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.

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

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.