Indexing the archive…
Your Universe of Digital Possibilities
Three doors, one prize. You pick; Vega — who knows where it is — opens an empty door and offers the switch. Every rule of her protocol is printed above the doors before the first round. Staying feels like a coin flip. Play it and count.
Bayes’ rule on the host’s protocol, not on the empty door: if your door hides the prize she opens either of the other two (½); if the other closed door hides it she has exactly one legal door (1). That forced hand carries the closed door to 2/3. A host who opened at random and happened to show an empty door would leave ½ · ½ — a different game.
The exact central interval for K, the count of wins in n independent plays at win probability p — computed from the rules alone, before a single play. A record inside it is the rules working; a record outside it happens one time in twenty, and the band says so in advance. When the chance of each play is fixed before it is played but differs from play to play, the same band comes from the Poisson-binomial sum.
The update is not about what you saw — an empty door — but about what her protocol made her do. Bayes’ rule with her likelihood (½ if your door hides the prize, 1 if the other one does) moves the closed door to 2/3. Its dark twin is The Envelope, where the seductive update is the wrong one; The Call is the same edition’s lesson in time — speak second, take the game. And the three-case enumeration you paid to see returns as the strategy tree of The Truel.