it reads as "somewhere between a big loss and a big win".
- Reading the interval as a promise. A 95% interval is built so that 95% of intervals made this way catch the true ROI. About 1 in 20 miss it, and you won't know which.
- Using the interval from your best system. If you tried 30 filters and kept the best, its interval is far too optimistic. Correct for the search first (Lesson 7.4).
- Treating your ROI as the likeliest true edge. Backtested ROIs are usually flattering, so the truth is more often in the lower half of the range than the upper. Plan around the lower end.
- Mixing results from before and after commission. Work out the interval on your ROI after Betfair's 2%. That's the number your bank sees.
Why a narrow interval matters so much: why good models still lose money.