In This Guide
What is the Poisson Distribution?
The Poisson distribution is a statistical tool that calculates the probability of a specific number of events occurring within a fixed time period, given an average rate of occurrence. Named after French mathematician Siméon Denis Poisson (1781-1840), it's perfect for modeling rare, independent events.
Why Poisson Works for Sports Betting
Low-Scoring Sports
Goals are rare enough to model individually
Independent Events
Each goal doesn't directly cause another goal
Fixed Time Period
90 minutes of soccer, 60 minutes of hockey
Not for Basketball
200+ points means too many events
Not for American Football
Scoring isn't truly independent (drives)
Tennis Has Issues
Variable game lengths break the model
Sport Suitability for Poisson Modeling
| Sport | Avg Events/Game | Poisson Suitable? | Notes |
|---|---|---|---|
| Soccer | 2.5-2.8 goals | Excellent | Ideal use case |
| Hockey (NHL) | 5.5-6.5 goals | Very Good | Works well, slightly higher variance |
| Baseball | 8-9 runs | Moderate | Can work for totals, pitching-dependent |
| NFL | 45-50 points | Poor | Too many scoring events, not independent |
| NBA | 220+ points | Poor | Far too many events |
The Poisson Formula Explained
Don't be intimidated by the math. The Poisson formula is straightforward once you understand what each part means:
P(k) = (λk × e-λ) ÷ k!
Probability of exactly k events, given average rate λ
Breaking Down Each Component
Probability of k events
The output - the probability that exactly k goals will be scored. For example, P(2) gives you the probability of exactly 2 goals.
Lambda (λ) - Expected Goals
The average number of goals expected. If a team averages 1.8 goals per game, λ = 1.8. This is the most important input to get right.
Euler's Number (e ≈ 2.71828)
A mathematical constant. You don't need to understand why it's here - just know it's approximately 2.71828. Your calculator handles this automatically.
k Factorial
k multiplied by every integer below it. So 3! = 3 × 2 × 1 = 6. And 0! = 1 by definition. This adjusts for the number of ways k events can occur.
Example: Probability of Exactly 2 Goals
Let's say a team has expected goals (λ) of 1.5. What's the probability they score exactly 2?
P(2) = (1.52 × e-1.5) ÷ 2!
= (2.25 × 0.2231) ÷ 2
= 0.502 ÷ 2
Building Your First Poisson Model
The Poisson distribution only tells you probabilities given an expected goal rate. The real skill is calculating accurate expected goals for each team. Here's the standard approach:
The 4-Step Process
Calculate Attack Strength
Team's goals scored ÷ League average goals
Calculate Defense Strength
Team's goals conceded ÷ League average goals
Calculate Expected Goals
Combine attack strength vs opponent's defense
Away xG = Away Attack × Home Defense × League Avg
Apply Poisson Distribution
Calculate probabilities for each possible scoreline
Complete Soccer Prediction Example
Let's predict the outcome of Arsenal (home) vs Chelsea using Premier League data.
Input Data (Season Averages)
Arsenal (Home)
Goals Scored: 2.1/game
Goals Conceded: 0.9/game
Chelsea (Away)
Goals Scored: 1.7/game
Goals Conceded: 1.3/game
League Average
Goals/Team/Game: 1.4
Step-by-Step Calculation
Arsenal Attack Strength
2.1 ÷ 1.4 = 1.50
Arsenal Defense Strength
0.9 ÷ 1.4 = 0.64
Chelsea Attack Strength
1.7 ÷ 1.4 = 1.21
Chelsea Defense Strength
1.3 ÷ 1.4 = 0.93
Arsenal Expected Goals (Home)
1.50 × 0.93 × 1.4 × 1.1 = 2.15 xG
Chelsea Expected Goals (Away)
1.21 × 0.64 × 1.4 = 1.09 xG
Poisson Output: Match Probabilities
58.2%
Arsenal Win
21.4%
Draw
20.4%
Chelsea Win
Most Likely Scorelines
2-1
14.8%
2-0
12.3%
1-1
11.2%
3-1
10.6%
1-0
10.1%
2-2
6.1%
Applying Poisson to Hockey
The Poisson model works excellently for NHL hockey, with a few adjustments for higher scoring rates and overtime/shootout considerations.
Hockey-Specific Adjustments
Higher Scoring = Higher Lambda
NHL averages ~3.0 goals per team per game vs soccer's ~1.4. The model still works, just use higher expected goals.
Account for Overtime
NHL games can't end in a draw. Calculate 60-minute probabilities, then distribute draw probability ~50/50 for moneyline or use 3-way markets for regulation only.
Empty Net Goals
Late-game empty net goals inflate totals. Some models exclude these from historical averages for cleaner predictions.
Goaltender Impact
Starting goaltender changes expected goals significantly. Adjust defense strength based on the starter's save percentage vs league average.
NHL Example: Maple Leafs vs Bruins
Toronto (Home)
Expected Goals: 3.2
Boston (Away)
Expected Goals: 2.8
45.3%
Toronto Win (Reg)
18.9%
Overtime
35.8%
Boston Win (Reg)
Finding Betting Value with Poisson
The Poisson model's real power isn't just predicting winners - it's finding value bets where the sportsbook's odds don't reflect true probabilities.
Converting Probabilities to Fair Odds
| Market | Your Probability | Fair Decimal Odds | Fair American Odds |
|---|---|---|---|
| Arsenal Win | 58.2% | 1.72 | -138 |
| Draw | 21.4% | 4.67 | +367 |
| Chelsea Win | 20.4% | 4.90 | +390 |
| Over 2.5 Goals | 61.8% | 1.62 | -161 |
| Score 2-1 | 14.8% | 6.76 | +576 |
Identifying a Value Bet
Your model says: Over 2.5 goals has 61.8% probability (fair odds: -161)
Sportsbook offers: Over 2.5 at -130 (implies 56.5%)
Edge: 61.8% - 56.5% = +5.3% edge
When your probability is higher than the implied probability of the odds, you have positive expected value. Use our Expected Value Calculator to quantify this precisely.
Model Limitations to Know
What Poisson Doesn't Capture
Team motivation - Cup finals, relegation battles, nothing-to-play-for scenarios
Injuries & suspensions - Key player absences change expected output significantly
Tactical matchups - How team styles interact specifically
In-game events - Red cards, early goals changing game state
Known Biases
Underestimates 0-0 draws - Basic Poisson slightly underpredicts scoreless matches
Goal dependence - Goals in a match aren't truly independent (team that scores may sit back)
Sample size issues - Early season data can be unreliable
Advanced Techniques
Improving Your Poisson Model
1. Use Expected Goals (xG) Instead of Actual Goals
xG measures shot quality, not just outcomes. Teams with high xG but low actual goals are due for regression upward. Use xG from sites like FBref or Understat.
2. Weight Recent Form More Heavily
Instead of season averages, use weighted averages that prioritize last 5-10 matches. Teams evolve throughout a season.
3. Separate Home/Away Data
Some teams have massive home/away splits. Calculate attack and defense strength separately for each venue.
4. Dixon-Coles Adjustment
An academic adjustment that corrects for the 0-0, 1-0, 0-1, and 1-1 scoreline biases in basic Poisson. Adds ~1-2% accuracy for correct score markets.
5. Time-Decay Weighting
Apply exponential decay so matches from 2 weeks ago count more than matches from 2 months ago. Typical half-life: 30-50 days.
Frequently Asked Questions
What is the Poisson distribution in sports betting?
The Poisson distribution is a statistical model that calculates the probability of a specific number of events (like goals) occurring in a fixed time period. In sports betting, it's used to predict soccer and hockey scores based on historical scoring data, helping bettors identify value in match odds, totals, and correct score markets.
Why does Poisson work for soccer and hockey but not football or basketball?
Poisson works best for low-scoring sports where goals are relatively rare, independent events. Soccer (2-3 goals per match) and hockey (5-6 goals per match) fit this criteria. Basketball (200+ points) and American football (40-50 points) score too frequently for Poisson to be accurate, and scoring in those sports is less independent.
How accurate are Poisson predictions for soccer betting?
Basic Poisson models can achieve 50-55% accuracy on match outcomes, which is competitive with sportsbook odds. Advanced models that incorporate factors like home advantage, team form, injuries, and expected goals (xG) data can improve accuracy further. The key is using Poisson probabilities to identify value rather than just predicting winners.
What data do I need to build a Poisson model?
At minimum, you need: average goals scored and conceded per game for each team, and the league average goals per team per game. For better accuracy, add: home/away splits, recent form (last 5-10 matches), head-to-head records, and expected goals (xG) data if available.
Can I use Poisson for live betting?
Yes, but you need to adjust the model for remaining time and current score. Reduce the expected goals proportionally to time remaining, then recalculate probabilities. For example, at halftime with 0-0, calculate expected goals for just the second half based on your original rates.
Ready to Build Your Model?
Use our Poisson Goal Predictor to start making data-driven predictions. Enter team statistics and instantly see match outcome probabilities, expected scorelines, and over/under odds.