Probability
Poisson Event Probability Calculator
Calculate Poisson probability masses, mean and standard deviation, compare observed with expected count, value expected impact, and test an alternative event rate and mitigation scenario.
Poisson mean lambda-
Probability of exactly zero events-
Probability of exactly one event-
Probability of exactly two events-
Probability of three or more events-
Poisson standard deviation-
Observed count minus model mean-
Expected impact before mitigation-
Impact per event after mitigation-
Residual expected impact plus fixed mitigation cost-
Expected impact less mitigated total cost-
Expected events at alternative rate-
Baseline minus comparison expected events-
Decision view
Poisson probability mass and event-count reference
Poisson probability mass and event-count referenceThe exact zero-, one-, and two-event masses sum with the remaining three-or-more mass, while the mean, standard deviation, observed count, and alternative-rate mean provide count references.
Exact scenario comparisonAverage events per exposure unit changes while all other entered assumptions remain constant.
| Average events per exposure unit | Poisson mean lambda | Probability of exactly zero events | Probability of exactly one event | Probability of exactly two events | Probability of three or more events | Poisson standard deviation | Observed count minus model mean | Expected impact before mitigation | Impact per event after mitigation | Residual expected impact plus fixed mitigation cost | Expected impact less mitigated total cost | Expected events at alternative rate | Baseline minus comparison expected events |
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How to use Poisson Event Probability Calculator
- Use a stable average event rate per exposure unit.
- Enter exposure and an observed count for comparison.
- Add impact, mitigation, fixed cost, and an alternative rate scenario.
Calculator guide
Understanding Poisson Event Probability Calculator
A Poisson count model starts with one mean event count lambda, not a percentage per trial. This calculator derives lambda from rate and exposure and displays exact probability mass for zero, one, two, and three-or-more events.
Detailed calculation process
Detailed Poisson probability calculation
The default case uses lambda=2 and reconciles the entire probability mass.
General formula: lambda=rxP(X=k)=exp(-lambda)lambda^k/k!P(X>=3)=1-P0-P1-P2sigma=sqrt(lambda)E[I]=lambda c The same lambda determines every exact-count probability, the mean, variance, and expected impact.
What each symbol means
Worked substitution with the default inputs
The four displayed probability categories sum to 100%, and the alternative rate gives 0.05*25=1.25 expected events.
Worked situations
Practical examples
- A rate of 0.08 across 25 units gives lambda=2 expected events.
- At lambda=2, zero, one, and two each have explicit probability mass and the remaining mass is three or more.
Better inputs
Useful tips
- Match the rate denominator to the exposure units.
- Check for clustering or changing rates.
- Do not interpret expected count as a guaranteed observed count.
Before relying on the result
Limitations and common mistakes
- Counts must be independent at a stable average rate.
- Overdispersion and seasonality are not modeled.
- Impact per event is treated as constant.
Reference
Key terms
- Lambda
- Poisson mean and variance for the selected exposure window.
- Probability mass
- Probability assigned to one exact integer count.
- Tail probability
- Combined probability of counts at or beyond a threshold.
Important note
Validate the stable-rate and independence assumptions before using the probabilities for operational or safety decisions.
Frequently asked questions
Why is standard deviation the square root of lambda?
The Poisson distribution has variance lambda.
Do the four displayed probabilities sum to 100%?
Yes, subject only to display rounding.
When is Poisson inappropriate?
Clustering, varying rates, excess zeros, or overdispersion call for another count model.