RE: Would you switch?
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The fact is, you have double the chances of winning the 1 million dollars if you switch doors once you are told one of the doors that has a penny behind it. You have a two out of three chance to win the million dollars if you switch doors every time while only having a one out of three chance if you stick with your first door.
This hasn't been proven to be 100% true. Its just one way of looking at things.
Another way would be:
Contestant Picks Door two out of the three choices given. This gives a 33% chance at winning $1 Million dollars. The other two doors each also have a 33% chance at containing 1 Million Dollars or Nothing.
The person making the offer then reveals Door number 3 has nothing in it and offers the contestant to switch doors.
There's now two doors to choose from not three making odds of winning 50/50. The contestant doesn't gain any advantage via switching doors.
This debate was a hot topic and debated frequently at one point. The game show Lets Make A Deal hightened the debate as they offered prices in the same fashion as your example. That being said, if I was every given such an offer I would change doors because ........ why the hell not.
Yet it has been proven true without a shadow of a doubt using math and computer simulations.
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There's still an ongoing debate about it but for the most part, it doesn't matter to me regardless as I would always pick to switch. The decision to switch is an easy one. If a person believes the unchoosen door now has 66% of winning then its an easy switch. If a person believes the two remaining doors makes the odds 50/50% of winning then switching does no harm.
There is an ongoing debate about almost everything, doesn't mean it's actually valid.
Many believe the Earth is flat. ¯\(ツ)/¯
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Things are always changing. New proof comes up showing old solutions and ways of thinking to be wrong time and time again. Its one of the reasons why I dislike the current school system.
As for your flat Earth example, there was a time that most thought Earth was flat and the people who thought it was round were the crazy ones.
In math things never change, once something has been proven it is true, and the probability is solidly in maths territory.