On A Similar Theme - Probability Again

[Cloverjo]

I remember reading about a problem which went something like this:
Mr Smith has a daughter named Tallulah. What is the probability of his other child being a girl?

The answer was very surprising and counter-intuitive and not the same as the answer in Factor-Fiction's related thread, but the author went through every step and it all made sense. Unfortunately, I can't find the book or remember its title. Does anyone recognise this, and tell me the title or author?

[jomifl]

Ths is going to end up at the three box conundrum...please not again...

[jim360]

So what was the answer the author gave?

[Prudie]

Is this not exactly the same question? More to the point what was the answer?

[Cloverjo]

I wish I could remember, jim. All I know is it was different to what we would expect, and that the fact one was called Tallulah did make a difference, while we would expect it not to.

[jim360]

If the name makes a difference then this is perhaps a problem in what I'd call "annoying logic", where the answer makes sense but draws on more than just pure mathematics.

[jourdain2]

After 2 boys, my niece was named 'Miranda Tallulah' (her dad insisted on the 'Tallulah') and that remained her pet name. From this, incredibly slight, evidence I would hazard that the other child is a boy!

[Gizmonster]

I don't know the book or the author; but I'm fairly sure that the answer is 1 in 3 and NOT 1 in 2, which everyone expects :)

[factor-fiction]

http://en.wikipedia.org/wiki/Boy_or_Girl_paradox
There's a reference here to what happens when more information is provided (in this case the day of the week one child was born rather than the name of the child, but the principle is the same) and this, it is argued, changes the odds.
The twist in all these is the answers depend on subtle differences in the wording and in how one came by the information.
I feel quite a few posters on the Sex thread dismissed alternative answers without considering them because our brains convince us we are right.

[Cloverjo]

For the few of you who might still be interested, the answer is 1/2. I haven't found the book I was looking for, but here is an explanation where the girl is called Florida instead of Tallulah.
I'm sure my book had a simpler explanation, though. I'll keep looking.
http://economics-files.pomona.edu/GarySmith/The%20Two-Child%20Paradox%20Reborn.pdf

[factor-fiction]

Thanks for this, Cloverjo. It just shows that the issues here are not as black and white as some might think. These paradoxes have been the cause of much debate among mathematicians for years and I have seen equally convincing arguments for 1/2 and 1/3 and even other figures.

[DTCwordfan]

What happens to the probability here if the baby is a 'true hermaprodite'?

[Cloverjo]

DTC, I think we can assume for the purposes of the exercise that each child is either a boy or a girl.

[BlackadderV]

Well, if we can make assumptions, that throws the mathematical laws of probability out of the window.

[jim360]

Interesting -- I guess this is the full analysis of what I was referring to when I mentioned possibly having to make a "humiliating" backtrack. What I did was note that if you start by drawing a probability tree for two-child families, and then add to the tree an event that amounts to being introduced to one child from the family, then you have the following events:

Two boys, introduced to eldest (boy)
Two boys, introduced to youngest (boy)
Two girls, introduced to eldest (girl)
Two girls, introduced to youngest (girl)
Boy and girl, introduced to eldest (boy)
Boy and girl, introduced to youngest (girl)
Girl and boy, introduced to eldest (girl)
Girl and boy, introduced to youngest (boy)

Which, it is clear, means that there are four scenarios in which you can be introduced to a boy, and two of these correspond to a two-boy family. At this point I stopped, since a) I wasn't sure that I could calculate the probability of these, and b) in my opinion it's actually a different question -- but at any rate the conclusion was, certainly, but once you start fussing about how the information changes then the problem also changes.

I've not checked the calculations in the link myself, yet, but I might try to do it at some point today. On the face of it it's the continuation of the analysis I'd started yesterday.

At the risk of looking horribly arrogant, my own summary might be that the question as originally posed is ambiguous, but those that overlook one answer or the other are in some sense equally mistaken, and it's important to understand where both come from.

[Prudie]

I don't really want to go into this again but if "These paradoxes have been the cause of much debate among mathematicians for years and I have seen equally convincing arguments for 1/2 and 1/3 and even other figures" is true then how come everyone who argued for 1/2 on the sex thread was deemed not only incorrect, but in some cases incorrect because they had missed obvious information or were even stupid???

[BlackadderV]

Correct, Prudie. I said it on the sex thread and I will repeat it here. Given that one of the children is a boy the probability of that event is 1 (certainty). The other child can only be a boy or a girl, the probability of which is 1/2. Using the mathematical laws of probability we arrive at 1 x1/2 = 1/2.

[jim360]

In maths it's important to get the right answer but it's if anything more important to get there using a correct method. So, so many people yesterday said "it's two boys or boy and girl, therefore 1/2". No calculation, nothing like that -- it's just a bit haphazard and, arguably, means that such people are missing the subtlety of the problem. The answer of 1/3 is never considered as even possibly right (and in some circumstances it certainly is right).

[Old_Geezer]

They were deemed incorrect because they were incorrect. They had instinctively made assumptions about the question that they should not have, which led them to a simplified version and thus an incorrect answer.

On that thread I suggested a simple piece of code that would prove the issue to all is they could get a high school kids to write it for them. I expect it could be simulated in a spreadsheet by someone who is a bit of a wiz at that sort of thing.

[Cloverjo]

Oh dear. Sorry. I didn't want to rehash the argument of the other thread. I was interested in the difference that knowing the unusual name of one of the children makes.

[Prudie]

Perhaps then OG you should write a paper that shows all the already written conjectures that 1/2 is the answer are completely incorrect.