Poisson Calculator

Understanding the Poisson Distribution Formula

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval when these events happen with a known average rate and independently of the time since the last event. The probability mass function is:

P(X = k) = (λk × e-λ) / k!

Where:
- λ (lambda) = expected average rate of occurrence
- k = number of occurrences we're calculating probability for
- e = Euler's number (approximately 2.71828)
- k! = k factorial (k × (k-1) × (k-2) × ... × 1)

Breaking down the components:

• λ^k: Represents the expected frequency scaled by the number of events
• e^-λ: Normalization factor ensuring all probabilities sum to 1
• k!: Accounts for the ordering of events being irrelevant

Worked Example - Soccer Goals: If a team averages λ = 1.8 goals per match, what's the probability they score exactly k = 2 goals?

P(X = 2) = (1.82 × e-1.8) / 2!
         = (3.24 × 0.1653) / 2
         = 0.5356 / 2
         = 0.2678 or 26.78%

Fair Decimal Odds = 1 / 0.2678 = 3.73
Fair American Odds = +273

Real-World Sports Betting Applications

The Poisson distribution excels at modeling low-frequency, independent events in sports. Here are common applications with typical λ values:

Soccer (Football) - Total Goals:

• Premier League average: λ = 2.7 goals per match
• Strong attacking team vs weak defense: λ = 3.2 goals
• Defensive matchup: λ = 1.8 goals

Example: Liverpool vs Brighton, expected 3.1 total goals
Calculate P(Over 2.5 goals) = P(X ≥ 3)

P(X ≥ 3) = 1 - [P(X=0) + P(X=1) + P(X=2)]
         = 1 - [(3.10×e-3.1)/0! + (3.11×e-3.1)/1! + (3.12×e-3.1)/2!]
         = 1 - [0.0450 + 0.1395 + 0.2163]
         = 1 - 0.4008
         = 0.5992 or 59.92%

Fair Decimal Odds = 1.67

Baseball - Home Runs:

• MLB team average: λ = 1.3 home runs per game
• Power-hitting lineup vs pitcher in hitter-friendly park: λ = 2.1

Example: Yankees expected to hit 2.0 home runs
P(Exactly 3 home runs) = (2.03 × e-2.0) / 3!
                       = (8.0 × 0.1353) / 6
                       = 0.1804 or 18.04%

Fair American Odds = +454

Soccer - Corner Kicks:

• Average match: λ = 10.5 corners per game
• Attacking team vs defensive team: λ = 11.8 corners

Hockey - Goals:

• NHL average: λ = 6.0 total goals per game
• High-scoring teams: λ = 6.8 goals

Basketball - Three-Pointers (Single Player):

• Sharpshooter prop: λ = 3.5 three-pointers made

When Poisson Works and When It Breaks Down

The Poisson distribution relies on two critical assumptions that must be evaluated for each betting application:

Assumption 1: Independence of Events

Events must occur independently—the occurrence of one event doesn't affect the probability of subsequent events. This assumption holds well for:

• Soccer goals in most situations (each possession is relatively independent)
• Baseball home runs by a team over a full game
• Corner kicks in soccer

The independence assumption breaks down when:

• Score effects: Leading teams become more defensive, trailing teams more aggressive (especially late in soccer matches)
• Red cards: A sending-off dramatically changes goal probabilities
• Momentum shifts: In basketball, hot shooting streaks violate independence

Assumption 2: Constant Average Rate

λ must remain constant throughout the interval. In reality, λ can fluctuate due to:

• In-game adjustments: Teams change tactics mid-game
• Weather conditions: Wind/rain reduce scoring rates over time
• Fatigue: Player stamina affects outcome rates late in games

When these assumptions fail dramatically, apply adjustments or switch models (e.g., bivariate Poisson for correlated team totals).

Converting Probabilities to Fair Odds

Once you've calculated Poisson probabilities, convert them to odds formats for comparison with sportsbook lines. Understanding this conversion is essential for identifying value bets.

Decimal Odds Conversion:

Decimal Odds = 1 / Probability

Example: P(X = 3) = 0.2240 (22.40%)
Decimal Odds = 1 / 0.2240 = 4.46

Payout on $100 bet = $100 × 4.46 = $446
Profit = $446 - $100 = $346

American Odds Conversion:

If Probability > 50% (Favorite):
American Odds = -(Probability / (1 - Probability)) × 100

If Probability ≤ 50% (Underdog):
American Odds = ((1 - Probability) / Probability) × 100

Example 1 - Underdog:

P(Exactly 4 goals) = 0.1680 (16.80%)
Since 16.80% < 50%:
American Odds = ((1 - 0.1680) / 0.1680) × 100
              = (0.8320 / 0.1680) × 100
              = +495

Interpretation: Fair odds are +495, so +450 or lower is -EV, +500 or higher is +EV

Example 2 - Favorite:

P(At least 1 goal) = 0.9298 (92.98%)
Since 92.98% > 50%:
American Odds = -(0.9298 / (1 - 0.9298)) × 100
              = -(0.9298 / 0.0702) × 100
              = -1325

Interpretation: Fair odds are -1325, so -1300 or longer (less negative) is +EV

Practical Workflow for Poisson Calculator Usage

Follow this systematic process to use the Poisson calculator effectively for sports betting:

Step 1: Determine Expected Rate (λ)

• Historical averages: Team/player average over last 10-20 games
• Matchup adjustments: Opponent strength, pace, defensive metrics
• Situational factors: Home/away splits, weather, injuries, fatigue
• Market implied: Reverse-engineer from betting lines if available

Example: Calculating λ for Manchester City goals
- City averages 2.3 goals per game (home)
- Opponent allows 1.8 goals per game (away)
- League average: 1.5 goals per team per game

Adjusted λ = 2.3 × (1.8 / 1.5) = 2.76 goals

Step 2: Identify the Proposition

• Exact prop: "Team scores exactly 2 goals" → use k = 2, exact probability
• Over/Under: "Over 2.5 goals" → use k = 3, at least probability
• Under: "Under 1.5 goals" → use k = 1, at most probability
• Range: "Between 2-3 goals" → P(X=2) + P(X=3)

Step 3: Calculate Fair Probability and Odds

Input λ and k into the calculator. For our City example with λ = 2.76:

P(Exactly 2 goals) = 25.13% → Decimal 3.98 / American +298
P(At least 2 goals) = 67.78% → Decimal 1.48 / American -209
P(At most 2 goals) = 57.35% → Decimal 1.74 / American -135

Step 4: Compare with Sportsbook Lines

Sportsbook offers: Man City "Exactly 2 goals" at +350
Fair odds: +298
Implied probability at +350 = 22.22%
Your calculated probability = 25.13%

Edge = 25.13% - 22.22% = +2.91% (positive EV!)

Step 5: Validate Assumptions

• Is the match likely to remain competitive? (independence check)
• Are there late-breaking news items affecting λ?
• Does the sport/prop suit Poisson? (avoid basketball totals, etc.)
• Is your edge large enough to overcome model uncertainty? (recommend ≥3% edge)

Step 6: Size Your Bet

Using Kelly Criterion with edge = 2.91%, odds = +350:
Kelly % = (0.2513 × 4.5 - 1) / 3.5 = 0.037 or 3.7% of bankroll

Conservative fractional Kelly (0.25x) = 0.9% of bankroll
On $10,000 bankroll = $90 bet