What you're talking about is called hypothesis testing. Basically, you can figure out exactly how unlikely a given event is under a certain set of assumptions

What you're talking about is called hypothesis testing. Basically, you can figure out exactly how unlikely a given event is under a certain set of assumptions (in this case, how unlikely it would be for a fair coin to show heads 10 times in a row). If it is sufficiently unlikely, where you and people reading your work decide what "sufficiently unlikely" means, you can claim that your initial assumptions were incorrect. I've been away from statistics for too long to go into specifics, but you can mathematically determine how large a sample size you need to be X percent confident that something weird is happening. I'd suggest reading http://en.wikipedia.org/wiki/Hypothesis_test http://en.wikipedia.org/wiki/Hypothesis_test for more info.