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Brilliant explanation and one I've not actually seen used before. There are many ways to walk someone through why switching is the best outcome but this is the most succinct I've seen. Well done sir.

No. It doesn't change any probabilities. You still have a 1in3 chance of picking the correct door at the beginning. Think of it like this... You have seen the show before and know the format, so you are starting the finale knowing you are going to stick or knowing you are going to swap...

The "solution" is to stop thinking about the second half of the problem as some sort of "random" event. There is no probability, only a strategy. There is NEVER a 50/50 chance anywhere in the puzzle (even though I said there was in my first reply - its been a while since I actually had to...

If you are right first time and you choose to stick you win. If you are right first time and you choose to swap you lose. If you are wrong first time and choose to stick you lose. If you are wrong first time and you choose to swap you win. These are the only 4 outcomes possible. Agree...

No. The prize cannot be moved, that would rig the game.

The gameshow gets cancelled if Monty starts revealing the prizes, so the floor manager obviously tells him which door to reveal. But yes. The maths doesn't require Monty to know, just that if an incorrect door is shown, you are better swapping. If X is the prize and Y is the choice made...

Always swap. Monty can only ever remove an incorrect door and so you chances increase from 1in3 to 50/50. Unless you are from Birmingham... In which case don't worry. The prize will be a speedboat and useless for you so drop your darts on the floor and leave.