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.

Relat­ed Con­tent:

Are You One of the 2% Who Can Solve “Einstein’s Rid­dle”?

Can You Solve These Ani­mat­ed Brain Teasers from TED-Ed?

John Cage Per­forms Water Walk on US Game Show I’ve Got a Secret (1960)

A Young Hunter S. Thomp­son Appears on the Clas­sic TV Game Show, To Tell the Truth (1967)

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.