Poisson Probability Calculator
Calculate the chance of observing events over a fixed interval when events happen independently at a steady average rate.
What is a Poisson probability calculator?
A Poisson probability calculator helps you find the likelihood of seeing a specific count of events in a fixed interval. It is commonly used when events are relatively rare, occur independently, and happen at an average rate that stays constant.
In practical terms, this tool answers questions like:
- What is the probability of getting exactly 3 website signups in the next hour?
- What is the chance of seeing at most 1 machine failure this week?
- How likely is 5 or more customer support tickets in a day?
The Poisson formula
The Poisson random variable is usually written as X ~ Poisson(λ), where λ is the average number of events per interval. The probability of exactly k events is:
P(X = k) = (e-λ × λk) / k!
Here:
- λ = expected events in the interval
- k = observed count (0, 1, 2, ...)
- e = Euler’s constant (~2.71828)
How to use this calculator
- Enter λ, your average event rate.
- Select the probability type (exactly, at most, at least, or between).
- Enter the event count value(s) for k.
- Click Calculate Probability.
The calculator returns both the decimal probability and percentage form, plus basic distribution facts (mean and standard deviation).
When Poisson distribution is appropriate
Use it when:
- Events occur one at a time.
- Events are independent of each other.
- The average rate is stable for the interval you care about.
- You are counting occurrences, not measuring magnitudes.
Examples of Poisson use cases
| Domain | Event Count | Typical λ Meaning |
|---|---|---|
| Operations | Machine breakdowns per month | Average breakdowns/month |
| Web Analytics | Clicks per minute | Average clicks/minute |
| Healthcare | Patient arrivals per hour | Average arrivals/hour |
| Customer Support | Tickets per day | Average tickets/day |
Interpretation tips
If your result is, for example, 0.1847, that means there is roughly an 18.47% chance the event count in your selected range occurs under the Poisson model.
Common mistakes to avoid
- Using a negative or non-integer value for k.
- Mixing time units (e.g., λ per day but asking for per hour counts).
- Using Poisson when the event rate changes significantly across the interval.
- Ignoring dependence between events (which violates assumptions).
Quick FAQ
Can λ be a decimal?
Yes. λ is an average, so decimal values are normal (e.g., 2.3 events/hour).
Is Poisson the same as binomial?
No, but Poisson can approximate binomial when trials are many and success probability is small.
What are mean and variance for Poisson?
Both are equal to λ. That is one defining property of the Poisson distribution.