Monty Hall, the Illegal Switch

From the Mind Blown track. It draws the three doors and the hundred-door version side by side and states the answer; this lesson is the proof the drawing only gestures at, plus the twist its footer doesn’t have room for - the host’s own rule is not a colourful detail, it is the entire trick, and the man the game is named after didn’t always follow it.

Before you start. You need nothing beyond fractions like “2 out of 3”. If you’ve read The Birthday Paradox you already know the habit this note leans on again: count every way the game can go, rather than guess.

The idea in 30 seconds

Switching wins 2/3 of the time and staying wins 1/3 - but only under a host who always opens a goat door and always offers the switch.
The host isn’t a bystander. Two-thirds of the time he’s forced onto the one door left, because he can never open the car.
Prior, likelihood, likelihood ratio, posterior: your starting chance, how likely the host’s exact move is under each story, how many times more likely one story is than the other, and your updated chance once that ratio is folded in.
The bigger the game - a hundred doors instead of three - the harder the illusion is to hold onto, because the host’s forced hand becomes impossible to miss.
The whole answer rides on the host’s rule, his protocol, not on the doors. Change the rule and the number changes with it.
The real Monty Hall said he didn’t always follow that rule, which means the actual game show wasn’t always the puzzle everyone thinks it is.

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