>>11834023>What is this problem again?First you pick 1 of 3 closed doors. The host at this point will always open 1 of the 2 remaining doors to show you what's behind whichever door said host chooses. Note that there are always 2 Goat Doors and 1 (winning) Car Door.
So first step you pick Door B for example and still don't know any of the doors' contents:
Door A - ?
Door B - ?
Door C - ?
Second step host reveals Door C:
Door A - ?
Door B - ?
Door C - Goat
Now you're down to A or B (stick with your initial pick or switch to Door A).
Here is where most people first hearing this insist A vs. B MUST be a 50/50 decision.
HOWEVER - There are subtle constraints in the host's behavior that change the odds.
The host CANNOT reveal a door with a car behind it at the second step. Doing this would end the game by giving you the answer (some anons will debate the host can do this but it's kind of a silly argument).
The host CANNOT reveal your own initial pick door was a goat at step 2; otherwise he'd reveal you are 100% in need of switching because your first pick is a loss.
Knowing host can never pick your own door or a door with a car, here are the (P)ossibilities using D1 for First Door Pick, HR for Host Reveal, and D2 for Second Door Pick):
P1
D1 - C
HR - G
D2 - G
P2
D1 - G
HR - G
D2 - C
P3
D1 - G
HR - G
D2 - C
Note now that D2 is your "switch" option every time. Out of 3 (P)ossibilities, switching gets you (C)ar 2 times (2/3 odds to win if you switch EVERY TIME).
You have 3 possible choices which are car, goat, or goat, and given the host's constrained behavior it's more likely (2/3 vs. 1/3) you picked one of the 2 goats than it is likely you picked the 1 car door, and the host's reveal will always be a bad goat pick regardless making the final odds entirely depend on how many goats vs car door/s there are in the beginning and whether or not you're going to switch after the inevitable host goat reveal.
