How to Calculate a Poisson Probability

For rare events happening λ times on average, calculates the probability of exactly k occurrences as e^(−λ) × λ^k ÷ k!. It fits counts such as customers per hour or accidents per day.

Explanation

The Poisson distribution describes rare events occurring at a steady average rate, and gives the probability of exactly $k$ of them.

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

Here $\lambda$ is the average number of events in the window you are looking at. A Poisson distribution has the unusual property that its mean and its variance are both $\lambda$ .

Example

A shop receives 3 customers an hour on average. What is the chance that exactly 2 arrive in a given hour, so $\lambda = 3$ and $k = 2$ ?

P(X = 2) = \dfrac{e^{-3} \times 3^2}{2!} = \dfrac{0.049787 \times 9}{2} = 0.2240

About 22.4%.

Adding the cases of 0, 1 and 2 customers gives a 42.3% chance of two or fewer.

When it applies

The average rate $\lambda$ stays constant
Events are independent, so one occurrence does not make another more likely
Two events cannot occur at exactly the same instant

Where it is used

Customers arriving per hour, calls reaching a switchboard
Accidents per day, machine failures per week
Typos per page, weeds per square metre

When a binomial distribution has a very large $n$ and a very small $p$ , with a moderate $np = \lambda$ , it converges to the Poisson distribution. It is the law of "many opportunities, each unlikely".