Bounds on probe lengths and number of rehashes with universal hashing
This does not seem like a good example for teaching purposes, as there's no clean analysis or useful conclusions you can draw.

In particular, from only the assumption that you have a weakly universal hash function, you can't draw any useful conclusions about the probability of a having less than K items in the bucket.  That would require some notion of K-wise independence.  Weak universality is a weakened form of 2-wise independence, and if $K&gt;2$, it gives you no information about K-wise independence.  In particular, the hash function could be arbitrarily bad, and the probability could be arbitrarily high.  You need stronger assumptions about the hash function to do any useful analysis.

Instead of assuming weak universality, if you want to get a clean analysis, I suggest you assume that the hash function is (or behaves like) a random function, chosen uniformly at random from the set of all possible functions with that signature.  Then the calculations get easy, since each output of the hash function is uniformly random and independent of all other outputs.