Relation Between Binomial and Negative Binomial

Question

I have come across a question in a text book that I can't seem to figure out. Firstly,

Let X be the number of faulty items in a batch of 10. What is the distribution of $X$, and what is $P(X = k)$?

This is obviously the binomial distribution, $\binom{10}{k} p^k (1-p)^{10-k}$

Let Y be the number of items you have to check to find 2 faulty ones. What is the distribution of $Y$ , and what is $P(Y = n)$?

Again, this is obviously the negative binomial distribution, $\binom{n-1}{1} p^2(1-p)^{n-2}$

However, the third part has stumped me

Give values of k and n for which $P(X ≥ k) = P(Y ≤ n)$

Anyone able to give me any pointers to be able to solve this? Thanks so much in advance.

Answer 1

Let us examine batches of $10$ one by one searching for the number of items that must be checked in order to find $2$ faulty ones.

Observe that the following statements concerning the first batch of $10$ to be examined are equivalent:

• At least $2$ items in this batch are faulty.
• To find $2$ items that are faulty it is enough to check this batch.