## The Idea in Brief The Secretary Problem is a famous puzzle in probability and decision theory. It models situations where someone must make an irreversible choice from a sequence of options, such as hiring, dating, or investment. The problem shows that even under uncertainty, a simple stopping rule maximises the chances of success. The optimal strategy is surprisingly elegant: reject roughly the first 37% of candidates, then choose the next one who is better than all before. This maximises the chance of picking the very best option, though success is never guaranteed. --- ## Key Concepts ### 1. The Setup - A decision-maker faces **n candidates**, arriving in random order. - Each candidate has a unique rank (from best to worst), but only relative comparisons are possible — the decision-maker cannot see absolute scores. - After each interview, the decision-maker must either **hire immediately** or reject forever. ### 2. The Optimal Strategy - The problem boils down to balancing **exploration** (learning about candidate quality) and **exploitation** (making the final choice). - The mathematically optimal strategy is: - Skip the first **n/e ≈ 37%** of candidates, no matter how good they seem. - Then, hire the next candidate who is better than everyone seen so far. - This is sometimes called the **37% rule**. ### 3. Probability of Success - Using this rule, the chance of hiring the very best candidate tends towards **1/e ≈ 37%**, regardless of how large n becomes. - No other strategy gives a higher probability. ## Implications The Secretary Problem illustrates the power of **optimal stopping theory** — how to act under uncertainty when decisions are irreversible. Its counterintuitive result (rejecting many good options early on is rational) makes it a classic example in probability, economics, and behavioural science.