A neat probability problem from James Tanton

Last month I bought a copy of James Tanton’s “Solve This:  Math Activities for Students and Clubs” on the recommendation of Fawn Nguyen.   As is true with all of Fawn’s recommendations (that aren’t related to college football), it has been a joy to go through.

I picked out a pretty challenging problem to try out for our Family Math today.  My goal was not to give the boys a complete understanding of the solution to the problem, but rather to show them a situation that they could understand and speculate about a little.    Although walking them through the clever solution at the end proved to be a little difficult, I’m really happy with how this problem engaged them and look forward to doing more problems from this book later in the year.

We started with a simple explanation of the problem and used several lego figures to help with the illustration.  Before diving in to the solution, we spent a few minutes just talking about what they thought the answer would be.

Next we tried a few examples.   Almost comically, every time we flipped the coin we got heads, so Unikitty kept falling off the cliff at the first step.  Finally we got a long sequence where we had more tails than heads for a long time – actually a really long time.   This long sequence was a lucky illustration of just how complicated this problem can get if you try to look at it case by case.   Sorry that it is hard to see heads / tails in the flips on the camera – I probably should have used something that had different colored sides rather than a coin.

Next we went to our whiteboard to try to work out the math.   It was interesting to me that both boys thought that the series 1/2 + 1 /4 + 1 / 8 + 1 / 16 + . . . . would show up in the solution.  I wanted to approach the problem with binary trees to show them that the series they were looking for doesn’t show up quite as easily as they thought it would.    I also wanted to illustrate this approach because we’d looked at binary trees last week:  https://mikesmathpage.wordpress.com/2014/09/07/binary-trees-and-pascals-triangle/

Though this part of walking through the problem wasn’t as clear as it could have been, I’m happy for the kids to see that you don’t always march right to the solution of a problem in a straight line.  I’m also happy for them to see that problems that are relatively easy to illustrate can sometimes be a little more complicated than they seemed when you start thinking about them more carefully.

Finally the clever mathematical solution.  The idea used in solving this problem is a little bit over their heads, but it is a great mathematical idea for them to see nonetheless.  The fun, and actually pretty amazing, part of this solution is the idea of finding a clever piece of symmetry in the problem.  That symmetry allows you to write down an equation whose solution is the probability that you are looking for.  Quite a remarkable idea – I don’t know what the probability is, but I know a quadratic equation that it has to solve!

As it took a while to walk through this solution, I chose to talk pretty informally.  At the end, I let my younger son see that his guess at the solution – i.e. that the probability we are looking for is 1 – does indeed solve the equation.  With my older son, I let him play with the quadratic equation a little and see that p = 1 was the only root.

Tough stuff, but hopefully still a fun example:

I’m really happy that Fawn recommend James Tanton’s book to me.   Despite the difficulty of this first project, the boys were engaged all the way through.  Can’t wait to try out a few more.

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