Quote
Java
As far as the raw math is concerned, there is absolutely zero difference between Isildur's scenario #1, and scenario #2.
This is false. Let's say that your prize is behind initial door. Then, the host always opens an empty door. Your initial likelihood of picking the right door is 1/3, so this is the likelihood of host opening empty door and prize being behind your right door.
If the prize is not behind your initial door (2/3 total probability), the host in scenario #1 always picks the empty door. In scenario #2, host picks the empty door in 1/3 of total cases and you lose instantly by host picking the prize door in 1/3 of cases.
So the probability distribution is this:
#1
Probability of winning by staying = 1/3
Probability of winning by switching = 2/3
#2
Probability of winning by staying = 1/3
Probability of losing by staying = 1/3
Probability of losing instantly = 1/3
If we postulate that the host randomly picks the the remaining empty door, the chances become
#2
Probability of winning by staying = 1/3
Probability of losing by staying = 1/3
After normalizing the probability, this becomes
Probability of winning by staying = 1/2
Probability of losing by staying = 1/2