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!

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