Poisson Distribution Calculator | Probability Tool

What is a Poisson distribution calculator?

A Poisson distribution calculator finds the probability that a certain number of events happen in a fixed period of time or space, given how often the events happen on average. It is used when events occur one at a time, independently of each other, at a steady average rate. Typical examples include the number of phone calls a call center gets in an hour or the number of typos on a page.

Poisson distribution formula

The Poisson distribution gives the probability of exactly k events when the average rate is λ (the Greek letter lambda):

$P(X = k) = \dfrac{e^{-\lambda} \lambda^{k}}{k!}$

Each symbol means:

• λ (lambda) — the average number of events per interval. This is the only input the distribution needs.
• k — the exact number of events whose probability is being calculated. It must be a whole number, 0 or higher.
• e — Euler's number, a fixed constant equal to about 2.71828.
• k! — "k factorial," the product of all whole numbers from 1 up to k. For example, 4! = 4 × 3 × 2 × 1 = 24. By convention, 0! = 1.

The formula only needs λ and k. It does not need a total number of trials, unlike the binomial distribution.

How to calculate a Poisson probability

1. Choose λ, the average number of events in the interval being studied.
2. Choose k, the exact number of events to find the probability for.
3. Raise e to the power of negative λ: this gives e^(−λ).
4. Raise λ to the power of k: this gives λ^k.
5. Find k!, the factorial of k.
6. Multiply the results of steps 3 and 4, then divide by the result of step 5.

The number that comes out is a probability between 0 and 1. Multiply it by 100 to express it as a percentage.

Worked example

A call center receives an average of 5 calls per hour (λ = 5). What is the probability it receives exactly 3 calls in a given hour (k = 3)?

1. e^(−5) ≈ 0.006738
2. 5^3 = 125
3. 3! = 3 × 2 × 1 = 6
4. Multiply: 0.006738 × 125 ≈ 0.842244
5. Divide by 6: 0.842244 ÷ 6 ≈ 0.140374

The probability of exactly 3 calls in that hour is about 0.140, or 14%.

More examples

Where the Poisson distribution is used

The Poisson distribution models counts of rare, independent events. Common uses include:

• Call centers and customer service: predicting how many calls or chats arrive per hour.
• Manufacturing: estimating the number of defects in a batch of products.
• Insurance: estimating how many claims will arrive in a given period.
• Biology and medicine: modeling mutation counts or the spread of rare diseases.
• Physics: describing radioactive decay, where each atom decays independently at a constant average rate.
• Traffic and network engineering: estimating vehicle arrivals at an intersection or requests hitting a server.

Conditions for using the Poisson distribution

The Poisson distribution only fits a situation when all of these hold:

• Events happen one at a time; two events cannot happen at exactly the same instant.
• Events happen independently — one event does not make the next one more or less likely.
• The average rate, λ, stays constant across the interval being studied.

If the rate changes over time or events tend to cluster together, the plain Poisson distribution will not describe the data well.

Poisson vs. related distributions

• Binomial distribution: used when there is a fixed number of trials and a known probability of success on each one, such as flipping a coin 20 times. The Poisson distribution is what the binomial distribution turns into when the number of trials grows very large and the success probability per trial shrinks toward zero, while their product stays around λ.
• Normal distribution: for large λ, roughly above 20, the Poisson distribution looks close to a bell-shaped normal distribution with mean λ and variance λ.
• Exponential distribution: while the Poisson distribution counts how many events happen in a fixed interval, the exponential distribution describes the waiting time between those events.

A note on the history

French mathematician Siméon Denis Poisson introduced the distribution in 1837, in a book about probability applied to court verdicts. It attracted little attention at first. Its use grew in the early 20th century, after statisticians such as Ladislaus Bortkiewicz and Ronald Fisher applied it to real count data, including a famous study of Prussian soldiers killed by horse kicks.

Frequently asked questions

What does lambda (λ) mean in a Poisson distribution? Lambda is the average number of events expected in the interval being studied. It is also the variance of the distribution, so a bigger λ means both a higher average count and more spread in possible outcomes.

Can k or λ be negative or a fraction? λ must be greater than 0 but can be a decimal, such as 2.5 events per hour. k must be a whole number of 0 or more, since it counts actual events; it cannot be negative or fractional.

How is the Poisson distribution different from the binomial distribution? The binomial distribution needs a fixed number of trials and a fixed success probability per trial. The Poisson distribution needs only an average rate and works for events that could, in principle, happen any number of times.

When should the normal distribution be used instead? For large λ, generally above about 20, a normal distribution with the same mean and variance approximates the Poisson distribution closely and is often easier to work with by hand.

Why does the calculator sometimes show a number in scientific notation? When the true probability is extremely small, such as 0.0000001, the calculator switches to scientific notation (for example, 1.0e-7) instead of showing a rounded value of 0.000000, which would hide how small the result really is.

Does the Poisson distribution allow for more than one event at once? No. The model assumes events happen one at a time. If two events routinely happen together, a different distribution fits the data better.

References

1. Poisson, Siméon Denis. Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Paris, 1837.
2. Haight, Frank A. Handbook of the Poisson Distribution. New York: John Wiley & Sons, 1967.
3. "Poisson Distribution." NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.19. https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm
4. Johnson, Norman L., Adrienne W. Kemp, and Samuel Kotz. Univariate Discrete Distributions. John Wiley & Sons, 2005.