Nontransitive short con #2: penny ante

Greg says:

I thought that if you liked the nontransitive dice, you might also like a nontransitive coin-tossing problem called "Penny-ante". Basically, your opponent chooses a series of three coin tosses (HTH, for example), and you choose another series (HHT). Then you flip a coin until one of these patterns shows up. So if we flipped HTTHHHT, you would win, because the pattern "HHT" appears at the end of the sequence. Seems fair, right? Well, it turns out that, no matter what your opponent chooses, you can always choose a sequence that's more likely to occur. In fact, your odds of winning *at worst* are 2-to-1. You choose the winner by choosing the opposite of the second position of your opponent's sequence, then tacking it in front of the sequence and ignoring the third position. So if your opponent chooses "THT", you choose "TTH".

Here's a good article about the game (PDF), And it's Puzzle 13 on this page.