"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Sunday, January 25, 2009

To change, or not to change?

The following scenario, known as "The Monty Hall Problem" derives its origin from an American game show from the 60's "Let's make a Deal!". The host's name was Monty Hall. There are two goats and a car behind three doors. Your objective is to point to a door where you believe the car may be, and Monty, the game show host, reveals one of the goats (behind one of the other doors) to make life easier for you. Then he makes a deal with you by giving you a chance to change your choice, if you wish to do so, or to remain with your original choice.

Informal proof of the result
What follows may be considered a proof of the Monty Hall outcomes, even for those with little knowledge of probability.
Only one case will be dealt with - why changing gives a 2/3 probability of winning. Realize that 1/3 winning and 2/3 loosing are synonyms in this particular case. So a 2/3 probability of winning upon changing is synonymous with, a 1/3 probability of loosing upon changing.
Also a 2/3 probability of winning upon changing, is synonymous with a 1/3 probability of winning upon staying, which then is equivalent to a 2/3 probability of loosing upon staying. So the above basically states that the following proof only needs to cover one scenario.

Here it goes:
Due to the nature of the content behind the doors we can deduce a priori that by randomly picking a door we're likely, 2 out of 3 to pick a Goat.
That is, the probability that we've picked a Goat on the first pick is 2/3.

I hope that is straightforward so far.

It follows from the above that if we were to change our choice, the probability that we would be changing from a Goat would also be 2/3. This is the key observation.

But, since Monty has already exposed the other Goat, that change would certainly be to a Car.

So to sum up, whenever we pick a Goat on the first pick, by changing we're certain to pick a Car. But since the probability of picking a Goat in the first pick is 2/3, so is the probability of picking a car upon changing. QED

For a fun simulation of the Monty Hall game show click here.

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