Poisson Distribution Calculator
How To Use This Calculator
The Poisson distribution calculator helps you estimate the probability of specific outcomes in sports events. Enter the expected average and an over/under line to see probabilities for totals betting.
Use historical data or sharp book implied totals to determine the expected average (λ).
Input the expected average and the over/under line from your sportsbook.
See the over, under, and exact probabilities for your line.
Review the bar chart to see the probability of each exact outcome (0-10).
What is the Poisson Distribution?
The Poisson distribution is a probability model developed by French mathematician Siméon Denis Poisson in 1837. It predicts the likelihood of a given number of events occurring in a fixed interval, given a known average rate.
In sports betting, it is one of the most widely used models for predicting goal totals in soccer and hockey, as well as player props like points, assists, rebounds, and strikeouts. By knowing the expected average, you can calculate the probability of any exact outcome or over/under total.
P(X = k) = (λ^k × e^−λ) / k!Best Sports for Poisson Modeling
Low-scoring, independent goals — ideal fit
Discrete goal events at steady rate
Run totals fit the Poisson model well
Points, assists, rebounds, strikeouts, shots
Step-by-Step Poisson Calculation
A soccer match has an expected total of 2.5 goals (λ = 2.5). The sportsbook offers Over 2.5 at -110. Is it a good bet?
Common Mistakes to Avoid
Basketball and American football scores are too high and too continuous for Poisson. The model works best for discrete, low-frequency events like goals in soccer or hockey.
Standard Poisson assumes independence between teams. In reality, a team trailing may push harder, affecting both teams' scoring rates. Consider bivariate Poisson for more accuracy.
Team form changes over a season. Using full-season averages can mask recent trends. Weight recent matches more heavily or use rolling averages of the last 6-10 games.
Poisson gives you a baseline model, not a crystal ball. Always compare your calculated probabilities to market odds and only bet when you find genuine value.
Frequently Asked Questions
What is lambda (λ) in the Poisson formula?
Lambda (λ) is the expected average number of events in a given interval. For soccer, this is the expected number of goals in a match. It is the single parameter that defines the entire Poisson distribution — both the mean and variance equal λ.
How do I calculate expected goals for Poisson?
For each team, calculate their average goals scored per game and their opponent's average goals conceded per game. Combine these with league averages to get an attack strength and defense strength rating. Multiply these by the league average to get each team's expected goals.
Can I use Poisson for correct score predictions?
Yes. Calculate the Poisson distribution for each team separately using their expected goals. Then multiply the probabilities together. For example, P(2-1) = P(Home scores 2) × P(Away scores 1). This assumes scoring independence between teams.
Why does Poisson underestimate 0-0 draws?
The standard Poisson model assumes independence. In reality, some matches have a “low-scoring” dynamic where both teams play defensively, making 0-0 more likely than the model predicts. Bivariate Poisson models can account for this correlation.
What is bivariate Poisson?
Bivariate Poisson extends the standard model by adding a correlation parameter between the two teams' scoring. This produces more accurate predictions for scorelines like 0-0 and 1-1 where team-level independence assumptions break down.
Is Poisson better than using market odds?
Market odds from sharp books are generally very efficient. Poisson is best used as a supplement — build your own model, compare it to market odds, and bet when your model disagrees with the market by a meaningful margin.
Can I use Poisson for player props?
Yes — player props are one of the best use cases for Poisson. Any discrete counting stat like points, assists, rebounds, strikeouts, or shots on goal can be modeled with Poisson. Use the player's season average or recent rolling average as your λ value and compare the Poisson probabilities to the sportsbook's line.
How do I account for home advantage?
Separate your historical data into home and away records. Use home attack/defense ratings for the home team and away ratings for the away team. Typically, home teams score about 0.3-0.4 more goals per game than away teams.
Pro Tip: Use Sharp Book Totals as λ
Don't have time to build your own expected goals model? Use the implied total from a sharp sportsbook like Pinnacle as your λ value. Their lines are highly efficient and give you a solid baseline. Then compare the Poisson probabilities to odds at softer books to find value.