People make that statistical error all the time. The question is whether to use one-tailed or two-tailed statistical tests. You are proposing using a one-tailed statistical test, but two-tailed is correct here.
The thing that seems anomalous is getting the same result six ties in a row. Whether that's six heads in a row, or six tails in a row, is irrelevant. What you should be testing for is the probability of getting the same result n times in a row, whether that is heads or tails.
http://www.ats.ucla.edu/stat/m... tells more.
But, wait: what if you are about to say "Well, if it had been Bernie that won six times in a row, I'd accept it as chance, but since it's Hillary, I am suspicious"? Wouldn't that be a reason to use one-tailed probabilities?
No. What you just did there was to insert your own bias. The whole point of statistics is to avoid bias.
A good rule is, if you can't decide if you should use two-tailed probability or one-tailed, always use two-tailed.
--in any case, though, if the Washington Post is accurate, there were over a dozen coin flips, and Sanders also won some of the tosses.