There's another version where a stats professor divides people into two groups. One group flips a coin 50 times and writes on a whiteboard the results. The other comes up with 50 results.

They leave the room and when they come back immediately know which is which. Specifically because of the over-alternation that yours mentioned. A string of HHHHHH is exactly as likely as HTHTHT, but people don't intuit that.