Poisson Distribution Calculator
Find the probability of k events given an average rate.
Results update as you type.
How it works
The Poisson distribution models the number of times an event happens in a fixed interval of time or space when events occur independently at a known average rate, and this calculator finds the probability of any specific count. You enter the average rate, often called lambda, which is the mean number of events expected, and the number of events you want the probability for; it returns that probability. Classic examples include the number of customers arriving at a shop per hour, emails received per day, typos per page, radioactive decays per second, or goals scored in a match. The distribution assumes events are independent and the average rate is constant over the interval. For an average of two events, the chance of seeing exactly three is about 18 percent. The Poisson distribution is the limiting case of the binomial distribution when trials are many and the success probability is tiny, which is why it describes rare events well. It is widely used in queueing theory, reliability engineering, insurance, telecommunications, and the sciences. The mean and the variance of a Poisson distribution are both equal to lambda. Enter the average rate and the number of events to compute the probability.
P(k) = e^(−λ) · λᵏ / k!
The Poisson probability of exactly k events equals e raised to the negative average rate, times the rate raised to the power k, divided by k factorial. The average rate lambda is the mean number of events expected in the interval. The distribution applies when events occur independently at a constant average rate, and it gives the chance of observing any specific count in that interval.
Source:Probability theory (Poisson distribution)
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Worked example
How to use this calculator
- 1
Enter your values
Enter average rate (λ) and number of events (k).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Probability of exactly k updates instantly as you type — no button to press.
Frequently asked questions
What is the Poisson distribution?
The Poisson distribution gives the probability of a given number of events occurring in a fixed interval of time or space, when the events happen independently at a constant average rate. It is defined by a single parameter, the average rate lambda. It is ideal for modelling counts of relatively rare, random events such as arrivals, failures, or occurrences per unit.
What is lambda in the Poisson distribution?
Lambda is the average rate, the mean number of events expected in the interval you are considering. It is the only parameter the Poisson distribution needs. Interestingly, both the mean and the variance of a Poisson distribution equal lambda, so a higher average rate also means more variability in the number of events you might observe.
When should I use a Poisson distribution?
Use it to model the count of independent events over a fixed interval when you know the average rate, such as calls to a call centre per minute, defects per batch, or accidents per month. The events should be independent and occur at a steady average rate. It is especially suited to rare events where each occurrence is unlikely but many opportunities exist.
How does it relate to the binomial distribution?
The Poisson distribution is the limiting form of the binomial distribution when the number of trials is very large and the probability of success on each is very small, with their product held at lambda. This makes Poisson a convenient approximation for binomial problems involving many trials and rare successes, and it explains why it models rare events so well.
Can the number of events be large?
Yes, the Poisson distribution allows any non-negative whole number of events, though the probability of counts far above the average rate becomes very small. The calculator computes the exact probability for whatever count you enter. For very large counts and rates, the Poisson distribution approaches a normal shape, which can be used as an approximation.