Poisson Probability Calculator
Calculate exact, cumulative, tail, and range probabilities for a Poisson random variable.
Tip: use non-negative integers for event counts. For very large λ, results use a normal approximation.
What is the Poisson formula?
The Poisson distribution models how many times an event happens in a fixed interval of time, space, area, or volume when events occur independently at an average rate. It is widely used in operations, quality control, call center planning, network traffic analysis, and reliability engineering.
Core equation
P(X = k) = (e-λ λk) / k!
- X: random number of events in the interval
- λ: expected number of events in that interval
- k: specific number of events (0, 1, 2, ...)
How to use this calculator
Step 1: Enter the average rate (λ)
If you observe about 6 arrivals per hour on average, then λ = 6 for a one-hour interval.
Step 2: Choose probability type
- Exact: probability of exactly k events.
- At most: probability of k or fewer events.
- At least: probability of k or more events.
- Range: probability events fall between a and b (inclusive).
Step 3: Enter event count(s)
Counts must be whole numbers. For range mode, provide both lower and upper bounds.
Example use cases
Customer support tickets
Suppose your team receives an average of 4 tickets per hour. What is the probability of exactly 6 tickets in the next hour? Set λ = 4, choose Exact, and enter k = 6.
Manufacturing defects
If a process averages 1.2 defects per batch, you can estimate:
- Probability of zero defects (first-pass quality)
- Probability of at least 3 defects (high-risk batch)
- Probability that defects stay in an acceptable range
When Poisson is appropriate
- Events happen independently.
- The average rate is roughly constant in the chosen interval.
- Two events cannot occur at exactly the same instant (or the probability is negligible).
- You are counting occurrences, not measuring a continuous value.
Poisson vs. other distributions
Poisson vs. Binomial
Use binomial when you have a fixed number of trials with success/failure outcomes. Use Poisson when you count events over continuous intervals and only know the average rate.
Poisson vs. Normal
For larger λ, Poisson can be approximated by a normal distribution. This calculator automatically switches to a normal approximation at high rates to keep calculations stable.
Interpretation tips
- Multiply probability by 100 for percent interpretation.
- Small probabilities are not impossible events; they are just unlikely in one interval.
- The expected value and variance of a Poisson variable are both λ.
- If your observed variance is much larger than the mean, data may be overdispersed and a different model may fit better.
Quick FAQ
Can λ be a decimal?
Yes. λ is an average and can be any non-negative real number.
Can k be negative or decimal?
No. Event counts are non-negative integers.
What does P(X ≥ k) mean?
It is the chance that the count is at least k, including k itself.
Does interval length matter?
Yes. Always define λ for the same interval you are analyzing. If arrivals are 2 per minute, then for 5 minutes use λ = 10.