The Monty Hall Dilemma

Imagine yourself on the Let's Make a Deal Show, which ran from 1975 to 1985.  The host, Monty Hall, shows you three closed doors.  Behind one is a great prize (let us say a new car), behind the others is nothing or something of less value (let us say a goat -- sorry, goat lovers).  You get to select one door.  You pick door # 3.  Rather than opening door # 3 and showing you whether you get the car or the goat, Monty opens door # 1 and shows you that there is a goat behind it.  Then he asks you whether you want to stick with door # 3 or switch to door # 2.  What should you do?

When I first heard this problem, my response was that you don't really have any good basis to answer this question unless you have more information.  My assumption was that Monty knew where the valued prize was and he did not always stop and offer you the option of switching (I never watched this show any longer than it took to change the channel to another station).  Given that assumption, your best strategy depends on whether Monty is more likely to offer you the option of switching if your initial choice was wrong or if he is more likely to offer you the option of switching if your initial choice is right.  Eventually another psychologist (Matthew Kelley at Lake Forest College) explained to me that Monty always stopped after the initial choice, always opened one of the other doors, that other door never revealed the valued prize, and Monty always offered the contestant the option of switching -- clearly Monty knew where the valued prize was.  Now that changes everything.

Let us figure the probability of winning.  There is a 1/3 chance that your initial choice will be correct and a 2/3 chance that it will be incorrect.  Accordingly, if you stick with your original choice, you will win 1/3 of the time, but if you switch, you will win 2/3 of the time.

For a more detailed explanation, see http://math.ucsd.edu/~crypto/Monty/montybg.html and http://mathforum.org/dr.math/faq/faq.monty.hall.html .  To play, see http://www.mredkj.com/javascript/prob_rules.html .  There are a lot of other simulators and pages on this out there too.

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