# Deriving the Negative Binomial Probability Mass Function

Today is going to kind of be a continuation off of yesterday’s post about the binomial. Today, however, we’re going to focus on the negative binomial and how you can derive the formula for the negative binomial from the binomial itself. Let’s do this with an example:

Scenario: A recent poll has suggested that 64% of Canadians will be spending money – decorations, Halloween treats, etc. – to celebrate Halloween this year. What is the probability that the 13th  Canadian random chosen is the 6th to say they will be spending money to celebrate Halloween?

Solving it the Binomial Way
This sounds like a binomial-type question, right? So your first thought might be as follows:

“I need to have a total of 6 ‘successes’ out of a total of 13 people asked. So n = 13, p = 0.64, and I’m looking for P(X = 6). I can apply the binomial formula like this:”

This is how we applied the binomial formula in a similar problem yesterday. But if you look closer at this question specifically, this isn’t quite asking for something that we’ve seen the binomial used for. The above binomial calculation works if you need a total of 6 successes from 13 trials and it doesn’t matter in which order they came.

This question, though, is stating that that 13th trial needs to be one of the successes—more specifically, it needs to be the 6th one. So how can we approach that?

Let’s divide the problem into two easier-to-solve parts. Based on the question, we know that the 13th trial needs to be a success. So ignore it for a second. If we ignore the 13th trial, we’re also ignoring the 6th success.

So what we’re left with is 12 trials and 5 successes.

The question doesn’t say anything about these 12 trials other than that 5 of them have to be successes. This allows us to use the binomial formula to figure out the probability that 5 of the 12 are successes: n = 12, p = 0.64, and we’re looking for P(X = 5):

That takes care of the first 12 trials. Let’s go back to the 13th one. We know, from the question, that the 13th one needs to be a success. We can find this easily: since it’s just one trial, and we know the trials are all independent and all have the same probability of success, the probability of this 13th trial being a success is just equal to 0.64, our probability of success.

Now we’ve got the two parts: the first 12 trials and the 13th trial. So how do we put this together to find the answer to the question? As I just mentioned, all the trials are all independent, so we can just multiply their probabilities:

There’s your answer!

Solving it the Negative Binomial Way
The negative binomial probability model is generally used to answer the question, “what is the probability that it will take me n trials to get s successes?”

Let n = the number of trials and s = the number of successes. To find P(X = n), or the probability that you’ll need n trials to get to a specified s number of successes, you find the following:

Let’s plug in our values from our question, which is asking for P(X = 13), so n = 13 and s = 6:

If we break up that 0.646 into (0.645)(0.641), you’ll notice that this equation is exactly the same as the one we used when solving this the binomial way.

This part was from our first 12 trials, this part was from our 13th trial and knowing it was a success. So if you solve this the binomial way, you actually are solving it using the negative binomial equation!

# Deriving the Binomial Probability Mass Function

Today I want to talk about binomial random variables. Specifically, I want to talk about how you can “derive” the binomial probability mass function (pmf) using a simple example.

I want to discuss these things in a way that someone who is completely unfamiliar with statistics can understand them, so let’s start from the beginning!

1. What is a binomial random variable?
2. What is a probability mass function?
3. What is the probability mass function for a binomial random variable?
4. How can you “derive” this probability mass function from a simple example?

1. What is a binomial random variable?

You can think of a binomial random variable as something that counts how often a particular event occurs in a fixed number of trials. In order for a variable to be a binomial random variable, the following conditions must be met:

• Each trial must be performed the same way and must be independent of one another
• In each trial, the event of interest either occurs (a “success”) or does not (a “failure”) (in other words, there must be a binary outcome in each trial)
• There are a fixed number, n, of these trials
• In each of the n trials, the probability of a success, p, is the same.

Here are some good “basic” examples of binomial random variables:

• Define a “success” as getting a “heads” on a coin flip. If you flip 10 coins and let X be the number of heads you get from those 10 flips, X is a binomial random variable (n = 10, p = 0.5)
• Define a “success” as rolling a 5 on a 6-sided die. If you roll a die 20 times and let X be the number of times you roll a 5, then X is a binomial random variable (n = 20, p = 1/6)
• The probability of any given person being left-handed is 0.15. If you randomly ask 50 people if they are left-handed or not, and let X be the number of people who are left-handed out of the 50, then X is a binomial random variable (n = 50, p = 0.15).

As you can see, there are really two values we need to know in order to define something as a binomial random variable (which, in reality, is defining the shape of the binomial distribution from which that random variable comes): n, the number of trials, and p, the number of successes. If X is a binomial random variable, we can express this as X~binom(n,p). This is read as “X follows a binomial distribution with n trials and a probability of success p.” In our previous three examples, we could express the X’s as follows:

• X~binom(10,0.5)
• X~binom(20,1/6)
• X~binom(50,0.15)
1. What is a probability mass function?

A probability mass function (pmf) is a lot less scary than it sounds. It is simply a function that gives the probability that a (discrete) random variable is exactly equal to some value. For example, suppose X is our random variable. Let x represent a specific value that X could assume. A pmf for X could give us P(X = x), or the probability that X is equal to that specific value x.

1. What is the probability mass function for a binomial random variable?

Let X be a binomial random variable. To find the probability that X equals a specific value x, we use the following formula:

where n is the number of trials, p is the probability of success, and

1. How can you “derive” this probability mass function from a simple example?

This pmf might look a little complicated. However, we can understand it and how it comes about by looking at a simple example and “working backwards” to get to the above pmf formula. So let’s do it!

Suppose you roll a fair die four times. Let X be the number of times you roll a 1. You want to find the probability of rolling exactly three 1s.

In this example, the number of trials is the number of rolls. So n = 4. The probability of success is the probability of rolling a 1 on any given roll. So p = 1/6. And what we want to find is the probability that exactly three of the four rolls will result in a success. So we want P(X = 3).

Let’s see if we can figure this out without the formula.

We want the probability of getting exactly three successes out of the four rolls, or P(three successes). Another way of stating this is that we want P(success and success and success and failure), or that we want three successes and one failure. Don’t worry about the order yet; we’ll deal with that later.

So. We know that the probability of success on any one roll is 1/6, which means that the probability of failure on any one roll is 5/6. We also know that the rolls are independent of one another, as the outcome of one roll does not affect the outcome of another (e.g., rolling  a 5 on the first roll will not affect the outcome of the second roll).

Because of this independence, our calculation of P(success and success and success and failure) is just going to be the probabilities of each of the four outcomes multiplied by each other, or

In our example, this is

Notice that this is the

part of the pmf formula when x = 3 and n = 4. This is the part that gives us the probability of three successes out of four trials. However, we’re not done yet! We also need to take into account the fact that these three successes and one failure can happen in different orders. Specifically, if we denote a success as “S” and a failure as “F,” these can occur in the following orders:

SSSF
SSFS
SFSS
FSSS

That is, there are four different ways we can get three successes and one failure when n = 4. So what we need to do now is combine this with our previous calculation above to get

That 4 is actually the number of combinations of three successes and one failure we can get when n = 4. Another way we could calculate this without writing out all the possible combinations? Use this formula:

Specifically:

Our final calculation, then, is

So the probability of rolling exactly three 1s out of four rolls is 0.01543, and we can see that in our process of figuring this out, we’ve actually “derived” our binomial formula of

Cool, huh?