I am going through some old exams in one of my courses, and I don’t have acces to solutions, and found a exercise I am not sure how to tackle. I am not looking for the answer but some help/guidance if possible.

Consider a class of hash funtions , $ \mathcal{H}$ , over a finite universe $ {U}$ into {0, … , n-1}. Where $ |U|$ , n > 1.

Show that for any family of hash functions: $ $ Pr[h(u_1) = h(u_2)] \geq \frac{1}{n} – \frac{1}{|U|} $ $ Here, the probability is over the choice of the hash function h drawn uniformly at random from the family H.