Saw this tweet from Nassim Taleb yesterday:
https://twitter.com/nntaleb/status/622133016400404480
It reminded me of an old and unfortunate mistake from the 1998 Minnesota state high school math contest. Here was that question:
You go to visit a friend who has two children. However, you cannot remember the gender of either child. When you arrive at the house, one of those children answers the door. That child is a boy. What is the probability that the other child is a boy. Warning, the answer is not 1/2.
And, yes, that warning was part of the original question.
We were a little tight for time this morning, so I decided to use these two problems for a quick Family Math project. Here’s my kids taking a look at Taleb’s question:
Here’s my version of the second question (without the warning) and the thoughts my kids had thinking through it:
Although it isn’t all that difficult to understand the statement of either of these two questions, understanding why the two situations are different is sometimes no so easy. Taleb compares the question in his tweet to the famous Monty Hall problem. Another potentially good comparison – though much harder to understand mathematically – come from a recent Andrew Gelman blog post:
Hey Guess What? There Really is a Hot Hand
So, even though this was a quick little project, it still is both fun and instructive!
Mike's Math Page 
It might help in the second scenario to describe the cases with triplets:
(value of first coin, value of second coin, value of coin shown). The list of cases would then be:
(H, H, H) first coin shown
(H, H, H) second coin shown
(H, T, H)
(H, T, T)
(T, H, H)
(T, H, T)
(T, T, T) first coin shown
(T, T, T) second coin shown
all with equal likelihood. We can throw out cases 4, 6-8 based on the information that we were shown a head. Then, it is pretty clear that 2/4 of the remaining cases are from value states HH.