HTH or HHT? Coin Patterns and Penney's Game

Flip a fair coin until you see HHT and count the flips. Then do it again waiting for HTH. Any three consecutive flips spell HHT with probability $1/8$ and HTH with probability $1/8$, so it is natural to guess the two waits are equal. The averages come out different:

The difference is in what a near miss leaves behind. Chasing HHT, once you hold HH you cannot lose ground: another H keeps you at HH, and a T finishes the job. Chasing HTH, a miss on the last flip (HTT) throws you all the way back to the start.

Counting over a long run gives the same answer. Each position starts a copy of either pattern with probability $1/8$, so both patterns turn up once every 8 flips on average, and the mean gap from one copy to the next is 8 for both. Right after an HHT you hold nothing useful, so the wait for the next HHT is the same as a fresh start, and the fresh wait is 8. Right after an HTH you already hold the final H, which can begin the next copy (HTHTH holds two copies sharing a middle H). The gap of 8 includes that head start. From a fresh start you first have to earn the H, which takes 2 flips on average, so the wait for HTH is $2 + 8 = 10$.