Trying out a challenging problem shared by Amy Hogan

Saw a neat tweet from Amy Hogan yesterday:

I found a series that gives the solution, but don’t know a closed form for the solution. Still, though, I thought sharing this problem with kids would be fun – some of the basic ideas you need to start down the road to the solution are accessible to kids. I’m really happy with how the discussion went today.

We started out by reading the problem and I asked the boys a few questions to make sure they understood how the game worked:

Next we started talking about the solution. My younger son had a little bit of a misconception initially, but the problem started to make a bit more sense for him after my older son suggested a different approach.

Still, it was interesting to hear the boys discuss whether the probability of winning in exactly 3 tosses was 1/4 or 1/8.

Now we tried to figure out how likely it would be to win with a string of exactly three heads at the end (so your final four flips would be THHH).

Finally, we tried to count the ways that you could win with final flips of THHHH – so winning by flipping four heads in a row. Their guess in the last video was that there would be 3 cases – but they realized fairly early on that there would be more than 3.

After we finished I showed them my code in Mathematica that calculated the sum that gives the final answer.  To 10 decimal places the answer is 0.7112119049.

Thank to Amy Hogan for sharing this problem – even though the exact answer is a little out of reach for young kids, it is still a terrific problem for them to study.