Mosteller's "Fifty Challenging Problems in Probability with Solutions" – Solving these gave me a very good grounding on how to think about probability. As with Newman's "A Problem Seminar", I suggest taking copious notes and to try generalizing your results.

My plan for this fall is to go through as many of the 50 challenging probability problems as we can. I don’t know how many are accessible to the boys, but hopefully most of them are with some help.

Today we tackled problem #1 – You have some red socks and some black socks in a drawer. When you pick two socks at random the probability of a red pair is 1/2. What is the smallest number of socks that could be in the drawer?

Here’s how I introduced the problem to the boys and their initial solution:

The second part of the problem asks what the minimum number of socks is assuming that the number of black socks is even. This problem gave the boys a bit more trouble, but was a great learning opportunity for them.

In the last video they showed that the number of black socks couldn’t be 2 or 4 with direct computation. Here they showed that it could be 6 using algebra. This was a nice opportunity for some factoring practice.

Finally, we went to the computer and wrote a short (and obviously inefficient) program to test out some solutions. It was fun to find a few more (sadly we were super rushed for time here, but still had a nice conversation).