Probability Of Two Boys Puzzle (Standard vs Tuesday Boy)

Question

For those that aren't familiar with Gary Foshee's probability puzzle/paradox from 4-5 years ago, you can find an analysis here: http://news.bbc.co.uk/2/hi/programmes/more_or_less/8735812.stm

While this puzzle has been discussed here on Stack Exchange: The Tuesday Birthday Problem - why does the probability change when the father specifies the birthday of a son?

(two or three times in fact), the focus has been on the solution and not the LOGIC/MECHANICS behind why the solutions between the two statements below are different.

Statement 1: "I have two children. One of them is a boy. What is the probability I have two boys?" -- OK, the explanation makes sense, it's 1/3.

Statement 2: "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?" -- Again, the explanation makes sense, it's 13/27

What are the mechanics behind the change in probability? Is it because the more specific he is about one of his children the less chance it could be either child he is referring to as opposed to one in particular? So there's some kind of probability "overlap" that is removed (sorry, I don't know a better way to put it).

I've seen this explained similarly here (though still not quite as satisfactorily as I would like): https://www.quora.com/What-is-the-reasoning-behind-Gary-Foshees-boy-born-on-a-Tuesday-problem-where-providing-additional-apparently-irrelevant-information-changes-the-resulting-probabilities

By adding specific information "You've effectively told people about one individual, not told them that one of your children is a member of a category."

Thank you!

Answer 1

The overlap (possibility that both children are of the specific kind mentioned) is the issue, I think. Here’s a picture that helped me understand. Consider a random point $(x,y)$ in the unit square.

If you know that one of the two coordinates is bigger than $1/2$ (in set $A$), you have narrowed down the possibilities to the solid red region in the left picture. The chance that the other coordinate is also bigger than $1/2$ (you’re in $A\times A$) is about $1/3$ (blue-spotted part of the red region). It’s significantly less than the size of $A$ ($1/2$), because $A\times A$ is a significant part of $A\times I\cup I\times A$, the red region you know you’re in.

However, if you know that one coordinate is not only in $A$, but also in $B$ (say between $0.67$ and $0.71$), you are in the cross-shaped region on the right ($B\times I\cup I\times B$). The blue part of that region ($A\times B\cup B\times A$) is pretty close to $1/2$ of the cross, because the overlap is not so significant.

Answer 2

Well, let's try it a little differently.

Question #1: Mr. Smith, I hear you have two children. Is either of them a boy? Answer : Why, yes. At least one is. Solution 1: condider four Mr. Smiths. One would have said "no" because he has two girls. Three would say "yes"; one because he has two boys, and two because they have a boy and a girl. So the probabilty that a Mr. Smith who says "yes" has two boys is 1/3.

Question #2: Mr. Smith, I hear you have two children. Is either a boy who was born on a Tuesday? Answer #2: What an odd question! But you guessed right, at least one is. Solution #2: Consider 196 Mr. Smiths. 169 will say "no," 49 of them because they have two girls. Of the 49 who have two boys, 13 will say "yes." Of the 98 who have one boy, 14 will say "yes." So the answer is (13)/(13+14)=13/27. Mechanism for the change: A father of two boys is nearly twice as likely to have one who was born on a Tuesday, than a father of one. That is, 13/49 to 14/98=7/49.

Question #3: Mr. Smith, I hear you have two children. Tell me an interesting fact that applies to at least one. Answer #3: Well, at least one is a left-handed, red-haired boy who was born on Tuesday and grew up to be chiropractor. Solution #3: No matter how many additional facts he adds, or leaves out, the probability that the other child is a boy remains 1/2. Consider the same four Mr. Smiths as in Question #1. One had to tell you about a boy, and it is irrelevant whether the additional facts apply to both. One had to tell you about a girl. Two had a choice, and only one would choose to tell you about the boy.