Poisson Distribution Calculator
Model sports outcomes with the Poisson distribution. Enter the expected average and an over/under line to see probabilities for every outcome.
Expected rate and line
λ = expected TDs/goals/points; line = posted total for over/under.
| λ (expected) | Over/under line |
|---|---|
Over
45.62%
Under
54.38%
Probability distribution (k = 0 to 10)
| k | P(X = k) | Distribution |
|---|---|---|
| 0 | 8.21% | |
| 1 | 20.52% | |
| 2 | 25.65% | |
| 3 | 21.38% | |
| 4 | 13.36% | |
| 5 | 6.68% | |
| 6 | 2.78% | |
| 7 | 0.99% | |
| 8 | 0.31% | |
| 9 | 0.09% | |
| 10 | 0.02% |
Frequently Asked Questions
What is the Poisson distribution in sports betting?
The Poisson distribution is a probability model that predicts how many times an event (like a goal or run) will occur in a fixed period. It only requires one input — the expected average (lambda) — and is widely used to model soccer goals, hockey goals, baseball runs, and more.
How do I find the expected goals (lambda) value?
Lambda is typically derived from team scoring averages, adjusted for opponent strength. For example, if a team averages 1.5 goals per game and the opponent concedes 1.3 per game against a league average of 1.2, you might estimate lambda as 1.5 × (1.3/1.2) = 1.625.
When does the Poisson model break down?
Poisson assumes events are independent and occur at a constant rate. It struggles with correlated events (e.g., momentum shifts), very high-scoring games, or sports where scoring patterns are bursty. It also ignores in-game context like red cards or injuries.
Can I use Poisson for totals and over/under bets?
Yes — Poisson is excellent for modeling totals markets. Enter the expected combined score as lambda and the posted total as the line. The calculator shows you the exact probability of going over or under, which you can compare to the implied odds from the sportsbook.
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