[electrical]

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Re: Monty. The way the problem is presented is of utmost importance. In order for you to have an advantage by switching doors, Monty must have shown you a door with a goat INTENTIONALLY and not RANDOMLY when he revealed one. Why? Because he knows where the prize is, and by showing you a goat he is essentially giving you a "tell". If he chooses the door at random, and happens to show a goat, it's still 33% that your door had the prize and 33% that the other door had the prize. What happened to the last 33%? The goat ate it! That is, the other 33% of the time Monty will have accidentally revealed the prize and that makes your choose moot.

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We know for a fact that Monte showed us a goat. There is no possibility that he shows us a prize instead. He shows us a goat. Whether he got there by accident or by choice, he has shown us a goat and we should switch.

Imagine the puzzle expanded to 1000 doors. You pick a door at random, and it has 0.1 percent chance of having the prize. Monte gets the other 999 doors. One by one, he exposes goats until there is only one door left. Do you think it is equally likely that you picked the prize initially, or that the prize is behind the final door under Monte's control?

I should point out that Monte's intention has nothing to do with it, only the 99.9 percent likelyhood that he had the prize under his control in the beginning of the game, and he has eliminated all but one door from consideration.

In the original puzzle it isn't the case that Monte could have shown you the prize one-third of the time. We know for a fact that he showed you a goat. He showed you (a) goat(s). You should switch.