The Monty Hall Problem: Always Switch

You're on a game show. Three doors: one hides a car, two hide goats. You pick door 1. The host — who knows where the car is — opens door 3, revealing a goat, and asks: "want to switch to door 2?" Most people shrug: two doors left, 50/50, switching changes nothing. Most people are wrong. Switching wins two times out of three, and this little puzzle once made a thousand PhDs write angry letters.

The intuition that finally works

Forget the opened door for a second. Your original pick captured the car with probability 1/3 — nothing the host does can retroactively change that. Which means the car is somewhere in the other two doors with probability 2/3. When the host (who never reveals the car) opens one of those two doors, he isn't generating new randomness — he's funneling that entire 2/3 into the one remaining closed door. Switching is simply collecting it.

Still itchy? Scale it up: 100 doors, you pick one, the host opens 98 goat doors and leaves door 47 conspicuously shut. Your first pick was right 1% of the time. Would you seriously stay?

The detail everyone misses

The answer depends entirely on the host's rules, not just his actions. Monty knows the car's location, always opens a goat door, and always offers the switch. If instead the host opened a random door that merely happened to show a goat, the odds genuinely would be 50/50. Same visible events, different information content — because a constrained choice reveals more than a lucky one. This is the puzzle's deepest lesson: how information was produced matters as much as the information itself.

The great pile-on of 1990

When Marilyn vos Savant published the correct answer in Parade, she received some ten thousand letters, roughly a thousand from people with doctorates, many dripping with condescension, nearly all wrong. Even the great Paul Erdős refused to accept it until shown a computer simulation. Take comfort: if this puzzle bends your brain, you're in Hall-of-Fame company.

Why it matters off-stage

Monty Hall is the cleanest known demonstration that evidence depends on process. A medical test result, a survived company in a success study, a suspiciously specific alibi — each means something different depending on the rule that produced it. Survivorship bias, publication bias and selection effects are all Monty in disguise: someone opened the goat doors before you arrived. The master question, on the game show and off: what could I have been shown instead, and why wasn't I?