What is the Poisson distribution in sports betting?

The Poisson distribution is a probability distribution that models the likelihood of a given number of events (like goals) occurring in a fixed interval. In sports betting, it's used to predict score probabilities in low-scoring sports like soccer and hockey based on expected goals or points.

How do you calculate goal probability using Poisson?

The Poisson probability formula is P(k) = (λ^k × e^(-λ)) / k!, where λ (lambda) is the expected average (like expected goals), k is the specific outcome (exact number of goals), and e is Euler's number (approximately 2.71828). For example, if a team's expected goals is 1.5, you can calculate the probability of them scoring exactly 2 goals.

How accurate is Poisson for predicting sports outcomes?

Poisson distribution provides a reasonable approximation for low-scoring sports like soccer, but has limitations. It assumes events are independent and occur at a constant rate, which isn't always true in sports. Goals often cluster due to momentum, injuries, or tactical changes. Most professional bettors use Poisson as one input among many in their models.

Can Poisson be used for high-scoring sports like basketball?

Poisson distribution is less suitable for high-scoring sports like basketball or American football because the normal distribution becomes a better approximation when expected values are high. Poisson works best when the average number of events per interval is relatively low (typically under 10-15).

Poisson Distribution Calculator for Sports Betting

What Is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming these events occur at a constant mean rate and independently of one another. Named after French mathematician Simeon Denis Poisson, who published it in 1837, the distribution has become one of the most important tools in statistical analysis.

In the context of sports betting, the Poisson distribution helps predict scoring outcomes by modeling goals, points, or other discrete events. It's particularly useful for low-scoring sports like soccer, hockey, and lacrosse, where the average number of goals per game is relatively small (typically between 1-4).

The Poisson Formula Explained

The probability mass function of the Poisson distribution is:

P(k) = (λ^k × e^-λ) / k!

Where:

P(k) = Probability of exactly k events occurring
λ (lambda) = The expected average number of events (e.g., expected goals)
e = Euler's number (approximately 2.71828)
k! = Factorial of k (e.g., 3! = 3 x 2 x 1 = 6)

For example, if a team has an expected goals (xG) of 1.5, we can calculate the probability of them scoring exactly 2 goals: P(2) = (1.5² × e^-1.5) / 2! = (2.25 × 0.223) / 2 = 0.251 or about 25.1%.

How Bookmakers Use Poisson

Professional sportsbooks and betting exchanges use Poisson-based models as one component of their pricing algorithms. While their models are far more sophisticated—incorporating team form, injuries, head-to-head records, and market movements—the Poisson distribution provides a mathematical foundation for pricing correct score, over/under, and match result markets.

Understanding this helps bettors recognize when a market might be mispriced. If your Poisson model consistently differs from the market's implied probabilities, it could indicate value—though it's essential to remember that markets aggregate information from thousands of bettors and sophisticated models. Learn more about how bookmakers create lines in our odds compilers and line setting guide.

Limitations of the Poisson Model

While useful, the Poisson distribution has several limitations for sports betting:

Independence assumption: Poisson assumes events are independent, but goals often cluster due to momentum shifts, red cards, or tactical changes
Constant rate assumption: It assumes the scoring rate is constant throughout the match, ignoring the higher goal rates in the last 15 minutes of soccer matches
No correlation: The model treats home and away goals as independent, but in reality, if one team scores, it affects the other team's tactics
Doesn't capture context: Weather, referee tendencies, travel fatigue, and other factors aren't modeled

For more advanced modeling, researchers often use bivariate Poisson or negative binomial distributions that account for correlation between teams' scores and overdispersion in the data.

Using Expected Goals (xG) with Poisson

Expected goals (xG) has revolutionized soccer analytics by quantifying the quality of scoring chances. Instead of using historical goal averages, you can input team xG values for more accurate predictions. Sites like Understat and FBref provide free xG data for major leagues.

When using xG with Poisson:

Season averages: Use at least 10+ matches of xG data for stable estimates
Home/away adjustments: Teams typically perform differently at home vs. away
Opposition strength: Adjust for the quality of opponents faced
Recent form: Weight recent matches more heavily if you believe form is important

Practical Applications

Beyond soccer, Poisson can be applied to other low-scoring sports:

Hockey (NHL): Average goals per team is around 2.8-3.2 per game
Baseball: Run scoring can be modeled, though pitcher matchups add complexity
Lacrosse: Similar goal-scoring patterns to hockey

For high-scoring sports like basketball or American football, the normal distribution is typically a better approximation than Poisson when expected values exceed 10-15 points per period.

Poisson Distribution Calculator for Sports Betting

⚽ Enter Expected Goals

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Probability of Each Team Scoring X Goals

Home Team Goals

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Total Goals Over/Under Probabilities

Correct Score Probability Matrix

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