HTH or HHT? It's Not 50/50

What to remember

• HHT and HTH each have probability $1/8$ at any position, but the expected waits are $8$ and $10$.
• For a fair coin, $E[T]$ adds $2^k$ for every $k$ where the first $k$ letters equal the last $k$, so self-overlap slows a pattern down (HHH: $14$, HTH: $10$, HHT: $8$).
• In a race HHT beats HTH with probability $2/3$; races depend on how the two patterns overlap each other.
• Penney's game is non-transitive: against $abc$, reply with the opposite of $b$ followed by $a$ and $b$, and you win at least $2/3$ of the time.
• The method for every pattern problem: write down the states of progress, condition on the next flip, and solve from the end (HHT: 8, 6, 2; HTH: 10, 8, 6).