>>13515869THE GAME WHERE THE HOST PICKS A DOOR AT RANDOM
So let's first understand all the possible outcomes that could happen from keeping your door or switching, BUT, in a world where the host picks a remaining door at RANDOM (i.e. NOT the ACTUAL Monty Hall game, but one where the host has no knowledge of where the car is and just opens a random door).
There are 3 doors. Let's say you always pick door 1 (it doesn't matter which label of door you pick first, because all other scenarios of initial door are equivalent).
You pick door 1.
Now, there are three possibilities of where the car is, because there are 3 doors. It could be in door 1 (the one you picked), door 2 or door 3.
You picked door 1, so the host will now open either door 2 or door 3 at RANDOM. Now let's make a table of the 3 possibilities of which door the car is behind, and for each of those possibilities, the outcome of whether a SWITCH will win or lose for each of the 2 random host door reveals.
BEHIND DOOR 1
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OPENS 2: Switching LOSES
OPENS 3: Switching LOSES
BEHIND DOOR 2
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OPENS 2: You LOSE as soon as the door opens
OPENS 3: Switching WINS.
BEHIND DOOR 3
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OPENS 2: Switching WINS
OPENS 3: You LOSE as soon as the door opens
There are 6 possible scenarios above, and switching can only win in 2 of those scenarios. That's a 1/3 chance of winning, and so switching in this different game MAKES NO DIFFERENCE (i.e. where the host can pick the door at random, and it's actually POSSIBLE to lose at the point the host opens the door).
Now let's look at what happens to the outcomes when the host does not allow you to lose by revealing the car to you.
