Game shows and probabilities

[ydoaPs]

Congratulations, you're on a game show with a chance to win $10 billion after taxes! The answer to this puzzle is related to just what that chance is. Here's the game:

 

You have three doors from which to choose. Behind one is the money, and there is nothing behind the others. After you select a door, the host will open one of the other two doors and then give you a chance to switch your choice before revealing what is behind every door. If the first door is opened and there is nothing behind it, should you switch your choice or keep with the door you chose originally? Why?

[GlassPilot]

Switch.

If you keep your door you have a 1 in 3 chance of winning. By switching you have a 2 in 3 chance of winning. Has to do with the host introducing known data into the problem after your initial selection.

Anyway. Absolutely switch.

[Pugdaddy]

This is interesting and maybe even counter-intuitive. In the first round I have 1 chance in 3 to chose the money door. Which also means that I have 2 chances in 3 that I picked the empty door. The key is that the host knows what door the money is behind and opens one of the two doors that were empty. I have two choices now stay or switch. If I stay, I can only have the money door if I picked it in the first round which is 1 chance in 3. If I switch, I can only win if I picked the empty door in the first round. Since my chances of choosing the empty door in the first round is 2 in 3, I also have a 2 in 3 chance by switching in the second round. If the host did not know what door the money was behind and opened a door randomly, then the odds would be the same for staying or switching.