Bananas pass by one at a time and you must pick the best — but once you reject one it’s gone, and you can’t know if a better one is still coming. What’s the strategy? Look at the first 37% and pick none, just remember the best so far. Then grab the first one that beats them all. This simple rule gives you a 37% chance of landing the single best banana — astonishingly good, and it’s the same math for hiring, apartments, and dating. Slide the look-phase to find the sweet spot.
The secretary (best-choice) problem: candidates of unknown rank arrive in random order; you must accept or reject each irrevocably and want the single best. The optimal policy is a threshold rule — reject the first r candidates, then take the first who exceeds all seen — and the success probability is maximized when r/n → 1/e ≈ 0.368. Remarkably, that same 1/e is also the WIN probability at the optimum. It is the foundational result of optimal-stopping theory. A fail-loud self-check throws unless the optimal cutoff sits at n/e and 1/e ≈ 0.368. ◆ real probability, node-verified.
The clean 37% assumes no recall, random order, one winner, and you only value THE best (no partial credit). Relax any of those — allow second-best to count, or repeated offers — and the optimal threshold shifts; the 1/e result is exact under the classic rules.