Hi, I've thought through this puzzle for a long time and changed the situation a bit to describe my question about it.
Instead of the guest vs host, there are three people choosing from the door.
A choose 1st B choose 2nd C choose 3rd
Each of the guest has 33.33% of winning.
B and C are friends so they decide to team up.
Their team has the probability of winning raised to 66.66%. No problems yet.
B checked the door but found goat. C decides to betray his friend
and disengage his alliance with B.
Now what is the chance of C winning? Is it the initial 33.33% (which turns into 50 50 against A), or 66.66% from the previous alliance he had with B?
Will A get a better chance winning if he exchange the door with C?
If B and C didn't team up from start, would A get a better chance winning if he exchange the door with C after B got it wrong?
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As usual with Monty Hall problems, the question is why B was the one who was offered the chance to check the door.
Assuming that Monty specifically picked B because he knew (and B didn't) that B had a goat, but picked one of the two goats at random, and also that Monty didn't know about the alliance (and therefore didn't change his strategy) then the answers are as follows.
Now what is the chance of C winning? Is it the initial 33.33% (which turns into 50 50 against A), or 66.66% from the previous alliance he had with B?
If A was the winning choice (1/3), then Monty would have chosen B or C. MONTY DiDN'T PICK C.
If B was the winning choice (1/3), then Monty would have chosen A or C. THIS DIDN'T HAPPEN.
If C was the winning choice (1/3), then Monty would have chosen A or B. MONTY DiDN'T PICK A.
All of these choices were at random.
So out of six choices, four of them didn't happen. Of the remaining equiprobable choices, in one case C wins, in the other A wins. So the chances of C winning are now 50/50 with A.
Will A get a better chance winning if he exchange the door with C?
No, same argument.
If B and C didn't team up from start, would A get a better chance winning if he exchange the door with C after B got it wrong?
No, same argument.
The key difference between this and traditional Monty Hall problems is that with one contestant, Monty is specifically trying to confuse and build suspense for that one contestant. With three contestants, either Monty doesn't care who wins, or he's specifically trying to confuse some specific person.
Edited because of mistake in prior analysis, reworking.
Followup to prior post. If we assume that Monty only knows about A and is playing the traditional game, then we have the following case analysis
If A was the winning choice, Monty will show either B or C. MONTY DID NOT SHOW C
If B was the winning choice, Monty will show C. MONTY DID NOT DO THIS
If C was the winning choice, Monty will show B.
So as usual, 3 cases out of six didn't happen, 1 out of six has A a winner, and 2 out of six has C a winner. So A wins by switching, and C wins by A not switching.