Relation between subspace union and probabilities union

[Damidami]

Today I was reading in a probabilities textbook that the probability of the union of two events is:

$$p(E_1 \cup E_2) = p(E_1) + p(E_2) - p(E_1 \cap E_2)$$

and reminded me of the similarity with the dimension of the union of two subspaces of a vector space:

$$dim(V_1 \cup V_2) = dim(V_1) + dim(V_2) - dim(V_1 \cap V_2)$$

Question is: is there a theory/generalization that makes two concepts a particular case of this more general theory? (they look very similar so there must be something common with those concepts)

Thanks,
Damián.

 

[fresh_42]

The only common concept is the avoidance of a double count. Since the intersection is part of both components, any measure of the components, be it counting, dimensions, volumes or whatever, would add it twice if we added the measures of the components, which is why we have to subtract one of them again.