Any reactions from other participants?wtf wrote: ↑Tue Jul 03, 2018 9:33 pmThe problem becomes clear once we explicitly state the assumptions.
* Monty knows what's behind each door.
* Monty must reveal exactly one door.
* Monty must reveal a door concealing a goat.
Now the puzzle is clear.
The player makes a choice.
1/3 of the time the player chooses correctly and the other two doors conceal goats. Monty reveals one of the goats. Switching picks the other goat and the player loses.
The other 2/3 of the time, the player has initially chosen a goat. There are two doors left, one concealing the car and the other concealing a goat. Monty must reveal the door concealing the goat. The remaining door contains the car. Switching wins the car.
So 1/3 of the time switching loses; and 2/3 of the time switching wins.
Let me repeat this. 2/3 of the time the player's initial guess is a door with a goat. That leaves one door with a car and one with a goat. Monty is required by the rules of the game to reveal the door with the goat. That means that switching guarantees we get the car. This case happens 2 times out of 3.
1 time out of 3 the player's initial guess is the door with the car. Switching loses.
So switching is the winning strategy in 2 out of 3 equally likely cases.
I hope someone now will answer the question of how much a gambling casino should charge a person to play the game. After that, I hope you'll be interested in looking at or presenting generalizations of the game: What the player should do to maximize the odds of winning, if there are more than three doors, and also if Monty leaves more than two unopened doors, one of which is the player's first choice.
To wtf: Posing the game with goats as losers and a car as the prize led to this anecdote: There was a farmer who played the game and never switched, because he wanted a goat more than the car.