Mathew, I think you are not seeing the difference between a pattern and an event. The probability of a particular pattern can depend on previous events, the probability of a single event does not.
Any single coin toss is 50/50, all day, every day.
A choice between 2 doors is 50/50, all day, every day.
To answer your initial post - is it in your interest to switch - what are your odds of picking correctly?
50/50.
You assume a pattern (event A may/does alter event B), when in this example, there is no relationship between event A and B that justifies viewing it as such, absent any other data.
The 'feel vs real' of statistics.
You would be right that you will always end up with two doors - but to say it was 1/2 would be wrong because you fail to look at the process which got you there. One door is really 1/3 chance of being the car and the other door 2/3 chance being the car. Its really that simple....
Just because there is two doors does not mean its 1/2. If I had two boxes and one contained a prize and I put it in one box 1/3 of the time and the other box 2/3 of the time and you knew this, it would no longer be a 1/2 chance.
However you would be correct if lets say after you picked your door and a goat was revealed, you ignored the pass actions and then tossed a coin to determine whether you switched or not. Then it would be a fifty fifty chance.... this would also be true of 100 doors....
Here are some webpages which explain it in further detail.
This is a good page which describes it in detail....
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