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What many people don't get about this is that the host will ONLY open a door that has a goat behind it and offer to let you switch for the unopened door. It would be useless to open the door with the car (which you've missed 2 times out of 3), because you would then know you had a goat, and would be pissed off.
So, the opening of the door with the goat isn't really random, and so the probabilities are affected because of that condition set on the opening.
IF THE HOST WERE SIMPLY TO OPEN A DOOR AT RANDOM (revealing the car 1 time in 3), THEN THE PROBABILITY OF GAINING BY SWITCHING IS 50%. BUT HE'S NOT, SO THE ODDS SWING TOWARDS THE SWITCH, BECAUSE HE'S ELIMINATING HALF THE BAD CHOICES (a conditional probability).
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The secret to great marksmanship is deciding what the target was AFTER you've shot.
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