Can a fair coin have dependent tosses? I am studying a bit of probability theory now and I want to know if a fair coin can have dependent tosses. I mean a coin has $50%$ chance each of getting heads or tails. Does it make sense to say the result of $n$'th coin toss has $70%$ chance of being the same as the result of $n-1$ 'th toss? Or when one says result of coin toss is dependent on previous results, can that coin be fair? Does it make sense to say something like that?
 A: By definition, a fair coin is when a sequence of independent Bernoulli trials has an equal probability 1/2 for either outcome per trial. Saying that the nth coin toss has a 70% chance of being the same as the n-1th, or is dependent on the n-1th toss contradicts the definition of a fair coin because each trial must be independent, and the probability of either outcome must be 1/2 for each toss. If you had a coin that did take previous tosses into account, this would be a biased coin. Thus, stating "a fair coin where each toss is dependent on previous results" is illogical.
A: I think it depends on what exactly you mean by the word "fair."
In its common usage, the word connotes that the tosses are independent and that each throw has an even chance of heads or tails. So if you mean the word in any way close to how it is universally understood, then no, the word "fair" cannot include the scenarios you described.
But that answer's no fun, of course. So let's run a thought experiment: if you were to distill it down to something axiomatic, what are the most sensible conditions to impose on the word "fair" (assuming that you don't want the default idea)? To me, "fairness" demands that heads and tails should on some level be equally common. I'd argue that having the same unconditional probability on a given throw being heads or tails is the more important property to keep. So perhaps your coin could just be the same throw, repeated infinitely often, after a first random throw. Perhaps you impose some strong inter-throw correlation like you described. Maybe each throw alternated with probability 1. But at the end of the day, calling any of these "fair" is quite a stretch and you'd be hard-pressed to find a neutral observer who would support those labels. 
