Statistics

Poisson Probability Calculator

Calculate exact and cumulative Poisson event probabilities.

Last validated: 2026-02-14

Poisson Probability Calculator evaluates probabilities for event counts over fixed intervals with known average rate. It is useful for queueing, defect counts, and arrival modeling. The tool gives exact probabilities for specific count outcomes. Use it when events are independent and rate is approximately stable.

Permalink

Input Pattern

Enter values in the left panel, keep units explicit, run the calculation, then copy or share the result. Invalid fields are highlighted immediately.

Poisson Inputs

Result

P(X = k): 0.1849589735

P(X ≤ k): 0.3208471989

How to use this tool

Enter average event rate and target event count.
Run calculation for exact and cumulative probabilities.
Compare multiple count thresholds for planning decisions.

Worked Example

Auto-generated from the tool's current default or entered inputs.

Example Inputs

Lambda: 3.5
K: 2

Expected Outputs

Lambda: 3.5
K: 2
Exact probability: 0.184959
Cumulative probability: 0.320847

Interpretation

Lambda is 3.5. Use this as a decision input, not as a standalone conclusion.
K is 2. Use this as a decision input, not as a standalone conclusion.
Exact probability is 0.18495897346170082. Use this as a decision input, not as a standalone conclusion.
Interpret this value alongside sample quality and assumptions to avoid overconfidence.

Scenario Compare (A vs B)

Use this to compare two input sets and quantify change in key outputs.

Confidence and limitations

Outputs are deterministic formula results and depend entirely on input quality and units.
Results are for planning and learning; verify with domain standards before safety-critical use.
Interpretation depends on assumptions (distribution, independence, sample quality).

Formula References

Frequentist estimation and hypothesis-test formulas used in standard textbooks.
Interpretation depends on sample quality and assumptions.

Assumptions

Outputs are deterministic formula results and depend entirely on input quality and units.
Results are for planning and learning; verify with domain standards before safety-critical use.
Interpretation depends on assumptions (distribution, independence, sample quality).

Explore more versions

Tailored guides for specific audiences, regions, and scenarios.

core