First if you start from the assumption that you know I have two children and what is the probability that I have two boys.
I can therefore have - B/B, B/G, G/B, G/G
Since the question is about the probability of two boys the G/G option can be eliminated. That leaves three scenarios of which one is B/B, so the probability of two boys is 1/3
Now looking at the original scenario using the same technique. Lets call a boy born on Tuesday BTu. and list all the scenarios.
1)BTu + Girl on any day of the week = seven possibilities
2)Girl born on any day of the week + BTu = seven possibilities
3)BTu + boy born on any day of the week = seven possibilities
4)Boy born on any day of the week + BTu = six possibilities (the situation where both boys were born on a Tuesday was counted in scenario 3 and must be discounted as we are looking for equally likely possibilities).
Adding all probabilities gives 27 in total ((7 x 3) + 6), which represents all equally likely combinations of children with specified gender and birth day. Of these combinations 13 are two boys . Therefore the probability is 13/27.
Consider that 13/27 is very different from 1/3 where the key point about day of the week is not considered. This seems quite remarkable and apparently the rarer the trait specified the closer the probability approaches to 1/2
Quite interesting, especially if your a mathematician.
