Poisson Distribution Calculator
Our free binary calculator solves poisson distribution problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
P(X=k) = (λ^k × e^(-λ)) / k!
The Poisson distribution gives the probability of observing exactly k independent events in a fixed interval, given that events occur at a known average rate λ. Mean and variance are both equal to λ.
Worked Examples
Example 1: Customer arrivals at a coffee shop
Problem:A coffee shop averages 4 customers every 10 minutes (λ = 4). What is the probability that exactly 6 customers arrive in the next 10 minutes?
Solution:P(X=6) = (4⁶ × e⁻⁴) / 6! = (4096 × 0.0183) / 720 ≈ 0.1042.
Result:P(X=6) ≈ 10.42%
Example 2: Rare defects in manufacturing
Problem:A factory finds an average of 1.5 defective units per 1,000 produced (λ = 1.5). What is the probability of finding 0 defects in the next batch of 1,000?
Solution:P(X=0) = (1.5⁰ × e⁻¹·⁵) / 0! = 1 × 0.2231 / 1 = 0.2231.
Result:P(X=0) ≈ 22.31% chance of a defect-free batch
Frequently Asked Questions
What kind of real-world situations does the Poisson distribution model?
The Poisson distribution models the count of independent events that occur at a known constant average rate within a fixed interval of time, distance, or space — customer arrivals at a store per hour, typos per page in a manuscript, radioactive decay events per second, server requests per minute, or potholes per mile of road. It's the natural choice whenever events happen randomly and independently, without a fixed upper limit on how many could occur.
How does the Poisson distribution relate to the binomial distribution?
The Poisson distribution is the limiting case of the binomial distribution when the number of trials n becomes very large and the success probability p becomes very small, while their product n×p stays fixed at λ. This is why the Poisson approximation works well for 'rare event' binomial problems, such as estimating defect counts in a large manufacturing batch where each individual item has a tiny defect probability.
Why is the variance of a Poisson distribution always equal to its mean?
This is a defining mathematical property of the Poisson distribution, derived directly from its formula — both the mean and variance equal λ. This unique 'mean equals variance' relationship is often used as a diagnostic check: if real-world count data has a variance much larger than its mean (called overdispersion), a Poisson model may not fit well, and a Negative Binomial distribution is often used instead.
How is the Poisson distribution used in queueing theory and staffing decisions?
Call centers, hospital emergency rooms, and web servers all use Poisson models to predict how many arrivals to expect in a given time window, which then feeds into staffing formulas (like the Erlang C formula) to determine how many staff or servers are needed to keep wait times acceptably short during that period.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy