Poisson Totals: Price an NHL or Soccer Total From the Mean
A 6.0 hockey total at -110 on both sides looks like a coin flip. Run the math and you find a different bet.
Take a game with a true mean of 6.0 goals. The Over wins 39.4% of the time. The Under wins 44.6%. The bet pushes 16.1%. Those three numbers sum to 100%, and the Over is the worse side by five points. The mean has to reach 6.16 before the Over and Under are equal.
This guide gives you the model behind those numbers. You get the formula, a probability table, the mean each line needs, a moneyline example, a soccer example and a calculator. You also get a list of the places where the model fails, with sources.
The Model in One Line
A Poisson model treats goals as random events arriving at a steady average rate. If a team averages λ goals in a game, the chance of exactly k goals is:
P(k) = e^(-λ) × λ^k / k!
Two facts make this model useful for totals.
- The mean and the variance are both λ. One number describes the whole distribution.
- If the home team scores Poisson(λ1) and the away team scores Poisson(λ2) independently, total goals follow Poisson(λ1 + λ2). You price a total from one number, the expected total.
Sports statisticians have used this setup for four decades. M. J. Maher fitted independent Poisson scores to 12 separate football leagues and reported in 1982 the model gives a reasonably accurate description of scores, with small systematic differences (Maher, Statistica Neerlandica, 1982). Dixon and Coles built a Poisson regression on English league and cup data from 1992 to 1995 and tested the model against bookmaker odds from 1995 to 1996. They report a positive return from a betting strategy based on the model (Dixon and Coles, Applied Statistics, 1997).
Hockey fits the same frame. Buttrey, Washburn and Price modelled NHL goals as a Poisson process whose rate depends on the two teams, home ice, and power play or short-handed status. They fitted every game of the 2008-09 season and report the model predicts adequately (Buttrey, Washburn and Price, Journal of Quantitative Analysis in Sports, 2011).
Table 1: Outcome Probabilities by Mean
The table fixes the line at 6.0 and moves the mean. Every figure comes from the formula above, so you recompute any cell with a calculator.
| Mean total | 5 goals or fewer | Exactly 6 | 7 goals or more | Over 5.5 | Over 6.5 |
|---|---|---|---|---|---|
| 5.5 | 52.9% | 15.7% | 31.4% | 47.1% | 31.4% |
| 5.8 | 47.8% | 16.0% | 36.2% | 52.2% | 36.2% |
| 6.0 | 44.6% | 16.1% | 39.4% | 55.4% | 39.4% |
| 6.2 | 41.4% | 16.0% | 42.6% | 58.6% | 42.6% |
| 6.5 | 36.9% | 15.8% | 47.3% | 63.1% | 47.3% |
Read the 6.0 row. Over 5.5 wins 55.4% of the time. The fair price is about -124. Over 6.5 wins 39.4%, so the fair price is about +154. Moving the line one goal changes the probability by 16 points. The exact-6 slice holds 16.1% of all games, and a whole-number line treats this slice as a refund.
Whole-Number Totals Push
On a 6.0 total, a game with exactly six goals returns your stake. You win on 7 or more. You lose on 5 or fewer. The push removes 16% of the games from the bet, so your decision depends on the other 84%.
Use this formula for the break-even probability on a pushing bet:
Win share = P(win) / (P(win) + P(lose))
At mean 6.0, the Over has P(win) = 39.4% and P(lose) = 44.6%. The share is 0.394 / (0.394 + 0.446) = 46.9%. A fair price for this Over is about +113.
A standard -110 bet needs a win share of 110 / 210 = 52.38% to break even. Table 2 shows the mean each Over line needs to reach the fair point and the -110 break-even point.
| Over line | Mean for a 50% bet | Mean to break even at -110 | Gap in goals |
|---|---|---|---|
| 5.5 | 5.67 | 5.81 | 0.14 |
| 6.0 (pushes) | 6.16 | 6.29 | 0.13 |
| 6.5 | 6.67 | 6.82 | 0.15 |
| 7.0 (pushes) | 7.16 | 7.30 | 0.14 |
Two lessons follow.
- The Poisson median sits about 0.17 goals above the half-goal line. A 5.5 line at even money implies a mean near 5.67, not 5.5. Do not read a half-goal line as the expected total.
- Vig costs you about 0.14 goals of mean at every line. Your projection has to beat the line by this gap before the bet earns money.
Worked EV example
Say your model projects 6.5 goals and the book offers Over 6.0 at -110.
- P(win) = P(7 or more) = 47.3%.
- P(lose) = P(5 or fewer) = 36.9%.
- P(push) = P(6) = 15.7%.
- Profit on a win at -110 is $0.909 per $1 risked.
- EV = 0.473 × 0.909 - 0.369 = +$0.061 per $1 risked.
You earn 6.1 cents per dollar. Change the projection to 6.2 and the same bet returns 0.426 × 0.909 - 0.414 = -$0.027. A 0.3 goal error in your mean flips the sign of the bet. Your projection needs more precision than most models deliver.
Moneylines From Two Means
You find a win probability when you have one mean per team. Sum the joint probabilities of every score where the home team scores more.
Example: home mean 3.2, away mean 2.9, total 6.1.
| Result after 60 minutes | Probability |
|---|---|
| Home leads | 46.5% |
| Away leads | 37.1% |
| Tied | 16.4% |
An NHL moneyline pays on the full game, so you must assign the tied games to a side. The books settle the standard two-way market after overtime. FanDuel states the standard two-way moneyline covers the full game including overtime and shootout, while the three-way moneyline settles on regulation only (FanDuel NHL betting guide). Check the rules at FanDuel and at any other book before you compare prices across markets.
You have two simple choices for the tie.
- Split ties in proportion to regulation wins. Home becomes 46.5 / (46.5 + 37.1) = 55.7%, a fair price near -126.
- Split ties evenly. Home becomes 46.5 + 8.2 = 54.7%, a fair price near -121.
Five points of price separate the two assumptions. Five points is a quarter of the vig on a standard 20-cent line. Test the tie split against your own overtime results before you trust a number built on either one.
A Soccer Example
Soccer scores are lower, so the tie has a larger share. Use home mean 1.5 and away mean 1.2, a total of 2.7.
| Market | Probability | Fair price |
|---|---|---|
| Home win | 44.1% | +127 |
| Draw | 25.5% | +292 |
| Away win | 30.4% | +229 |
| Over 2.5 goals | 50.6% | -102 |
| Score 0-0 | 6.7% | +1,390 |
| Score 1-1 | 12.1% | +727 |
Maher noted the independence assumption leaves small systematic differences and reported a bivariate Poisson model with a correlation of 0.2 between the two scores improves the fit (Maher, 1982). Karlis and Ntzoufras proposed bivariate Poisson models for football and water-polo data on the same grounds (Karlis and Ntzoufras, The Statistician, 2003). Compare the 25.5% draw rate with the draw rate in your league before you price a three-way market.
Where the Model Breaks
The model fails in five known ways. Each one points to a bet or a skip.
- Independence. Goals for the two teams are not independent. The bivariate work above exists because of this. Ties and low scores are the first cells to misprice.
- Constant rate. Hockey teams do not score at one steady rate for 60 minutes. Thomas fitted a semi-Markov model to the times between NHL goals and moved past the plain Poisson process and found a goal shortens the remainder of the game (Thomas, Journal of Quantitative Analysis in Sports, 2007).
- Game state. Buttrey and co-authors make the rate depend on power play and short-handed status. The authors list the goalie pull decision as a use of the model. A single pregame mean cannot see an empty net.
- Settlement. Action Network reports the overtime or shootout winner receives credit for the game-winning goal, and this goal counts toward the total (Action Network NHL over/under guide). A model built on 60 minutes misses the extra goal in every game past regulation. Team total rules differ by book. Action Network reports FanDuel limits team totals to regulation, while BetMGM includes overtime and shootouts (Action Network team total rules).
- Variance. Poisson forces variance to equal the mean. Compute the variance and the mean of total goals for your own sample. If the variance runs above the mean, the tails are heavier than the table shows and a negative binomial model fits better.
Calculator
Enter your projected total, the line and the Over price. The tool applies the formulas above.
How to Use This
- Convert every line to a mean before you compare each with your projection. A 5.5 at even money means 5.67. A 6.0 at even money means 6.16.
- Subtract 0.14 goals for the vig. If your projection does not clear the line by this gap, pass.
- On whole-number lines, judge the bet by the win share, not by the raw win probability. The raw number hides the push.
- Check the settlement rules for overtime, shootouts and team totals at your book before you bet.
- Record the model tie rate and your observed overtime rate. A large gap means your goal means are wrong, even when your totals look right.
Sources
- Maher, M. J. (1982). Modelling association football scores. Statistica Neerlandica 36(3), 109-118.
- Dixon, M. J. and Coles, S. G. (1997). Modelling association football scores and inefficiencies in the football betting market. Applied Statistics 46(2), 265-280.
- Karlis, D. and Ntzoufras, I. (2003). Analysis of sports data by using bivariate Poisson models. The Statistician 52(3), 381-393.
- Buttrey, S. E., Washburn, A. R. and Price, W. L. (2011). Estimating NHL scoring rates. Journal of Quantitative Analysis in Sports 7(3).
- Thomas, A. C. (2007). Inter-arrival times of goals in ice hockey. Journal of Quantitative Analysis in Sports 3(3).
- Action Network. NHL over/under betting guide.
- Action Network. Do NHL team total goals count after overtime and shootouts.
- FanDuel. How to bet on the NHL.
All probabilities, fair prices and EV figures in the tables come from the Poisson formula with the inputs shown. Recompute any of them to check.