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The key to calculate the probability is continuance. Continuance in this case meaning that the winning door is decided before the game begins and is not changed during the game. If the winning door could be changed during the game, your chance of winning would be completely independent of probability and entirely up to the gamehost.
Due to the continuance, your first choice is actually completely irrelevant. The gamehost could easily skip it and just ask you: Would you like to open all but one door looking for the prize or would you like to open just one door?
Lack of continuance is - for instance - the reason that the silly roulette trick where you keep betting on one color and double your bet if you lose, does NOT work in the long run. One spin doesn't affect the other and hitting red 10 times in a row, won't increase the chance of black on the 11th spin. THAT is still 50/50.
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