I do need to take another light week of writing I’m afraid. There’ll be the Theorem Thursday post and all that. But today I’d like to point over to Gaurish4Math’s WordPress Blog, and a discussion of the Galton Board. I’m not familiar with it by that name, but it is a very familiar concept. You see it as Plinko boards on The Price Is Right and as a Boardwalk or amusement-park game. Set an array of pins on a vertical board and drop a ball or a round chip or something that can spin around freely on it. Where will it fall?

It’s random luck, it seems. At least it is incredibly hard to predict where, underneath all the pins, the ball will come to rest. Some of that is ignorance: we just don’t know the weight distribution of the ball, the exact way it’s dropped, the precise spacing of pins well enough to predict it all. We don’t care enough to do that. But some of it is real randomness. Ideally we make the ball bounce so many times that however well we estimated its drop, the tiny discrepancy between where the ball is and where we predict it is, and where it is going and where we predict it is going, will grow larger than the Plinko board and our prediction will be meaningless.

(I am not sure that this literally happens. It is possible, though. It seems more likely the more rows of pins there are on the board. But I don’t know how tall a board really needs to be to be a chaotic system, deterministic but unpredictable.)

But here is the wonder. We cannot predict what any ball will do. But we can predict something about what every ball will do, if we have enormously many of them. Gaurish writes some about the logic of why that is, and the theorems in probability that tell us why that should be so.

Do you ever think about why stuff dissolves? Like, why a spoon of sugar in a glass of water should seem to disappear instead of turning into a slight change in the water’s clarity? Well, sure, in those moods when you look at the world as a child does, not accepting that life is just like that and instead can imagine it being otherwise. Take that sort of question and put it to adult inquiry and you get great science.

Peter Mander of the Carnot Cycle blog this month writes a tale about Jacobus Henricus van ‘t Hoff, the first winner of a Nobel Prize for Chemistry. In 1883, on hearing of an interesting experiment with semipermeable membranes, van ‘t Hoff had a brilliant insight about why things go into solution, and how. The insight had only one little problem. It makes for fine reading about the history of chemistry and of its mathematical study.

In other, television-related news, the United States edition of The Price Is Right included a mention of “square root day” yesterday, 4/4/16. It was in the game “Cover-Up”, in which the contestant tries making successively better guesses at the price of a car. This they do by covering up wrong digits with new guesses. For the start of the game, before the contestant’s made any guesses, they need something irrelevant to the game to be on the board. So, they put up mock calendar pages for 1/1/2001, 2/2/2004, 3/3/2009, 4/4/2016, and finally a card reading . The game show also had a round devoted to Pi Day a few weeks back. So I suppose they’re trying to reach out to people into pop mathematics. It’s cute.

Comic Strip Master Command had the regular pace of mathematically-themed comic strips the last few days. But it remembered what the 14th would be. You’ll see that when we get there.

Ray Billingsley’s Curtis for the 11th of March is a student-resists-the-word-problem joke. But it’s a more interesting word problem than usual. It’s your classic problem of two trains meeting, but rather than ask when they’ll meet it asks where. It’s just an extra little step once the time of meeting is made, but that’s all right by me. Anything to freshen the scenario up.

Tony Carrillo’s F Minus for the 11th was apparently our Venn Diagram joke for the week. I’m amused.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 12th of March name-drops statisticians. Statisticians are almost expected to produce interesting pictures of their results. It is the field that gave us bar charts, pie charts, scatter plots, and many more. Statistics is, in part, about understanding a complicated set of data with a few numbers. It’s also about turning those numbers into recognizable pictures, all in the hope of finding meaning in a confusing world (ours).

Brian Anderson’s Dog Eat Doug for the 13th of March uses walls full of mathematical scrawl as signifier for “stuff thought deeply about’. I don’t recognize any of the symbols specifically, although some of them look plausibly like calculus. I would not be surprised if Anderson had copied equations from a book on string theory. I’d do it to tell this joke.

And then came the 14th of March. That gave us a bounty of Pi Day comics. Among them:

And Missy Meyer’s informational panel cartoon Holiday Doodles mentions that besides “National” Pi Day it was also “National” Potato Chip Day, “National” Children’s Craft Day, and “International” Ask A Question Day. My question: for the first three days, which nation?

Edited To Add: And I forgot to mention, after noting to myself that I ought to mention it. The Price Is Right (the United States edition) hopped onto the Pi Day fuss. It used the day as a thematic link for its Showcase prize packages, noting how you could work out π from the circumference of your new bicycles, or how π was a letter from your vacation destination of Greece, and if you think there weren’t brand-new cars in both Showcases you don’t know the game show well. Did anyone learn anything mathematical from this? I am skeptical. Do people come away thinking mathematics is more fun after this? … Conceivably. At least it was a day fairly free of people declaring they Hate Math and Can Never Do It.

Hi again. I was hesitant to look at this month’s statistics, as I pretty much fell off the face of the earth for a week there, but I didn’t have the chance to do the serious thinking that’s needed for mathematics writing. The result’s almost exactly the dropoff in readership I might have predicted: from 440 views in October down to 308, and from 220 unique visitors down to 158. That’s almost an unchanged number of views per visitor, 2.00 dropping to 1.95, so at least the people still interested in me are sticking around.

The countries sending me the most viewers were as ever the United States, then Austria (hi, Elke, and thank you), the United Kingdom and then Canada. Sending me a single visitor each were Bulgaria, Cyprus, Czech Republic, Ethiopia, France, Jordan, Lebanon, Nepal, New Zealand, Russia, Singapore, Slovenia, Switzerland, and Thailand. This is also a drop in the number of single-viewer countries, although stalwarts Finland and the Netherlands are off the list. Slovenia’s the only country making a repeat appearance from last month, in fact.

The most popular articles the past month were:

How Many Trapezoids I Can Draw, the attempt to classify how many different sorts of trapezoids there are, something of obvious steady interest to the public.

And I apologize for not having produced many essays the past couple weeks, and can only fault myself for being more fascinated by some problems in my day job that’ve been taking up time and mental energy and waking me in the middle of the night with stuff I should try. I’ll be back to normal soon, I’m sure. Don’t tell my boss.

There was a new pricing game that debuted on The Price Is Right for the start of its 42nd season, with a name that’s designed to get my attention: it’s called “Do The Math”. This seems like a dangerous thing to challenge contestants to do since the evidence is that pricing games which depend on doing some arithmetic tend to be challenging (“Grocery Game”, “Bullseye”), or confusing (“The Check Game”), or outright disasters (“Add Em Up”). This one looks likely to be more successful, though.

The setup is this: The contestant is shown two prizes. In the first (and, so far, only) playing of the game this was a 3-D HDTV and a motorcycle. The names of those prizes are put on either side of a monitor made up to look like a green chalkboard. The difference in prize values is shown; in this case, it was $1160, and that’s drawn in the middle of the monitor in Schoolboard Extra-Large font. The contestant has to answer whether the price of the prize listed on the left (here, the 3-D HDTV) plus the cash ($1160) is the price of the prize on the right (the motorcycle), or whether the price of the prize on the left minus the cash is the price of the prize on the right. The contestant makes her or his guess and, if right, wins both prizes and the money.

There’s not really much mathematics involved here. The game is really just a two-prize version of “Most Expensive” (in which the contestant has to say which of three prizes and then it’s right there on the label). I think there’s maybe a bit of educational value in it, though, in that by representing the prices of the two prizes — which are fixed quantities, at least for the duration of taping, and may or may not be known to the contestant — with abstractions it might make people more comfortable with the mathematical use of symbols. x and all the other letters of the English (and Greek) alphabets get called into place to represent quantities that might be fixed, or might not be; and that might be known, or might be unknown; and that we might actually wish to know or might not really care about but need to reference somehow.

That conceptual leap often confuses people, as see any joke about how high school algebra teachers can’t come up with a consistent answer about what x is. This pricing game is a bit away from mathematics classes, but it might yet be a way people could see that the abstraction idea is not as abstract or complicated as they fear.

I suspect, getting away from my flimsy mathematics link, that this should be a successful pricing game, since it looks to be quick and probably not too difficult for players to get. I’m sorry the producers went with a computer monitor for the game’s props, rather than — say — having a model actually write plus or minus, or some other physical prop. Computer screens are boring television; real objects that move are interesting. There are some engagingly apocalyptic reviews of the season premiere over at golden-road.net, a great fan site for The Price Is Right.

I don’t understand why, but an awful lot of the advice I see about blogging says that it’s important not just to keep track of how your blog is doing, but also to share it, so that … numbers will like you more? I don’t know. But I can give it a try, anyway.

For June 2013, according to WordPress, I had some 713 page views, out of 246 unique visitors. That’s the second-highest number of page views I’ve had in any month this year (January had 831 views), and the third-highest I’ve had for all time (there were 790 in March 2012). The number of unique visitors isn’t so impressive; since WordPress started giving me that information in December 2012, I’ve had more unique visitors … actually, in every month but May 2013. On the other hand, the pages-per-viewer count of 2.90 is the best I’ve had; the implication seems to be that I’m engaging my audience.

My most frequent commenters, “recent”, whatever that means, are Chiaroscuro and BunnyHugger (virtually tied), with fluffy, elkelement, MJ Howard, and Geoffrey Brent rounding out the top six.

The most common source of page clicks the past month was from the United States (468), with Brazil (51) and Canada (23) taking silver and bronze. And WordPress recorded one click each from Portugal, Serbia, Hungary, Macedonia (the Former Yugoslav Republic), Indonesia, Argentina, Poland, Slovenia, and Viet Nam. I’ve been to just one of those countries.

A friend who’s also into The Price Is Right claimed to have noticed something peculiar about the “Any Number” game. Let me give context before the peculiarity.

This pricing game is the show’s oldest — it was actually the first one played when the current series began in 1972, and also the first pricing game won — and it’s got a wonderful simplicity: four digits from the price of a car (the first digit, nearly invariably a 1 or a 2, is given to the contestant and not part of the game), three digits from the price of a decent but mid-range prize, and three digits from a “piggy bank” worth up to $9.87 are concealed. The contestant guesses digits from zero through nine inclusive, and they’re revealed in the three prices. The contestant wins whichever prize has its price fully revealed first. This is a steadily popular game, and one of the rare Price games which guarantees the contestant wins something.

A couple things probably stand out. The first is that if you’re very lucky (or unlucky) you can win with as few as three digits called, although it might be the piggy bank for a measly twelve cents. (Past producers have said they’d never let the piggy bank hold less than $1.02, which still qualifies as “technically something”.) The other is that no matter how bad you are, you can’t take more than eight digits to win something, though it might still be the piggy bank.

What my friend claimed to notice was that these “Any Number” games went on to the last possible digit “all the time”, and he wanted to know, why?

My first reaction was: “all” the time? Well, at least it happened an awful lot of the time. But I couldn’t think of a particular reason that they should so often take the full eight digits needed, or whether they actually did; it’s extremely easy to fool yourself about how often events happen when there’s a complicated possibile set of events. But stipulating that eight digits were often needed, then, why should they be needed? (For that matter, trusting the game not to be rigged — and United States televised game shows are by legend extremely sensitive to charges of rigging — how could they be needed?) Could I explain why this happened? And he asked again, enough times that I got curious myself.

Putting together links to all my essays about trapezoid areas made me realize I also had a string of articles examining that problem of The Price Is Right, with Drew Carey’s claim that only once in the show’s history had all six contestants winning the Item Up For Bids come from the same seat in Contestants’ Row. As with the trapezoid pieces they form a more or less coherent whole, so, let me make it easy for people searching the web for the likelihood of clean sweeps or of perfect games on The Price Is Right to find my thoughts.

I have one last important thing to discuss before I finish my months spun off an offhand comment from The Price Is Right. There are a couple minor points I can also follow up on, but I don’t think they’re tied tightly enough to the show to deserve explicit mention or rate getting “tv” included as one of my keywords. Here’s my question: what’s the chance of winning an average pricing game, after one has got an Item Up For Bid won?

At first glance this is several dozen questions, since there are quite a few games, some winnable on pure skill — “Clock Game”, particularly, although contestants this season have been rotten at it, and “Hole In One … Or Two”, since a good miniature golfer could beat it — and some that are just never won — “Temptation” particularly — and some for which partial wins are possible — “Money Game” most obviously. For all, skill in pricing things help. For nearly all, there’s an element of luck.

I’m not going to attempt to estimate the chance of winning each of the dozens of pricing games. What I want is some kind of mean chance of winning, based on how contestants actually do. The tool I’ll use for this is the number of perfect episodes, episodes in which the contestant wins all six pricing games, and I’ll leave it to the definers of perfect such questions as what counts as a win for “Pay The Rent” (in which a prize of $100,000 is theoretically possible, but $10,000 is the most that has yet been paid out) or “Plinko” (theoretically paying up to $50,000, but which hasn’t done so in decades of playing).

One week, it seems, isn’t enough to tell the difference conclusively between the first bidder on Contestants Row having a 25 percent chance of winning — winning one out of four times — or a 17 percent chance of winning — winning one out of six times. But we’re not limited to watching just the one week of The Price Is Right, at least in principle. Some more episodes might help us, and we can test how many episodes are needed to be confident that we can tell the difference. I won’t be clever about this. I have a tool — Octave — which makes it very easy to figure out whether it’s plausible for something which happens 1/4 of the time to turn up only 1/6 of the time in a set number of attempts, and I’ll just keep trying larger numbers of attempts until I’m satisfied. Sometimes the easiest way to solve a problem is to keep trying numbers until something works.

In two weeks (or any ten episodes, really, as talked about above), with 60 items up for bids, a 25 percent chance of winning suggests the first bidder should win 15 times. A 17 percent chance of winning would be a touch over 10 wins. The chance of 10 or fewer successes out of 60 attempts, with a 25 percent chance of success each time, is about 8.6 percent, still none too compelling.

Here we might turn to despair: 6,000 episodes — about 35 years of production — weren’t enough to give perfectly unambiguous answers about whether there were fewer clean sweeps than we expected. There were too few at the 5 percent significance level, but not too few at the 1 percent significance level. Do we really expect to do better with only 60 shows?

The obvious thing to do is test: watch a couple episodes, and see whether it’s nearer to 1/6 or to 1/4 of the winning bids come from the first seat. It’s easy to tally the number of items up for bid and how often the first bidder wins. However, there are only six items up for bid each episode, and there are five episodes per week, for 30 trials in all. I talk about a week’s worth of episodes because it’s a convenient unit, easy to record on the Tivo or an equivalent device, easy to watch at The Price Is Right‘s online site, but it doesn’t have to be a single week. It could be any five episodes. But I’ll say a week just because it’s convenient to do so.

If the first seat has a chance of 25 percent of winning, we expect 30 times 1/4, or seven or eight, first-seat wins per week. If the first seat has a 17 percent chance of winning, we expect 30 times 1/6, or 5, first-seat wins per week. That’s not much difference. What’s the chance we see 5 first-seat wins if the first seat has a 25 percent chance of winning?

Let’s accept the conclusion that the small number of clean sweeps of Contestants Row is statistically significant, that all six winning contestants on a single episode of The Price Is Right come from the same seat less often than we would expect from chance alone, and that the reason for this is that whichever seat won the last item up for bids is less likely to win the next. It seems natural to suppose the seat which won last time — and which is therefore bidding first this next time — is at a disadvantage. The irresistible question, to me anyway, is: how big is that disadvantage? If no seats had any advantage, the first, second, third, and fourth bidders would be expected to have a probability of 1/4 of winning any particular item. How much less a chance does the first bidder need to have to get the one clean sweep in 6,000 episodes reported?

Chiaroscuro came to an estimate that the first bidder had a probability of about 17.6 percent of winning the item up for bids, and I agree with that, at least if we make a couple of assumptions which I’m confident we are making together. But it’s worth saying what those assumptions are because if the assumptions do not hold, the answers come out different.

The first assumption was made explicitly in the first paragraph here: that the low number of clean sweeps is because the chance of a clean sweep is less than the 1 in 1000 (or to be exact, 1 in 1024) chance which supposes every seat has an equal probability of winning. After all, the probability that we saw so few clean sweeps for chance alone was only a bit under two percent; that’s unlikely but hardly unthinkable. We’re supposing there is something to explain.

One assumption for a binomial distribution are that we have some trial, some event, which happens many times. Each episodes is the obvious trial here. The outcome we’re interested in seeing has some probability of happening on each trial; there is indeed some probability of a clean sweep each episode. The binomial distribution assumes that this probability is constant for every trial, that it doesn’t become more or less likely the tenth or hundredth or thousandth time around, and this seems likely to hold for The Price Is Right episodes. Granted there is some chance of a clean sweep in one episode; what could be done to increase or decrease the likelihood from episode to episode?

If the probability of having one or fewer clean sweep episodes of The Price Is Right out of 6,000 aired shows is a little over one and a half percent — and it is — and we consider outcomes whose probability is less than five percent to be so unlikely that we can rule them out as happening by chance — and, last time, we did — then there are improbably few episodes where all six contestants came from the same seat in Contestants Row, and we can usefully start looking for possible explanations as to why there are so few clean sweeps. At least, that’s the conclusion at our significance level, that five percent.

But there’s no law dictating that we pick that five percent significance level. If we picked a one percent significance level, which is still common enough and not too stringent, then we would say this might be fewer clean sweeps than we expected, but it isn’t so drastically few as to raise our eyebrows yet. And we would be correct to do so. Depending on the significance level, what we saw is either so few clean sweeps as to be suspicious, or it’s not. This is why it’s better form to choose the significance level before we know the outcome; it feels like drawing the bullseye after shooting the arrow the other way around.

The last important idea missing before we can judge this problem about The Price Is Right clean sweeps of Contestants Row is the significance level. Whenever an experiment is run — whether it’s the classic probability class problems of flipping coins or rolling dice, or whether it’s watching 6,000 episodes of a game show to see whether any seat produces the most winners, or whether it’s counting the number of red traffic lights one gets during the commute — there are some outcomes which are reasonably likely, some which are unlikely, and some which are vanishingly improbable.

We have to decide that some outcomes have such a low probability of happening naturally that they represent something going on, and are not just the result of chance. How low that probability should be is our decision. There are some common dividing lines, but they’re common just because they represent numbers which human beings find to be nice round figures: five percent, one percent, half a percent, one-tenth of a percent. What significance level one picks depends on many factors, including what’s common in the field, how different outcomes are expected to be, even what one can afford. Physicists looking for evidence of new subatomic particles have an extremely high standard before declaring something is definitely a new particle, but, they can run particle detection experiments until they get such clear evidence.

To be fair, we ought to pick our significance level before we’ve worked out the probability of something happening, but this is the earliest I could discuss it with motivation for you to read about it. But if we take the five percent significance level, we see we know already that there’s a little more than a one and a half percent chance of there being as few clean sweeps as observed. The conclusion is obvious: all six winning contestants in an episode should have come from the same seat, over 6,000 episodes, more often than the one time Drew Carey claimed they had. We can start looking for explanations for why there should be this deficiency.

We became suspicious of the number of clean sweeps in The Price Is Right when there were not the expected six of them in 6,000 episodes. The chance there would be only one was about one and a half percent, not very high. But are there so few clean sweeps that we should be suspicious? That is, is the difference between the expected number of sweeps and the observed number so large as to be significant? Is it too big to just result from chance?

This is significance testing: is whatever quantity we mean to observe dramatically less than what is expected? Is it dramatically more? Is it at least different? Are these differences bigger than what could be expected by mere chance? For every statistician’s favorite example, a tossed fair coin will come up tails half the time; that means, of twenty flips, there are expected to be ten tails. But there being merely nine or as many as twelve is reasonable. Three or fifteen tails may be a little unlikely. Zero or twenty seem impossible. There’s a point where if our observations are so different from what we expect then we have to reject the idea that our observations and our expectations agree.

It’s not enough to say there’s a probability of only 1.5 percent that there should be exactly one clean sweep episode out of 6,000, though. It’s unlikely that should happen, but if we look at it, it’s unlikely there should be any outcome. Even the most likely result of 6,000 episodes, six clean sweeps, has only about one chance in six of happening. That’s near the chance that the next person you meet will have a birthday in either September or November. That isn’t absurdly unlikely, but, the person betting against it has the surer deal.

We worked out the likelihood that there would be only one clean sweep, with all six contestants getting on stage coming from the same seat in Contestants Row, out of six thousand episodes of The Price Is Right. That turned out to be not terribly likely: it had about a one and a half percent chance of being the case. For a sense of scale, that’s around the same probability that the moment you finish reading this sentence will be exactly 26 seconds past the minute. It’s pretty safe to bet that it wasn’t.

However, it isn’t particularly outlandish to suppose that it was. I’d certainly hope at least some reader found that it was. Events which aren’t particularly likely do happen, all the time. Consider the likelihood of this single-clean-sweep or the 26-seconds-past-the-minute thing happening to the likelihood of any given hand of poker: any specific hand is phenomenally less likely, but something has to happen once you start dealing. So do we have any grounds for saying the particular outcome of one clean sweep in 6,000 shows is improbable? Or for saying that it’s reasonable?

When last we talked about the “clean sweep” of winning contestants coming from the same of four seats in Contestants Row for all six Items Up For Bid on The Price Is Right, we had got established the pieces needed if we suppose this to be a binomial distribution problem. That is, we suppose that any given episode has a probability, p, of successfully having all six contestants from the same seat, and a probability 1 – p of failing to have all six contestants from the same seat. There are N episodes, and we are interested in the chance of x of them being clean sweeps. From the production schedule we know the number of episodes N is about 6,000. We supposed the probability of a clean sweep to be about p = 1/1000, on the assumption that the chance of winning isn’t any better or worse for any contestant. The probability of there not being a clean sweep is then 1 – p = 999/1000. And we expected x = 6 clean sweeps, while Drew Carey claimed there had been only 1.

The chance of finding x successes out of N attempts, according to the binomial distribution, is the probability of any combination of x successes and N – x successes — which is equal to (p)^{(x)} * (1 – p)^{(N – x)} — times the number of ways there are to select x items out of N candidates. Either of those is easy enough to calculate, up to the point where we try calculating it. Let’s start out by supposing x to be the expected 6, and later we’ll look at it being 1 or other numbers.

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.

So, we calculated that on any given episode of The Price Is Right there’s around one chance of all six winners of the Item Up For Bid coming from the same seat. And we know there have been about six thousand episodes with six Items Up For Bid. So we expect there to have been about six clean sweep episodes; yet if Drew Carey is to be believed, there has been just the one. What’s wrong?

Possibly, nothing. Just because there is a certain probability of a thing happening does not mean it happens all that often. Consider an analogous situation: a baseball batter might hit safely one time out of every three at-bats; but there would be nothing particularly odd in the batter going hitless in four at-bats during a single game, however much we would expect him to get at least one. There wouldn’t be much very peculiar in his hitting all four times, either. Our expected value, the number of times something could happen times the probability of it happening each time, is not necessarily what we actually see. (We might get suspicious if we always saw the expected value turn up.)

Still, there must be some limits. We might accept a batter who hits one time out of every three getting no hits in four at-bats. If he got no runs in four hundred at-bats, we’d be inclined to say he’s not a decent hitter having some bad luck. More likely he’s failing to bring the bat with him to the plate. We need a tool to say whether some particular outcome is tolerably likely or so improbable that something must be up.