Interesting variant of Monty Hall

Question

I was wondering how we could think about the following variant of the classic Monty hall theorem,

Suppose now that the host actually does not remember what is behind any of the doors, once you choose one of the three doors, he will open at random one of the other two doors. If he reveals the prize the show ends, if he does not reveal the prize, should you still switch doors?

I was trying to formulate it but I am just not sure if I am making correct assumptions.

For example,

say we choose door $A$, then the probability that he opens door $B$ is equal to the probability he chooses door $C$ is equal to $0.5$.

The probability that the prize is behind door A, B and C is 1/3

the probability that Monty ends up opening any given door is also $1/3$

So should one just condition on something else now?

Looking forward to hearing any opinions on this,

Thanks

Answer 1

Suppose you choose door 1. And Monty reveals door 2, but Monty has no prior information.

Then the conditional probability that the prize is in fact behind door one, given that it is not behind door 2 is $50\%$

If Money has no information, then not enough information gained by the reveal, to justify switching.

Answer 2

There is a long discussion of this on The Straight Dope by Cecil Adams. I will add the link here:

http://www.straightdope.com/columns/read/916/on-lets-make-a-deal-you-pick-door-1-monty-opens-door-2-no-prize-do-you-stay-with-door-1-or-switch-to-3

Answer 3

1. We pick right at first (happens 1/3 of the time).

If that happens, then Monty will for sure show a goat and we will win 100% if we stay, 0% if we switch.

2. If we pick wrong at first (happens 2/3 of the time).

   1. Monty now ends the game with 50% chance and neither loss or win.
   2. The other 50% we will benefit switching 100% of the time.

So with 1/3 we will lose for sure on switching if a goat is shown. With 2/3 if a goat is shown we win by switching. If a goat is not shown there is nothing for us to choose.

There is 1/3 stolen away if Monty shows the car. But that is nothing we can do anything about. The only thing we can consider is our initial 1/3 vs 2/3 and then given that a goat is shown, we will lose or win 100% respectively, so the conditionals are clear.