Poisson Models for Football Betting (Step‑by‑Step)

Poisson Models for Football Betting (Step-by-Step)

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What Is the Poisson Distribution?

The Poisson distribution is a probability model that estimates how often an event occurs within a fixed interval — like goals in a football match.

It assumes events (goals) happen independently and at a known average rate (λ, lambda).

In simple terms:

If you know how many goals a team scores on average, you can use Poisson to estimate how likely they are to score 0, 1, 2, 3… goals in a given match.

Formula

P(x;λ)=e−λλxx!P(x; λ) = \frac{e^{-λ} λ^x}{x!}P(x;λ)=x!e−λλx​

Where:

• P(x) = probability of x goals
• λ (lambda) = expected goals
• e = Euler’s number (≈ 2.71828)
• x! = factorial of x (e.g., 3! = 3 × 2 × 1)

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Step 1: Collect Team Data

You’ll need at least one season’s worth of results to build averages.

Data Needed	Description
Goals scored at home	Team’s offensive strength
Goals conceded at home	Team’s defensive weakness
Goals scored away	Team’s attack rating on the road
Goals conceded away	Team’s defensive rating on the road

Example (per match averages from Premier League 2023–24):

• Home goals per match = 1.55
• Away goals per match = 1.25

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Step 2: Calculate Attack and Defence Strengths

For each team: Attack Strength=Team’s Avg Goals ScoredLeague Avg Goals (Home or Away)\text{Attack Strength} = \frac{\text{Team’s Avg Goals Scored}}{\text{League Avg Goals (Home or Away)}}Attack Strength=League Avg Goals (Home or Away)Team’s Avg Goals Scored​ Defence Strength=Team’s Avg Goals ConcededLeague Avg Goals (Home or Away)\text{Defence Strength} = \frac{\text{Team’s Avg Goals Conceded}}{\text{League Avg Goals (Home or Away)}}Defence Strength=League Avg Goals (Home or Away)Team’s Avg Goals Conceded​

Example:

• Arsenal average 2.1 home goals vs 1.55 league avg → Attack = 1.35
• Spurs concede 1.3 away vs 1.25 avg → Defence = 1.04

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Step 3: Estimate Expected Goals (λ)

For the home team: λhome=Home Attack Strength×Away Defence Strength×League Avg Home Goalsλ_{home} = \text{Home Attack Strength} × \text{Away Defence Strength} × \text{League Avg Home Goals}λhome​=Home Attack Strength×Away Defence Strength×League Avg Home Goals

For the away team: λaway=Away Attack Strength×Home Defence Strength×League Avg Away Goalsλ_{away} = \text{Away Attack Strength} × \text{Home Defence Strength} × \text{League Avg Away Goals}λaway​=Away Attack Strength×Home Defence Strength×League Avg Away Goals

Continuing the Arsenal v Spurs example:

• Arsenal expected goals: 1.35×1.04×1.55=2.181.35 × 1.04 × 1.55 = 2.181.35×1.04×1.55=2.18
• Spurs expected goals: 1.10×0.90×1.25=1.241.10 × 0.90 × 1.25 = 1.241.10×0.90×1.25=1.24

So our expected goals (λ):

• Arsenal λ₁ = 2.18
• Spurs λ₂ = 1.24

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Step 4: Calculate Goal Probabilities

Now apply the Poisson formula to find the probability of each team scoring 0–5 goals.

Goals (x)	Arsenal (λ = 2.18)	Spurs (λ = 1.24)
0	0.113	0.289
1	0.247	0.358
2	0.269	0.222
3	0.195	0.092
4+	0.176	0.039

These probabilities should sum close to 1.00 (100%).

🧮 Use our Poisson Calculator to get these instantly for any teams.

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Step 5: Build the Score Matrix

Now multiply team probabilities to get joint match outcomes.

Arsenal ↓ / Spurs →	0	1	2	3	4
0	3.3%	4.1%	2.5%	1.0%	0.4%
1	7.4%	9.0%	5.5%	2.2%	0.9%
2	8.0%	9.7%	5.9%	2.4%	1.0%
3	5.8%	7.1%	4.3%	1.8%	0.7%
4	3.3%	4.1%	2.5%	1.0%	0.4%

Sum the diagonal cells (0-0, 1-1, 2-2…) for draw probability.
Sum the upper and lower triangles for home and away win probabilities.

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Step 6: Convert Probabilities to Odds

Fair Odds=1Probability\text{Fair Odds} = \frac{1}{\text{Probability}}Fair Odds=Probability1​

Example:
If Arsenal win probability = 55%,
→ Fair odds = 1 ÷ 0.55 = 1.82

Compare that with bookmaker odds to identify value bets.

🎯 For value detection, use the Expected Value Calculator.

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Step 7: Derive Over/Under and BTTS Probabilities

Add up goal combinations:

• Over 2.5 = all outcomes where total goals ≥ 3.
• Under 2.5 = total ≤ 2.
• BTTS = all outcomes where both teams score ≥ 1.

Example:

• Over 2.5 = 56% (implied odds 1.79)
• BTTS = 60% (implied odds 1.67)

Compare with market prices to see if a bet is mathematically justified.

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Step 8: Test, Track & Refine

The Poisson model is simple but powerful. Its accuracy depends on good data.
You can refine it by:

• Updating stats weekly.
• Including home/away weighting.
• Adjusting for team form or xG data.
• Factoring injuries, red cards, or tactical shifts.

⚙️ Advanced bettors combine Poisson with expected goals (xG) data for more realistic forecasts.

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Poisson Model Limitations

• Assumes independence: real matches have momentum.
• Ignores situational factors: red cards, pressure, tactics.
• Not great for extreme scorelines.
• Better for totals and draws than high-variance markets.

Still, as a baseline, it beats guessing — and it’s transparent and reproducible.

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How Bookmakers Use Poisson Models

Bookmakers also use Poisson-based frameworks (plus proprietary data layers) to price markets, especially for football, tennis, and baseball.

By understanding their approach, you’re effectively reading their pricing language — and can identify when odds diverge from real probabilities.

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Example: Applying Poisson to a Real Match

Let’s say Manchester City host Aston Villa:

• City expected goals: 2.6
• Villa expected goals: 0.9

Compute probabilities:

• City win ≈ 72% (odds 1.39)
• Draw ≈ 17% (odds 5.88)
• Villa win ≈ 11% (odds 9.09)

If a bookmaker offers City at 1.55, that’s slightly longer than your model’s 1.39 — potential value.

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Poisson Model in Practice

Use the model to:

• Identify fair odds across markets.
• Detect mispriced totals or correct scores.
• Manage variance and bankroll effectively.

→ Read next: Bankroll Management for Bettors
→ See also: Value Betting Strategies

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Summary Table

Step	Action	Tool
1	Gather team goal data	Results/Stats feed
2	Calculate attack & defence strengths	Spreadsheet
3	Estimate expected goals (λ)	Formula
4	Apply Poisson distribution	Poisson Calculator
5	Combine outcomes	Matrix table
6	Convert to odds	EV Calculator
7	Compare with bookmaker prices	Odds comparison sites