Category: Math puzzle (probability, conditional probability, Bayesian reasoning)
Three doors. Behind one: a car. Behind the other two: goats. You pick a door — say Door 1. The host, who knows where the car is, opens one of the other doors to reveal a goat. He offers you a chance to switch. Do you?
Switch. Always switch.
Switching wins the car 2/3 of the time. Staying wins 1/3. This is not a coin flip.
When you first picked Door 1, you had a 1/3 chance of being right. The other two doors collectively had a 2/3 chance. The host opening a losing door doesn't change your 1/3 — it concentrates the full 2/3 onto the one remaining unchosen door. You're essentially choosing between your original 1/3 guess and the "other two doors" option that now has nowhere else to hide.
The rigorous treatment uses conditional probability and Bayes' theorem. See the full write-up on Wikipedia: Monty Hall problem.
The problem is mathematically equivalent to the earlier three prisoners problem and the much older Bertrand's box paradox.
In September 1990, Marilyn vos Savant published the correct answer in her "Ask Marilyn" column in Parade magazine. What followed was one of the most remarkable episodes in the public history of mathematics.
Approximately 10,000 letters poured in, the vast majority telling her she was wrong. Nearly 1,000 came from people with PhDs, many in mathematics or statistics. Robert Sachs, a math professor at George Mason University, wrote to say she had "blew it big" and offered to help her understand. He later sent a second letter: "After removing my foot from my mouth I'm now eating humble pie."
The problem was informally renamed "Marilyn and the Goats."
Over the following two years, acceptance among general readers shifted from 8% to 56%, and among academics from 35% to 71%. Stanford statistician Persi Diaconis noted: "Our brains are just not wired to do probability problems very well."
Paul Erdős, one of the most prolific mathematicians in history, refused to believe the answer. When told the solution, he said: "That's impossible."
A colleague arranged a formal proof. Erdős grew angry. Only after watching a computer simulation produce hundreds of trials overwhelmingly favoring switching did he concede he was wrong. (Source)
In 1991, Monty Hall sat down with New York Times reporter John Tierney at his Beverly Hills dining room and conducted live experiments. He confirmed the 2/3 answer under idealized rules — but then demonstrated what he actually did on the show:
"You're sure you want Door No. 2? Before I show you what's behind that door, I will give you $3,000 in cash not to switch to it."
Hall called it the "Henry James treatment" — "The Turn of the Screw." He would use escalating cash offers and psychological pressure to manipulate contestants. The real game show was not the idealized puzzle. As he put it: "If the host is required to open a door all the time and offer you a switch, then you should take the switch. But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood."
(Tierney, "Behind Monty Hall's Doors," The New York Times, 1991; Wikipedia)
In Season 9, Episode 20 ("Wheel of Mythfortune," November 22, 2011), MythBusters tackled the problem head-on.
YouTube clip: Should You Really Switch Doors In The Monty Hall Problem? | MythBusters