I don't know if this has been covered before, or if there's even anything to say about this.
On the first hand, given to events $A,B$ with probability $P(A),P(B)$, we can prove from the axioms of probability that (for events not necessarily disjoint) the probability of $A\cup B$ is
$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$
On the other, we have that given two subspaces $F,G$ of a finite-dimensional vector space $E$, we can prove what's known as Grassman's formula, which can be written as
$$\text{dim}(F+G)=\text{dim}(F)+\text{dim}(G)-\text{dim}(F\cap G).$$
This two formulae look incredibly similar, but even after proving both I don't understand how is it that they look so similar. Is there any argument, relation, bijection or analogy that makes this intuitive?