In “A Random Walk through Middle Land” [Skeptic], Michael Shermer presents the following scenario: you are a contestant on Let’s Make a Deal and are shown three doors. Behind one is a new car; the others hide goats. You choose a door, and host Monty Hall reveals a goat behind a different door. Shermer then posits that you have a two-thirds chance of winning by switching your choice because there are only three possible door configurations (good, bad, bad; bad, good, bad; and bad, bad, good), and with the latter two you win by switching. But he fails to recognize that the second configuration has been taken off the table. You have a 50 percent chance.
SHERMER REPLIES: In nearly 100 months of writing the Skeptic column, I have never received so many letters as I did disagreeing with my description of the so-called Monty Hall Problem. James Madison University mathematics professor Jason Rosenhouse, who has written an entire book on the subject—The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser (Oxford University Press, 2009)—explained to me that you double your chances of winning by switching doors when three conditions are met: 1) Monty never opens the door you chose initially; 2) Monty always opens a door concealing a goat; 3) When your initial choice was correct, Monty chooses a door at random. “Switching turns a loss into a win and a win into a loss,” Rosenhouse says. “Since my first choice is wrong two thirds of the time, I will win that often by switching.”
At the beginning you have a one-third chance of picking the car and a two-thirds chance of picking a goat. Switching doors is bad only if you initially chose the car, which happens one third of the time, and switching doors is good if you initially chose a goat, which happens two thirds of the time. Thus, the probability of winning by switching is two thirds. Analogously, if there are 10 doors, initially you have a one-tenth chance of picking the car and a nine-tenths chance of picking a goat. Switching doors is bad only if you initially chose the car, which happens one tenth of the time. So the probability of winning by switching is nine tenths—assuming that Monty has shown you eight other doors with goats.
Still not convinced? Google “Monty Hall Problem simulation” and try the various computer simulations. You will see that you double your actual wins by switching doors. One of my skeptical correspondents ran his own simulation more than 10,000 trials, concluding that “switching doors yields a two-thirds success rate while running without switching doors yields a one-third success rate.” (Go to http://tinyurl.com/bu9jl for the simulation.)
Note: This article was originally printed with the title, "Letters to the Editors".