Print Story Mathematics anxiety and two kinds of panic
Logic & Maths
By Alan Crowe (Fri Dec 24, 2010 at 08:23:44 AM EST) landsburg, sex ratio, expectation (all tags)
In a far away land the natives don't much like children, but must have a male heir. So each married couple keeps having children until they have a boy, then they stop. Does this skew the demographics?

Every couple has a boy, but half get a boy first time: one boy, no girl. Clearly the stopping rule skews the demographics in favour of boys. On the other hand I've falsely spun the stopping rule as pro-boy. The rule is actually "keep having lovely girls until a yucky boy comes along, then give up in disgust". Many families consist of many girls and one boy. Clearly the stopping rule skews the demographics in favour of girls. On the third hand the demographics are fixed by the 50:50 ratio of boys to girls. The stopping rule is irrelevant.

So there is the kind of panic when you cannot tell whether your answer is correct or not. Steve Landsburg is blogging the second kind of panic. Terror inside.



Consider trying to solve the problem by writing a computer simulation. In one run the Yings get a boy on the first try, while the Yangs get a girl and then a boy. The program prints out +1 boys. In another run the Yings get two girls before they get a boy and the Yangs also get two girls before they get a boy. The program prints out -2 boys. We can add up and divide by the number of runs to get an estimate of the expected surplus of boys over girls. Without the stopping rule the surplus will be zero. Does the stopping rule shift the expected value away from zero?

That is not the only way to write the program. It could print out the proportion of boys in the population. Then the first run would print 2/3 boys, followed by 1/3 boys for the second run. With no stopping rule the expected value of the proportion will come out to be 1/2. Does the stopping rule shift this expected value away from one half.

Now it is obviously obvious that it is entirely obvious that these two questions are the same and must have the same answer. Sadly Steven Landsburg is a fully paid up member of the awkward squad and asks "Oh really? Are you sure?" Where are your social skills Steve? My blatant over use of the word obvious makes it clear that I haven't a clue and will panic if pressed so you shouldn't have asked.

I'm trying to decide if the stopping rule skews the demographics. Am I trying to decide if the stopping rule nudges the expected value of the number of surplus boys away from zero? Am I trying to decide if the stopping rule nudges the proportion of boys away from one half? The first kind of mathematical panic is when you cannot decide if your answer is correct. The second kind of mathematical panic is when you cannot decide if you have understood the question.

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It's a stupid question by gazbo (2.00 / 0) #1 Fri Dec 24, 2010 at 10:58:42 AM EST
Or rather, it's a good question asked in a stupid way.  The original phrasing that Google used implies we're modelling this with an infinitely large number of families.  As such, the 50/50 population split answer is fine.

Then this guy's come along and said "fuck it - let's lose that assumption and instead assume a finite number of families."  Which at the face of it may sound reasonable until you realise that the answer then is "I don't know, it depends how many families there are in each country".

Here's an alternative way of asking the question he's asking: "Taking at random a single family using this stopping rule, what's the probability they have more boys than girls?"

That should yield the answer he's after (I think) but without confusing people by making them interpret the problem.  Oh, and it has the benefit that the answer is then an actual number, not a range of values dependent on an unknown variable (number of families) that he's decided to introduce.


I recommend always assuming 7th normal form where items in a text column are not allowed to rhyme.

Expectation of ratios by Alan Crowe (2.00 / 0) #2 Fri Dec 24, 2010 at 02:43:25 PM EST
My suspicion is that taking expected values of ratios is usually a bad idea. Little numerator and little denominator get weighted the same as big numerator and big denominator, while in the real world the big over big is more important. But I want to play with some examples first. Perhaps I'll find time and do that in the new year.

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