I found this thread after watching TED talk by Peter Donnelly: How juries are fooled by statistics.

And I have this comment on your explanation in favor of Alice. You state that : "In contrast, for Alice to "win" (that is, observe the sequence HT), she must first toss a head. If the next toss shows a tail, she wins. But if is a head, she doesn't start over. In fact, she is already halfway to winning on the subsequent toss"

My conclusion is that if the first toss is H and the next one is again H, indeed she is already halfway, but Bob is DONE! He wins, the game end there. Or am i missing something?