Poisson Probability Calculator
Calculate exact-count and cumulative probabilities for events with a specified mean per interval. Compare below, at-most, at-least and above probabilities.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Model independent events at a constant mean rate for one fixed interval. Evaluate the Poisson mass in log space and sum the numerically smaller tail by recurrence to reduce cancellation.
P(X=k)=exp(−λ)λ^k/k!; E[X]=variance=λ; standard deviation=√λ
Worked example
For mean 2 events per interval, P(X=3)=0.180447044, P(X≤3)=0.857123460 and P(X≥3)=0.323323584.
Assumptions and input details
Mean is 0–10000; observed count is an integer 0–100000. At mean zero, zero events has probability one. The model assumes independent events with a constant rate and no simultaneous compound arrivals; it does not establish that a real process satisfies those assumptions. Mean and observed count must refer to the same time, area or other interval. Floating-point probabilities may round or underflow to zero for extreme tails. The graph/table show up to 103 sampled integer counts near the mean and omit the rest; cumulative results use the full distribution, not just displayed samples.
References: NIST Poisson distribution
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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