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 When the contestant has chosen, the host theatrically opens one of doors she did not choose, to reveal a goat.
-There are now two doors left; one concealing a goat, one concealing the car. Assume the host will never open the door the contestant chose, nor the one concealing the car, and the car will not move.
+“Knowing what you know now, would you reconsider?”}}
-Should the contestant stick with her original choice, change to the other door, or should she be indifferent?}}
+If you have not seen it before, intuitively you may say, well, at the beginning each door carried an equal probability — 1/3 — and the remaining doors still do after the reveal — 1/2 — so while the player’s odds have ''improved'', either choice remains even. It diesn’t matter whether she sticks or twists, so she should be indifferent.
-If you have not seen it before, intuitively you may say, well, each door carried an equal probability at the beginning — 1/3 — and after the reveal — 1/2 —so while the player’s odds are now ''better'', they are the same whether she sticks or twists, so she should be indifferent.
+Bayesian probability theory shows this intuition to be ''wrong''.
-Bayesian probability theory shows this intuition to be ''wrong''.
+Staying put is to commit to choice you made then the odds were worse. So its odds remain the same. You have no more information about your original choice: you already knew it may or may not contain the car. You do, however, know something new about one of the doors you ''didn’t'' choose. The odds as between the other two doors change, from 1/3 each to 0/3 for the open door — it definitely ''doesnʼt'' hold the car — and 2/3 for the closed one, which still might.
-The new information tells you nothing about your original choice: you already knew it may or may not contain the car. It does, however, tell you something about the doors you didn’t choose. The odds as between the other two doors change, from 1/3 each to 0/3 for the open door — we now know it definitely doesnʼt hold the car — and 2/3 for the closed one, which still might.
+The probabilities for the remaining options are therefore 1/3, for your original choice, and 2/3 for the other remaining door.
-The probabilities for your remaining options are therefore 1/3 for your original choice and 2/3 for the other remaining door.
+Oddly, a new person who now arrives and is presented the choice without that prior information, would calculate the probability at 50:50. The probabilities are a calculation based upon what you know. The calculation would be wrong because an important assumption in calculating probabilities - that the car and goat were randomly, normally distributed between two doors - is wrong. A third door has been unrandomly eliminated.
 So you ''should'' switch doors. You exchange a 1/3 chance of being ''right'' for a 1/3 risk of being wrong. This proposal outrages some people, at first. Apparently, even statisticians. But it is true.