This is the graded finale for the whole course. It pulls the entire arc together: the exact secretary problem and the look-then-leap rule, why both the optimal look fraction and the win probability converge on 1/e (about 37%), the general explore-then-commit pattern under irreversibility, the shift to a falling reservation-value threshold once you care about expected value instead of only the single best, the real-world variations (recall, known distributions, search costs, unknown N, two-sided search), and — just as important — the honest limits of the model. Several questions carry deliberate traps drawn from the most common misreadings of the 37% rule, so read each stem carefully before you lock in an answer.
How this exam works
This is a final, one-way exam. Questions come one at a time, and submitting an answer locks it for good — there is no going back, no retry, and no restart. Your score stays hidden until the very end, when you will see whether you passed. The pass mark is 70%. Some questions are marked select all that apply and need every correct option checked (and no wrong ones) to earn the point. Take your time on each question, because you only get one shot at it.
What class of problem does "optimal stopping" describe?
Select an answer to continue.
Course Recap
Big picture
Optimal Stopping & the Secretary Problem — the whole course
- Optimal Stopping & the Secretary Problem
- The secretary problem & the 37% rule
- N options seen one at a time, accept/reject irreversibly with no recall, goal = the single best; look-then-leap rejects the first ~37%, remembers that sample's best, then takes the first later option that beats it.
- Both the optimal look fraction and the win probability converge on 1/e ≈ 37% as N grows; e appears from maximising the win-probability trade-off at r/N = 1/e; by hand, N=3 with r=1 wins 50%.
- Explore-then-commit
- The general pattern: an explore phase that only CALIBRATES a bar (best-of-sample) and commits to nobody, then a commit phase that takes the first option over the bar.
- The explore/exploit trade-off is sharpened by no-recall irreversibility; the optimal threshold depends on how much looking remains — picky when many options are left, relaxed as they run out.
- Expected value & the reservation value
- The pure 37% rule optimises a harsh single-best goal; once near-misses count (cardinal payoffs), the optimal look shrinks BELOW 37%.
- The optimal policy becomes a DECLINING reservation value: hold out for excellence early, lower the bar as options deplete, and accept anything on the last option.
- Variations that matter
- Recall lets you look longer (free recall → just compare all); a known distribution → use a reservation value with no sampling; search costs → stop earlier.
- Unknown N → use a TIME threshold (explore the first 1/e of the time); rejectable/two-sided search → aim lower and commit earlier; the 37% peak is broad, so a rough N is fine.
- Transfer & honest limits
- Applies to hiring, house-hunting and selling (reservation value, descending threshold), knowing when to stop researching, and the "37% then commit" dating meme with heavy caveats.
- Limits: optimises the single best (wrong objective for most choices), assumes strict irreversibility and no recall, assumes ranks-only with unknown distribution, assumes known N and random order — and do not fetishise the exact number.
- The secretary problem & the 37% rule
Key takeaways — the whole course
Optimal stopping is the art of choosing when to stop searching once options arrive one at a time and commitments are hard to undo. The secretary problem is its cleanest form: reject the first ~37% to calibrate a bar, then leap on the first option that beats it — a rule where both the ideal look fraction and the win rate converge on 1/e ≈ 37%, and where e falls out of a genuine trade-off, not a notation trick. Zoom out and this is just explore-then-commit: an explore phase that only sets a bar under no-recall irreversibility, then a commit phase that takes the first option over it, with a threshold that must relax as options run out. Change the goal to expected value and the optimal look shrinks below 37%, replaced by a declining reservation value — picky early, forgiving late, accept anything last. Real life then bends the rule: recall lets you look longer, a known distribution replaces sampling with a reservation value, search costs push you to stop earlier, an unknown N sends you to a time threshold (explore the first 1/e of the time), and two-sided search means aiming lower and committing sooner. Above all, respect the model’s honest limits — it optimises for the single best (rarely your true objective), assumes strict irreversibility and no recall, and assumes ranks-only with an unknown distribution and known, randomly ordered N. Keep the strategy shape, not the exact percentage: calibrate, then commit, and let a falling bar carry you home.