Two Tails (Solution)

There’s a really neat trick to solve this kind of problem.

On each coin flip, you make a fair bet of $1 that the next flip is going to be tails.

• If it’s T, you lose your $1. You bet $1 again on the next flip that it will be T. Repeat.
• If it’s T, you do 2 things:
  • You double down on this bet. You take your now $2 (since you won your last bet) and bet that the next flip is going to be T. If you win, the game will be over and you’ll have $4 from this bet.
  • You also start the process over again, by which I mean you make a brand new $1 bet that the next flip is T.

So here’s the trick. When you finally get TT, you know how much money you’re going to win.

• 2 flips ago, you’ll have started a betting streak of TT that will have just paid off $4
• 1 flip ago, you’ll have started a betting streak of T that will have also just won, so that’s $2 extra dollars.

So when you get TT, you know you’ll have just won $6. Notice that on every single coin flip, you placed a new $1 fair bet. So if you win $6 at the end, and all your bets were fair, you should have placed (on average) 6 bets of $1, meaning that on average this process should take 6 flips!

If you want to try to make sure you understand this procedure, try to figure out how many flips you’d expect until you see HHHH. It should be 30.