## JH van ‘t Hoff and the Gaseous Theory of Solutions; also, Pricing Games

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 $\sqrt{DAY}$. 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.

• #### Marta Frant 5:27 am on Thursday, 7 April, 2016 Permalink | Reply

Questions, questions, questions… The constant ‘why’ is what makes the world go around.

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• #### Joseph Nebus 2:07 am on Saturday, 9 April, 2016 Permalink | Reply

‘Why’ is indeed one of the big questions. ‘What’ and ‘The Heck?’ are also pretty important.

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## Reading the Comics, November 1, 2015: Uncertainty and TV Schedules Edition

Brian Fies’s Mom’s Cancer is a heartbreaking story. It’s compelling reading, but people who are emotionally raw from lost love ones, or who know they’re particularly sensitive to such stories, should consider before reading that the comic is about exactly what the title says.

But it belongs here because in the October 29th and the November 2nd installments are about a curiosity of area, and volume, and hypervolume, and more. That is that our perception of how big a thing is tends to be governed by one dimension, the length or the diameter of the thing. But its area is the square of that, its volume the cube of that, its hypervolume some higher power yet of that. So very slight changes in the diameter produce great changes in the volume. Conversely, though, great changes in volume will look like only slight changes. This can hurt.

Tom Toles’s Randolph Itch, 2 am from the 29th of October is a Roman numerals joke. I include it as comic relief. The clock face in the strip does depict 4 as IV. That’s eccentric but not unknown for clock faces; IIII seems to be more common. There’s not a clear reason why this should be. The explanation I find most nearly convincing is an aesthetic one. Roman numerals are flexible things, and can be arranged for artistic virtue in ways that Arabic numerals make impossible.

The aesthetic argument is that the four-character symbol IIII takes up nearly as much horizontal space as the VIII opposite it. The two-character IV would look distractingly skinny. Now, none of the symbols takes up exactly the same space as their counterpart. X is shorter than II, VII longer than V. But IV-versus-VIII does seem like the biggest discrepancy. Still, Toles’s art shows it wouldn’t look all that weird. And he had to conserve line strokes, so that the clock would read cleanly in newsprint. I imagine he also wanted to avoid using different representations of “4” so close together.

Jon Rosenberg’s Scenes From A Multiverse for the 29th of October is a riff on both quantum mechanics — Schödinger’s Cat in a box — and the uncertainty principle. The uncertainty principle can be expressed as a fascinating mathematical construct. It starts with Ψ, a probability function that has spacetime as its domain, and the complex-valued numbers as its range. By applying a function to this function we can derive yet another function. This function-of-a-function we call an operator, because we’re saying “function” so much it’s starting to sound funny. But this new function, the one we get by applying an operator to Ψ, tells us the probability that the thing described is in this place versus that place. Or that it has this speed rather than that speed. Or this angular momentum — the tendency to keep spinning — versus that angular momentum. And so on.

If we apply an operator — let me call it A — to the function Ψ, we get a new function. What happens if we apply another operator — let me call it B — to this new function? Well, we get a second new function. It’s much the way if we take a number, and multiply it by another number, and then multiply it again by yet another number. Of course we get a new number out of it. What would you expect? This operators-on-functions things looks and acts in many ways like multiplication. We even use symbols that look like multiplication: AΨ is operator A applied to function Ψ, and BAΨ is operator B applied to the function AΨ.

Now here is the thing we don’t expect. What if we applied operator B to Ψ first, and then operator A to the product? That is, what if we worked out ABΨ? If this was ordinary multiplication, then, nothing all that interesting. Changing the order of the real numbers we multiply together doesn’t change what the product is.

Operators are stranger creatures than real numbers are. It can be that BAΨ is not the same function as ABΨ. We say this means the operators A and B do not commute. But it can be that BAΨ is exactly the same function as ABΨ. When this happens we say that A and B do commute.

Whether they do or they don’t commute depends on the operators. When we know what the operators are we can say whether they commute. We don’t have to try them out on some functions and see what happens, although that sometimes is the easiest way to double-check your work. And here is where we get the uncertainty principle from.

The operator that lets us learn the probability of particles’ positions does not commute with the operator that lets us learn the probability of particles’ momentums. We get different answers if we measure a particle’s position and then its velocity than we do if we measure its velocity and then its position. (Velocity is not the same thing as momentum. But they are related. There’s nothing you can say about momentum in this context that you can’t say about velocity.)

The uncertainty principle is a great source for humor, and for science fiction. It seems to allow for all kinds of magic. Its reality is no less amazing, though. For example, it implies that it is impossible for an electron to spiral down into the nucleus of an atom, collapsing atoms the way satellites eventually fall to Earth. Matter can exist, in ways that let us have solid objects and chemistry and biology. This is at least as good as a cat being perhaps boxed.

Jan Eliot’s Stone Soup Classics for the 29th of October is a rerun from 1995. (The strip itself has gone to Sunday-only publication.) It’s a joke about how arithmetic is easy when you have the proper motivation. In 1995 that would include catching TV shows at a particular time. You see, in 1995 it was possible to record and watch TV shows when you wanted, but it required coordinating multiple pieces of electronics. It would often be easier to just watch when the show actually aired. Today we have it much better. You can watch anything you want anytime you want, using any piece of consumer electronics you have within reach, including several current models of microwave ovens and programmable thermostats. This does, sadly, remove one motivation for doing arithmetic. Also, I’m not certain the kids’ TV schedule is actually consistent with what was on TV in 1995.

Oh, heck, why not. Obviously we’re 14 minutes before the hour. Let me move onto the hour for convenience. It’s 744 minutes to the morning cartoons; that’s 12.4 hours. Taking the morning cartoons to start at 8 am, that means it’s currently 14 minutes before 24 minutes before 8 pm. I suspect a rounding error. Let me say they’re coming up on 8 pm. 194 minutes to Jeopardy implies the game show is on at 11 pm. 254 minutes to The Simpsons puts that on at midnight, which is probably true today, though I don’t think it was so in 1995 just yet. 284 minutes to Grace puts that on at 12:30 am.

I suspect that Eliot wanted it to be 978 minutes to the morning cartoons, which would bump Oprah to 4:00, Jeopardy to 7:00, Simpsons and Grace to 8:00 and 8:30, and still let the cartoons begin at 8 am. Or perhaps the kids aren’t that great at arithmetic yet.

Stephen Beals’s Adult Children for the 30th of October tries to build a “math error” out of repeated use of the phrase “I couldn’t care less”. The argument is that the thing one cares least about is unique. But why can’t there be two equally least-cared-about things?

We can consider caring about things as an optimization problem. Optimization problems are about finding the most of something given some constraints. If you want the least of something, multiply the thing you have by minus one and look for the most of that. You may giggle at this. But it’s the sensible thing to do. And many things can be equally high, or low. Take a bundt cake pan, and drizzle a little water in it. The water separates into many small, elliptic puddles. If the cake pan were perfectly formed, and set on a perfectly level counter, then the bottom of each puddle would be at the same minimum height. I grant a real cake pan is not perfect; neither is any counter. But you can imagine such.

Just because you can imagine it, though, must it exist? Think of the “smallest positive number”. The idea is simple. Positive numbers are a set of numbers. Surely there’s some smallest number. Yet there isn’t; name any positive number and we can name a smaller number. Divide it by two, for example. Zero is smaller than any positive number, but it’s not itself a positive number. A minimum might not exist, at least not within the confines where we are to look. It could be there is not something one could not care less about.

So a minimum might or might not exist, and it might or might not be unique. This is why optimization problems are exciting, challenging things.

Niklas Eriksson’s Carpe Diem for the 1st of November, 2015. I’m not sure how accurately the art depicts bedbugs, although I’m also not sure how accurately Eriksson should.

Niklas Eriksson’s Carpe Diem for the 1st of November is about understanding the universe by way of observation and calculation. We do rely on mathematics to tell us things about the universe. Immanuel Kant has a bit of reputation in mathematical physics circles for this observation. (I admit I’ve never seen the original text where Kant observed this, so I may be passing on an urban legend. My love has several thousands of pages of Kant’s writing, but I do not know if any of them touch on natural philosophy.) If all we knew about space was that gravitation falls off as the square of the distance between two things, though, we could infer that space must have three dimensions. Otherwise that relationship would not make geometric sense.

Jeff Harris’s kids-information feature Shortcuts for the 1st of November was about the Harvard Computers. By this we mean the people who did the hard work of numerical computation, back in the days before this could be done by electrical and then electronic computer. Mathematicians relied on people who could do arithmetic in those days. There is the folkloric belief that mathematicians are inherently terrible at arithmetic. (I suspect the truth is people assume mathematicians must be better at arithmetic than they really are.) But here, there’s the mathematics of thinking what needs to be calculated, and there’s the mathematics of doing the calculations.

Their existence tends to be mentioned as a rare bit of human interest in numerical mathematics books, usually in the preface in which the author speaks with amazement of how people who did computing were once called computers. I wonder if books about font and graphic design mention how people who typed used to be called typewriters in their prefaces.

• #### ivasallay 11:13 pm on Wednesday, 4 November, 2015 Permalink | Reply

I wouldn’t have seen any of these without your blog. Thank you for including all of them.
Mom’s Cancer is sad but appears to be slightly improving. I hope for remission.
Adult Children makes an awesome point.

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• #### Joseph Nebus 1:16 am on Friday, 6 November, 2015 Permalink | Reply

I’m glad you enjoy. (I’m assuming enjoy.) Part of what’s fun about doing these, besides that it provokes me to write about stuff I didn’t plan ahead of time to do, is that I get to read a great diversity of comic strips. And sometimes introduce people to comics they had no idea existed.

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• #### sheldonk2014 11:01 am on Monday, 16 November, 2015 Permalink | Reply

Hey Joseph
Thank you for visiting
As always Sheldon

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• #### Joseph Nebus 4:07 am on Tuesday, 17 November, 2015 Permalink | Reply

Quite welcome. Glad to see you again.

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## The Mathematics Of A Pricing Game

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.

• #### elkement 5:38 pm on Wednesday, 9 October, 2013 Permalink | Reply

I have read an interesting book on our abilities to assess simple math – called The Science of Fear (as concerned with the assessment of risk and probabilities). I would not be surprised if the simple calculation involved in this show would make a difference. In this books psychological evidence was given that, for example, 3 of 100 ias perceived intuitively in a different way than 3% although we know these are the same.

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• #### Joseph Nebus 2:53 am on Friday, 18 October, 2013 Permalink | Reply

I’m interested and glad to see that book’s in the library. I figure to borrow it next time I’m there.

I’ve been fascinated informally with how framing a problem makes people more or less likely to solve it ever since long ago I noted that nobody in my family had any problem setting the VCR to record stuff, and someone else pointed out that we talked about “setting” the machine instead of “programming” it to record, the way most people did. Whether our general ability followed from thinking of it as an easy thing to do or vice-versa I couldn’t answer but the correlation interested me.

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• #### elkement 7:23 am on Friday, 18 October, 2013 Permalink | Reply

Thanks for sharing this anecdote. So probably if you consider ‘programming’ as something very interesting (as I do) you might intimidate readers by using such ‘geeky’ / ‘technical’ language although just wanted to share your enthusiasm. I will try to take that into account when writing about something abstract the next time.

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• #### Joseph Nebus 11:30 pm on Tuesday, 22 October, 2013 Permalink | Reply

I don’t actually know there’s a connection, but it does feel intuitively like there’s probably a link between terms that sound like jargon and people feeling they can’t follow it. I suppose that’s similar to the book-publishing lore that every equation cuts book sales in half. That lore seems far too pat to be literally true but it does seem qualitatively to be on to something.

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## Solving The Price Is Right’s “Any Number” Game

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.

## Why Someone Should Not Take That Deal

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• #### Joseph Nebus 9:47 pm on Friday, 31 August, 2012 Permalink | Reply

I’m sorry; I was unnecessarily confusing there. The video shows a contestant who managed to pick off the top four prizes, eliminating the biggest payout, in descending order, so the possible offers from the Banker (or the amount in the suitcase) could be assumed to have shrunk as rapidly as possible. The contestant was remarkably unlucky not in picking the lowest-value suitcase (that’s as likely as picking the highest-value suitcase, after all) but in picking the highest-possible-value suitcase for elimination four times in a row.

The question I put out about the five unrevealed suitcases was intended to be a separate problem, not the one faced by the video’s contestant.

I don’t know whether by the construction of the game the Banker or the Host know which suitcase holds the greatest or the lowest values. My suspicion is that they would not because the fewer people know the money amounts and the less this information is communicated, the harder it is to rig the show’s outcome.

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• #### Chiaroscuro 3:37 am on Friday, 31 August, 2012 Permalink | Reply

Well, for the purely mathematical, playing-the-odds answer: The total money up for grabs is $67511. Divided by 5, that means the average money left in each case is$13502.20. So the contestant should hold out, given that $13502.20 is larger than$11750.

–Chi

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• #### Joseph Nebus 9:50 pm on Friday, 31 August, 2012 Permalink | Reply

You’re correctly calculating the expectation value, or in other terms, the average payoff for the Contestant who in this situation sticks to her suitcase rather than taking the sure thing.

Arguably, therefore, sticking with the suitcase is the correct thing to do.

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• #### fluffy 5:17 am on Friday, 31 August, 2012 Permalink | Reply

I’m with Chiaroscuro on this one.

Also, I want to be on Deal Or No Deal solely so that I can run through the numbers as quickly and purely-randomly as I can, because I can’t stand watching that show which is nothing but people talking about their supersitious reasons why they’d be taking one number over another. If they insist on patter then I’d come up with reasoning about how much I HATE each number and why none of them could possibly be a winning number. like, SCREW 13, that’s how old I was when my grandma who taught me chess died!

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• #### Joseph Nebus 9:57 pm on Friday, 31 August, 2012 Permalink | Reply

Ah, now … I don’t like the post-Millionaire trend of giving contestants unlimited time to reason out their answer, but I also understand that for a show like Deal or No Deal, where there’s literally nothing to do other than for the Contestant to say why she or he is picking this, that the explanation is … well, no, the suspense is the meat of the show, but there’s not suspense unless there’s something the audience is waiting to see revealed, and if you rush things there’s no waiting.

I suspect that the show’s producers would be happy with you coming up with lunatic reasons for each number selection. (I’m reminded of a comic who said that if he were to be on Millionaire he’d want to come up with ludicrously wrongheaded chains of reasoning before picking the correct answer, eg, concluding Pearl Harbor was in 1941 because he remembers it was before Lincoln was shot but after the invention of Tamagotchis.) But rushing through just wouldn’t do; the audience has to get to know you through the explanations.

(I also don’t like the Millionaire-influenced design of game show sets as dark, technophobia-induced pits with heavy bass playing. The Australian Deal or No Deal set better fits the tone of being a lighter, warmer show. It should be more cozy than oppressive.)

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## About Chances of Winning on The Price Is Right

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.

• #### Joe Fix It 2:21 pm on Saturday, 21 April, 2012 Permalink | Reply

Did you watch all 6000 episodes yet???

Ya gotta put this stuff into an e-book!!!

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• #### Joseph Nebus 6:47 pm on Saturday, 21 April, 2012 Permalink | Reply

Not even nearly six thousand episodes. I’d be surprised if I’ve managed two thousand episodes lifetime, and that’d be impossible without the modern conveniences of Tivo and online streaming.

I have thought about e-books, but certainly haven’t got enough material for it yet.

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## So If You Can’t Win The Clock Game You Should Feel Bad

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). • #### Joseph Nebus 6:05 pm on Sunday, 19 February, 2012 Permalink | Reply Tags: binomial distribution ( 13 ), clean sweeps ( 7 ), expectation value ( 9 ), game show ( 13 ), Matlab ( 8 ), Octave ( 5 ), probability ( 73 ), significance level ( 3 ), significance testing ( 6 ), television ( 7 ), The Price Is Right ( 21 ), tv ## Proving Something With One Month’s Counting 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? • #### Chiaroscuro 6:48 am on Monday, 20 February, 2012 Permalink | Reply Impressive how looking at this smaller occurence (By bid rather than by show) yields such a more ready result than considering all the shows Like • #### Joseph Nebus 7:23 am on Monday, 20 February, 2012 Permalink | Reply It’s a neat effect. It comes about from looking at something, the first bidder winning, that’s just so enormously likely to happen compared to the clean-sweep, though. One or two successes, more or less, doesn’t substantially change the fraction of wins for the first seat out of these 120 items up for bid. One or two successes, more or less, would be a big change in the number of clean-sweep episodes out of 6000. So, roughly, it’s easier to tell the difference between something happening and just luck when it’s easy to get a lot of examples of the thing happening. Like • #### Joseph Nebus 4:16 am on Thursday, 16 February, 2012 Permalink | Reply Tags: binomial distribution ( 13 ), clean sweeps ( 7 ), Drew Carey ( 10 ), expectation value ( 9 ), game show ( 13 ), Matlab ( 8 ), Octave ( 5 ), probability ( 73 ), significance testing ( 6 ), television ( 7 ), The Price Is Right ( 21 ), tv ## What Can One Week Prove? We have some reason to think the chance of winning an Item Up For Bids, if you’re the first one of the four to place bids — let’s call this the first bidder or first seat so there’s a name for it — is lower than the 25 percent which we’d expect if every contestant in The Price Is Right‘s Contestants Row had an equal shot at it. Based on the assertion that only one time in about six thousand episodes had all six winning bids in one episode come from the same seat, we reasoned that the chance for the first bidder — the same seat as won the previous bid — could be around 17 percent. My next question is how we could test this? The chance for the first bidder to win might be higher than 17 percent — around 1/6, which is near enough and easier to work with — or lower than 25 percent — exactly 1/4 — or conceivably even be outside that range. 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? • #### Joseph Nebus 8:24 pm on Sunday, 12 February, 2012 Permalink | Reply Tags: ansatz ( 2 ), assumption, assumptions ( 3 ), clean sweeps ( 7 ), Drew Carey ( 10 ), expectation value ( 9 ), game show ( 13 ), penalty, probability ( 73 ), television ( 7 ), The Price Is Right ( 21 ), tv ## Figuring Out The Penalty Of Going First 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. • #### Chiaroscuro 5:40 am on Monday, 13 February, 2012 Permalink | Reply A much nicer explanation of the sort of thing I just did with a fair amount of the [1/x] button in the windows XP calculator and some messing around. Indeed, it’s some very rough assumptions made; but we’ve got to start somewhere, and this is a good place to start. Like • #### nebusresearch 4:42 am on Wednesday, 15 February, 2012 Permalink | Reply Oh, ew, you worked it out from trying out different percentages until you found one that matched? Actually, that’s a respectable numerical-solution technique, called “regula falsi”, that I should probably explain since it’s powerful, simple, and works. I’ll make a note of that. Like • #### Chiaroscuro 4:22 am on Thursday, 16 February, 2012 Permalink | Reply Oh goodness no. I *estimated* a few times to get to the proper neighborhood, then figured a way to reverse what I was doing. “So 23% yields… and how about 20%.. hmmm. lower. How about 16.6%.. too low. okay, then we do this in reverse and start with 1/6000…” It is a wonderful method for ‘ballpark figures’, quite true. Like • #### Joseph Nebus 7:33 am on Monday, 20 February, 2012 Permalink | Reply Ah, OK, I follow now. I think I can tie this in to something I’d wanted to talk about anyway, too, so I appreciate the hook. Like • #### Joseph Nebus 3:05 am on Wednesday, 8 February, 2012 Permalink | Reply Tags: binomial distribution ( 13 ), Bob Barker ( 2 ), Chuck Woolery, clean sweeps ( 7 ), Drew Carey ( 10 ), probability ( 73 ), significance testing ( 6 ), television ( 7 ), The Price Is Right ( 21 ), tv, wheel of fortune ## Interpreting Drew Carey If we’ve decided that at the significance level we find comfortable there are too few clean sweeps of any position in Contestants Row, the natural question is why there are so few. We estimated there should have been six clean sweeps, based on modelling clean-sweep occurrences as a binomial distribution. Something in the model went wrong. Let’s try to reason out what it was. 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? • #### Chiaroscuro 5:56 am on Wednesday, 8 February, 2012 Permalink | Reply And I didn’t want to interrupt. Of course, we might estimate how much more likely the advantage given to the last bidder is- let’s say it’s roughly 31% over time (Figuring it could raise to a rough 1/3, but there’s exact bids, and$1 bids which are sometimes too low, which might lower it from that). Making the big assumption that the other three bidders have a roughly equal chance as well (Which is also a big assumption, given that the second bidder knows the first bid, the third knows the first and second, but it’s pretty good for me) Then we’ve got the last seat at 31%, and the other three seats at 23%.. From THERE we’d have to solve for the odds of a clean sweep in one show, and see where that lies, and I suspect that’s enough to throw the odds just towards harder enough to make it a bit more towards 1 in 6,000.

But I’m not going to do that math, offhand.

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• #### nebusresearch 2:54 am on Thursday, 9 February, 2012 Permalink | Reply

I’m not sure that, with the given information, there’s really a way to say how much of an advantage the last bidder gets over the ones who go first. We can come up with decent reasons to think it’s one thing or another, but I think the limits of calculation on this data, the one clean sweep in 6,000 shows, are estimating how big a disadvantage the person going first has.

Of course, someone tracking every episode to see which number bidder — first, second, third or fourth — as opposed to which seat could probably make a pretty good estimate within a couple of months.

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• #### Chiaroscuro 6:08 am on Thursday, 9 February, 2012 Permalink | Reply

A very rough calculation would lead to one in six thousand being the expected result of each round, the person winning being the first bidder is 17.6%. (A 20% chance would yield one in 3,125.) You are correct though in that it does not matter what breaks a clean sweep for this- if the second, third, or fourth bidder wins the item up for bid, that breaks the clean sweep. It doesn’t matter which has the higher advantages, just how low the first bidder’s is.

(To note, my original estimate of 23% put the odds at one in 1,555.)

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• #### nebusresearch 12:26 am on Sunday, 12 February, 2012 Permalink | Reply

You’ve got just the right calculation, yes, and I make it out to be the same estimate.

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• #### BunnyHugger 8:47 pm on Wednesday, 8 February, 2012 Permalink | Reply

I miss when they used to shop for prizes. Everyone remembers the dalmatian statue that was almost always in the prize showcase, but there was an end table shaped like an elephant that recurred often and which I liked as a kid.

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• #### nebusresearch 2:57 am on Thursday, 9 February, 2012 Permalink | Reply

I miss the shopping for prizes too, even if they did bring the flow of gameplay to a stop. (I suppose I might search YouTube for ancient episodes to see what I think of the pacing now.) Between all the special spots on the wheel and the toss-up puzzles it can be hard spotting the clean lines of the original game underneath all the cruft now.

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## Finding, and Starting to Understand, the Answer

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.

• #### BunnyHugger 4:38 am on Sunday, 5 February, 2012 Permalink | Reply

Good explanation.

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• #### nebusresearch 4:46 am on Monday, 6 February, 2012 Permalink | Reply

Aw, thank you kindly.

Don’t worry. I can drag it out to the point nobody cares what I was explaining anymore.

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• #### snidelytoo 8:23 pm on Sunday, 18 March, 2012 Permalink | Reply

“declaring as the result of something suspicious what is actually only chance” … that’s not as transparent a phrase as I’d like to see in its position. Hmmm, “declaring something to be a suspicious result when it is actually only chance”? I’m not sure I’ve come up with anything better, but I hope you can see my problem.

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• #### Joseph Nebus 1:00 am on Tuesday, 20 March, 2012 Permalink | Reply

I agree with your assessment, that it’s not a very clear phrase. Unfortunately it’s hard to get the concept just right.

The classic example is probably this: imagine flipping a fair coin over and over and over and over, and it comes up tails every single time, from now to the end of time. It’s impossible that this should happen if the coin is fair and the toss isn’t rigged in any way. Yet, it’s really not impossible, since there’s no reason a fairly flipped coin shouldn’t come up tails, and that it’s come up tails the last million times doesn’t mean it shouldn’t come up tails again.

It can be put reasonably precisely in the form of theorems and proofs, but that just changes the kind of difficult-to-read statements that one looks at.

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## The Significance of the Item Up For Bids

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.

Or …

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