The 37% Rule: When to Stop Looking and Make a Choice

Suppose you’re interviewing ten candidates for a job. You must decide whether to hire each person immediately after their interview, and anyone you pass over is gone for good. How many people should you meet before making your first offer?

It sounds reckless to reject good candidates on purpose. Yet if your goal is to hire the single best person in the group, a little deliberate waiting gives you a much better chance than choosing at random. This is the secretary problem, a mathematical model of deciding when to stop searching.

First, make the decision precise

The model makes some strong assumptions. You know there are exactly ten candidates, but not how good they’ll be. They arrive in random order. After each interview, you can tell whether this candidate is better than everyone you’ve met so far, but you must accept or reject them on the spot. Your aim is to choose the best of all ten—not merely someone good enough.

Real hiring rarely works quite like that. But those rules let us isolate a familiar dilemma: choose too early, and you may miss someone better; wait too long, and the best person may already have left.

A simple strategy divides the interviews into two phases. First, interview a set number of people without hiring anyone. Use them to establish a standard. Then hire the first later candidate who beats everyone you interviewed during that first phase. If no one does, hire the last candidate.

Try rejecting the first three

For ten candidates, the best version of this strategy is to pass over the first three, then take the next candidate who is better than all three. “Better” here means better than the entire group you’ve seen so far: a new personal best, not just someone stronger than the third interviewee.

Imagine the candidates’ true rankings, from best to worst, are hidden from you. If the best candidate appears among the first three, you cannot win; you’ve already rejected them. That happens 3 times in 10. But when the best candidate arrives later, you have a chance—provided the strongest person who came before them was in your first three interviews.

Why? If the earlier front-runner appeared after interview three, you would have hired that person before the true best showed up. If the earlier front-runner was in your sample of three, you would still be waiting when the best arrived.

Work out the odds

Take the case where the best candidate appears fourth. That position has a 1-in-10 chance. The strongest of the first three must be in the first three—which is guaranteed—so you win with probability 1/10.

Now suppose the best appears fifth. Among the first four candidates, the strongest must be one of your first three. Random order makes that a 3-in-4 chance. The probability of winning in this particular way is therefore (1/10) × (3/4).

The same reasoning works for every possible arrival of the best candidate. If they appear in position j, the strongest of the previous j − 1 people must fall within your initial sample of three. Your total chance of success is:

(1/10) × (3/3 + 3/4 + 3/5 + 3/6 + 3/7 + 3/8 + 3/9) ≈ 39.9%.

That is nearly four times the 10% chance you’d get by selecting a candidate at random. Sampling four people instead gives a slightly lower chance, about 39.8%, so three is the better cutoff for a group of ten.

Where does 37% come from?

With a much larger group, the best sampling period approaches about 37% of the total. For 100 candidates, that suggests observing roughly the first 37 without choosing anyone, then accepting the next new best.

Here is the mathematical reason. Let x be the fraction you sample. The success calculation we used for ten candidates becomes, for a large group, approximately x ln(1/x). The symbol ln means the natural logarithm; here it comes from adding fractions such as 1/37 + 1/38 + 1/39 and approximating that sum with an area under a curve.

The slope of x ln(1/x) is ln(1/x) − 1. At the highest point, the slope is zero, so ln(1/x) = 1. That means 1/x = e, where e is approximately 2.718. Thus x = 1/e ≈ 0.368. At that cutoff, the chance of choosing the single best candidate also approaches about 36.8%.

Use the rule, but know what it promises

The 37% rule does not mean you have a 37% chance of making a good choice. It means that, under the model’s strict conditions, you have about a 37% chance of selecting the one best option. You might still choose an excellent second-best option—or lose the best one during your sampling period.

For apartments, jobs, or purchases, the conditions deserve scrutiny. You may not know how many options will appear, their order may not be random, and an earlier option may remain available. If you have a clear minimum standard—an apartment you can afford that meets your needs—finding one that passes it may matter more than identifying the absolute best in hindsight.

The useful takeaway is the shape of the decision: spend an early stretch learning what’s available, then decide using a standard you set before the pressure of the next option arrives. The surprising part isn’t that waiting helps. It’s that, when you cannot go back, waiting forever has a cost you can calculate.

Generated by AI · OpenAI · gpt-6-sol