Interesting discussion. I agree decisions by different schools are not independent events, but there does seem to be a high degree of noise, or whatever the right term would be, in admissions decisions for the schools admitting low percentages.
For example, of 2145 admits to the Stanford class of 2018, only 214 were also admitted to Harvard. I don’t know the corresponding numbers for Harvard but assume they’re broadly similar, as both schools have ~80% yield (of course the ~ 20% not going to Stanford or Harvard go to a variety of other schools). Assuming there is considerable overlap in the applicant pools, this would seem to say that even if the process for determining who’s a competitive applicant is similar at these two schools, that determining yes vs. no among that competitive applicant group leads to different decisions in a lot of cases. I suspect the same is true in looking at decisions at MIT vs. Caltech, Yale vs. Princeton, etc.
Now there are some students where there’s an obvious reason why they were admitted to one school but not another similar school, such as their parents are prominent, active alums at one school and that doesn’t particularly help at another school. I also think though that at places with very low admissions rates, those making decisions have to work pretty hard to try and differentiate among applicants, and then hard to predict factors start to come into play like, the first application reader at one school really liked an applicant’s essays, where the first reader at a peer school didn’t.