Let's imagine that you are a contestant on a game show of which I am the cheesy host.  Let's say you have completed three out of four challenges.  Your final challenge is a game of chance.  You are presented with three doors.  Two of these doors have goats behind them.  The third door has a brand new luxury car behind it.  You are asked to pick one of these doors.  Which one would you like to pick?

So, which do you think is the better strategy:

1. Always stick with your first choice.
2. Always change your choice.

Believe it or not, it is twice as likely for you to win if you change your choice the second time around.  Why?  Let's think about it this way.  What is the chance that your first choice will be a goat?  Two out of three right?  Whereas the chance that you will pick the car the first time around is one out of three.  On the other hand, since I will always reveal one of the doors that you didn't choose but has a goat behind it, I am actually eliminating one of the bad choices.  So, if you originally picked a door which had a goat behind it (the odds being that you will do this two thirds of the time) and then you switch, you will end up switching to the door with the car behind it.  This means that if you always switch your choice, you will win this game approximately two thirds of the time.  On the other hand, if you stick with your first choice, you will only end up winning approximately one third of the time.

For example, let's say that the car is behind door #2.  Right off the bat, you have a two in three chance of picking a door with a goat behind it.  On the other hand, you have a one in three chance of picking the door with the car behind it.  If you pick either door #1 or #3, I, the host, will reveal the other door that has the goat behind it.  In these two situations, when you switch your choice, you will end up picking door #2, because the other door with the goat has been eliminated.  The only way that you would lose by switching, or conversely win by sticking, would be if you first chose door #2.  In this case, when you switch your choice, you will unfortunately pick the door with the goat behind which I didn't reveal.  Of course, if you are the type of person to always stick with your first choice, this would be the only scenario in which you would come out victorious.  Thusly, you will win approximately two thirds of the time by switching your choice, whereas you only have a one in three chance of winning if you stick with your original choice.

Although you may be able to see the logic behind it, you may still wonder if this will prove true.  Well, let's test it out with some trial runs.  I have written some JavaScript code which will be executed after you click the button below.  The first simulation will have the player always stick with his/her first answer for 10,000 games.  The second simulation will have the player always switch his/her answer for 10,000 games.

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