36.79%

Suppose you have an opening for a secretary (does anyone actually have secretaries any more?).  Further suppose that 100 people have applied for the job.  You're the only one who can interview them, but you have conflicting goals: you would like to choose the very best candidate for the job, but you would like to interview as few of the candidates as possible.  What's your best strategy?

It turns out to be the following:

1. Interview 37 (36.79% of 100), rating them as objectively as possible and giving each candidate a score.
2. Remember the score of the candidate who ranked highest.
3. Keep interviewing candidates, but hire the first one you find who ranks higher than the previous highest candidate.

By following that strategy, there's a 36.79% probability that you will select the very best candidate of the entire lot.

Go read Nick Berry (at DataGenetics) as he proves it...