The last couple of puzzles have clearly been too easy, so let's crank it up a notch. You and a friend agree to play the following game. You have a fair coin, and each of you is going to pick a sequence of length three of some combination of heads and tails. Once each of you has picked your sequence, you'll start flipping the coin and whoever's (consecutive) sequence comes up first wins.
In a gesture of generosity, you allow your buddy to choose and announce his sequence first. Once he's done that, you'll pick yours, at which point the flipping starts.
This leads to two questions:
- (Easy) Once your friend has selected his sequence, what strategy should you use to pick yours?
- (Not so easy) Based on the fact that you get to choose second and you choose wisely, what's the average probability that you can win this game?