The Famously Controversial “Monty Hall Problem” Explained: A Classic Brain Teaser

When the news broke last week of the death of game-show host Monty Hall, even those of us who couldn't quite put a face to the name felt the ring of recognition from the name itself. Hall became famous on the long-running game show Let's Make a Deal, whose best-known segment "Big Deal of the Day" had him commanding his players to choose one of three numbered doors, each of which concealed a prize of unknown desirability. It put not just phrases like "door number three" into the English lexicon but contributed to the world of stumpers the Monty Hall Problem, the brain-teaser based on the much-contested probability behind which door a contestant should choose.

Let's Make a Deal premiered in 1963, but only in 1990, when Marilyn vos Savant wrote one of her Q&A columns about it in Parade magazine, did the Monty Hall Problem draw serious, frustrated public attention.

"Behind one door is a car; behind the others, goats," went the question, setting up a Let's Make a Deal-like scenario. "You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch your choice?" Yes, replied the unhesitating Savant and her Guinness World Record-setting IQ, you should switch. "The first door has a 1/3 chance of winning, but the second door has a 2/3 chance."

This logic, which you can see broken down by University of California, Berkeley statistics professor Lisa Goldberg in the Numberphile video at the top of the post, drew about 10,000 letters of disagreement in total, many from academics at respectable institutions. Michael Shermer received a similarly vehement response when he addressed the issue in Scientific American eighteen years later. "At the beginning of the game you have a 1/3rd chance of picking the car and a 2/3rds chance of picking a goat," he explained. "Switching doors is bad only if you initially chose the car, which happens only 1/3rd of the time. Switching doors is good if you initially chose a goat, which happens 2/3rds of the time." Thus the odds of winning by switching becomes two out of three, double those of not switching.

Useful advice, presuming you'd prefer a Bricklin SV-1 or an Opel Manta to a goat, and that the host opens one of the unselected doors every time without fail, which Hall didn't actually do. When he did open it, he later explained, the contestants made the same assumption many of Savant and Shermer's complainants did: "They'd think the odds on their door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered. By opening that door we were applying pressure." Ultimately, "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" — a rare consideration in anything related to mathematics, but when dealing with the Monty Hall problem, one ignores at one's peril the words of Monty Hall.

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Based in Seoul, Colin Marshall writes and broadcasts on cities and culture. His projects include the book The Stateless City: a Walk through 21st-Century Los Angeles and the video series The City in Cinema. Follow him on Twitter at @colinmarshall or on Facebook.

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