4.4 Negative binomial distribution

Geometric distributions describe the probability of observing the first success on the \(n^{th}\) trial. The **********negative binomial distribution************ is more general, describing the probability of observing the \(k^{th}\) success on the \(n^{th}\) trial.

💡 Is it negative binomial? Checklist
- Trials are independent - Each trial outcome can be classified as success or failure - The probability of success is the same for each trial - The last trial *must* be a success

Example

The probability of a success is 0.8. What is the probability that it takes 6 trials to get 4 successes? These are the 10 ways that the 4th success could be achieved on the 6th trial. All of these have 2 failures and 4 successes, so the probability is the same: \(0.2^2*0.8^4=0.016384\). Thus, the probability that the 6th trial contains the 4th success is \(10 * 0.016384 = 0.16384\)

We broke the problem into two parts:

1. Find the probability of a single sequence
2. Multiply it by the number of possible sequences

How can we easily find the number of possible sequences?

\[ \binom{n-1}{k-1}=\frac{(n-1)!}{(k-1)!((n-1)-(k-1))!}=\frac{(n-1)!}{(k-1)!(n-k)!} \]

This is the number of ways we can order \(k - 1\) successes and \(n - k\) failures in \(n - 1\) trials.

💡 Negative Binomial Distribution The negative binomial distribution describes the probability of observing the \(k^{th}\) success on the \(n^{th}\) trial, where all trials are independent: \(\large P(k^{th}\text{ success on }n^{th} \text{ trial}) = \binom{n-1}{k-1}p^k(1-p)^{n-k}\) where \(p\) is the probability that an individual trial is a success.