On September 9, 1990 the following letter was published in the “Ask Marilyn” column of an American magazine called Parade:
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you: ‘Do you want to pick door #2?’ Is it to your advantage to switch your choice of doors? —Craig F. Whitaker, Maryland
If you’re like me, you couldn’t read this question without pausing and pondering the answer on your own. The situation sounds familiar – I have a vague memory of watching a similar conundrum on Australian TV in the ’80s. Three doors, a prize, and a gloating authority figure offering ambiguous encouragement. The dramatic climax of a show where beaming Guy Smiley host transforms into dark manipulator, toying with the contestant’s dreams in the name of entertainment.
If they’re your dreams you must know the game is rigged, that the host is out to get you. A matador to your bullish goal, twirling and goading on his abstract perch of perfect information. But what if it wasn’t up to him? What if, no matter which door you picked first, he must show you a goat behind another door, then offer to let you switch?
As it happens, this was a question that Marilyn vos Savant was able to answer, in a reply that sparked thousands of angry letters from readers of all stripes. Discarding the host’s motivations by forcing him to offer the switch allows us to boil the question down to one of simple (though counter-intuitive) probability. In this form it is commonly known as the Monty Hall problem, named after the host of a long-running game show called Let’s Make a Deal.
The answer? You switch!
Although it seems at first that switching might give you even odds at best, the truth is that it would always give you a 2/3 chance of taking the prize. If you don’t switch you’ll be stuck with a 1/3 chance, which is exactly the same as you had before any doors were opened. To illustrate, let’s say there are three doors – A (with fabulous prize), B (with goat), and C (with goat). If you can trust me here, and always agree to switch, then this can play out in three ways…
- You pick door A (prize). Monty opens door B or C to show a goat. You switch to the other door, which is also a goat. FAIL.
- You pick door B (goat). Monty opens door C to show a goat. You switch to door A, which holds the prize. WIN!
- You pick door C (goat). Monty opens door B to show a goat. You switch to door A, which holds the prize. WIN!
See if you play it like this, then the only way you can lose is if you pick the prize door to begin with. So would you have stayed? Don’t worry, we’re not alone. Many of the letters Parade received on the topic were apparently from professional mathematicians and scientists. Some of these challenged her assumption on the ‘always offer‘ behaviour of the host, but many others simply chastised her for faulty probability. Here’s an example from a Virginian professor of mathematics:
You blew it! (…) Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice – neither of which has any reason to be more likely – to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and, in the future, being more careful.
To his credit, this gentleman wrote back later to apologize and retract his complaint, citing “intense professional embarrassment”. An earlier version of this problem had been analyzed in 1959 by Scientific American columnist Martin Gardner, who concluded that ”in no other branch of mathematics is it so easy for experts to blunder as in probability theory.”