I am trying to determine if the probability of a given outcome in a series of poker hands should be based on previous outcomes. Let's say that I play an infinite number of poker hands along with 8 other players using a standard 52 card deck. Each player receives 2 hole cards and there are 5 community cards.
I am dealt two flushes of the same suit in consecutive hands. Given the number of possible outcomes, common sense, along with the experience of playing thousands of hands of poker, suggests that it is very unlikely that I would be dealt a third consecutive flush of the same suit. However, the only math I tried seemed to suggest otherwise.
I tried using basic probability for consecutive outcomes in multiple events and the math seems to indicate that my odds of being dealt the same flush in consecutive hands actually improves with each completed flush in such a way that my odds on the third hand would be the same as my odds for any given hand. Is this the case or does it require a more complex form of math because I am considering the probability for the same outcome in three consecutive hands out of an infinite number of hands played?