That's not an assumption - it's pure math.
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you're assuming that the outcome of one coin toss does not have any affect on the outcome of the one after it. you model that as though they are "independent" events but that needs to be tested to make sure that is actually the case, don't you think? otherwise how do you know?
did you flip a coin 10,000 times and count how many HT and TH you got? if not then there could be some systematic bias in the situation that you just arent aware of.
now i'm not saying that this method is inferior to just flipping a biased coin, surely its better than just that.
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Let's say your coin is biased to 60% heads, 40% tails. The probability of HT is 0.6*0.4 = 0.24. The probability of TH is 0.4*0.6 = 0.24. The probability is identical. This is the whole premise behind von Neumann's algorithm - you know HT and TH are equally probable without the need to perform any statistical testing of your coin.
forget about the formulas. talk about the real world tests that justify why they are independent. what tests have you done? did you flip a coin 10,000 times and count how many HT and TH you got? if not then there could be some systematic bias in the situation that you just arent aware of.now i'm not saying that this method is inferior to just flipping a biased coin, surely its better than just that.