Let's Make a Deal!

Our family started watching the 2008 movie "21" starring Kevin Spacey, Kate Bosworth, and Jim Sturgess the other night.  The movie is based upon the book Bringing Down the House and tells the story of how a group of MIT students won a lot of money playing Blackjack in Las Vegas.  We didn't make it to the end, but one of the opening scenes caught my attention.  The MIT mathematics professor played by Spacey challenges one of his students (played by Sturgess) with the classic "Monty Hall Problem", which is the subject of today's post.

The "Monty Hall Problem" is a brain teaser that is named in honor of Monty Hall, the host of the television game show "Let's Make a Deal".  The problem was originally submitted by Steve Selvin to the journal American Statistician in 1975, but it became famous from a letter sent by reader Craig Whitaker to Marilyn vos Savant (best known for having the highest known IQ reported by the Guinness Book of Records) for her weekly column "Ask Marilyn" in Parade magazine in 1990.  Here is the question as posed by Whitaker:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. 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?

Let's break this down using simple rules of probability.  At the beginning of the game, the contestant has a 1/3 chance of winning a car, which means that the probability of winning a goat is 2/3.  When the game show host reveals a goat behind one of the two remaining doors (not selected by the contestant), the contestant can increase the probability of winning a car to 2/3 if he or she switches to the other door!  

When vos Savant first explained this veridical paradox, approximately 10,000 readers, including 1,000 who had a PhD in mathematics, wrote back to the magazine to tell her that she was wrong.  The famous mathematician Paul Erdős (read about the so-called Erdős Number here) remained unconvinced until someone showed him a simulation of the problem.  If a simulation worked for Erdős, it will probably work for all of you too!

Let's look at all the different permutations of this game, which gives us a total of 9 different scenarios:

If the contestant (in this case, you) adopts the strategy of not switching, he or she wins 3 times out of 9 total chances, or 1/3.  However, if the contestant does indeed switch doors, he or she will increase the chance of winning a car to 6/9, or 2/3!

The "Monty Hall Problem" has made its way into pop culture.  For example, the problem was featured in a 2011 episode of the television show "MythBusters", in which the hosts confirmed that when posed with a version of the "Monty Hall Problem", most people will not switch their choices (when the better strategy, again confirmed by the hosts, would be to switch their choice).  The television show "Survivor" also used a version of the "Monty Hall Problem" in seasons 41 and 42.  Surprisingly, in both cases, the contestant chose not to switch and still won the game!  It just goes to show that playing the odds is not always a winning strategy!  Next time, I want to talk about one more famous problem in probability, and then I promise I will explain what all of this has to do with leadership!