In the mapping class group G of a finite-type surface, each mapping class is either periodic, reducible or pseudo-Anosov. One can ask which category has the largest proportion in G. For example, when you generate a length-n word using a finite generating set of G as an alphabet, what will be the typical outcome? Or, in a large metric ball on the Teichmüller space or the Cayley graph of G, what is the number of mapping classes in each category? In this talk, I will explain some results and methods about this question. 

If time allows, I will describe an analogous problem for the outer automorphism group of a free group F_n and explain why a typical outer automorphism is not geometric (i.e., induced from a pseudo-Anosov mapping class).