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Poisson Distribution
Common Core: S.MD.1

Poisson Distribution

Author: Sophia Tutorial

This lesson will explain the Poisson Distribution.

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What's Covered

This tutorial is going to teach you a little bit about what's called the Poisson distribution. It's a special case of the binomial distribution. Specifically you will focus on:

  1. Poisson Distribution


This is an example of where you would use the Poisson distribution.

You have a particularly busy intersection. That intersection averages about 4.3 accidents per week. That's not a whole lot of accidents per week for a busy intersection, there's a lot of cars passing through it. The probability of having an accident at that intersection is fairly small. All other things being equal, what's the probability that the intersection experiences a week with just one accident?

This distribution can be solved using something called the Poisson distribution. It's a distribution that works well for rare events. Rare meaning the probability of success is very low, but the number of trials is very high. You end up with, out of a very large number of trials, only a few successes.

Term to Know

Poisson Distribution

A distribution used to calculate the probability of a given number of events happening in a fixed interval when the events occur independently and the average rate of occurrence is known

The Poisson distribution is nice because it doesn't require the use of n and p. All you need is the typical rate of occurrence. In this example, it was the 4.3. If the average number of successes in a given time frame is this letter lambda-- a Greek letter lambda-- and x is the number of potential successes that you could have. The probability that you get exactly k successes is equal to lambda to the k-- so the expected number to the number of successes-- times the number e to the negative lambda divided by k factorial.


k factorial means that you start from k and multiply by k minus 1, k minus 2, all the way down to 1. 4 factorial would be 4 times 3 times 2 times 1.

The example that you had was lambda, the expected rate of occurrence, was 4.3. The number of successes that you wanted this week was 1.

Simplify it out, and you get 0.058, meaning there's about a 6% chance of just one accident. If you had chosen zero accidents, which you could do, 0 factorial is 1. Be aware of that. The 0 factorial is 1.


The Poisson distribution gives probabilities from situations that arise from rare events. The number of trials, n, is high, and p, the probability of success, is low. If you know the average rate of occurrence, you can figure out the probability of some exact number of successes.

Good luck.

Source: This work adapted from Sophia Author Jonathan Osters.

Terms to Know
Poisson Distribution

A distribution used for rare events. It can find the probability of exactly a certain number of successes within a given timeframe, assuming that events occur independently.