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The three door Monty Hall problem and variable change?

Okay we all know the three door problem or Monty Hall and how percentage increases if you switch doors. For those who don’t know what I’m talking about please just google “Monty Hall” or wiki it. My question is what would the percentage be if there was another person that has chosen a door. Let’s say the Game Show Host knows which door holds the car and which ones hold goats. You thought you were the only one playing. Until the Game Show Host reveals that another person was playing the game and he has selected one of the other doors. Well, since GSH has revealed the door with the goat leaving behind your door and the other person’s door, and the other person doesn’t know if he has the car or not. Now, GSH tells “both” of you that it is in your best interest to switch doors. First of all would you switch and why? Do percentage still increase for “both”?

I hope you guys understand this problem and my question?

There are 3 possible scenarios:
Case 1:
A(car), B(goat), C(goat)
Case 2:
A(goat), B(car), C(goat)
Case 3:
A(goat), B(goat), C(car)

Let’s assume you always pick door A and the second person always picks door B. (It could be done randomly, but the results are the same, just more complicated to explain).

Now the host comes along. He wants to show the goat door that neither of you have picked.

Here’s where this differs from the original Monty Hall problem. The host can only do this if you have case 1 or case 2.

So given that the host has shown you a goat door, you know that case 3 couldn’t have been the original set-up.

In this revised scenario, you have the same chance of getting the car whether you stay or whether you switch.

It’s a little paradoxical, because you can’t get there if both of you have picked goats (Case 3). In other words, the chance that you picked the car is not 1/3, but actually 1/2 *given that the host was able to show both of you the goat door*. With this additional given, your odds don’t change either way.

That makes sense because you wouldn’t want the odds to show that you have a better chance *and* the other player has a better chance. That would be totally paradoxical.

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