Poisson Distribution Calculator: Exact, Cumulative, At Least, and Between Probability
Use this Poisson distribution calculator to calculate exact probability, cumulative probability, at least or at most probabilities, greater-than or less-than tails, and between-count ranges from an average event rate. It works for calls per hour, defects per item, arrivals, rare events, service requests, Excel POISSON.DIST checks, and statistics homework problems.
- Exact probability Calculate Poisson probability P(X = k) for a given average rate and whole-number event count.
- Cumulative tails Find P(X <= k), P(X < k), P(X >= k), or P(X > k) for count-based questions.
- Range probability Calculate the probability that the event count falls between two whole numbers.
- Distribution table Review exact probabilities, cumulative values, mean, variance, standard deviation, and Excel-style outputs.
Calculate Poisson Probability
Enter the average number of events in the interval and the count you care about. The calculator assumes events are independent, the event count is a whole number, and the average rate is stable across the interval.
Inputs
Results
For an average rate of 3 per hour, the probability of exactly 4 events is about 16.80%.
The highlighted bar shows k = 4.
| Formula | P(X = 4) = e^-3 * 3^4 / 4! |
|---|---|
| Probability | P(X = 4) = 0.1680 |
| Interpretation | About 16.80% of intervals would have exactly 4 events. |
Poisson Probability Table
The table shows exact probabilities and cumulative probabilities for the current average rate.
| Events k | P(X = k) | P(X <= k) | P(X >= k) |
|---|
Poisson Distribution Formula
The Poisson formula uses the average event rate and a whole-number count. In this distribution, the mean and variance are both equal to the average rate.
| Exact probability | P(X = k) = e^-lambda * lambda^k / k! |
|---|---|
| Excel or Sheets exact | POISSON.DIST(k, lambda, FALSE) |
| Excel or Sheets cumulative | POISSON.DIST(k, lambda, TRUE) |
| Mean | mean = lambda |
| Variance | variance = lambda |
| Standard deviation | standard deviation = sqrt(lambda) |
Common Mistakes
- Using a Poisson model for events that are not independent.
- Using the wrong time or space interval for the average rate.
- Mixing intervals, such as using a rate per hour when the question asks about a 30-minute period.
- Entering a percentage as the average rate instead of the expected event count.
- Using Poisson when the rate changes heavily across the interval.
- Entering a non-whole number for the event count. Poisson counts must be 0, 1, 2, 3, and so on.
Poisson Distribution Calculator Examples
These examples match common statistics homework wording and can be checked with Excel or Google Sheets using POISSON.DIST.
| Question type | Setup | Answer |
|---|---|---|
| Exactly 4 events when lambda = 3 | P(X = 4), average rate = 3 per hour | 0.1680, or about 16.80% |
| At most 4 events when lambda = 3 | P(X <= 4), average rate = 3 per hour | 0.8153, or about 81.53% |
| At least 4 events when lambda = 3 | P(X >= 4), average rate = 3 per hour | 0.3528, or about 35.28% |
| Between 1 and 5 events when lambda = 3 | P(1 <= X <= 5), average rate = 3 per hour | 0.8663, or about 86.63% |
How to Calculate Poisson Distribution Probability Step by Step
A Poisson probability problem usually gives an average rate, called lambda, and asks for the probability of a whole-number event count. For example, if a call center receives an average of 3 calls per hour, lambda = 3 for one hour.
- Identify lambda, the average number of events in the interval.
- Identify k, the event count you want to calculate.
- Choose exact, at most, at least, greater than, less than, or between probability.
- Use P(X = k) = e^-lambda * lambda^k / k! for exact probability.
- Add exact probabilities together when the problem asks for cumulative or range probability.
- Write the answer as a decimal and percentage, then interpret it in the problem context.
When This Calculator Helps
Statistics homework
Use it when your assignment asks for exactly, at most, at least, fewer than, more than, or between probabilities for event counts over an interval.
Real-world counts
Use it for arrivals, calls, defects, rare events, website errors, accident counts, service requests, or any count process with a stable average rate.
Poisson Distribution FAQs
When should I use the Poisson distribution?
Use it for counts of events in a fixed interval when events happen independently and the average rate is stable, such as calls per hour or defects per page.
What is lambda in a Poisson problem?
Lambda is the expected number of events in the interval. It is also the mean and variance of a Poisson distribution.
What is the difference between exact and cumulative Poisson probability?
Exact probability is one count, such as P(X = 3). Cumulative probability adds multiple counts, such as P(X <= 3) or P(X >= 3).
Is Poisson the same as binomial?
No. Binomial uses a fixed number of trials and a success probability. Poisson models event counts over an interval and uses an average rate.
How do I calculate Poisson distribution probability?
Use the formula P(X = k) = e^-lambda * lambda^k / k!, where lambda is the average number of events in the interval and k is the whole-number event count.
How do I calculate Poisson probability in Excel or Google Sheets?
Use POISSON.DIST(k, lambda, FALSE) for exact probability and POISSON.DIST(k, lambda, TRUE) for cumulative probability P(X <= k).
What does at least mean in a Poisson probability problem?
At least k means P(X >= k). It includes k and all larger event counts. For example, at least 4 means 4, 5, 6, and so on.
Related Statistics Resources
If the problem moves from probability to testing, assumptions, or written interpretation, these pages are good next steps.
Need Help With a Poisson Probability Problem?
Statskan can help you choose the right distribution, calculate the probability, show the formula, and write the interpretation clearly for your statistics homework.