Game One: You are given a choice of three doors. You pick number one. The host opens one of the other two doors, having been given instructions that, if you pick the car, the host is to open one of the other doors, and if you pick a goat, the host opens the other door with a goat. Stalemate. It is a predetermined outcome.
Game Two: The prior game’s outcome stands. The new choice you have is do you keep door number one, or do you switch?
How do you have a 2/3 chance of winning if you switch?
In: 107
Think of it this way: imagine it’s *ten* doors.
You pick one.
What are the chances you picked correctly? 10%
That means there’s a 90% chance the prize is behind one of the doors you did not pick.
Now Monty reveals what’s behind 8 of the other doors (it’s all goats and not random, he knows where the prize is hidden). However there is *still* a 90% chance you picked wrong at the very beginning! The fact that you have been shown what was behind 8 of the other doors doesn’t change that.
By revealing what’s behind the 8 other doors, Monty has given you new information. You already knew there was a 90% chance you were wrong, but you now know that in the event you were wrong (which is 90% likely, remember) that there is only *one* possible place the prize could be.
Or, alternately, imagine you picked a door, then were told “Okay, you can keep what’s behind that door, or you can open up all the other doors and keep whatever you want that’s behind them.
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