Examining the “Monty Hall Dilemma.” You solved the game show problem of whether or not to...

Examining the “Monty Hall Dilemma.” You solved the game show problem of whether or not to switch your choice of three doors, one of which hides a prize, after the host reveals what is behind a door that is not chosen. (Despite the natural inclination of many to keep one’s first choice, the correct answer is that you should switch your choice of doors.) This problem is sometimes called the “Monty Hall Dilemma,” named for Monty Hall, the host of the popular TV game show Let’s Make a Deal. In Thinking & Reasoning (July, 2007), Wichita State University professors set up an experiment designed to influence subjects to switch their original choice of doors. Each subject participated in 23 trials. In trial 1, three (boxes) representing doors were presented on a computer screen; only one box hid a prize. In each subsequent trial, an additional box was presented, so that in trial 23, twenty-five boxes were presented. In each trial, after a box was selected, all of the remaining boxes except for one either (1) were shown to be empty (Empty condition), (2) disappeared (Vanish condition), (3) disappeared, and the chosen box was enlarged ( Steroids condition), or (4) disappeared, and the remaining box not chosen was enlarged (Steroids2 condition). Twenty-seven subjects were assigned to each condition. The number of subjects who ultimately switched boxes is tallied, by condition, in the following table for both the first trial and the last trial, These data are saved in the MONTYHALL file.

Based on Howard, J. N., Lambdin, C. G., and Datteri, D. L. “Let’s make a deal: quality and availability of second-stage information as a catalyst for change.” Thinking&Reasoning , Vol. 13, No. 3, July 2007.

a. For a selected trial, does the likelihood of switching boxes depend on condition?

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b. For a given condition, does the likelihood of switching boxes depend on trial number?

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c. On the basis of the results you obtained in parts a and b, what factors influence a subject to switch choices?

Results of Smoking Survey

“Let’s Make a Deal.” Marilyn vos Savant, who is listed in the Guinness Book of World Records Hall of Fame as

having the “Highest IQ,” writes a weekly column in the Sunday newspaper supplement Parade Magazine . Her column, “Ask Marilyn,” is devoted to games of skill, puzzles, and mind-bending riddles. In one issue ( Parade Magazine , Feb. 24, 1991), vos Savant posed the following question:

Suppose you’re on a game show and you’re given a choice of three doors. Behind one door is a car; behind 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 then says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice?

Marilyn’s answer: “Yes, you should switch.The first door has a 1/3 chance of winning [the car], but the second has a 2/3 chance [of winning the car].” Predictably, vos Savant’s surprising answer elicited thousands of critical letters, many of them from Ph.D. mathematicians, that disagreed with her. Who is correct, the Ph.D.s or Marilyn?