this post was submitted on 03 May 2025
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If the host were to pick a door randomly, there would be 6 equally likely possibilities.
First, you pick either Goat 1, Goat 2, or Car. In the first case (1/3 chance), the host picks either Goat 2 or Car. In the second (1/3 chance), the host picks either Goat 1 or Car. In the last (1/3 chance), the host picks either Goat 1 or Goat 2.
Out of these 6 possibilities, two of them result in the host revealing a car, which would end the game early. Eliminating those two possibilities, so the host always reveals a goat, leaves 4 possibilities. This is the "new information" that is used by the host.
In the first case (1/3 chance), switching gives you the car. In the second case (1/3 chance), switching gives you the car. In the last case (1/3 chance), switching gives you a goat.
Thank you! This did it.
Breaking down the how the blocks of potential outcomes stemming from the host's choice, and how the reveal of the goat in two choices leaves only the car as a possible outcome for those two choices, was the key information here that filled in the blanks for me.