So I have to figure out that in the case of the secretary problem(http://en.wikipedia.org/wiki/Secretary_problem) what is the probability that I hire exactly one applicant over the course of going through the whole algorithm, given that the applicants come in random order.
According to the algorithm, you automatically hire whoever is the current best. So if you wind up only hiring 1 person, then that means that the first person you interviewed had to be the very best out of all n choices.
So, is it fair to rethink of this question as "what is the probability that the best candidate is the first one you interview?" In which case, would the answer be (n-1)!/n! because there are n! total permutations but if you assume the best candidate is in slot 1, you can permute the other (n-1) as many ways as you want.
This sort of makes sense to me but it seems wrong at the same time and I can't quite pinpoint why.