Annual Monty Hall Game Show Thread 2015

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Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

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I have to disagree with the "always switch" conclusion. The correct answer about the probability is that it is 50/50.

The problem is not in the math--it is that the setup and explanation of the problem always given is a cheat. The problem is changed in the middle, but the original probabilities are not. That is why it is so counterintuitive.

Those who discuss the probability in terms of 33% and 66% after the first door is eliminated are carrying forward into the new problem (where only two doors exist) information from the original problem, and thus not establishing anything more earth shattering than the fact that we originally had 66% goats and 33% cars to choose from.

In fact, the reason people commonly say the answer is 50/50 is that most people intuitively accept the initial premise that a door is simply removed from the problem--leaving two doors, known to hide a goat and a car. This is clearly a 50/50 choice.

You can appreciate this truth if you consider the person who comes into the game at the point where the contestant is told that there is goat behind one of the doors he didn't pick. That door is effectively removed from the game, as if it never existed. The new person faces simply two doors, with no knowledge about what has gone on before, and the certainty that there is one goat and one car. It does not matter which of the two doors he picks (or whether he "picks" one then "switches", or doesn't switch.) In the end, he chooses one door--and has a 50/50 shot at getting it right. Thus, for him, it doesn't matter if he switches or not--his probability is always 50/50.

Again, the problem with this riddle is that the mathematical explanations always start with the premise that you must carry forward the old 33/66 probabilities from the first problem into the second. That is the cheat--you don't. The Monty Hall problem really is a 33/66 probability problem changed into a 50/50 problem, but discussed mathematically (incorrectly)after the basic premise has been changed as if it is still a 33/66 problem.