If I draw a graph (the kind with vertices and edges), it is straightforward to determine if the graph is connected or not. But what if I tell you I have a graph in front of me, and ask you to guess if it is connected without seeing it? That is, how likely is it that a typical graph is connected?

One can ask similar questions in other fields: Is a typical group infinite? Is a typical 3-manifold hyperbolic? Of course, the answers to these types of questions depend strongly on what is meant by "typical."

This talk will be a survey of "random" constructions in three areas of mathematics: graph theory, group theory, and 3-manifold topology. I will give a definition of a random object from each of those fields, as well as a few interesting results relating to each one.