We stumbled on this problem in the book my older son is studying over the summer:

A game involves flipping a fair coin up to 10 times. For each “head” you get 1 point, but if you ever get two “tails” in a row the game ends and you get no points.

(i) What is the probability of finishing the game with a positive score?

(ii) What is the expected win when you play this game?

The problem gave my son some trouble. It took a few days for us to get to working through the problem as a project, but we finally talked through it last night.

Here’s how the conversation went:

(1) First I introduced the problem and my son talked about what he knew. There is a mistake in this part of the project that carries all the way through until the end. The number of winning sequences with 5 “heads” is 6 rather than 2. Sorry for not catching this mistake live.

(2) Next we tried to tackle the part where my son was stuck. His thinking here is a great example of how a kid struggling with a tough math problem thinks.

(3) Now that we made progress on one of the tough cases, we tackled the other two:

(4) Now that we had all of the cases worked out, we moved on to trying to answer the original questions in the problem. He got a little stuck for a minute here, but was able to work through the difficulty. This part, too, is a nice example about how a kid thinks through a tough math problem.

(5) Now we wrote a little Mathematica program to check our answers. We noticed that we were slightly off and found the mistake in the 5 heads case after this video.

I really like this problem. There’s even a secret way that the Fibonacci numbers are hiding in it. I haven’t shown that solution to my son yet, though.