My older son and I looked at problem 19 from the 2014 AMC 10 b this morning:

He told me that he’d solved the problem correctly this morning and it seemed like a neat problem to talk about, so I wanted to hear his solution. It turned out that he had some good intuition, but not exactly a full solution:

So, after the rough argument that the answer was 1/3 I wanted him to try to find a more complete argument. Finding the right argument proved a little tough, so I asked him to go back to the information that came with the original problem. Using this information helped him find a more complete solution:

It was interesting to me to see the “hand waving” argument that showed the probability was likely to be around 1/3, and also instructive to see how much more work was required to find the complete solution. The whole process today gave me some insight into how kids think about the difference between thinking something is true and knowing something is true.