Download the appBy Outlier · Updated
About the Poisson Calculator
The Poisson model estimates how often a count, such as the number of goals, will occur. Enter the expected average count for a set period to see exact-count and over/under chances.
How to use this calculator
- Enter the expected average count for the period you want to study.
- Choose a count to see the chances of exactly, at least, or at most that many events.
- Or choose an over/under line to see win, loss, and push chances.
- Read the estimated chances and fair odds for each result.
Formula and definitions
- Chance of exactly k events: P(X = k) = e^(−λ) × λᵏ ÷ k!.
- Chance of at most k events: add the exact-count chances from 0 through k.
- Chance of at least k events: 1 − chance of at most (k − 1) events.
- Fair decimal odds with pushes = (1 − push chance) ÷ win chance.
- λ (lambda)
- The expected average count for the chosen period.
- X and k
- X is the event count. k is the count you want to check.
- e
- A mathematical constant, about 2.71828.
- Factorial (!)
- k! multiplies the whole numbers from 1 through k. For example, 3! = 6; 0! = 1.
- Fair odds
- Odds based on the estimated chance with no sportsbook margin.
Worked example
- Set the expected average count to 2.5.
- The chance of exactly 3 events is about 21.38%.
- The chance of at least 3 is 45.62%. The chance of at most 3 is 75.76%.
- Over 2.5 needs at least 3 events, so its win chance is also 45.62%.
How to read your result
Exactly, at least, and at most can include the same results. Do not add those percentages together.
The over/under view separates wins, losses, and pushes. Its fair odds have no sportsbook margin.
Assumptions and limitations
- The model assumes independent events, a steady average rate, and variance equal to that average. Variance measures how spread out the counts are.
- Changing playing time, opponents, linked events, or more zero counts than the model expects can make the estimate unreliable.
- Average counts can range from 0 to 200; event counts go up to 1,000. Very small chances may round to 0.00%.
Common mistake to avoid
Over 3 needs 4 or more events to win. Exactly 3 is a push. By contrast, at least 3 includes exactly 3 as a success.
Put your result to work
Try a related calculator, or read our sports betting strategy guides to learn more.
- Expected Value Calculator — Average profit or loss
- Implied Probability Calculator — Break-even win chance
- No-Vig Fair Odds Calculator — Odds without sportsbook margin
Poisson FAQs
What is lambda in a Poisson calculator?
Lambda is the expected average count for the period you are studying. Use an average for that same period. A full-game average should not be used unchanged for one quarter.
How do whole-number over/under lines work?
If the count equals the line, the bet pushes. The fair odds allow for that stake refund. A half-point line cannot push when the count is a whole number.
How results are calculated
Results use the formulas and assumptions above. Money is rounded to cents. Odds and percentages are rounded to make them easier to read.
The calculator uses the odds and win chances you enter. It does not load live odds.