system
Given the following post, respond as if you're commenting on Reddit:
human
About the Monty Hall problem, what if the host doesn't know which door has the prize?

Let's say the director of the show wants to diminish the chance of the contestant to win the game and don't tell the host where the prize is. After the contestant picks a door, the host opens another, at random. If the host randomly opens the door with the prize, they have to retake, after shuffling everything, otherwise the game continues as usual. For the contestant, changing the door still increases his probability to win?
gpt
If Monty doesn't know where the prize is, then 1/3 of the time he will open the door with that prize.  By specifying that in those cases there is a retake, this puts the odds to 50/50.

Let's look at the Keep strategy:

1/3 of the Time the contestant chooses the prize.  Monty's guaranteed to choose a Goat. The contestant then keeps their door and wins.

2/3 of the Time the contestant chooses a goat.  Half the time Monty will choose a goat too. So, in 1/3 (2/3*1/2) of the cases the game continues, the contestant keeps their door, and loses.  But, in the other half the time (1/3 of all games) Monty chooses the Prize and the game resets.

Since resetting doesn't change the odds (including the odds of resetting again and again), that gives equal odds of winning and losing for the Keep strategy and thereby equal odds for the Switch strategy.  Of course, it does draw out the game, and there will be some times when filming takes *days*.