Take a thousand families, with two children, where one of the children was a boy born on a Tuesday.
I don't mean a thousand theoretical families. I mean, lets say you straight up took one thousand real families, that matched the above constraints, straight out of the census. No joke, you break out the SQL.
When you check the gender of the other child, you are going to see the breakdown of gender being 50% male, 50% female.
Now, I know there's a lot of fun handwaving going on. Here's the flaw, in a nutshell. There are indeed three possibilities, when one child is constrained to be a boy:
The mistake -- and it is a mistake, because when you actually run the experiment, the hypothesis is invalidated -- is thinking that each of the above cases is equally likely. Specifically, order of birth has been incorrectly elevated as a determining factor. So we see:
boy, girl: 33%
girl, boy: 33%
boy, boy: 33%
When we really should be seeing:
boy, boy: 50%
boy, girl: 25%
girl, boy: 25%
Or, more accurately:
same-gender, both male: 50%
boy first: 25%
girl first: 25%
Another way to frame the query, with similar results, is to say:
Select the gender of all second children where the first child was born on a Tuesday and the first child was male.
Select the gender of all first children where the second child was born on a Tuesday and the second child was male.
You'll note the girl, girl families will show up in neither result set. So they can do nothing to skew the numbers.
The results of both queries will, predictably, be 50/50 male and female.
This is a good example of why framing a problem correctly is so difficult and critical. It's only because this problem is so amenable to experimental formulation that it's easily defensible.
(Note that the use of Tuesday was an excellent DoS against math geeks.)
(Note also, by the way, this is the exact opposite of the Monty Hall problem. In that problem, people are expecting:
Door 2: 50%
Door 3: 50% ...when, really, we have:
Host Told You Where The Car Was: 66%
Was Behind 3, Therefore Exposed 2: 33%
Was Behind 2, Therefore Exposed 3: 33%
Host Didn't Tell You Where The Car Was: 33%
Randomly Exposed 2: 16.5%
Randomly Exposed 3: 16.5%
If you modify the Monty Hall problem, such that he opens a random door *which might actually expose the car*, then when he opens the door and you see a goat, it doesn't matter whether you switch or not.)