Poisson Probability Calculator
Compute exact and cumulative probabilities for a Poisson random variable.
where λ is the average event rate and k is the number of events.
What Is the Poisson Probability Distribution?
The Poisson distribution models the number of times an event occurs in a fixed interval of time, distance, area, or volume, assuming the event happens independently and at a roughly constant average rate.
Common use cases include calls arriving per minute, website errors per day, defects per batch, or traffic accidents at a specific intersection each month. If you know the average rate (λ), the Poisson model helps you estimate the probability of observing 0, 1, 2, or more events.
How to Use This Poisson Calculator
- Enter the average rate λ (lambda).
- Select the probability type you need.
- Enter k (or a and b for a range).
- Click Calculate Probability.
The calculator supports:
- P(X = k): exact probability of exactly k events
- P(X ≤ k): probability of at most k events
- P(X ≥ k): probability of at least k events
- P(a ≤ X ≤ b): probability of a bounded range of counts
Poisson Distribution Formula and Interpretation
Exact probability
The exact probability is:
P(X = k) = e-λ λk / k!
Here, λ is both the mean and the variance of the distribution. That means:
- Expected value: E[X] = λ
- Variance: Var(X) = λ
- Standard deviation: √λ
Why this matters
If your process has an average of 4 events per hour, the Poisson model gives a full probability map for event counts in one hour: the chance of 0 events, 1 event, 2 events, and so on.
Practical Examples
Example 1: Support tickets
Suppose a team receives an average of 6 tickets per hour. You can compute:
- P(X = 10) to evaluate overload risk in a single hour
- P(X ≤ 3) to estimate unusually quiet periods
- P(X ≥ 8) for staffing alert thresholds
Example 2: Manufacturing defects
If defects occur at an average of 1.2 per batch, the Poisson distribution helps estimate the chance of zero-defect batches, or the chance of seeing 2 or more defects in a lot.
When the Poisson Model Is Appropriate
Use a Poisson model when these assumptions are reasonably true:
- Events are independent.
- The average event rate is stable over the interval.
- Two events are unlikely to occur at exactly the same instant.
- You are counting events, not measuring continuous values.
Common Mistakes to Avoid
- Using a changing rate: If λ varies by time (e.g., rush hour vs off-peak), one single Poisson may be inaccurate.
- Using non-integer k: Event counts must be whole numbers.
- Ignoring overdispersion: If variance is much larger than mean, consider negative binomial models.
- Confusing P(X ≥ k): This is a tail probability, not just 1 - P(X = k).
Quick FAQ
What does λ mean?
λ is the average number of events in the chosen interval.
Can λ be zero?
Yes. If λ = 0, then the process produces zero events with probability 1.
Is Poisson only for time?
No. You can use it for space, area, volume, or any fixed interval where random count events occur.
Bottom Line
A Poisson probability distribution calculator is a fast way to quantify event-count risk and expectation. Whether you are working in operations, quality control, analytics, reliability, or finance, it turns average rates into actionable probabilities for planning and decision-making.