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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.
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