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?