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Poisson Distribution Calculator

Model event probabilities for totals markets using the Poisson distribution.

Inputs

Probabilities

P(X = 3) — Exactly 3 22.0468%
P(X < 3) — Less than 3 49.3624%
P(X ≤ 3) — At most 3 71.4092%
P(X > 3) — More than 3 28.5908%
P(X ≥ 3) — At least 3 50.6376%
Fair Odds: Over 2.5 -103
Fair Odds: Under 2.5 +103

Probability Distribution

k = 0
6.7206%
k = 1
18.1455%
k = 2
24.4964%
k = 3
22.0468%
k = 4
14.8816%
k = 5
8.0360%
k = 6
3.6162%
k = 7
1.3948%
k = 8
0.4708%
k = 9
0.1412%
k = 10
0.0381%

How It Works

The Poisson distribution models the probability of a given number of events occurring in a fixed interval, given a known average rate. It is widely used in sports betting for totals markets (goals, runs, points).

The probability mass function is: P(X = k) = (e^(-λ) × λ^k) / k!, where λ is the expected average and k is the target value.

To avoid numerical overflow with large values, this calculator uses iterative computation: start with P(0) = e^(-λ), then P(k) = P(k-1) × λ / k.

Fair odds are derived by converting cumulative probabilities to American odds format. Compare these fair odds against the sportsbook's posted lines to find value.

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