# Gambler's Fallacy, or Monty Hall Problem?

Assume a case where there are 30 doors. 29 have goats, and 1 has a car. You begin to chose doors one by one until there are only two doors left. All the doors you have chosen have been goats, leaving just one goat door, and one car door.

Now two cases:

1. Assuming you have selected a "special" door from the start in your head, and have been picking every other door but that one, what is the probability that the "special" door will be a car.
2. Assuming you have just been picking randomly up until now, what is the probability that that door will be a car?

Now if I understand correctly, I believe the answer to #1 will be 1/30, like the Monty Hall Problem, because when you first selected it the probability was 1/30. I think the answer to #2 is 50%, thus the Gambler's Fallacy because the probability of you picking one then is 50%, independent up to now.

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No, the answer to #1 is also 50%. You didn't know that you don't accidentally open the door with the car. – Daniel Fischer May 16 '14 at 19:59
The Monty Hall effect comes from the fact that Monty knows where the car is and purposefully opens the/a goat door. – Hagen von Eitzen May 16 '14 at 20:02
Thanks, but I don't understand why it is any different if Monty shows the goats, than if the player shows the goats? – Dane Bouchie May 16 '14 at 20:03
The difference in this case is that you were "lucky" to choose 28 consecutive goats. If at any stage you were to have chosen the car, the process would have ended. Or you would have had to start over with a new set of 30 doors. – Paul Hurst May 16 '14 at 20:18
I've always thought the Monty Hall problem makes an unreasonable assumption about the game. If a player were to choose a goat door on the first choice, Monty Hall would just opened that door and they win a lovely goat. If the player were to choose the car door, then Monty Hall would show them one of the goats and then let them change if they want. In this scenario, you would get a goat every time you change doors. A more useful question would be to ask what is the probability that a goat will be more valuable than a car in the next 100 years. – Paul Hurst May 16 '14 at 20:22