The optimal stopping calculator above solves the classical secretary problem exactly. Given the number of options you will see, it returns the number you should reject outright while you calibrate, and the probability that the resulting rule picks the single best option in the set. It does not use the famous 37% shortcut to produce that answer — it maximises the exact probability over every possible cutoff, then shows you what the shortcut would have given.
Arb Digital builds free calculators that state their assumptions instead of hiding them behind a memorable headline number. That is unusually important here, because the 37% rule is one of the most quoted and most misapplied results in decision mathematics. The arithmetic is beautiful and the assumptions behind it are almost never true of a real decision, and both halves of that sentence belong on the same page.
What This Optimal Stopping Calculator Does
Enter n, the total number of options you will see. The tool evaluates the success probability of every possible cutoff from 0 to n − 1, picks the best one, and reports it. The strategy it describes has two phases. In the first, you reject everyone, no matter how good they are, and simply record the best you have seen. In the second, you accept the first option that beats every option from the first phase. If none ever does, you are left with the last option by default.
The headline is the cutoff. The grid shows the exact success probability, the cutoff as a share of n, and how a strategy of your choosing compares — the n ÷ e shortcut, rejecting the first half, rejecting the first quarter, or simply picking one at random. The bars underneath plot the success probability across a range of cutoffs so you can see how flat the peak is, which turns out to be the most practically useful thing on the page.
An optional window field converts the cutoff into a point in time. If you are viewing 20 flats over 6 weeks, the tool will tell you that the cutoff falls at roughly week 2.1, which is a more usable instruction than a count when options arrive on a schedule.
How to Use It
- Enter the number of options you expect to see in total. Be honest rather than optimistic — the rule is built on this number.
- Read the cutoff. That is how many you will look at and turn down on principle.
- If your search runs over a period, enter the window length and unit to get the cutoff as a date rather than a count.
- Choose a comparison strategy to see what a simpler rule of thumb would cost you.
- Look at the bar chart before you commit. If the peak is flat, the exact cutoff matters far less than the assumptions do.
The Formula / How It's Calculated
Let r be the position of the first option you are willing to accept, so you reject the first r − 1. The probability that this rule selects the single best option out of n is:
P(r) = ((r − 1) ÷ n) × Σi = r to n 1 ÷ (i − 1), with P(1) = 1 ÷ n.
The reasoning behind it is short. For the rule to succeed, the best option must appear at some position i at or after r — probability 1 ÷ n for each position — and the best of the first i − 1 options must fall inside the rejected block, which happens with probability (r − 1) ÷ (i − 1). Multiply and sum over every position i the best option could occupy.
The calculator evaluates that sum for every r and keeps the maximum. For n = 100 the best is r = 38, meaning reject the first 37, and the success probability is 37.10%. For n = 10 the best is reject the first 3, with a success probability of 39.87%. For n = 5 it is reject the first 2, with 43.33%. For n = 4 it is reject the first 1, with 45.83%. Those small-n answers are worth having, because they are exactly the cases where the 37% shortcut is furthest from right: 37% of 5 rounds to 2, which happens to agree, but 37% of 4 rounds to 1 only by luck of the rounding, and the honest answer at small n comes from the sum, not the shortcut.
As n grows, the optimal cutoff converges on n ÷ e and the success probability converges on 1 ÷ e, or about 36.79%. That is where 37% comes from. Note that both numbers converge on the same figure, which is the coincidence that makes the result so quotable: you reject about 37% of the options and win about 37% of the time.
The Four Assumptions That Make It Optimal
The 37% rule is provably optimal, but only for a very specific problem. The secretary problem as classically stated makes four assumptions, and every one of them is doing real work.
You know n in advance. The size of the pool is fixed and known before you start. If you do not know how many flats you will see, there is no defined cutoff to compute.
Options arrive in random order. No option is more likely to be excellent because of where it sits in the sequence. Real sequences are rarely random: applicants who apply first are often the keenest, and a letting agent shows the best property first on purpose.
You have ranks only, not scores. You can say that this candidate is better than that one, but you cannot say by how much, and you have no external standard of "good enough". This is the assumption that fails most often, and it fails in your favour: in real life you usually can tell that an option is outstanding in absolute terms.
Decisions are final and there is no recall. Reject an option and it is gone; accept one and the search ends. No going back, no holding two in play, no negotiating.
There is a fifth, quieter assumption in the objective itself: the rule maximises the probability of getting the single best option and treats every other outcome as an equal failure. Ending up with the second best is scored exactly the same as ending up with the worst. Almost nobody actually feels that way, which is why the rule is a poor guide when a very good outcome is nearly as valuable as the best one.
How This Differs From Our Probability Tools
The probability calculator handles the probability of events and their combinations; it has no notion of a sequence or a decision, so it cannot implement a stopping rule. The expected value calculator works with cardinal payoffs and their probabilities, which is precisely the information the secretary problem assumes you do not have — if you can score your options in money or utility, expected value is the better framework and this page is the wrong tool. The permutation calculator counts the orderings this problem averages over. This page is the only one of the four that answers the question "when should I stop looking?".
What the Flat Peak Actually Tells You
The most useful practical finding on this page is not the cutoff but the shape of the curve around it. Success probability rises steeply from a cutoff of zero, peaks, and then falls away gently. For n = 100, rejecting 37 gives 37.10%, rejecting 30 gives 36.47%, and rejecting 45 gives 36.21%. Being eight or nine off the exact cutoff costs well under a percentage point.
That has a direct implication. Precision in the cutoff is worth almost nothing, while the assumptions are worth a great deal. If you are unsure whether you will see 20 options or 30, do not agonise: the difference in your cutoff is a couple of options and the difference in your odds is negligible. If, on the other hand, you can actually score options rather than only rank them, you should abandon this rule entirely rather than tune it, because a threshold rule using real scores does much better.
Why Real Decisions Break the Rule
Applying this literally to hiring, house hunting or dating goes wrong in predictable ways, and it is worth naming them.
You almost always have absolute information. You know what a fair rent is in the area before you view anything. That external standard means you do not need to spend the first 37% of your search learning what "good" looks like, and a threshold rule — accept anything above a pre-set bar — beats the secretary rule whenever such a bar exists.
The pool is rarely fixed. Applications keep arriving; new listings appear weekly. When the pool is open-ended, the problem becomes one of search cost against expected improvement, which is a different model with a different answer — and it is usually better handled with the arithmetic of a weighted average calculator or a straightforward cost comparison than with a stopping rule.
And rejection is not always one-sided. In hiring and in dating, the option you choose can decline you. The two-sided version of the problem has its own literature and its own, different cutoffs. A course-note treatment of the standard derivation, including the sum used on this page, is available in Subhash Suri's optimal stopping notes at UC Santa Barbara.
Arb Digital designs and builds free interactive calculators that state their assumptions, show their working, and earn links because they are genuinely useful. Browse what we have already published, or tell us what your audience keeps searching for.
Browse the Free Tools Hub Talk to Arb DigitalCommon Mistakes to Avoid
- Using 37% of n at small n. For fewer than about 15 options, compute the exact cutoff. The shortcut is an asymptotic result and rounding it can be a whole option out.
- Applying it when you have absolute standards. If you already know what a good option looks like in absolute terms, a threshold rule beats this one. The secretary rule exists precisely because that information is missing.
- Treating the 37% success rate as a guarantee. It is the probability of landing on the very best option. It also means the rule misses the best option about 63% of the time.
- Forgetting the default outcome. If no later option beats the calibration block, you end up with the last option in the sequence, whatever it is. That is part of the rule, not a failure of it.
- Chasing the exact cutoff. The peak is flat. Getting the assumptions right matters far more than getting the cutoff right to the nearest option.
Related Free Tools From Arb Digital
For probabilities of events rather than sequences, use the probability calculator. When you can attach payoffs to outcomes, the expected value calculator is the right framework. The permutation calculator and binomial coefficient calculator handle the counting behind results like this one, the birthday paradox calculator covers another famously counter-intuitive probability, and the weighted average calculator is the practical option when you can score choices on several criteria. Everything else is on the free online tools hub.
Frequently Asked Questions
Reject roughly the first 37% of your options while learning what is on offer, then accept the first option that is better than all of them. For large pools the optimal cutoff is n divided by e, and the rule lands on the single best option about 37% of the time.
For n = 4 reject the first 1, with a 45.83% success rate. For n = 5 reject the first 2, at 43.33%. For n = 10 reject the first 3, at 39.87%. For n = 100 reject the first 37, at 37.10%. Below about 15 options the shortcut can be a whole option out, so compute rather than estimate.
Four: you know the total number of options in advance, they arrive in random order, you can only rank them against each other rather than score them, and every decision is final with no recall. The objective also counts only the single best option as a success.
Because both the cutoff and the success rate converge on the same constant, 1 divided by e. The chance of picking the best of 100 million is essentially the same as the chance of picking the best of 100, which is the most surprising feature of the result.
You are left with the last option in the sequence by default. That outcome is built into the probability the calculator reports, not an exception to it.
Not literally. Real decisions break every one of the four assumptions, and in particular you usually do have absolute standards, which makes a simple threshold rule better. Treat this as a well-defined model that shows why a calibration phase helps, not as instructions.
Very little. At n = 100, rejecting 30 gives 36.47% and rejecting 45 gives 36.21%, against 37.10% at the optimum. The peak is flat, so the assumptions matter far more than the exact number.
Then the rule has no defined cutoff. Use your best honest estimate and accept that the answer is approximate, or switch to a model built for an open-ended search, where the trade-off is search cost against expected improvement.
This page computes a mathematical result under stated assumptions. It is not advice about hiring, housing, relationships or any other real decision, and real decisions violate the model's assumptions in ways that change the right answer.
