The Secretary Problem — The 37% Rule

What just happened?

You review $n$ candidates one at a time, in random order, and after each you must immediately decide “take” or “reject forever” — with no going back. You want to pick the best. A surprisingly simple and elegant optimal strategy: reject the first ~37% of candidates, merely observing them and remembering the best one as a “benchmark”, and then pick the first one who surpasses all seen so far. The scene shows two phases — observation (rejection) and selection — split by a threshold, plus the success-probability curve with a clear maximum at 37%.

The optimal cutoff threshold:

— you reject the first $n/e$ candidates (where $e \approx 2.718$ is Euler’s number), which for large $n$ gives a threshold converging to 37%.

The chance of choosing the best:

This strategy gives a ~37% chance (exactly $1/e$) of landing on the absolute best candidate — regardless of whether there are 10, 100, or a million of them! This is remarkable, because naively one would think that, seeing each only once, the chance should be negligible ($\approx 1/n$).

Why exactly 1/e? (the invariant)

The magic number $1/e$ (the reciprocal of Euler’s number) appears here naturally from probability: maximizing $P(r)$ over the threshold $r$, we get a condition whose solution tends to $r/n = 1/e$. At that point the chance of success itself is also $1/e$. The panel on the right shows both bars glued to the $1/e$ line — and that regardless of $n$.

History and applications

The secretary problem (also known as the marriage problem, the sultan’s dowry problem, or the “best choice problem”) is a classic optimal-stopping problem, popularized by Martin Gardner in 1960. It has real applications in decision theory, economics, and even in life choices (hence the playful “37% rule” for dating — after seeing 37% of the options, pick the next better one). It is a beautiful example of how mathematics gives a concrete, optimal strategy in a seemingly hopeless situation.

Why this matters for your exam:

This is an extension of the topic of probability and the number $e$. It shows that even with irreversible “live” decisions a simple threshold rule beats random choice — and that the constant $e$ pops up in places where you would never expect it.

Remember: observe and reject the first $n/e \approx 37\%$, then take the first one who is better. Chance of the best: $1/e \approx 37\%$.