Why would you stop making that assumption when "mathematical proof" is mentioned? We can mathematically prove what happens in the average case as easily as we can analyze the worst case, and frequently do so. Furthermore, as I already demonstrated, programmers usually care more about the average case than the worst case.
And yes, average can mean several different things. I was slightly sloppy about that, but not so sloppy that I think it would cause any real confusion. However I stay away from "expected" with a lay audience because I worry that laypeople are likely to misunderstand "expected" as "median". Instead I'd lean towards "amortized".