You can read Aaron's answer, Dave, to quickly understand the Monty Hall setup. It's genius.
But another example is having three people sit in a chair, and then one of them getting up, walking away.
Would you say that only two chairs are remaining or that three chairs are remaining (now with two people in total in them)?
The Monty Hall problem is about people removing that door altogether, while in reality the equation remains one of three doors. There will always be two doors without the prize behind them, and showing that fact for one of them does not change that equation. The doors remain three doors.
The first pick remains one-third chance, so not switching means you are sticking with one-third chance.
The other two doors combined have two-third chance, and the game host showed you one of the two not holding the prize. That means that the remaining door (of the two) holds two-third chance.
This also means that winning the prize is not guaranteed. There is indeed one-third chance you got it right the first time. But you double your chances by switching.
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In the UK, voters are very familiar with strategic voting. It is not just a cultural thing, the Brits are smarter and tell each other the truth, the whole truth, and they do not lie with words when it comes down to the voting public. They inform each other.
Here, in the USA, people tell each other things that sound good. They stick it out with their personal political truth, and they ignore the larger political game of power.
As such, they remove the chair that is now empty and focus on the two people sitting in two chairs. Voters ignore that political power has at minimum three chairs. And they don't talk about it.
The Monty Hall problem is therefore about two things:
1. Reacting when there is nothing to react (removing the door altogether).
2. Not reacting when they should react (refuse to believe that their mind is tricking them into an incorrect idea).
With not-reacting when they are mistaken, folks do not talk this out among each other. The silence is the mistake. The silence is the Monty Hall problem.
Power and truth are two distinct parts in politics. Power is central. Truth is secondary. When you know your vote is not going to matter (because you know you are voting for the loser), then switch it to a third-party candidate.
