Probability thread part II - what are the odds?

[Lweji]

www.mumsnet.com/Talk/am_i_being_unreasonable/3803785-to-ask-if-you-can-answer-a-question-re-probability-Maths-question?watched=1&msgid=93458537#93458537

In a country where people keep having babies until they get a girl, and then they stop, what is the expected ratio of boys to girls in this country?

Not enough information.

In any case, even with a 73,000,000:1 chance, if there were 10x the number of families with two babies in the world, it could happen 10x.

Yes, and of course, because the authorities would investigate such cases there's a good chance you end up in court even though innocent. There would be 10 innocent defendants, and maybe 1 guilty person who did murder their babies; like the medical test, a lot of false positives.

@SerendipityJane

What other information do you need?

DadDad thank you taking the time to explain the Tuesday born boy using the visual concept of kids in a room, sitting or standing, I found that really useful.

Thanks, @OffToTheMoon - I'm a former maths teacher, so I'm glad I can occasionally still deliver the goods...

SerendipityJane What other information do you need

Starting split male/female in the population for a start. Plus number of woman capable of bearing a child and what age to assume it stops.

Assumed age of first childbirth (and indeed start of sexual activity m/f).

That's before we assume that any given live birth has a 50/50 chance of being M/F. I would immediately question that too.

Anyone who answers the question you put without asking for more details is wrong.

@SerendipityJane

I wasn't being sarky, I was genuinely interested in what details you wanted.

Okay. Let's assume this policy gets adopted now. Let's say that there are a large enough number of women to have an impact on the population, they can keep having babies until they get the girl. The probability of M/F is indeed 50/50, what is the distribution among the babies born after this policy has been introduced?

@iseetodaywithanewsprintfray , @SerendipityJane - I'm guessing you also need to make an assumption that there are no terminations allowed on the basis of the baby's sex (to rule out mothers having a tendency to terminate boys to avoid a large family).

(or indeed, terminating a girl because they want the opportunity to have a boy first).

@DadDadDad - yes, make those assumptions. No terminations, you get a boy, you say, oh well, back into the bedroom we go.

If we wait until the n mothers have all had their girl, then the distribution of the number of boys has a mean of n (so on average, we would expect the number of boys to equal the number of girls) and a variance of 2n. (Yes, I really did do some analysis of the sum of n negative binomial distributions ).

For large n, this distribution will be close to a Normal distribution (the ratio of boys to girls will have mean 1 and standard deviation sqrt(2/n).)

There's some series summations going on with that question.

If you think about it as an average per woman, then the average number of girls is:
1/2 + 1/4 + 1/8.... = 1 (obviously because they stop at 1 girl so by definition the average number of girls is one!)

But the corollary for the number of boys is:
0x1/2 + 1x1/4 + 2x1/8 + 3x1/16...
= Sum[(n-1)/(n+1)^2]

Only problem is I can't work out what that sums to!

Oh that's very wrong...

The boys is Sum[(n-1)/2^n]

And I think that sums to 1 as well. So you'd still expect equal numbers of girls and boys.

whatsthat

Yes, if you call S = 0/2 + 1/4 + 2/8 + 3/16 + ...
then 2S = 0 + 1/2 + 2/4 + 3/8 + ...
Now subtract these two lines:
2S - S = 1/2 + 1/4 + 1/8 + .... = 1 as you've already proved
S = 1.

I wasn't being sarky, I was genuinely interested in what details you wanted.

Don't worry - I dish out enough that if someone can sneak it past me they deserve full kudos

That said all you've done is make an interesting maths question for an exam somewhere, rather than any even vague attempt to model the real world (which is where my interest gets piqued). For example not every woman has the same chance of carrying to term. Not every child will make it to sexual maturity and so on. You're also assuming that everyone has the same underlying fertility and that no woman dies in childbirth.

But at least you'll understand why I'm not popular when politicians start to spout bullshit numbers and stats.

"So you'd still expect equal numbers of girls and boys."

Yes! That totally blew my fragile little mind.

@SerendipityJane - how do those examples change the expected outcome? I can't see that the distribution would look any different if those things are taken into account. I'd love to see the maths, if you have it.

But at least you'll understand why I'm not popular when politicians start to spout bullshit numbers and stats

I kept my mouth shut at work recently - when people were very excited about an 80% Good rating after a survey. Of 5 clients.

Yes, and of course, because the authorities would investigate such cases there's a good chance you end up in court even though innocent. There would be 10 innocent defendants, and maybe 1 guilty person who did murder their babies; like the medical test, a lot of false positives.

Good thing we don't still have the death penalty...

A fair coin is tossed a hundred times. What is the probability that there are no two consecutive tails in the sequence?

How could this be calculated?