A Mathematic Advantage

Hello again, SighFye here.

I had a thought today about an old game show that had a 3 door competition at the end of it. The contestant was presented with 3 doors. Behind 1 of these doors is a prize. They get the constant to select a door. When the constant has selected they open one of the doors the contestant did not choose and show no prize behind that door. The contestant is then asked whether they wish to stick to the door they selected or switch to the other door. After that, if the contestant selected the correct door, they win, if not, they lose.

There is however a way have a higher chance of winning. Choosing to switch to the other door with increase your chances of winning. But how can that be? Each door has 1/3 chance of winning. Surely this doesn't change. Technically it doesn't no, but let me explain.

This is the true percentage. There is a 66.7% chance of loosing, on your first selection. Say you select Door 1, the host will not open Door 3 as its the winning door. So he opens Door 2. The choice now are...

 

You can take Door 2 out as this one has been opened. So we have...

   
To choose Door 1 or 2 is 66.7% loss. Door 2 is not available so Door 1 is 66.7% loss. Meaning you have a greater chance of winning if you switch to Door 3.

So now I can hear you saying, what if you choose Door 3. Then you will loose. But look at this mathematically. There are three selections at the start, so there are only 3 scenarios.

Scenario 1:
You select Door 1, they open Door 2 and you switch to Door 3 and win!
Scenario 2:
You select Door 2, they open Door 1 and you switch to Door 3 and win!
Scenario 3:
You select Door 3, they open either Door 1 or Door 2, you switch to the other door and lose.

So playing the switch you have a 2/3 chance of winning. Odd are good in my eyes.

Let now look at sticking scenarios. You still only have 3.
Scenario 1:
You select Door 1, they open Door 2 and you stick to Door 1 and loose.
Scenario 2:
You select Door 2, they open Door 1 and you stick to Door 2 and loose.
Scenario 3:
You select Door 3, they open either Door 1 or Door 2, you stick to Door 3 and win.

So playing at sticking, you have 1/3 chance of winning.

So there you have it. How to increase your chances of winning in the three door game. Try it out for yourself here. http://www.theproblemsite.com/games/monty_hall_game.asp#gameTop

Happy Gaming Geeks.

SighFye