I was thinking about how to calculate a metric on a probability space.

One thing that makes sense is Arccos( P(A|B) P(B|A)) . This is the metric you get if you view events as vectors in a Hilbert space and look at the angle between the two vectors, angle, of course, being a metric. It generalizes to the space of random variables in general, which is where I first discovered it. There you get Arccos ( E(XY)^2/ E(X^2) E(Y^2) )

Just on probability events, I think one thing that also makes sense is - Log (P(A|A or B)P(B|A or B)). This should be a metric and should have geodesics in the space of events. The geodesic between A and B passes through (A or B). But I don't have as clear an argument as to why this works.

So your idea isn't actually that far from correct, if you look at my angle idea.