Poisson Distribution
Poisson Distribution Calculator: Teams Average 2.6 Goals, What Are the Odds of Over 2.5?
The Poisson distribution gives the probability that a low-frequency event happens k times in a fixed period, and it is the standard model for soccer and hockey goals, corners and similar counts. The formula is P(X = k) = e^−λ × λ^k ÷ k!, with λ the average. If two teams combine for 2.6 goals per game, the chance of 3 or more (over 2.5) is 48.2%. Enter λ and k and the Poisson calculator returns the exactly, at least and at most probabilities to compare with a sportsbook's total.
Poisson Distribution Calculator
Poisson DistributionEstimate the hit rate on totals and over/unders from average scoring
P(X = k) = e^−λ × λ^k ÷ k!At least k = 1 − Σ P(X = i) for i from 0 to k − 1.
How to use the Poisson distribution calculator
Enter your estimate of the average as λ, most often the two teams' combined goals per game, and the count you want as k. To price over 2.5, enter k = 3 and read 'at least 3'; to price under 2.5, enter k = 2 and read 'at most 2'. The calculator also lists the full distribution from 0 to 6.
Estimating λ is the real work. The simplest method averages the home team's goals scored with the away team's goals allowed for the home expectation, does the same for the away team, and adds the two for the game total. Adjusting for the league average sharpens it further; the Mysports.AI model breakdown explains that step.
The calculator converts the 'at least k' probability into fair odds. If a sportsbook prices the over above the fair price, the over has value; confirm it with the expected value calculator before you bet.
Where the Poisson distribution breaks down
Poisson assumes events are independent and the rate is constant, but soccer scoring rises in the second half and a leading team sits back, so Poisson tends to misjudge the extremes: real 0-0 draws are a little more common than the model says, and so are blowouts.
It is not built for high-scoring sports. NBA totals run well above 100 points, where the Poisson assumption that variance equals the mean fails; a normal distribution is the usual choice. MLB run totals sit in the single digits and Poisson works reasonably well there, with extra innings the main complication. NHL goal totals are a natural fit.
Example: λ = 2.6, pricing over 2.5 goals
- P(0) = e^−2.6 = 7.4%; P(1) = 19.3%; P(2) = 25.1%.
- At most 2 goals = 7.4 + 19.3 + 25.1 = 51.8%; at least 3 goals = 100 − 51.8 = 48.2%.
- Fair odds on over 2.5 = 1 ÷ 0.482 = 2.07 (+107); fair odds on under 2.5 = 1 ÷ 0.518 = 1.93 (-108).
If a sportsbook posts over 2.5 at +115 (2.15), that beats the fair 2.07 and the over has value with an EV of about +3.6%. At -105 (1.95) the same over is −EV.
FAQ
What is the Poisson distribution?+
A probability distribution that gives the chance an event occurs k times in a fixed period, using a single parameter λ, the average. Bettors use it for low-frequency counts such as soccer and hockey goals or corners, where scoring is rare and roughly random.
How do you estimate λ?+
The simplest approach adds the two teams' average goals per game. A better one averages the home team's goals scored with the away team's goals allowed for the home expectation, does the same for the away side, and adds the two together.
Can the Poisson distribution price a correct score?+
Yes. Compute the probability the home team scores i goals and the away team scores j goals separately, using each team's own λ, then multiply them. The product is the probability of an i-j final score.
Does Poisson work for NBA totals?+
No. NBA totals run well above 100 points, and the Poisson assumption that the variance equals the mean does not hold at that scale. Basketball totals are usually modeled with a normal distribution instead.
Why does Poisson underestimate 0-0 draws?+
When both teams play cautiously the scoring rate drops during the match, which violates Poisson's constant-rate assumption. As a result, 0-0 and other very low-scoring results happen slightly more often in real games than the model predicts.
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