A factor you don’t mention is correlation between characteristics.
Now, a very typical Harvard student would be rich AND white AND smart AND snobbish. But observe that a given student has a smaller probability of being all of these than of just, say, being rich. If you add enough majority characteristics, eventually the “typical” student will become very rare. Even if there’s a 99% probability of having any one of these characteristics, 0.99^n ->0 as n goes to infinity.
If those characteristics are positively correlated with each other, then the percentage doesn’t drop as fast as 0.99^n; for example I’d expect “snobbish” to be correlated with “rich”, and maybe “white” to be correlated with “smart” (from what I’ve heard of the American system (I never set foot in an American university myself), SAT requirements for minorities are lower than for whites).
Of course, this can be counterbalanced by negative correlations; still in this case, I’d expect non-smart students to be more likely to be rich, and non-rich students more likely to be smart.
A factor you don’t mention is correlation between characteristics.
If those characteristics are positively correlated with each other, then the percentage doesn’t drop as fast as 0.99^n; for example I’d expect “snobbish” to be correlated with “rich”, and maybe “white” to be correlated with “smart” (from what I’ve heard of the American system (I never set foot in an American university myself), SAT requirements for minorities are lower than for whites).
Of course, this can be counterbalanced by negative correlations; still in this case, I’d expect non-smart students to be more likely to be rich, and non-rich students more likely to be smart.