Monty Hall with N doors
Move the door count and the number of doors the host opens. The model puts the advantage of switching at (N−1)/R — it tracks the reveals, not the door count.
0.667the model's switch win rate at three doors
Motion
This section is about things that move, and about how badly the intuition for them scales. A bicycle drivetrain, a thrown ball, a car on an open road: in each case the quantity you care about is not the one on the dial, and the gap between the two is where the interesting answer lives.
Take the most familiar of them. Time saved by going faster is governed by the reciprocal of speed, not by speed, so a fixed increase returns a shrinking amount. The same twenty kilometres an hour is worth fifteen minutes over ten kilometres at 20 km/h and half a minute at 140 — a factor of twenty-eight across one slider. Reciprocals do that quietly, which is why a question about speed is worth a dial rather than a worked example.
Some of what belongs here runs a standard formula over coefficients taken from published tables, which makes it a different sort of thing from the probability pages. A probability model is true or false given its rules. A drag model is an approximation fitted to measurements that scatter, and it can be applied perfectly correctly and still miss the real number by a wide margin. So every coefficient is sourced to the document it came from, and each page names what its formula has no term for.
A drag model gets built accordingly: integrate a quadratic drag term, draw the vacuum solution behind it as a ghost line, and put the textbook 45° and the model's own optimum on one pair of axes — along with how little separates them. Not everything here is fitted, though. A drivetrain is one ratio of tooth counts multiplied by one length, so no measured coefficient enters at all — and even then a geometric assumption does, and calling two gears duplicates is a threshold somebody has to choose rather than a fact about the bicycle. What such a model carries no term for is effort: distance per pedal revolution says nothing about what that costs.
Where a coefficient shifts with conditions and the conclusion would need a judgement call, the tool stays unbuilt rather than dressing a guess up as a calculation.
Finished elsewhere on the site
The writing above is this section. Below are the tools running on the site today, from wherever they sit.