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:
- 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.
- 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.
- 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.
- 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%.
- 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.
- 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.
- 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.