Chance of two people picking same numbers

Question

Lets say two people pick $5$ different numbers out of a subset of $35$) numbers, what is the chance these $2$ people have $2$ or more picked numbers in common?

This is the chance of $1 - p$ (have nothing in common) $- p$ (have $1$ in common)

However how do I calculate $p(0)$ and $p(1)$?

Thx for the help.

Answer 1

By symmetry, you can just assume the first person picked $1,2,3,4,5$ and ask how many of those the second person has picked. The chance the second person has picked $k$ of them is $$\frac {{5 \choose k}{30 \choose 5-k}}{35 \choose 5}$$

Answer 2

Nothing in common: $\frac{{35 \choose 5} \times {30 \choose 5 }}{{35 \choose 5}^2} = \frac{30 \choose 5}{35 \choose 5}$

$1$ in common: $\frac{{35 \choose 5}\times {5 \choose 1}\times {30 \choose 4}}{{35 \choose 5}^2} = \frac{{30 \choose 4} \times {5 \choose 1}}{{35 \choose 5}}$