The Poisson distribution: how likely is a given number of random events?

The Poisson distribution gives the probability of seeing exactly k events in a fixed interval when events happen independently at a steady average rate λ. If a help desk averages 4 calls an hour, it tells you the chance of a quiet hour with no calls (about 1.8%) or a busy one with eight or more (about 5.1%). I built the Poisson distribution calculator to do the arithmetic, and this guide explains what it is doing.

What the Poisson distribution describes

Plenty of everyday quantities are counts of events in a stretch of time or space: emails per hour, customers per ten minutes, typing errors per page, goals per football match, potholes per kilometre of road. When those events arrive one at a time, independently of each other, at an average rate that stays constant, the count follows a Poisson distribution. It needs only one number, λ (lambda), the average count per interval, and it is both the mean and the variance of the distribution, so the standard deviation is √λ.

The distribution is named after the French mathematician Siméon Denis Poisson, who published it in 1837. One of its most famous early uses came in 1898, when Ladislaus Bortkiewicz showed that the number of Prussian soldiers accidentally killed by horse kicks each year fitted it well: rare, independent, unpredictable events, which is exactly the Poisson pattern.

The formula, step by step

The probability of exactly k events is P(X = k) = λᵏ × e^(−λ) ÷ k!, where e ≈ 2.71828 and k! is k factorial (4! = 4 × 3 × 2 × 1 = 24). Three pieces do the work:

• λᵏ grows with k, favouring larger counts;
• k! grows even faster, so very large counts become unlikely;
• e^(−λ) is a constant that makes all the probabilities add up to exactly 1. It is also the chance of zero events.

Cumulative questions, such as "at most 2" or "at least 8", are answered by adding up the individual probabilities, which is tedious by hand and instant in the calculator.

A worked example: calls per hour

Suppose a support line receives an average of λ = 4 calls per hour. The chance of no calls in a given hour is e^(−4) ≈ 0.0183, or 1.83%. The chance of exactly 4 calls is 4⁴ × e^(−4) ÷ 4! = 256 × 0.0183 ÷ 24 ≈ 19.54%. Exactly 3 calls is also 19.54%; whenever λ is a whole number, k = λ and k = λ − 1 are equally likely and share the peak.

• 2 calls or fewer: 23.81%
• 7 calls or fewer: 94.89%
• 8 calls or more: 5.11%

That last line is the useful one for planning: if staff can handle 8 calls an hour, about 98% of hours will be covered. The rate also scales with the interval. Over half an hour λ becomes 2, so the chance of 30 quiet minutes is e^(−2) ≈ 13.5%; over two hours λ is 8. To compare the long-run average with the cost of each outcome, the expected value calculator takes over, and Expected value: the long-run average of a game, bet or decision explains that step.

When the model applies

The Poisson model is a good fit when four things are roughly true: events occur one at a time; they are independent, so one call does not make the next more likely; the average rate is constant across the interval; and the count has no fixed upper limit. Arrivals at a counter in a steady period, radioactive decays over a short time, and typos in a long document are classic examples. If you only need a single probability from a known chance, the probability calculator may be simpler.

Poisson and the binomial distribution

The binomial distribution counts successes in a fixed number of trials. When the number of trials is large and each success is rare, the binomial is very close to a Poisson distribution with λ = n × p, which is sometimes called the law of rare events. With 1,000 trials at a 0.3% chance each, the binomial probability of exactly 2 successes is 22.42%; the Poisson value with λ = 3 is 22.40%. In practice that means you can use the simpler formula when n is big and p is small.

When it breaks down

Real counts often vary more than the model predicts, a problem called overdispersion. A shop has rush hours, calls cluster after an outage, and goals in a match are not quite independent. A quick check is to compare the mean and variance of your data: in a Poisson process they should be similar. If the variance is much larger, the model will understate the chance of extreme counts, and statisticians usually switch to a negative binomial model. The standard deviation calculator gives you the variance to compare, and Standard deviation explained: what it measures, how it's calculated, and the 68–95–99.7 rule explains what it measures.