## At Least One Daughter Exists

In the class I’m teaching we’ve entered probability. This is a fun subject. It’s one of the bits of mathematics which people encounter most often, about as much as the elements of geometry enter ordinary life. It seems like everyone has some instinctive understanding of probability, at least given how people will hear a probability puzzle and give a solution with confidence. You don’t get that with pure algebra problems. Ask someone “the neighbor’s two children were born three years apart and twice the sum of their ages is 42; how old are they?” and you get an assurance of how mathematics was always their weakest subject and they never could do it. Ask someone “one of the neighbor’s children just walked in, and was a girl; what is the probability the other child is also a girl?” and you’ll get an answer.

But it’s getting a correct answer that is really interesting, and unfortunately, while everyone has some instinctive understanding and will give an answer as above, there’s little guarantee it’ll be the right one. Sometimes, and I say this looking over the exam papers, it seems our instinctive understanding of probability is designed to be the wrong one. I’m happy that people aren’t afraid of doing probability questions, not the way they are afraid of algebra or geometry or calculus or the more exotic realms, though, and feel like it’s my role to find the most straightforward ways to understanding which start from that willingness to try.

Some of the rotten track record people have in probability puzzles probably derives from how so many probability puzzles start as recreational puzzles, that is, things which are meant to look easy and turn out to be subtly complicated. I suspect the daughters-question comes from recreational puzzles, since there’s the follow-up question that “the elder child enters, and is a girl; what is the probability the younger is a girl?” There’s some soundness in presenting the two as a learning path, since they present what looks like the same question twice, and get different answers, and learning why there are different answers teaches something about how to do probability questions. But it still feels to me like the goal is that pleasant confusion a trick offers.

## Illicitly Counted Coins

The past month I’ve had the joy of teaching a real, proper class again, after a hiatus of a few years. The hiatus has given me the chance to notice some things that I would do because that was the way I had done them, and made it easier to spot things that I could do differently.

To get a collection of data about which we could calculate statistics, I had everyone in the class flip a coin twenty times. Besides giving everyone something to do besides figure out which of my strange mutterings should be written down in case they turn out to be on the test, the result would give me a bunch of numbers, centered around ten, once they reported the number of heads which turned up. Counting the number of heads out of a set of coin flips is one of the traditional exercises to generate probability-and-statistics numbers.

Good examples are some of the most precious and needed things for teaching mathematics. It’s never enough to learn a formula; one needs to learn how to look at a problem, think of what one wants to know as a result of its posing, identify what one needs to get those results, and pick out which bits of information in the problem and which formulas allow the result to be found. It’s all the better if an example resembles something normal people would find to raise a plausible question. Here, we may not be all that interested in how many times a coin comes up heads or tails, but we can imagine being interested in how often something happens given a number of chances for it to happen, and how much that count of happenings can vary if we watch several different runs.

## One Goat Short

I like game shows. Liking game shows is not one of the more respectable hobbies, compared to, say, Crimean War pedantry, or laughing at goats. Game shows have a long history of being sneered at by people who can’t be bothered to learn enough about game shows to sneer at them for correct reasons. Lost somewhere within my archives is even an anthology of science fiction short stories about game shows, which if you take out the punch lines of “and the loser DIES!” or “and the host [ typically Chuck Woolery ] is SATAN!”, would leave nearly nothing, and considering that science fiction as a genre has spent most of its existence feeling picked-on as the “smelly, unemployed cousin of the entertainment industry” (Mike Nelson’s Movie Megacheese) that’s quite some sneering. Sneering at game shows even earned an episode of The Mary Tyler Moore show which managed to be not just bad but offensively illogical.

Nevertheless, I like them, and was a child at a great age for game shows on broadcast television: the late 1970s and early 1980s had an apparently endless menu of programs, testing people’s abilities to think of words, to spell words, to price household goods, and guess how other people answered surveys. We haven’t anything like that anymore; on network TV about the only game shows that survive are Jeopardy! (which nearly alone of the genre gets any respect), Wheel of Fortune, The Price Is Right, and, returned after decades away, Let’s Make A Deal. (I don’t regard reality shows as game shows, despite a common programming heritage. I can’t say what it is precisely other than location and sometimes scale that, say, Survivor or The Amazing Race do that Beat The Clock or Truth Or Consequences do not, but there’s something.) Now and then something new flutters into being, but it vanishes without leaving much of a trace, besides retreading jokes about the people who’d watch it.

All that is longwinded preliminary to one of those things that amuses mostly me. On the Thursday (27 October) episode of Let’s Make A Deal, they briefly looked like they might be playing The Monty Hall Problem.