Poisson Probability Calculator

This Poisson probability calculator works out how likely a given number of events is when those events happen independently at a known average rate. You enter the average number of events expected in your interval, written as lambda, and the count k you are interested in, and the tool returns the probability of exactly k events and the cumulative probability of k or fewer events. The Poisson formula is lambda to the power k, times e to the power minus lambda, divided by k factorial. The cumulative figure adds up that probability for every value from zero up to k. The Poisson model fits situations like the number of calls to a help desk in an hour, the number of faults on a length of cable, customers arriving at a counter, or rare disease cases in a region, as long as the events are independent and the average rate is steady. Analysts, operations planners, students and quality engineers use it to set staffing, judge whether a count is unusually high, and model arrivals and defects. Three tips help you apply it well. First, make sure lambda matches the same interval as the count you care about, so if your rate is per day but you want a weekly count, multiply lambda by seven. Second, the distribution is right skewed for small lambda and becomes more symmetric as lambda grows. Third, the cumulative probability is the natural tool for tail questions, since the chance of more than k events is simply one minus the chance of k or fewer. Always confirm the independence assumption holds before trusting the result.

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0.2240
P(X = k)
P(X <= k)0.4232
P(X = k) as %22.40%

P(k) = lambda^k * e^-lambda / k!. Estimate only, not financial or tax advice.

How it works

The tool raises lambda to the power k, multiplies by e to the power minus lambda, and divides by k factorial to get the chance of exactly k events. The cumulative probability sums that formula for every count from zero up to k. Both are decimals between zero and one.

Worked example

With lambda of 3 and k of 2, the exact probability is 3 squared times e to the minus 3, divided by 2 factorial, which is 9 times 0.049787 divided by 2, about 0.2240. Adding the probabilities for 0, 1 and 2 events gives a cumulative probability of 0.4232.

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Frequently asked questions

What is lambda in the Poisson distribution?

Lambda is the average number of events you expect in the chosen interval, such as an hour, a day or a square metre. It is both the mean and the variance of the distribution. The rate must stay roughly constant for the Poisson model to fit.

How do I find the chance of more than k events?

Work out the cumulative probability of k or fewer events, then subtract it from one. For example the chance of more than 2 events is one minus the chance of 0, 1 or 2 events. This tail probability is often what you need for planning capacity.

When does the Poisson model apply?

It applies when events occur independently, one at a time, at a steady average rate over time or space. Examples include call arrivals, equipment faults and rare events in a fixed region. If events cluster or the rate changes a lot, a different model is more suitable.