Had I stated the problem as "Given that a man chosen at random has two children, at least one of which is a boy born on a Tuesday, what is the probability that he has two boys?" then the answer would be 13/27, as I derived. But that is not how I stated it. Rather I used a slight rewording of the problem as I came across it, which I believe was the wording with which puzzle master Gary Froshee presented is at the recent Gathering for Gardner conference. I said: "I tell you I have two children, and (at least) one of them is a boy born on a Tuesday. What probability should you assign to the event that I have two boys?"
As my statistician correspondents observed - and I agree - with that wording, things are a lot more murky. Likewise for the simpler version I discussed, that makes no mention of the day of birth, where the "correct" answer is 1/3. At issue is the precise circumstance under which I come to impart the information I do. Why did I say what I did, and what other statements might I have made?
For example, suppose I come from a culture where it is obligatory to speak about one's elder child before making mention of any younger siblings? Factor that additional information into the arguments I gave last time and you will find you arrive at very different answers: 1/2 in each case.
Or how about the following scenario. You know I have two children, but that's all you know about my family. We meet in the street and I happen to have one of my children with me. I say, "Meet George, a Tuesday child." The only rational value you can put on the probability that my other child is a boy is 1/2. (The reason being, it is, as far as you know, purely random that George is the child who accompanies me. Of course, if I came from a culture where fathers are seen in public only with sons, and in such cases all their sons, ...)
Now, people who like probability puzzles (and those of us who love to hate them, or do I mean hate to love them?) know about these complications, but we learn to read them a certain way. That way is very much part of the code of pure mathematics in general, and probability questions in particular. We dress them up as real life scenarios involving one or two people, usually the puzzle poser and her or his target, to personalize them, and avoid always talking about "randomly selected individuals from a population." But it's just part of a genre of dressing up mathematical questions in imaginary scenarios.
It's a genre with a long history, going back to the mathematical puzzles posed by the ancient Greeks, by the Indian and Arabic-speaking mathematicians in the first millennium, and by the European mathematicians that came after them, starting with Fibonacci in the thirteenth century, whose famous rabbit colony grew in a fashion that I doubt any biologist has ever observed. (For two further examples of unrealistic "real-world scenarios" with long histories, see the birds problems in my December column.)
Those scenarios may all use familiar words and everyday activities, but they are far from realistic. They work provided everyone knows the code and is prepared to play the game.
When such problems are used in elementary educational settings, considerable caution is required, since the recipient in such cases in all likelihood does not yet know the code. In fact, as I argue in just a moment, I think such problems should be banned from the lower grades.
But for mature students, the kind who might typically be faced with some probability theory, there is little danger. If they have gotten that far in mathematics, they have surely learned the code. And then the problem can provide an excellent launching pad to dig deep into the very issues it is designed to illustrate - namely, how do you compute probabilities (and, more fundamental, what does probability mean)? For instance, the instructor can ask the class to formulate the children's gender problems in ways that lead to different, specific answers, and then discuss those formulations.
Case in point, one of my correspondents suggested that to get the 13/27 answer, imagine you are a witness in a court of law, and the judge, having established that you have two children, asks you, under oath, if at least one of your children is a boy born on a Tuesday. Well, maybe I'd better add the condition that the judge also says you that you must answer, giving a simple yes or no. (Though even then it is not hard to imagine how this could still be over-ridden by a further twist to the story to give a different answer.)
While the unrealistic nature of these spuriously "real-world" word problems can provide an excellent pedagogic entry point for more mature audiences, they can have potentially disastrous consequences for younger students learning mathematics for the first time.
I cringe whenever I see an elementary school textbook present a problem such as "If a quarter of a pound of ham costs $2, how much will three pounds cost?" Any child who has accompanied a parent shopping for groceries knows that things are cheaper per pound when you buy a greater quantity. As a result, though the child may eventually learn to solve such problems the way the textbook wants, the real lesson being imparted is that mathematics is a stupid, arbitrary subject having no relevance to the real world.
Or, consider, for a moment, the following actual exchange, recorded in an elementary school classroom:
TEACHER: Alright class, here is a ratio problem for you. In order to paint a certain wall pink, a painter uses a gallon of white paint mixed with three drops of red paint. How much white and red paint would he use to paint a wall three times that size?[Reported in The Word Problem As Genre In Mathematics Education, by Susan Gail Gerofsky.]
PUPIL: Teacher, I know! My parents run a painting company, so I learned this from them. If you paint a really big wall, you have to mix the color a little bit darker, because the sunlight falling on a large wall will make the color appear lighter. And you would have to mix up the first gallon, and then mix the other batches to a chip, because there might be a slight color difference in different job lots of paint from the factory. In any case, you wouldn't mix up three batches of paint all at once, because the colors would start to separate before you were ready to use them. You're usually better to trust your eye than just to go by the measurements anyway ...
TEACHER: Alright! Enough! What you have to realize is that we're not talking about painting here, we're talking about ratio!
Now, I suppose that, faced with such a response, a truly gifted teacher, with a lot of knowledge of mathematics, paints, painting, the physics of light, and color-cognition, who also has sufficient time available, might use the child's fascinating response to start a highly educational investigation, but it would be hard to pull off, especially on the fly. Personally, I'd rather not introduce the problem in the first place, and thereby avoid having to crush the enthusiastic interest of the pupil who answered. (What are the chances that the child in question never again volunteered an answer to a question in the math class? Other than - perhaps - the unrealistic answer the child knows the teacher is looking for.)
Of course, many students do eventually learn to "play the game," albeit in many cases at the cost of coming to view mathematics as having no relevance to the real world. Then we hit them with problems about how many 40-seat buses does it take to transport 500 men, and slap them down again when they answer 12 1/2. (In this case, the "right" answer is usually 13 buses. But in today's energy conscious world, the contractor might well send down 12 40-seat buses and an additional 20-seat bus, so 12 1/2 is not at all unrealistic.)
"Oh, so we should ignore some real-world knowledge, but not all of it. Tell me, teacher, which parts should we ignore?" The only answer, of course, is that you ignore the parts that would give an answer other than the one I want you to produce. Which is fine if you already know the math, but hopeless as a pedagogic device to help students gain their initial understanding. I suspect the only people who survive this educational process are the ones who go on to teach math.
Talking of how we should be teaching math, I hope that by now any math teachers in my audience have seen the Youtube video of math teacher Dan Meyer's TED talk last March. If not, please do so right now. You'll find it below. Dan is by no means unique in his approach, though sadly he is most definitely not the norm. What singles him out is his use of online-posted home-video to broadcast his thoughts. (I hope math textbook publishers have seen Dan's speech as well. By all appearances, they have set their sights on becoming the math ed equivalent of Fox News (sic). This not in the best long term interests of the United States.)