I noticed, several times, that there seems to be some “homogeneity principle” (like in physics) when computing conditional probabilities. It seems that most of the time, formula implying sums of conditional probabilities with different conditionning events will be ill-formed or wrong.
More specifically, if $(\Omega, \Sigma, \mathbf P)$ is probability space, $A$, $B$, $X$, $Y$ events s.t. $A$ and $B$ have nonzero probability, one will rarely write sums such as $\mathbf P(X) + \mathbf P(Y|A)$ or $\mathbf P(X|A) + \mathbf P(Y|B)$. Equalities such as $\mathbf P(X) = \mathbf P(Y|A)$ or $\mathbf P(X|A) = \mathbf P(Y|B)$ are also rare, except in very specific cases (e.g. when $A$ and $B$ happen to be equal or independent). However, as in physics, products of conditionnal probabilities with different conditionning events are allowed, which makes me think about “homogeneity”.
This result seems intuitive to me, in particular because if $X$ and $Y$ are disjoint events, one can get $P(X|A) + P(Y|B) > 1$, which would somehow be shocking. Furthermore, I would say that $(\Omega, \Sigma, \mathbf P)$, $(\Omega, \Sigma, \mathbf P(\cdot|A))$ and $(\Omega, \Sigma, \mathbf P(\cdot|B))$ are distinct mesure spaces, and different kinds of measures cannot be added (as in physics, once again).
But is there a deeper reason to this statement? Are there contexts where adding differently-conditionned probabilities would be sensible?
Do you think that I can say to my highschool students that they must not add differently-conditionned probabilities, unless they are absolutely certain of what they are doing?
Thanks in advance for your replies.