Answer by Michael Lamar:
In a way, your question gets at the heart of the difference between probability and statistics. In probability, we pick a probability model to describe some experiments and then we use that model to make predictions about the outcome of such experiments. A natural choice for a model to describe a coin flip is the 50/50 heads/tails model with independent flips. Using this model, we predict that, in an experiment in which n consecutive flips have come up heads after tossing the coin n times, there is still a 50% chance that the next flip will show a head.
In statistics, we try to understand what model makes sense based on the outcomes of previous experiments. So if we see n heads in the first n flips, we infer that some model other than the 50/50 independent model might be the right one.
In other words, if your ask a probabilist what is the probability that the 100th toss of a fair coin will be heads if the first 99 tosses were all heads, he would correctly answer "50%." If you ask the same question to a statistician, she would correctly respond, "There is no way that your coin is really fair."