The Monty Hall Problem

The Monty Hall problem is one of the most interesting probability puzzles. It is a veridical paradox because the result appears impossible but is demonstrably true. The problem is following:

Suppose you are on a game show. You are asked to choose one of three doors: behind one door is a car and behind the other two are goats. Your goal is to get the car. After you pick a door, the host, who knows what’s behind the doors, opens another door, which has a goat (since the two doors that you did not choose must have a goat). The Monty Hall problem asks, “Should you stick to your original choice or should you choose the remaining unopened door or does it make any difference?”

 

Now most people will say that it makes no difference whether you switch your choice or not, since you know that one of the closed door has a car, the other has a goat, and therefore, the chance of getting the car is 50% either way. But, the answer could not be more wrong. The Monty Hall problem is a puzzle about probability. The problem is simple to understand but the answer is counter intuitive, which makes it a great puzzle to study.

Let’s do a probability analysis. There are two possible choices: not switching and switching.

Not switching: When you first choose a door the probability of getting the car is 33%, since there are 2 goats and 1 car. If you do not switch, the probability does not change because you made the decision before knowing what is on the opened door.

Switching: There are two scenarios to consider. If you originally picked a car then switching will definitely give you a goat, and  if you originally picked a goat then switching will definitely give you a car (since the host has to reveal the only remaining goat). You have a 66% chance of picking the goat on your original choice, and therefore, there is a 66% chance of winning the car if you switch.

So, it is better to change your original choice. The Monty Hall problem is criticized to be wrong by many mathematicians. In fact, one of the most famous mathematicians Paul Erdős remained unconvinced until he was shown a computer simulation confirming the predicted result.

[Inspired by an episode of Numb3rs]

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2 thoughts on “The Monty Hall Problem

  1. John Sirach September 24, 2013 at 12:49 pm Reply

    This was also demonstrated for example in an episode of mythbusters on the discovery channel, and other programs where it showed a higher change to “win” when switching.

    At mythbusters they tested an always and a never switching situation. The always switching test showed a lot more “wins”.

    But what i wanted to say is that you have a very nice blog with interesting articles. I will keep an eye on it.

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