Poisson Probability Calculator
Estimate the chance of observing a particular count in a fixed interval when the average rate is known. The tool shows the method and key assumptions so you can inspect how the input values lead to the estimate.
How to use this calculator
- Enter expected events per interval (λ) using the unit and definition shown beside the input.
- Enter exact event count (k) using the unit and definition shown beside the input.
- Check that all values refer to the same time period and scenario.
- Review the outputs alongside the assumptions before using them in a decision.
Formula and assumptions
P(X=k)=e⁻λ λᵏ/k!; cumulative probability is the sum from 0 through k.
- The Poisson model assumes a stable average event rate and independent events in the interval. It may be inappropriate when events cluster or the rate changes over time.
- The estimate is calculated from values you enter. It does not connect to external accounts, obtain live prices, or validate local regulations or engineering specifications.
Worked examples
Worked example
If λ=3.2, this calculator reports both P(X=4) and P(X≤4).
Enter the expected count for one interval and the integer count of interest.
Interpret the model only when its assumptions fit the event process.
Understanding the result
The Poisson model assumes a stable average event rate and independent events in the interval. It may be inappropriate when events cluster or the rate changes over time.
Change one input at a time to see how sensitive the output is to that assumption.
Important notes
The Poisson model assumes a stable average event rate and independent events in the interval. It may be inappropriate when events cluster or the rate changes over time.
Frequently asked questions
What formula does the Poisson Probability Calculator use?
P(X=k)=e⁻λ λᵏ/k!; cumulative probability is the sum from 0 through k.
What should I verify before relying on the result?
The Poisson model assumes a stable average event rate and independent events in the interval. It may be inappropriate when events cluster or the rate changes over time.
Can I compare different scenarios?
Yes. Change one assumption at a time, keep units and definitions consistent, and save the inputs used for each comparison.
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Related explanations
Learn the method behind the number or compare the terminology before using the calculator.
- How to Calculate Probability — Learn basic probability formulas, complementary probability, and how to calculate simple event chances.
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This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.