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Poisson Distribution
Common Core: S.MD.1

Poisson Distribution

Author: Jonathan Osters

Calculate the probability of a poisson distribution.

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This tutorial is going to teach you a little bit about what's called the Poisson distribution. It's sort of a special case of the binomial distribution. So let's take a look at an example where we would use the Poisson distribution.

Suppose that you have a particularly busy intersection. And that intersection averages about 4.3 accidents per week. So that's not a whole lot of accidents per week.

For a busy intersection, there's a lot of cars passing through it. So the probability of having an accident at that intersection is fairly small. And all other things being equal, what's the probability that the intersection experiences a week with just one accident?

Well, this distribution can be solved using something called the Poisson distribution. And it's a distribution that works well for rare events. So rare meaning the probability of success is very, very low, but the number of trials is very high. Meaning you end up with, out of a very large number of trials, only a few successes.

And so the Poisson distribution is nice, specifically because it actually doesn't require the use of n and p. All we need is the typical rate of occurrence. So in this example, it was the 4.3.

So if the average number of successes in a given time frame is this letter lambda-- this is 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. This is not pronounced k with the exclamation point. k factorial means that you start from k and multiply by k minus 1, k minus 2, all the way down to 1. So 4 factorial would be 4 times 3 times 2 times 1.

So let's do an example. The example that we had was lambda, the expected rate of occurrence, was 4.3. And the number of successes that we wanted this week was 1. So we put 1 here and here, 4.3 here and here. Simplify it out, and you get 0.058, meaning there's about a 6% chance of just one accident.

If we had chosen zero accidents, which we could do, 0 factorial is 1. So just be aware of that. The 0 factorial is 1.

So to recap, the Poisson distribution gives probabilities from situations that arise from rare events. So n, the number of trials, is high, and p, the probability of success, is low. Whereby, if we know the average rate of occurrence, we can figure out the probability of some exact number of successes. And so that's the reason that we use the Poisson distribution. Good luck, and we'll see you next time.

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.

Formulas to Know
Poisson Distribution

P left parenthesis X equals k right parenthesis space equals space fraction numerator lambda to the power of k e to the power of negative lambda end exponent over denominator k factorial end fraction
k space i s space t h e space g i v e n space n u m b e r space o f space e v e n t space o c c u r r e n c e s
lambda space i s space t h e space a v e r a g e space r a t e space o f space e v e n t space o c c u r r e n c e s