Hi. This tutorial covers the Poisson distribution. So let's just start by defining it. So a Poisson distribution or a Poisson probability distribution is 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.
So here is the formula that helps you calculate probabilities of a certain number of events occurring. In this formula, we have two variables. We have k. And we also have lambda. So k is the given number of event occurrences. And lambda-- this is the Greek letter lambda-- is the average rate of event occurrences. Lambda will generally be given to you in these types of problems.
So what the formula is is that the probability of k, so the probability of a certain number of event occurrences, equals lambda to the power of k times e to the power of negative lambda. So e is that mathematical constant. It's about 2.7. And then you divide that by k factorial. So the exclamation point there is factorial. So there's a lot of math going on here. It's a somewhat complicated formula, but I'll work you through the formula in the next example here.
I have the formula down here that we'll use when we actually calculate a probability. So suppose an appliance store knows that the average number of washing machines under warranty sold by the store that will fail is 3 per year. So out of all the washing machines they sell that are under warranty, three of them will fail. And they'll have to bring those washing machines into the store. And the store will have to replace them.
So now, what is the probability of five washing machine failures in a year? So now, we're looking for, well, what's the probability that we're two above average? So a Poisson distribution can be used to calculate this probability. So it's a pretty simple problem with a pretty complex formula.
So let's notate a couple of things first. So the first thing we know is that k is going to equal 5. k is going to be the given number of event occurrences. So we want to know how many time-- or what's the probability of getting five occurrences? And then let's also notate the 3. The 3 is lambda. So lambda is the average number of occurrences.
So let's just start applying the formula. So it's going to be the probability of 3-- excuse me-- the probability of 5 equals-- now, it's a fraction. So I'm going to make my fraction bar. Now it's lambda to the k So it's going to be 3 to the power of 5 times again that mathematical constant e to the negative lambda to the negative third over k factorial. So 5 factorial.
So what we're going to do now is basically I'm just going to put all of this into my calculator. So let's start with the 3 to the power of 5. So I'll do 3 to the power of 5. So that's 243. Now, I need to multiply it by e to the power of negative 3. So I'm going to get e on my calculator like that. So it puts in that mathematical constant e, e to the negative third. So I hit Enter. And that's going to be my numerator. So my numerator is about 12.098.
But now, I'm going to divide that by 5 factorial. So I divide by 5, and then I'm going to get my factorial symbol out of the Probability menu. And hit Enter there. So that'll end up being just about 0.1. So we'll say that this is approximately equal to about 0.101 if we round to the nearest thousandth there.
So basically, that tells me that there is about a 10% chance that five washing machines will fail in a particular year if on average three fail. A lot of times, this calculation is built into a calculator. So I'm going to go to my calculator. And I'm going to go to a menu. And I'm going to pick this function Poisson PDF.
And what I'm going to do is I'm going to put in my value of lambda, and then I'm going to put it in my value of k. And that's going to give me the same value I got when I did the calculations by hand. So I just wanted to show you that a lot of times instead of doing a lot of this complex math, they'll just use a function either built into a calculator or into a computer.
And a couple other things about the Poisson distribution-- an advantage of the use of the Poisson distribution is that the number of trials is not needed. So we don't really need to worry about the number of washing machines they're selling per year or anything like that. All we just need is that average, average number or the average rate of event occurrences.
And then just to reiterate this, like I said before, Poisson probabilities are generally found using technology like we calculated. So that has been your tutorial on the Poisson distribution Thanks for watching.