I've been wondering something myself, maybe the guys around here can help me out with it.
Is everyone familiar with the Monty Hall problem? To review: You're on a game show. Let's make a deal. You must choose one of three curtains and you win what is behind that curtain. One curtain is hiding a brand new converible, the other two are hiding toasters. You pick curtain number one. Monty, the host, says "wow, great choice! But I'll tell you what, I'll show you what's behind curtain number two." He opens up curtain number two and it's a toaster. He then asks you if you want to switch to curtain number three or stay at curtain number one. What do you do? (Monty always opens up a curtain regardless of whether or not you picked the correct one originally. He never opens the one you picked, and he never opens the one with the car.) The answer is you switch of course, because it's twice as likely that it's in curtain number three.
My question: There are 100 curtains, one with a car, 99 with toasters. You choose curtain number 1, he shows you that curtains 4-100 all are hiding toasters. So you now obviously should switch to curtain two or three. Let's say you switch to curtain two. Now he shows you that curtain three is also hiding a toaster and gives you the option to switch again. Do you go back to curtain one or stay at curtain two?
My understanding, btw, is you should switch back to curtain one for 50.5% chance of winning
The problem with your formulation is that it does not unambiguously determine the algorithm the host is using. Neither does the formulation of the original problem, but it is implicitly understood to be the following by virtue of the fact that it is how the real-life game show worked:
1. Contestant chooses a door.
2. Host chooses a door at random that the contestant has not chosen and that contains a goat. Host reveals this door.
3. Contestant is given a choice to switch or stay.
Now, for your new question, the algorithm could be either of the following:
A.
1. Contestant chooses a door.
2. Host chooses 97 random doors that the contestant has not chosen and that contains toasters. Host reveals them.
3. Contestant chooses to switch.
4. Host chooses a random door that contestant has not currently chosen and that contains a toaster. Host reveals it.
5. Contestant chooses to switch or stay once more.
In this case, you should
switch back to door number 1, with a 50.5% chance of winning.
B.
1. Contestant chooses a door.
2. Host chooses 97 random doors that the contestant has not chosen and that contains toasters. Host reveals them.
3. Contestant chooses to switch.
4. Host chooses a random door that contestant has never chosen this game and that contains a toaster. Host reveals it.
5. Contestant chooses to switch or stay once more.
In this case, you should
stay with door number 2, with a 99% chance of winning.