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.

Just before assigning the project it struck me: we always count heads. So I told everyone to count the number of tails which came up, instead. It’s an equivalent problem, of course, as long as the coins are fair, and in probability-and-statistics puzzles we live in a universe where all coins are fair, all dice are evenly balanced, and all decks of cards are excellently shuffled, at least until we get to the part about hypothesis testing when we want to say what the odds are that things aren’t fair.

It felt odd calling for tails rather than heads, and I had to catch myself several times when describing what they were doing or talking about what the results were. Isn’t that bizarre? It’s not surprising that there are a few experiments called on to teach probability. It’s easy to create an experiment that has so many results it’s impossible to identify all possible outcomes, and identifying all possible outcomes is the first step to figuring out the probability of events happening. Coin tosses, die rolls, card selections, and drawing marbles out of bags hit that lovely spot where one can have an intuitive feel for what’s being asked and can still work it out on paper or, if need be, get a bag and some marbles and try it out until you believe the answer. The dice, the cards, and the bags of marbles are never provided; most of my students had coins, and I had enough loose change that I could cover the students who somehow didn’t even have a dime in their pockets.

It’s some kind of locking-in effect: someone in the misty dawn of time found this good example of counting how many times heads came up when one flipped a coin enough times, and people saw the example and copied it, and went on to teach other people using the count-of-heads, and students copied that, and on and on until it feels vaguely illicit to count the number of tails which come up instead. Probably many people came up with the same coin-flipping idea; but why count heads rather than tails? I don’t know. Maybe it’s easier to think of the side of the coin with the face on it. Maybe it’s just that heads comes first in the alphabet. I leave the question for the hungry psychology ABD.

Occasionally an example can wear out to the point it becomes embarrassingly dated. I remember one probability textbook with the question of figuring out the chance that three out of five transistors in a radio would fail, given a certain chance of any one failing. Five is a peculiar number of transistors to have in a radio, and transistor failure is almost as peculiar an idea: solid-state electronics, to a first approximation, fail the first time they’re turned on or never. But if we imagine, instead of transistors, the problem with vacuum tubes, we find what was an excellent example when it was written. Five was a reasonable number of vacuum tubes for a radio to have, and one or more tube failing a routine occurrence. I regret I don’t remember which book had this wide-lapel polyester jacket of a problem; I’m curious whether current editions have updated it to failed microchips in an MP3 player.

On the other hand, does it matter whether the examples are worn out? If a student only encounters them a few times in the introduction of the subject, and goes through the introduction just one or two times, they’re not likely to notice every book and every course has the same set of examples. It’s for the instructor to notice that heads get counted, dice get the number of times they come up less than three or else an even number tallied, that Jacks or hearts are picked out of decks in an apparently endless series of Statistics 1 classes, and that bags are full of red and black balls.

I think one of these times I may just go against tradition and find an actual bag and some marbles to bring in to class, though.