Monty Hall and Lets Make a Deal Problem

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When I was younger I read the Parade Magazine that came in the Sunday paper. My favorite column was “Ask Marilyn” who is Marilyn vos Savant considered the women with the worlds highest IQ. I also have one of her books which talks about logic, statistics and other topics. Back in 1990 she wrote a column about a question that was asked by a reader. I was reminded of the column while reading my twitter feed, seeing a tweet by Scientific American @sciam about  “Let’s make a deal: Revisiting the Monty Hall problem”.

This isn’t a new problem, but one that is over a hundred years old. It’s easy to relate to today or to most people because it happens all the time in the game show Let’s Make a Deal. Here is the initial question:

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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

The correct answer, contrary to what most people commonly think is that yes it is to your advantage to switch doors 2/3 of the time. But there are a lot of people, smart people, that say your chances of winning are 50-50. The reason that they are wrong is that one of the problems constants is that the host is avoiding the door with prize when opening the second door. His choice raises your probabilities to 2 out of 3 assuming you pick the other door and the host opens a door with a goat.

That the host opens a door with a goat is a constant, because the premise of the game is that there is only 1 prize, the car, and two goats. So the host will always show you the other door with the goat to make you think you have maybe already picked the door with the car. The game and choice would be worthless if he showed you the door with the car if you not picked it already. Removing that premise then having the host opening a door at random is a completely different problem.

It’s an interesting problem, but you have a 2/3 chance of winning if you switch, and it has been proved by tests. Please take the time to read the articles and Wikipedia citations that I have referenced above. Feel free to comment. This is the type of problem that I love to think on.


2 thoughts on “Monty Hall and Lets Make a Deal Problem

    Gregg Gordon said:
    April 16, 2011 at 1:10 pm

    Interesting that SA has to bring this up yet again. I’m not all that great at probability, and yet the proper explanation has always made sense to me. I see on SA that people are trying to argue it out using math and logic, so I just made a comment there that points out what’s always been obvious to me… probability theory assumes random variables, and Monty is not acting randomly, thus you can’t just plug his actions into the formula.

    Ray Ebersole responded:
    April 16, 2011 at 4:08 pm

    Right on point Gregg. That is how I tried to portray it here too. When you take into consideration that Monty isn’t being random, that throws out probabilities altogether. Makes total sense, did back in 1990 when I first read it and it still does today.

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