Probablity that at least n selections from one sample exist in another

Question

Have a probability question I'm not sure how to solve. Hopefully this makes sense:

1) Lets say we have a population of 1000 unique books.

2) First, we take a random sample (without replacement) of 500 of those 1000 books. Lets call this S1.

3) Second, we take another random sample (without replacement) of 750 of those same 1000 books. Lets call this S2.

What is the probability that >=100 books from S2 exist in S1?

What is the probability that <=100 books from S2 exist in S1?

What is the probability that exactly 100 books from S2 exist in S1?

Answer 1

The distribution of the number of items in both samples is the hypergeometric distribution. Suppose we let $\mathcal{S}_1$ and $\mathcal{S}_2$ denote the two samples from your initial population of $N=1000$ items, and let the operation $| \cdot |$ denote the size of a set. Since the samples were formed independently via simple random sampling without replacement with $|\mathcal{S}_1| = 500$ and $|\mathcal{S}_2| = 750$ you have:

$$\mathbb{P}(|\mathcal{S}_1 \cap \mathcal{S}_2| = s) = \frac{{750 \choose s}{250 \choose 500-s}}{{1000 \choose 500}} \quad \quad \quad \text{for all } 250 \leqslant s \leqslant 500.$$

(Note that there must be at least $250$ items in both samples.) So, you have:

$$\begin{equation} \begin{aligned} \mathbb{P}(|\mathcal{S}_1 \cap \mathcal{S}_2| \geqslant 100) &= 1, \\[6pt]\mathbb{P}(|\mathcal{S}_1 \cap \mathcal{S}_2| \leqslant 100) &= 0, \\[6pt] \mathbb{P}(|\mathcal{S}_1 \cap \mathcal{S}_2| = 100) &= 0. \\[6pt] \end{aligned} \end{equation}$$