### Status of the Bets

As promised, I am providing an update on the status of the various bets that got placed here last week.

The problem is this: In a certain country, each family continues having children until it has a boy, then stops. In expectation, what fraction of the population is female?

The answer is: not 1/2. A more precise answer depends on your additional assumptions — the number of families, the mortality rate, whether the parents count, whether there are grandchildren, etc. It might be possible to engineer those additional assumptions in some way that makes the answer come out to 1/2, but it would surely be impossible to argue that those ad hoc assumptions, whatever they are, are implicit in the statement of the problem.

Pretty much everyone seems to understand this now. (Here‘s one more learning aid, if you’re still struggling.) What, then, are we betting about? Here is a list of all the open and closed bets:

1. In an example with four families (and without including the parents) I calculated the answer to be just a hair under 44%, and offered to bet that a simulation would reveal it to be less than 46.5%. Larry offered to take the bet provided the parents were included in the count. I rashly agreed, but failed to specify that with the parents included, I’d want to revise my prediction upward a bit (though not to 50%). With the parents counted and the original cutoff of 46.5%, this is a bet I’d be very likely to lose.

Larry and I never actually disagreed about anything. As far as I know, we agree and have always agreed that in the four-family country, the answer is about 44% without the parents included and somewhat higher (though not 50%) with the parents included. In view of this, Larry agreed to settle on generous terms. He’ll be receiving a package from me in the near future.

2. Jim Davis and I seem to be on for \$1000; we’ve been negotiating the details in email. He and I seem to be in complete agreement on the mathematics. We agree, for example, that with four families and the parents not counted, the answer is about 44%. Jim’s position, though, as I understand it, is that the “natural interpretation” of the problem — that is, the interpretation that would come naturally to a professor of statistics — requires you to take a limit as the size of the country increases, giving an answer of 1/2. I find this an extraordinarily strange interpretation, since the problem specifies “a certain country”, not “a limit of what happens in a sequence of increasingly large countries”. We’ve had some very polite discussions about this and have agreed (I think) to submit the question to a panel of statistics professors for ajudication.Note that Jim does not claim that the answer is 50% in any particular country; just that the result approaches 50% in a sequence of ever-larger countries. Thus we have no real substantive disagreement; nevertheless, it looks like \$1000 is going to change hands. At Jim’s excellent suggestion, the winner will donate to a charity in the loser’s name.
3. Paul Hyden was initially in for \$1000, but after a few rounds of friendly email, he decided (as I understand him) that we did not substantially disagree, and I cheerfully released him from his offer to bet.
4. David McFadzean was initially in for \$100, but changed his mind after a few rounds of friendly email. He and I seem to be in complete agreement about the answer, though he expressed some reservations about whether that answer differs “significantly” from 50%.
5. Alex wanted in, but (as it developed in email) only subject to a “terminal boy scenario” that significantly changes the problem. When I specified that I intended to stick to the original problem, Alex allowed as how we have no disagreement. This bet, too, is therefore cheerfully discharged.
6. Martin Maier is tentatively in for \$100. It appears I neglected to send him a followup “Are you really you?” email. I’ve just now sent that email and things will unfold from here.
7. The infantile troll Scoot_AO agreed to a bet, after making the following remarkable claim:

Regardless, the average of 3/3, 3/3, 3/3, and 0/12 isn’t 75%.

Unfortunately, he’d been using a fake email address and I’ve been unable to contact him.

Finally, Lubos Motl, whose money I’d have most gladly taken, is unwilling to put any money — or any argument — behind his preferred answer of 50%. An argument would consist of a list of clearly stated assumptions about mortality rates, etc, together with a proof that these assumptions lead to an answer of 50%. Perhaps Lubos can provide that, though so far he hasn’t tried. Even so, he’d have shown, at best, that some set of assumptions yields the answer 50% — which is a long way from his claim that any reasonable set of assumptions yields that answer.

The take-away from all of this? Just because the expected number of girls equals the expected number of boys, it does not follow that the expected fraction of girls is 50%. Readers who were mistaken about this fell into two camps: Those who made arguments with mistakes in them (something most of us do on a daily basis, and nothing to be ashamed of) and those who thought that a temper tantrum could substitute for an argument (the “It does too follow! Neener neener!” camp). The latter group are the ones whose money I wish I’d taken.

One more followup: Littered through the comments of last week’s threads is an extraordinary discovery by our commenter Tom, with further explication by Thomas Bayes, Neil and others, that does get the answer 50% to a modified version of the problem. This is very cool and I learned something from it. In the next few days, Tom and/or Thomas will be guest-posting here to explain their discovery.

Thanks to all who participated here and to all who forced me to improve my explanations. We’ll be on to new topics soon.

#### 21 Responses to “Status of the Bets”

1. 1 1 curious

Are you still taking bets?

2. 2 2 Steve Landsburg

Curious:

Are you still taking bets?

Only from people who have provided working email addresses, which seems to exclude you.

3. 3 3 Alan Wexelblat

In one of his responses, Motl provided two programs (in Mathematica, iirc) that he appeared to claim substantiated his position. Are you able to evaluate those bits of code and point out the flaw? If not, would running those bits of code satisfy your conditions?

4. 4 4 Steve Landsburg

Alan Wexelblat: That code makes *my* assumptions and yields *my* answer; presumably Lubos provided this as evidence that he understood my assumptions and my answer. He is explicitly *not* claiming that his code gives the right answer to the problem. His claim is that my assumptions are not legitimate, but he still hasn’t given a concrete example of the a set of assumptions he’d consider legitimate.

(He has also given alternative code that correctly computes E(G)/E(G)+E(B), but this has nothing to do with the problem. He might as well have just provided a one-line program that says “PRINT 1/2″.)

5. 5 5 Cos

I think Jim Davis is more right than you, if I understand your summary correctly. When you pose this question about a “country” it puts a certain scale in mind. It’s not just “a certain set of families”, it’s an implication that you’re talking about a very large set of families. Furthermore, talking about a “country” implies you’re interested in the practical outcome of your math, not just the math itself. The outcome for “a certain country” obviously can’t be determined precisely – part of your expectation is the level of statistical uncertainty. For the scale of a “country”, in this case, the level of uncertainly means that the limit is a more relevant number than is the difference between the limit and the exact middle of the range for that specific population.

This is not a question of math, of course. But it sounds like you and Jim agree about the math. But I think you like stating mathematical questions in the form of scenarios in English, without actually paying attention to the stories you posit; you’ve still got the original math problem in mind, but other readers see what you’ve written. I think you often don’t understand what it is you’re actually communicating to people with what you have written, since you think you were just trying to translate a pure math problem into English in a readable way.

6. 6 6 Jonathan Campbell

Can I place a bet that, if the statistics professors know that the outcome of your bet with Jim Davis is based on their adjudication, they will not want to issue a definitive declaration on whether you or Jim is correct (unless, I suppose, they know that the proceeds of the bet go to charity, and they want to redistribute wealth from you and Jim to the charity’s constituents)?

7. 7 7 Silas Barta

Off topic, but when do you plan to do the promised follow-up to this post about the counterintuitive theorem that says you can predict perfectly from limited data? Because I see a number of ways it just can’t be right, even with the caveat about it being computationally impractical. (For example, simple information-theoretic bounds on the mutual information between what you’re predicting and what you know.)

8. 8 8 Roger Schlafly

It is not correct that Lubos failed to give an argument. He gave arguments on his own blog, and in the comments on yours. It is true that he never provided a “list of clearly stated assumptions about mortality rates, etc”, but then neither did you.

9. 9 9 Ken B

Lubos Motl has a post under the title Slippery Lube as well in which he asserts — or seems to in the limit where k is infinite — the SL is disallowing those comments which disagree with him (SL). That is pretty obviously false. (He then goes on to miss the point entirely.)

Or perhaps this is a cunning conspiracy by Motl and Landsburg to gin up traffic!

10. 10 10 Steve Landsburg

Silas:

when do you plan to do the promised follow-up …

Thanks for the reminder. I’ll get to this within the next couple weeks, maybe much sooner.

11. 11 11 Will A

@ Ken B

In an infinite population how is the fraction 1/2. As was also posted, what is infinity/(infinity+infinity)?

If infinity/(infinity+infinity) is not 1/2, then Lubos’s assumption that the population is infinite doesn’t match his answer of 1/2.

Lubos also asserted that these populations go to 0. If 0/(0+0) is not 1/2, this is another assumption that doesn’t match his answer.

What assumptions did he make that matches his answer of 1/2?

12. 12 12 Mark

Jim seems to have a fair chance of winning his bet; I too would see this as a limit problem, with exact answer 1/2, for the following reasons.

After the problem is stated I would make a decision as to what kind of answer is desired – is this a realistic, ‘engineering’ problem or a theoretical one? As soon as I heard all the families in a particular country wanted a boy, I would immediately realize we are in La-La land. Of course after they get a boy, then they stop reproducing! Pure fantasy. If I think about it a little while I realize that some of these moms may need to have hundreds of kids to fulfill their end of the puzzle. After this final absurdity is heaped on, I am just not ready to give any kind of ‘realistic’ answer to this puzzle that relies on items like the actual male-to-female birth ratio, the realistic number of kids a woman can have, the twin/triplet/xlet problem, etc. I realize some people have gone in this direction but I would not.

So I hack through the puzzle and realize I need to make a determination about the number of ‘families’ in the ‘country’; should it be finite or infinite? Well, we are already in La-La land… I’ve just swallowed three ridiculous assumptions, why stop now? We may as well be talking about reproducing ‘angels’ on the head of a ‘pin’. I choose to stay the course and make the number of families infinite.

Now things make sense – I see that, based on my crazy assumptions, the answer requires summing an infinite series (or never counting the ‘last boy’ if you’re into that kind of thing). I answer 1/2, and cleverly think I’ve dodged a bullet by not stating some huge percentage of boys, which was the obvious trap the interviewer wanted me to fall into. I’m especially glad I didn’t say something like ‘a banana’.

Note that I also think ‘not 1/2′ is a good answer too; you merely need to accept the first three ridiculous assumptions and then assume the number of families in the countries is finite. That’s ok; that’s allowed. I think Jim is betting that the statisticians will either come to the answer by faulty reasoning (summing the wrong infinite series which coincidentally gives the same result), or stay the course of the first three absurdities and then set the number of families to infinite as well.

13. 13 13 Will A

Mark,

This is a question poised by Google. I would think that an answer that could be simulated in (probably) java might be more appropriate.

As it relates to the puzzle, “some of these moms may need to have hundreds of kids to fulfill their end of the puzzle” probably would not be a valid.

The odds of having 20 girls first are ~ 1 in a Million.

14. 14 14 Mike H

When I first heard this puzzle, I said “initially 50%”, making the error that @Steve points out. [Aside : in any reasonable-sized country, 50% is within any sensible confidence interval of the correct answer]. However, if someone has a genetic predisposition towards female offspring, they will tend to have more kids than their neighbours. If citizens only marry citizens, this won’t actually change the expected number of girls vs boys for given values of G+B. However, if unwed citizens unlucky in love import spouses from an effectively infinite pool of foreigners, I suspect the proportion of female citizens would gradually increase over time. I wonder if, in the limit, it would approach 100%?

15. 15 15 Steve Landsburg

Mark: Assuming an infinite population won’t help. In that case, E(G/G+B) is undefined.

16. 16 16 Mikhail Gorbachev

I like the idea of looking at the problem as a heads vs. tails problem. It boils down the math and allows people to think about the math. I think Steve will argue that we shouldn’t care about children vs. coins, but many do.

N amount of people flip coins. They all flip until the get heads, what is the proportion of tails for all the flips?

P(h,t)= .5

Now relate the proportion to N.

I understand this takes the parent out of th equation, but it makes it so in my opinion we can won’t have all these stupid model arguments.

17. 17 17 Mark

@Steve

from Thomas Bayes

http://www.thebigquestions.com/2010/12/27/win-landsburgs-money/

“If you let K be infinite, then the birth sequence will never stop and there will be no chance that any census based on a finite number of children will include the final boy, so the expected proportion of girls (or boys) will be 1/2.”

18. 18 18 Steve Landsburg

Mark:

“If you let K be infinite, then the birth sequence will never stop and there will be no chance that any census based on a finite number of children will include the final boy, so the expected proportion of girls (or boys) will be 1/2.”

As stated, this seems incorrect to me. I suspect that what Thomas meant was “If you let K be large…..the expected proportion…will be arbitrarily close to 1/2.”

19. 19 19 Thomas Bayes

Mark and Steve:

I was thinking about it this way:

If you had K coins and tossed each one until you saw heads, then you might see something like this:

coin 1: H
coin 2: T T H
coin 3: T H
.
.
.
coin K: T T H

If you put them together in a long sting, you’d see this:

H T T H T H . . . T T H

If you counted the number of heads in the first M results and divided by M, then I believe the expected value of this ratio would be

1/2 + 1/(4K)Pr[complete]

where Pr[complete] is the probability that all K heads would show in the first M tosses. (If the actual sequence terminated before M tosses, then you’d replace M by the number of tosses.)

If K is infinite and you look at the first M results, then the expected ratio would be 1/2 because there is no chance of seeing all of the K heads.

This is a fabricated example that probably doesn’t apply to the concept of an infinite population. In short, I don’t know what it should mean to have an number of families in the population. I guess if we mean an infinite string of births that never terminates, then my example would be appropriate, but in that situation the birth policy would matter for each ‘family’ but would appear to be meaningless for the country. My preference is to avoid the issue of an infinite population, and just talk about a large number of families. In that case, the expected proportion of girls will be close to, but different from, 1/2.

20. 20 20 Will A

Mikhail,

I don’t think phrasing it as a coin example would help. People would make statements like:
“If you has N people to flip coins, some will refuse to flip coins and some of the coins will get lost under a couch or a storm drain. Therefore the answer is 50%”

The issue as I saw it is that people were arguing answers that didn’t match their assumptions. E.g.
“This is an ambiguous puzzle, therefore the only answer is to the puzzle is 1/2″
“Well it is never exactly 50%, therefore the answer is 1/2″
“The population goes to 0, therefore the answer is 1/2″

21. 21 21 Ken B

@Steve:
“At Jim’s excellent suggestion, the winner will donate to a charity in the loser’s name.”

Were you bamboozled? The winner pays? Is it really THAT easy to win Landsburg’s money (for charity at least)?