Saturday, July 03, 2010

Lies, Gender, and Damned Statistics

On Slashdot recently I encountered another version of Martin Gardner's two-children puzzle. The original problem is this:
I have two children. One of them is a boy. What are the odds the other one is also a boy?
If you're a human living in the world, three things are probably true of you vis-a-vis this puzzle. 1) You've heard it before, 2) you got it wrong the first time you heard it, and 3) the correct answer still seems wrong to you.

The correct answer, if you've never encountered it, is based on the following a priori: there are four equally-likely ways to have two children:
  • a boy followed by a girl
  • a boy followed by another boy
  • a girl followed by a boy
  • a girl followed by another girl
You look at that graph, you find all of the rows where both children are boys (1), and divide it by the number of rows where at least one of the children is a boy(3), and you get the answer: 1 in 3.

Mrs. Transient Gadfly will tell you that Mr. Transient Gadfly's position on all questions of this nature is that it is not a math question, it is a language question. And, moreover, it is an ill-posed one. The nature of how poorly this question is posed is laid bare by the variation linked above:
"I have two children, one of whom is a boy born on a Tuesday. What's the probability that my other child is a boy?"
If you follow the logic of the original problem (which, being that I am a human in whose true nature you will find the compunction to write this blog, I did) you'll write out all the days of the week your first child could be born, followed by all the days of the week your second child could be born, look at all of those that have a boy born on a Tuesday in them, count the number of those that have a second boy, and come up with the answer (it's 13 in 27, if you write out the table. Do not write out the table). If you are literally anyone else in the world, you will come up with a much better answer: 1 in 2. The crux of the issue, which the linked article almost hits on but then fails to, is that there is no universe in which the given answer (13 in 27) is correct. It would require the asker of the question to randomly chose a day of the week and a gender, and then only pose the question if he or she had a child that matched those criteria.

(Here is one of those moments where Transient Gadfly has an existential crisis about the nature and purpose of The Odds Are One: should I explain why what I just said is true? It would take, like, seven paragraphs and still nobody reading would understand the logic. I'm not going to do it this time. You'll just have to take my word on this one).

If you're anyone else, you look at that question and understand the only way someone would pose the question: he or she randomly chose one of his or her children, and listed two characteristics of that child: his gender, and the day of the week of his birth. And you will come up with the correct answer to the question, because when you randomly chose one of your two children, the gender of the other one is a coin-flip. So, you might well ask, what is the difference between the original question posed by Martin Gardner and the question involving the day of the week? And the answer is, absolutely nothing. There is no way to tell, from the way it is stated, whether the asker, a parent of exactly two children, randomly chose one of his progeny and told you his gender, or a parent of exactly two children, of whom at least one is a boy, told you that fact. And it matters, because in the former case it's a 50% shot that the other child is a boy, and in the latter it's a 33% chance.

I leave you with a link to an XKCD cartoon, because it's literally impossible to make this point better than he has here.

1 comment:

Anonymous said...

"I have two children. One of them is a boy. What are the odds the other one is also a boy?"

First, "odds" are not the same as "probability." "Odds" are what a bookie will pay on a bet. They can be expressed as (1-P)/P:1. So if the probability is 1/2, the odds are 1:1. If the probability is 1/3, the odds are 2:1.

Second, the answer to this question is P=1/2, not P=1/3. Martin Gardner himself retracted his answer of P=1/3 six months after posting it, but nobody seems to recall that.

It is true that 1/3 of all 2-child families with at least one boy have two. But that is not the same condition as being told by the parent that there is a boy, because the parent of a boy and a girl is equally likely to tell about the girl, as to tell you about the boy. So, while each of the four family types {BB, BG, GB, GG} has a 1/4 chance of existing, the chances the case exists AND you will be told about a boy are 1/4, 1/8, 1/8, and 0, respectively. This makes the answer (1/4)/(1/4+1/8+1/8)=1/2.

If you had said "I was selected to address you *because* I have two children, and one of them is a boy" then the answer is 1/3. But that is not what you asked. The difference is whether "one boy" is a requirement for selection, or an observation after selection.

"I have two children, one of whom is a boy born on a Tuesday. What's the probability that my other child is a boy?"

Also 1/2. You can get this answer either intuitively, or by writing out your table *AND* letting the parent of different types choose what to tell us randomly. The reason people object to your answer changing from 1/3 to 13/27, is that they view the information "born on a Tuesday" as an additional observation made after selection. Not a requirement. The reason your answer, which is based on a requirement, changes, is because a two-boy family is nearly twice as likely to have one born on a Tuesday as a one-boy family.