Saturday, July 16, 2011

The Monty Hall Paradox: What are the Odds?

One subject in mathematics always confuses people. No, not Calculus. This is the one some kids call "easy." Probability and Statistics is so difficult for us to understand. Let's look at one of my favorite problems that is famous in the world of game theory: The Monty Hall Paradox.

Suppose you are a contestant on the old TV show Let’s Make a Deal®, hosted by Monty Hall. Monty shows you three doors. Behind two of the doors is trash; behind one of them is a new car. You choose a door, and Monty then opens one of the other doors, revealing trash (he can always do this). You are then given a chance to switch your choice to the other door. What do you do?

The average person will stay because they don't like to be wrong. People also have a strong tendency to go with their gut instinct. Mathematically, the average person also stays because well, it's 50-50 right?

Turns out this is not the case. Switching doors doubles your chance of getting the car, bringing the odds from 1/3 to 2/3. This fact was impossible for me to understand, but I eventually figured out a reason, and understood why humans don't understand statistics.

Let's bring it to the basics, experimental probability. You used these principles to solve June's problem of the week (I hope you did it!!). We will look at all the possible scenarios.

1)You pick the first door and the second door has the car (the second door will always have the car for us). Monty reveals that the third door has trash. Since we said it's better to switch, we switch to door two and get the car.

2) You choose the second door, and Monty reveals that the third one has trash. As a math enthusiast, you know it is best to switch, so you move to door one and see a big pile of junk. Plain, dirty junk.

3) You go to the third door and Monty reveals the first. You then switch to the second door and drive home in a brand new car.

There were three different possibilities, and two got you the car. This got me satisfied. If you have any other reasonings for the people who aren't convinced, please post them! Isn't that cool?!

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