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Last year 639 people applied, sometimes even more. Which is to say, yes the odds are quite long, but these are fantastic positions. The current fellows can be viewed here: https://sf.princeton.edu/people/current-fellows … Past fellows here: https://sf.princeton.edu/people/past-fellows … FYI: Princeton Ph.D.s are not eligible.
I don't think those odds (~1/120) are so much longer than the normal TT history of science job market, even at places less prestigious than Princeton... "Hence academic life is a mad hazard..." — Max Weber
