@csdad2 I think we can agree that the “probabilities” (if that is the right word) of admission to colleges is, in most cases, unknown, and is perhaps unknowable from our point of view.
I think what you’re referring to is the concept of Bayesian probability, which is an interpretation of probability based on states of belief. In fact there is an entire Wikipedia article on interpretations of probability ([here](https://en.wikipedia.org/wiki/Probability_interpretation)).
Here’s another scenario (however unrealistic) where admissions to two colleges are provably independent:
Suppose I apply to two colleges P and Q (I’ll avoid A and B since they are overused). P and Q do not communicate in any way, and the amount of information each college has is constant. In particular, the amount of information P has does not depend on whether I applied to or got accepted by Q.
P and Q each have their own functions in terms of GPA, test scores, letters of recommendation, etc. that each output a real number between 0 and 1 (representing the probability of my acceptance). Maybe they’re monotonic in GPA, but that doesn’t matter. Call these numbers p and q. Then college P simulates a truly random probability p event, and accepts me iff the event holds. Same with Q.
In this case, I don’t actually know p or q, and the events of acceptance are likely correlated. But still Pr(Q|P) = q, and Pr(P and Q) = pq, i.e. acceptances to P and Q are independent.