Betting mathematics
Value betting: how to spot a value bet
A value bet is a bet whose odds are higher than the fair odds for your probability estimate, that is p × odds > 1. If you estimate the home team wins 50% of the time, the fair odds are 2.00, so odds of 2.10 carry +5% value. Value exists only if your estimate p is more accurate than the market, and that can only be checked over many bets.
Contents
Key terms
- Value bet
- A bet whose odds are higher than the fair odds for your estimate. Also called a +EV bet or a bet with positive expected value.
- Value betting
- An approach where you only place value bets, whether the outcome is a favourite or an underdog.
- Fair odds
- The odds with no margin for your estimate: 1 / p. For p = 0.50 the fair odds are 2.00.
- Edge
- How far your estimate sits above the price in the odds. Here it is expressed as value, in percent of the stake.
What is a value bet?
The odds are a price, and the estimate p is your opinion of the probability. A bet has value when the price is better than the opinion: the odds are higher than 1 / p. It is the same condition as positive expected value, only written in terms of the odds.
value = p × odds − 1
value > 0 when odds > 1 / pThe match odds are 2.10 / 3.40 / 3.60. Your estimate for a home win is p = 0.50.
- fair odds for your estimate: 1 / 0.50 = 2.00;
- value:
0.50 × 2.10 − 1 = +0.05, or +5% per unit staked; - the no-margin market gives the same home win 45.43% (see How to remove the margin), so your estimate differs from the market by 4.57 percentage points.
A value of +5% means that over many identical bets of 100 each, the average result would be +5.00 per bet, if p really is 0.50.
Which odds are value odds?
Every estimate has fair odds (zero value) and the odds that give the edge you want. For a 5% edge the odds must be at least 1.05 / p.
| Estimate p | Fair odds | Odds for +5% |
|---|---|---|
| 0.30 | 3.33 | 3.50 |
| 0.40 | 2.50 | 2.63 |
| 0.50 | 2.00 | 2.10 |
| 0.60 | 1.67 | 1.75 |
| 0.70 | 1.43 | 1.50 |
With favourites the gap between the fair odds and the value odds is small, so even a small error in the estimate wipes out the value. At p = 0.70 the gap is only 0.07 in the odds.
Where does value come from?
- An estimate better than the market: a model or an analysis sees something the odds have not priced in yet.
- Differences between bookmakers: the same bet has different odds at different bookmakers, so the best price can sit above the market's fair odds.
- Slow odds: after news (an injury, a line-up) some odds move more slowly than others.
Bookmakers take a margin, but they also have the most information just before kick-off. The AI model performance page shows that the no-margin odds had a better Brier score than the AI model on 1X2 (see AI vs bookmaker). Value therefore needs an estimate that beats exactly those odds, not just one that beats the average.
Why can a value bet still lose?
Value is an average over a long run. A single value bet loses every time the outcome does not happen, and at p = 0.50 that is every other time.
A run of value bets at odds 2.10 when the true probability of each is 0.50 (value +5%):
- after 100 bets the expected result is +5 units, but in 30.9% of cases the result is negative (47 wins or fewer);
- after 1,000 bets the chance of a loss falls to 6.9% (476 wins or fewer).
The numbers come straight from the binomial distribution and hold only if p really is 0.50.
False value: when the estimate is wrong
The same odds of 2.10, but the true probability of a home win is 0.45, not 0.50:
0.45 × 2.10 − 1 = −0.055
A bet that looked like +5% actually loses 5.5% of the stake on average. If the market's estimate (45.43%) is the right one, the value is −4.59%, which is exactly the margin on that outcome.
So value is not proven by a single win but by measurement: are the estimates calibrated (Probability calibration) and do the bets get better odds than the closing odds on average (CLV)?
Limitations
- Value depends on the estimate p, and the true probability is never seen directly. An error of a few percentage points flips the sign.
- The examples ignore tax on winnings, stake limits and accounts being restricted by bookmakers.
- The odds can change before you place the bet, so value at the time of analysis is not value at the time of the bet.
- Even a long run of value bets does not rule out a loss: after 1,000 bets at +5% a loss still has a 6.9% chance.