Probability vs Density

Part One: The PMF (Probability Mass Function)URL copied

Let's tackle a simpler cousin of probability density. In a world of discrete possibilities (Yes/No, 1-2-3-4-5-6, one-of-52-cards) we can compute the discrete probability for each possibility. We then plot them up and pat our backs, since we now have a map of what's possible and by how much.

Here, all the heights sum to 1. And every possibility must have a non-negative probability

It is all well and good until we face a common issue. What if there are infinite possibilities? How many probabilities are we going to compute?