About the Poisson Distribution Calculator
The Poisson distribution models the number of times something happens in a fixed period or space when the events are rare and independent: calls to a help desk per hour, typos per page or cars arriving at a toll booth per minute. Give the average number of events, λ, and a count k, and this calculator returns the probability of exactly k events and of fewer or more than k. Type the average rate λ and the count k and the Poisson probability appears at once, with a few related figures underneath, and you can copy the result with one tap. The sums run in your browser, so nothing you enter is uploaded.
Updated October 2026
How to use the Poisson Distribution Calculator
- 1Enter the average rate λ and the count k.
- 2Pick the units or options from the menus if the Poisson probability tool offers them.
- 3Read the Poisson probability in the large result box.
- 4Look at the extra rows for related figures.
- 5Press Copy result to keep the answer.
When you need the Poisson probability
Businesses use it for staffing and stock, hospitals for arrivals, quality teams for defects per unit, and insurers for claims. Students use it as the classic example of a distribution for counts.
The mean and the variance of a Poisson distribution are both λ. When λ is large, the shape looks like a bell curve. The distribution is a good approximation of the binomial when there are many trials and each has a small chance.
How the Poisson probability is worked out
P(X = k) = λᵏ × e^−λ ÷ k!. The cumulative probabilities add up P(X = 0), P(X = 1) and so on. If a shop gets 3 customers a minute on average, the probability of exactly 2 in a minute is 22.4 percent, the chance of 2 or fewer is 42.3 percent, and the chance of more than 2 is 57.7 percent. For λ = 4 and k = 6, P(X = 6) is 10.4 percent.
Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.
- Exact, at most, at least
- Complement probabilities
- Rates up to 700
Mistakes to avoid with Poisson probability
The model assumes that events happen at a steady average rate and independently of each other. Rush hours, or events that cluster, break these assumptions. The tool accepts λ up to 700 and k up to 2,000.
On phone and computer
On a phone, type into the boxes one after another and read the Poisson probability underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the Poisson probability move. The poisson distribution calculator works the same on both. Poisson counts events in a fixed time, and other models fit other cases. Try the Binomial Distribution Calculator for fixed trials, the Normal Distribution Calculator for a bell curve, and the Expected Value Calculator for average results.
Features
- Rare independent events
- Mean equals variance
- Check the steady-rate assumption
- Clear messages when the Poisson probability inputs make no sense
- Copy the result with one tap
- Private: nothing is uploaded
Frequently asked questions
How do I calculate Poisson probability?
P(X = k) = λᵏ × e^−λ ÷ k!. The cumulative probabilities add up P(X = 0), P(X = 1) and so on. If a shop gets 3 customers a minute on average, the probability of exactly 2 in a minute is 22.4 percent, the chance of 2 or fewer is 42.3 percent, and the chance of more than 2 is 57.7 percent. For λ = 4 and k = 6, P(X = 6) is 10.4 percent.
What is the formula for Poisson probability?
P(X = k) = λᵏ × e^−λ ÷ k!. The cumulative probabilities add up P(X = 0), P(X = 1) and so on. The poisson distribution calculator applies it exactly and rounds only the displayed answer to six significant figures.
What does λ mean?
The average number of events in the period you are looking at.
What is the probability of zero events?
e raised to the power of −λ. For λ = 3 it is 5 percent.
When is Poisson not suitable?
When events are not independent, or the rate changes with time, such as arrivals that bunch up at rush hour.
Is this poisson distribution calculator free and private?
Yes. It is free, needs no account and runs in your browser, so the numbers you type are not uploaded or stored.