I actually built this on a whim after watching that episode of TNG where Data locks everyone out of the Enterprise.
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.
The secret? I distracted myself with the website text and kept pressing the buttons quickly without thinking about it.