# 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 0.64^{6} into (0.64^{5})(0.64^{1}), 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!