Chance

The problems in this section have something uncomfortable in common: the arithmetic is short enough to do on paper, and the answer still feels wrong once you get it. None of them need advanced probability. What they need is giving up an assumption you did not know you were making.

That is why they are worth a slider rather than a paragraph. Reading that switching doors wins two games in three convinces almost nobody — the objection that two closed doors must mean even odds is too strong to be argued away. Watching a simulation run ten thousand games and settle onto 0.667 does something a proof does not. And once the parameter is on a slider you can push it somewhere the textbook never goes, and find out whether the result you just accepted actually generalises. Often it does not, or not for the reason you assumed.

Each tool here keeps two things apart that are easy to conflate: what the model computes, and what the model assumes. The exact probability is drawn as a dashed line and the simulated running average as a solid one, so you can see them meet. The writing underneath says which rule was fixed to get that number — who knows what, what is chosen at random, what is held constant — because in every one of these problems the surprising answer depends on a rule that gets stated once and then forgotten.

These are all the arithmetic kind of model: given the stated rules, the answers are true or false, not matters of judgement. Where a variant changes the answer — a host who opens doors at random, groups pooled at different sizes, counters that idle while one queue waits — I name it rather than leave it as an exercise.