kitchen table math, the sequel: pop quiz, part 4

## Tuesday, December 28, 2010

### pop quiz, part 4

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?

Catherine Johnson said...

I just noticed: this is a question that comes up in Google interviews --

Catherine Johnson said...

I wonder if that can possibly be true.

Allison said...

This is a good example of how experts and novices think differently.

This is a sloppy problem. An expert would have no trouble working this live in an interview--he would state his simplifications and assumptions to make the problem solvable, write down his model, solve the problem, and then talk about how the model relates to the problem (and how it doesn't.) He could talk about different assumptions, and how that would change the answer.

Someone who is not an expert but learned the basic discrete math to solve this problem would immediately see what model to use to solve the problem, but probably would need help to see the implicit assumptions he made and how they might not actually be warranted. talking it through, though, he'd probably see that.

A novice would be jumbled up. They'd probably know the model they were supposed to use, but they'd be a bit confused in the application--can they really make assumptions about fair coins, or constant probabilities? how many runs do they need to consider? do they need to think about the expectation for each run, or just one general one? When questioned, they might see the connection between the model they picked and this question, or it might confuse them more and make them doubt they'd done it right.

But using a problem like this to TEACH how to solve this problem to a novice would be cruel. A novice needs to learn from a clear model where everything is spelled out. Over time and experience, they'll learn how to relate the simplifications of a model to reality.

I'll write down the clean simple problem in another comment.

Allison said...

A clear problem for a novice would be

You have a coin with probability p of coming up heads. Event E is a run of independent coin flips where the coin is flipped until it comes up with the first tails. We say E_i is the event of a run of (i-1) heads tosses and then tails on the ith toss. So E_1 means you get a tails on the first (and therefore last) toss: T. E_2 is the event you get HT, etc.

What is the expected number of times you need to toss before you get Heads?

Wow, the OP of the puzzle at that link is an idiot. And, sadly, he's written a book to mislead more people. Even more sadly, he appears to be a professor in some field involving mathematics (economics, it appears).

Read the comments. He thinks you can average ratios by adding them and dividing by the number of ratios. So he thinks that the average of 4/4, 4/4, 4/4, and 0/12 is (1+1+1+0)/4 = 0.75, instead of (4+4+4+0)/(4+4+4+12) = 0.50.

He should be sued for mathematical malpractice.

Anonymous said...

Everyone wants to have a son but that doesn't mean that everyone is part of a reproducing couple. In fact, it seems that it is hard to get married in countries like China, where this is a common preference.

ari-free

Allison said...

Yeah, the original post is a disaster. Worse, the comments are so typical of discrete math folks. Instead of writing down the MATH to show people how to do the problem, they argue about how clever and smart they are.

Discrete math/combinatorics/prob theory gives mathematics a bad name. In the rest of mathematics, cleverness is not rewarded without clarity and precision being rewarded too.

What's even weirder about their solution is that they never ever discussed coin tosses or bernoulli trials. There was one clear model to elucidate the problem best, and they avoided it? Didn't know it?

Anonymous said...

Granted that the OP is an idiot, it does not follow that everyone who does discrete math is. There are plenty of good people doing discrete math, combinatorics, and probability.

There are plenty of idiots spouting nonsense about continuous math also.

Actually, the initial assumption of 50% probability of having a boy is wrong. I believe that the current probability is more like 0.51, but that it may depend on environmental factors as well.