Three Doors
Switching wins two games in three. Play it yourself on a lit stage and your two bars drift toward 1/3 and 2/3; then play ten thousand games in a few seconds and watch both rates lock onto 33.3% and 66.7% inside their 95% bands. With a hundred doors the answer becomes obvious, and with a host who is only guessing it disappears.
You are on a game show. There are three doors: a car behind one, a goat behind each of the others. You pick door 1. The host, who knows where the car is, opens door 3 and shows you a goat. Then he asks whether you want to switch to door 2. The statistician Steve Selvin posed the puzzle in a two-paragraph letter in 1975, named after Monty Hall, the host of Let's Make a Deal, and answered it: switch.
Almost nobody believes that. Two doors are left, so it feels like a coin flip. When Marilyn vos Savant gave the same answer in her Parade magazine column in 1990, around ten thousand readers wrote in to say she was wrong, nearly a thousand of them with PhDs, many on university letterhead. Paul Erdős, one of the most prolific mathematicians who ever lived, is said to have stayed unconvinced until he watched a computer simulation. That is what the panel at the bottom of this page is for.
The argument fits in one line. Your first pick is right one time in three, and staying only wins in those games. In the other two games out of three you picked a goat, and the host, who may not open your door or the car, has to leave the car closed for you. Switching wins exactly when your first pick was wrong. Turn the doors up to a hundred and it stops feeling like a trick: you pick one door, the host opens ninety-eight goats and leaves one door shut. You would switch.
The host's knowledge is what makes it work. Switch the host to forgetful and he opens doors at random. Sometimes he reveals the car and the game is spoiled. In the games where he happens to show only goats, the two closed doors really are a coin flip, even though what you see on the stage is identical. The same trap waits in medical tests, courtrooms and A/B tests: to read a piece of evidence correctly, you have to know the process that produced it.
Six ways to carve a maze, each with its own handwriting — then a wave of coloured light floods it and a gold thread snaps to the exit.