So here’s my homework problem: On the original WiiFit there were five activities for testing mental and physical agility, one of which I really disliked. Two of the five were chosen at random each day. On WiiFitPlus, there are two sets of five activities each, with one exercise drawn at random from the two disparate sets, each of which has a test I really dislike. Am I more likely under the WiiFit or under the WiiFitPlus routine to get a day with one of the tests I can’t stand? Here, my reasoning.
We got a WiiFit, and a Wii, for Christmas in 2008, and for me, at that time, it was just what I needed to lose an extraordinary amount of weight. As part of the daily weighing-in routine it offers a set of challenges to your mental and physical agility. This is a pair drawn from, in the original release, five exercises. One is the Balance Test, measuring whether you can shift a certain percentage of your weight to the left or right and hold it for three seconds; the balance board, used for each of these tests, measures how much of your weight is where, left or right, front or back of the board. One is the Steadiness Test, about how still you can stand for thirty seconds and is trickier than it looks. (Breathe slowly, is my advice.) One is the Single Leg balance Test, trying to keep your balance within a certain range of centered for thirty seconds (and the range narrows at ten, twenty, and twenty-five seconds in). One — the most fun — is the Agility Test, in which you swing your body forward and back, left and right to hit as many targets as possible. And the most agonizing of them is the Walking Test, which is simply to take twenty footfalls, left and right, and which reports back how incredibly far from balanced your walk is. The game almost shakes its head and sighs, at least, at how imbalanced I am.
To work out the task of figuring out whether it was plausible that there had been only one “clean sweep”, of all six contestants winning the Item Up For Bid on The Price Is Right coming from the same seat, we had started a little into the binomial distribution. The key ideas included that we have “Bernoulli trials”, a number of independent chances for some condition to happen — in this case, we had about 6,000 such trials, the number of hourlong episodes of The Price Is Right — and a probability p of successfully seeing some event occur on any one episode. We worked that out to be somewhere about p = 1/1000, if every seat is equally likely to win every time. There is also a probability of 1 – p or 999/1000 of the event failing to see this event, that is, that one or more contestants comes from a different seat.
To find the probability of seeing some number, call it x since we don’t particularly care what it is, of successes out of some larger number, call it N because that’s a convenient number, of trials, we need to figure out how many ways there are to arrange x successes out of N trials. For small x and N values we can figure this out by hand, given time. For large numbers, we’d never finish if we tried by hand. But we can solve it, if we attack the problem methodically.