Find the probability function until you get to non defective lights

Question

A shipment of 2500 car headlights contains 200 which are defective. You choose from this shipment without replacement until you have 18 which are not defective. Let X be the number of defective headlights you obtain. Find the probability function $f(x)$.

This seems to be hypergeometric distribution

$N = 2500$, $r = 200$, $x$ is the variable. $r$ is the "successes" which means the number of defective, since we are after defective.The $n$ here, which is the number we choose is tricky, how do I go about it?

Answer 1

What you are looking for is the Negative Hypergeometric distribution (NH) ( by analogy with the Negative Binomial distribution). Let $X$ be the number of defective headlights you choose until you obtain a given number of non-defective headlights ($m$). Then, $X$ is NH-distributed, i.e. $$ P \left( X=k \right) = \frac{ \binom{k+m-1}{k} \cdot \binom{N-m-k}{M-m} }{\binom{N}{M}} $$ Here, $N$ is the total number of headlights, i.e. $N=2500$; $M$ is the number of non-defective headlights, i.e. $M=2300$; and $m$ is 18. Plugging this into the above formula, we obtain: $$ P \left( X=k \right) = \frac{ \binom{k+17}{k} \cdot \binom{2482-k}{2282} }{\binom{2500}{2300}} $$