For the San Francisco Chronicle, Nami Sumida shows admission rates at University of California campuses, categorized by public and private high schools:
Admissions for UCLA and Berkeley, the most competitive of the nine undergraduate UCs, follow a similar trend. Private school seniors were 20 percentage points more likely to apply to the two campuses than their peers at public schools. But unlike systemwide admissions, UCLA and Berkeley admitted public and private school students at about equal rates.
You can download the full UC dataset, which dates back to the 1994 freshman college class. With time, and several categories, it seems like a fun dataset to poke at.