Let's move from the bathroom into the kitchen, since that's a socially acceptable location to have more than one person at the same time. In the kitchen, you may not want to have everyone in there at once. In fact, you probably want to limit the number of people you can have in the kitchen (too many cooks, and all that).

Let's say you don't ever want to have more than two people in there simultaneously. Could you do it with a mutex? Not as we've defined it. Why not? This is actually a very interesting problem for our analogy. Let's break it down into a few steps.