The ex post probability is either 0 or 1

The things you think of while doing the dishes. At the Maths Faculty in Cambridge, there are cunningly no dishwashers. Allegedly there are showers, a proven idea-generating tool. But I think they are there for other reasons.

Intuitively, it feels as if “after the fact” the probability of any event should either be 0 or 1. Either it happened or it didn’t. Well, except if your last name is Clinton.

Framing this in terms of rigorous axiomatic probability theory: consider the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in I},\mathbb{P})$. Consider real-valued random variables on this space and use the notion of conditional expectations. Define conditional probability in terms of conditional expectation as follows: for $A \in \mathcal{F}$ and $\mathcal{G} \subseteq \mathcal{F}$ a sub-$\sigma$-algebra we have $\mathbb{P}(A | \mathcal{G}) := \mathbb{E}(\mathbf{1}_A | \mathcal{G})$.

Now take $A \in \mathcal{F}_t$ – we are thinking of $t$ as being the current time, and we are looking at the event $A$ which occurred at or before the current time. Since $A$ is $\mathcal{F}_t$-measurable, $\mathbf{1}_A$ is an $\mathcal{F}_t$-measurable random variable. Thus we can “take out what is known”: $\mathbb{P}(A | \mathcal{F}_t) = \mathbb{E}(\mathbf{1}_A | \mathcal{F}_t) = \mathbf{1}_A$.

But indicator functions are always either 0 or 1. Hence the result.

This is a (very, very simple) 01 law. And you don’t really need maths for it.