Moderate# Winning Something in the Lottery

COMBIN-YEDNN9

A lottery player selects 5 different numbers from the set $\{1, 2, \ldots, 20\}$. The lottery organizers then randomly select 5 different numbers from the same set. The player wins a prize if at least one of her numbers matches a number selected by the organizers.

What is the probability that she wins a prize?