> It's amazing how many people trip up on this...
I think you're tripping up slightly.
The chances of getting any other alternative pattern, e.g all heads, all tails, or HTHTHT or THHTHHTHH... or any other should be the same.
To put it in binary terms, if heads is 0 and tails 1, so our sequence describes a binary number from 0, which is all heads, to, something like, 2 to the power of 10 billion minus 1, which is all tails, and everything in between, describes the possible sequences after 10 billion throws.
If each bit has a 50/50 chance of appearing, any of the results is equally likely [or unlikely] Thus all heads is not surprising, or alternatively, if you think it is, then any sequence you get should surprise you in the same way. If it doesn't, then you've tripped up.
Lots of heads in a row, is no more significant or unlikely than 1 2 3 4 5 6 in a lottery draw is.