Friday, December 15th, 2017

The Monty Hall Problem

July 10, 2013 by  
Filed under Newsletters

Lawyers make decisions based on probabilities all the time, but we may not understand how they truly work.  Consider the Monty Hall Problem.

It’s named, of course, after Monty Hall and the game show he hosted for years, Let’s Make a Deal.  You remember Let’s Make a Deal, right? At the end of the show, the final prize would be a beautiful shiny new car behind one of three doors.  Behind the other two doors were goats.  The contestant, Carrie Contestant, would pick one of the doors, then Monty, with great fanfare of course, would open one of the other two doors to reveal a goat.  He would then offer the contestant the opportunity to switch doors.  Great angst and hand-wringing along with shouts from the crowd would follow.

Should our Contestant switch?  And, of course, why?

The answer is yes she should switch.  Surprised?  Most are.  Many people think that her odds are now 50/50 after Monty opened one of the doors she didn’t pick, and so her chances are equal with each door.

Here’s why she should switch.  Each door has a one third chance of holding the car.  The door she picked had a one in three chance of the car being behind it when she picked.  The other two doors, at one third each, have a combined two out of three chance of holding the car.

The fact that Monty revealed where one of the goats was, and eliminated that door does not change the fact that the contestant has a one out of three chance, it’s just that the other two doors, which combined had a two out of three chance, now only has one door.  That one door now has the two out of three chance of holding the car.

In other words, if she doesn’t switch, it is as if Monty Hall never opened one of the other doors and she still has her original one out of three chance.  The other door will have a two out of three chance.

If you still think it is 50/50 odds, it is important to realize that since Monty Hall knows the placement of goats and car, he interferes with the randomness when he reveals a door.  He is only going to open a door that reveals a goat and not the car.  Monty’s actions are not independent of their placement.  After his interference, the odds are no longer random odds.  When a contestant choses a door which Monty knows has a goat behind it he is forced to reveal the only other remaining goat.

Remember, Carrie Contestant’s initial pick was random, so one third of the time she will have picked the car.  If she were to switch, the only time she would lose is the one third times in which she already had picked the door with the car behind it.  The other two thirds time, she had chosen wrongly and (because Monty Hall has now eliminated the other goat), she should switch.  Since there are two goats out of three doors, she will initially pick a goat two out of three times.  Thus, she will win the car two thirds of the time if she switches, versus one third of the time if she doesn’t.

As we live more and more in a digital world, the actual reality of our world will become more and more observable through the collection of data.  It will be an increasingly important skill to understand the true relationship of possible outcomes.

Kevin C. Coleman is a professional mediator and attorney who performs mediation services throughout the San Francisco Bay Area, including San Francisco, Marin County, Alameda, Contra Costa, Sonoma and outlying counties and throughout the state.

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