Let's play a game.

I'll choose a possible pair of sucessive coinflips: HH, HT, TH, TT. You choose one of the other three.

We then flip a coin until one of our pairs occurs. So if I picked HT and you picked HH, then we'd keep flipping until I flipped a tail following a head, or two heads in a row. In the former case I'd win, and in the latter you'd win.

What is your optimal strategy if I pick HH?

Assuming you'll always pick the pair that maximises your chance of winning, what pair should I pick? (Remember I pick first.)

 

How do things change if instead we both pick a triple (e.g. HHH, or THT?)