“What we are looking at for each student is the specific probability of acceptance of a student with a particular profile to a specific college…Of the 100 unhooked students with 4.0 GPAs, SATs of 1600, and three international awards, and two leadership positions who are applying to Harvard, 30 will be accepted this year, giving each a 30% chance of being accepted”
@MWolf That’s not the right way to think about college admissions. Harvard doesn’t put all those people with a particular set of qualifications in a hat and draw randomly. What you have is a sampling problem much more like the question of who will win either Michigan or Pennsylvania in the presidential election. You win or you don’t, and Michigan voters are looking for specific things, Pennsylvania voters some of the same things but maybe other things too or they may weight them differently. The probability of winning one state depends on how well you can estimate what the admissions officers (or voters) are looking for. So if you have a 1600 SAT then historical evidence may suggest 30% of those people have been admitted by Harvard in past years, but that’s really just saying you believe the yes/no decision is closer than average (“X is more likely to win in Michigan because many voters like policy Y”). Though unfortunately we can’t conduct an opinion poll with admissions committee members to improve our estimates.
And then as Nate Silver pointed out, polling uncertainties (voter intentions) in different states are correlated so you can’t treat the probabilities as independent and multiply the probabilities for each college or state (“if you win in Michigan you are more likely to win in Pennsylvania”). Likewise the minds of admissions officers.
In summary the “probability” of admission is attempting to measure how well you know the mind of an admissions officer/committee (or that of the voters), rather than suggesting that colleges (or voters) pick randomly amongst equally qualified candidates. If you knew their minds perfectly then the probability would be 1 or 0 for each college (or state). Much like the question of whether Schroedinger’s cat is alive or dead.
“Lottery tickets are actually nor really independent. It is sampling without replacement, so with every lottery ticket bought, the probability of getting a winning a prize with the next ticket increases. Not by a lot, but by more than 0”
That’s the situation for Powerball (where you can buy all of the different numbers in theory, as some people have tried to do after multiple rollovers, though the tickets are still independent if you pick numbers randomly). But you may end up sharing the proceeds even if you buy all possible combinations, which is why you have to wait for rollovers (where prior losers have contributed to the pot).
However it’s not the case for a raffle of Super Bowl tickets, where each additional ticket sold reduces the probability of the previous ticket winning and it’s easy to see that 2/(n+1)-1/n < 1/n so the second ticket has a lower expected incremental probability of winning than the first one you buy.