Dr. MontyHall (or, How I came to stop worrying and love the irrationality)

I swear that the Monty Hall problem proves that there is a deep logical inconsistency in the universe. The problem goes as such: A contestant is on a game show, and can pick from one of three doors. Two doors have a goat behind them, and one has a car. The contestant picks a door, and one of the doors which contains a goat (and is not the door which he picked) is opened. He can then either stay with his door, or switch to the other non-revealed door. Which is he better off doing? For an example, there are doors A, B, and C, and Contestant Qof (Roman and Greek letters are so restrictive in mathematics). Qof picks door A. Door C is revealed to have a goat. Should Qof stick with A, or switch to B? It seems that with one door down, Qof has a 50/50 chance, right? You'd LIKE to think that. Turns out that reality has a surprise for us (and by reality, I mean mathematics). Here's how it really works. Qof has a 1/3 chance of getting things right with his/her (we really need a neutral, but personal, 3rd personal pronoun) choice of A. Now, once C has been opened, Qof could stay with that 1/3 chance of A. Alternatively, you could say that Qof had 2/3 a chance of missing with picking A. Given this and the fact that C has been shown to be a goat, B must be the prize, and there is a 2/3 chance that by switching to B Qof will be riding home in style. It seems like there is something fishy here. What about the earlier argument? Why doesn't that work, as switching doors could be considered to be equivalent to picking one of two doors? Alternatively, it seems as though one could end up with two different probablities for the same door, depending on whether one starts with it and stays with it, or picks another door and then switches to it. However, after running a few million trials on a computer program (quite literally), it turns out that the solution is that switching doors doubles one's chances of winning. This brings one down to an ultimatum: Either accept the mathematical explanation of the Monty Hall problem, or face the fact that reality is inconsistent at its core. Given the obviousness of this choice, I think that maybe I should invest more of my time in Continental philosophy.