The assumption of independence underlying the math you're doing is probably outright false (albeit mathematically simple). As one example, imagine the case where all n schools use the exact same admissions criteria. Unless you're applying with a random application to each school, your math is shot; you will either get into every school or none of them. I won't even go into the genetic independence issue.
Every mathematical model is a false representation of reality. The predictive accuracy must be validated by experiment. And using the link I reference, you'll see there's actually some pretty strong data to start from in this case.
