Prepared by J. Laurie Snell, Bill Peterson and Charles Grinstead, with help from Fuxing Hou, and Joan Snell.

Please send comments and suggestions for articles to

jlsnell@dartmouth.edu.

Back issues of Chance News and other materials for teaching a Chance course are available from the Chance web site:

http://www.dartmouth.edu/~chance

Chance News is distributed under the GNU General Public License (so-called 'copyleft'). See the end of the newsletter for details.

Chance News is best read using Courier 12pt font.

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It has now been proved beyond doubt that smoking is one of the leading causes of statistics.

Fletcher Krebel

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**Contents of Chance News 8.08**

- 1. Notes on Chance Videos and web sites.
- 2. The Bible Codes are cracked.
- 3. When hope wins over science.
- 4. Mozart sonata's IQ impact: Eine Kleine oversold?
- 5. Marilyn vos Savant on the "Mozart effect."
- 6. Playing poker when you can see the future.
- 7. Colleges by the numbers.
- 8. How were Scott Berry's home run predictions?
- 9. The shaky science of predicting tremors.
- 10. The diameter of the World Wide Web.
- 11. New hope for cloud seeding.
- 12. Ranking cities for safety of children.
- 13. Two articles on bus safety.
- 14. Marilyn hears the famous two-bomb problem.
- 15. Fathom: Dynamic statistics software.
- 16. Chance Rules: a great book for all ages.
- 17. What is Random? A book to make you ponder.

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>>>>>==============>

Note: As you know, we put a series of Chance Lectures on the
Chance web site. These are lectures by experts in areas of
probability and statistics that occur regularly in the news. We
have also made a CD-ROM containing these lectures. Viewing these
lectures with the CD-ROM still requires a web browser, but does
not require an internet connection. If you would like one of
these CD-ROMs send an e-mail message to jlsnell@dartmouth.edu
giving the address where it should be sent.

David Howell at the University of Vermont uses our Chance Videos
in his course "Lies, Damn Lies and Statistics" and is interested
in corresponding with others on ways to use these videos in class.
His email address is David.Howell@uvm.edu. You can see his course
Psychology 95.

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Ge Groenewegen suggested
this web site which is maintained by Juha Puranen in the Department of
Statistics, University of Helsinki. It has a remarkably complete
list of links related to statistical education.

Bill Peterson has a web site for his Chance course at Middlebury. This is a first-year seminar that combines chance and writing. See Bill's web site FS 026, Fall 1998: CHANCE.

Another Chance-like course is being taught by Dan Schult at Colgate College. See Math 102 Course Information.

Finally Kenneth Steele suggested a web site that critically analyzes the performance of Marilyn vos Savant. It is Marilyn Fails at the Polls.

As readers have probably gathered from our accounts of Marilyn's
problems, we are not as critical of her solutions as some.
However, there are lots of interesting remarks about her problems
and their solutions at this web site.

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>>>>>==============>

In Chance news 7.04
we discussed an experiment carried out by
11-year-old Emily Rosa and published in the New England Journal of
Medicine. Emily designed and carried out a very simple experiment
which suggested that therapeutic touch may be less effective than
its proponents claim. The 1999-2000 version of Paul Velleman's
ActivStats
includes a brief video of Emily carrying out her experiment. The
associated web sites contain further links related to this
experiment. An interesting site is
Does Therapeutic Touch work? where you will find links to skeptics
and believers.

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>>>>>==============>

Bible codes mystery explained

Statistical Science press release

The next issue of Statistical Science will contain the paper: "Solving the Bible Code Puzzle", by Brendan McKay, Dror Bar-Natan, Maya Bar-Hillel, and Gil Kalai. This paper is available at Bible Codes debunked in Statistical Science.

Statistical Science recently provided a press release related to this paper in which included the following introduction to the paper written by Robert E. Kass, former Executive Editor of Statistical Science.

One of the fundamental teachings in statistical training is that probability distributions can generate seemingly surprising outcomes much more frequently than naive intuition might suggest. For good reason, experienced statisticians have long been skeptical of claims based on human perception of extraordinary occurrences. Now that computer programs are widely available to help nearly anyone mine available data, there are wonderful new possibilities for discovering misleading patterns.

In this context, when the article Equidistant Letter Sequences in the Book of Genesis, by Witztum, Rips and Rosenberg, was examined by reviewers and editorial board members for Statistical Science, none was convinced that the authors had found something genuinely amazing. Instead, what remained intriguing was the difficulty of pinpointing the cause, presumed to be some flaw in their procedure, that produced such apparently remarkable findings. Thus, in introducing that paper, I wrote that it was offered to readers as a challenging puzzle.

Unfortunately, though perhaps not so surprisingly, many people outside our own profession interpreted publication of the paper as a stamp of scientific approval on the work. However, even though the referees had thought carefully about possible sources of error, no one we asked was willing to spend the time and effort required to reanalyze the data carefully and independently. Rather, we published the paper in the hope that someone would be motivated to devote substantial energy to figuring out what was going on and that the discipline of statistics would be advanced through the identification of subtle problems that can arise in this kind of pattern recognition.

In this issue, Brendan McKay, Dror Bar-Natan, Maya Bar- Hillel and Gil Kalai report their careful dissection and analysis of the equidistant letter sequence phenomenon. Their explanations are very convincing and, in broad stroke, familiar. They find that the specifications of the search (for hidden words) were, in fact, inadequately specific: just as in clinical trials, it is essential to have a strict protocol; deviations from it produce very many more opportunities for surprising patterns, which will no longer be taken into account in the statistical evaluation of the evidence. Choices for the words to be discovered may seem innocuous yet be very consequential. Because minor variations in data definitions and the procedure used by Witztum et al. produce much less striking results, there is good reason to think that the particular forms of words those authors chose effectively tuned their method to their data, thus invalidating their statistical test.

Considering the work of McKay, Bar-Natan, Bar-Hillel, and Kalai as a whole it indeed appears, as they conclude, that the puzzle has been solved.

There has not yet been much press response. Science (Bible Code
Bunkum, Science 1999 September 24; 285: 2057a in Random Samples)
had a brief discussion based on this press release but the most
complete discussion we found was the following:

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>>>>>==============>

The Torah Codes, Cracked

Slate, (Online magazine) posted 6 Oct. 1999

Benjamin Wittes

Wittes is an editorial writer for the Washington Post. This is a well-written discussion of why Statistical Science believes the Bible Codes puzzle has been solved. We will not try to summarize it since Wittes article is available on the web. You will find there also a response from Michael Drosnin which will sound all too familiar to those who have followed this controversy.

Of course, this is not the end of the Bible Code. The movie "The Omega Code" will be opening October 15 in a theater near you. From its synopsis we read:

<<<========<<A prophetic code hidden within the Torah. A sinister plot sealed until the end of the Age. Two men caught up in an ancient supernatural struggle to determine the fate of the next millennium...

For thousands of years, mystics and scholars alike have searched for a key to unlock the mysteries of our future. The key has been found. The end is here.

Never before has our distant past so collided with our coming future. Never before has modern technology uncovered such profound mathematical complexities as revealed within the Bible Code. Never before has the world seen so many ancient prophesies falling into place.

>>>>>==============>

Hope for sale.

New York Times, October 3, 1999, Sec. 1 p 1

Gina Kolata and Kurt Eichenwald

In Pediatrics, a Lesson in Making Use of Experimental Procedures

New York Times, October 3, 1999, Sec. 1 p 40

Gina Kolata and Kurt Eichenwald

Researchers have tried for years to conduct a clinical trial to determine if bone marrow transplants can save lives of patients with an advanced stage of breast cancer. Women who believe that this is their only chance are reluctant to leave this decision to chance and so refuse to sign up for clinical trials.

High-dose chemotherapy accompanied by transplant involves harvesting bone marrow or stem cells from the patient prior to chemotherapy. The patient then receives massive doses of cell- killing drugs, which also destroy bone marrow. After chemotherapy, the stem cells are replaced and the bone marrow restored with the hope that the drugs have killed the cancer cells and the bone marrow will grow back before the patient dies of infection. It is a difficult procedure that seriously compromises quality of life and has health risks of its own.

The procedure was started in 1979 by Dr. Hortobagyi. Early attempts were not successful but by the late 1980's the results looked very promising. Patients with advanced breast cancer who were given transplants had remission rates of 50 to 60 percent compared with the 10 to 15 percent remission rates achieved by conventional means.

This led to pressure from Congress and elsewhere for insurance companies to pay for this experimental treatment and also led to businesses established to provide the very expensive treatment.

As these businesses thrived and the procedure became widely used, some cancer specialists including Dr. Hotobagyi began to have second thoughts. He realized that the women whom he and others had chosen in the early days of this treatment were carefully selected to be younger than 60 and to not have other illnesses such as heart disease or emphysema that would make the procedure even more risky. On the other hand the success rate had been compared with the general population of women with advanced breast cancer who were treated with conventional therapies. When Hotbagyi looked again at the data, he discovered that the success rate for the women he had treated with transplants was not significantly better than the success rate for a similar group with advanced breast cancer but otherwise healthy and treated by conventional methods.

Last May, results of five studies--two large clinical trials in the United States, two large trials in Europe, and a small trial involving 154 women in South Africa--were reported at the annual meeting of the American Society of Clinical Onocology. In four of the five clinical trials, there was no difference in survival rates for women who were randomly assigned to have transplants and those who were assigned to have conventional therapy. Only in the South African study did those with transplants do better than those with conventional therapy. But a further look at the South African data showed that those who had transplants lived about as long as those in the other studies, while those who had the conventional treatment did far worse, raising some doubt about the validity of this study.

The National Breast Cancer Coalition response was that the data speak for themselves: bone marrow transplants have been tested and failed. On the other hand, the American Society of Clinical Oncologists put out a press release saying that the papers reported presented "mixed early results" and more years of study are required. However, as remarked earlier researchers are finding it difficult to find subjects for such experiments.

The authors observe that, while Federal rules require that new drugs or devices (such as a heart valve) should be proven safe and effective before being sold to the public, medical procedures (such as bone marrow transplants and surgical procedures) are not regulated by the government. The authors suggest that this reflects the government's reluctance to interfere with doctors' practice of medicine.

The second article explains how the pediatric oncologists have solved this problem for pediatric cancer. These oncologists have organized and have refused to offer experimental treatments outside of clinical trials. They sponsor only credible experiments. With their high quality-control they have been able to persuade insurance companies to pay for the treatments. While only one percent of adult cancer patients enroll in clinical trials, 60% of pediatric cancer patients do. The article documents some success stories that resulted from the pediatric oncologists being able to quickly carry out clinical trials leading to definitive results.

DISCUSSION QUESTIONS:

(1) What kind of ethical problems does this situation raise, both in the case of adult patients and in the case of children?

(2) Do you think the government should treat new medical
procedures the way they do new drugs?

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Mozart sonata's IQ impact: Eine Kleine oversold?

Washington Post, 30 August 1999, A9

Rick Weis

Prelude or requiem for the 'Mozart effect'?

Nature, 26 August 1999,p 826

Scientific Correspondance

Christopher F. Chabris

Kenneth M. Steele et al.

Francis Rauscher et al.

In an article in Nature (Vol.365, p. 611, 1993) Francis Rauscher, a psychologist at the University of Wisconsin at Oshkosh, and her colleagues reported a study showing that listening to a Mozart piano sonata produced a temporary improvement in performance on one portion of a standard I.Q. text. This improvement related to what are called spacial-temporal tasks. An example of such a task is: given a series of paper figures with cuts and folds indicated, choose which of a set of figures would be the result of this cutting and folding.

This and related results by Rauscher and her colleagues led to a self-help book and CD-Roms to help children shine on the barrage of tests they must take these days. The governor of Georgia decreed that every new-born child be given a state-purchased cassette or CD of classical music. Catalogs offered stethoscope- like devises to allow pregnant women's prenatal children to listen to Mozart in the womb.

However, as time evolved a number of other researchers had trouble replicating the Rauscher result. See the remarks by Kenneth Steele in Chance News 6.06. Steele, a psychologist at Applalachian State College, and his colleagues carried out three experiments attempting to replicate the experiment. The most recent study was reported in Psychological Science, 10, 366-369 1999. In their correspondence to Nature they write:

We tested the performance of subjects on the same task (a 16- or 18- item paper folding and cutting task) after listening to the same Mozart music as in the original experiment. Control conditions were either the same or chosen to broaden the comparison set, and consisted of silence, relaxation instructions, minimalist music (Music with Changing Parts by P. Glass) or relaxation music (The Shining Ones by P. Thorton). ...The results showed that listening to music made no differential improvement in spatial reasoning in any experiment.

Christopher Chabris, a Psychologist at Harvard, reports the results of a meta study consisting of 16 studies of the effect of Mozart's music on performance on intelligence tests. They write:

Exposure to Mozart's music does not seem to enhance general intelligence or reasoning, although it may exert a small effect on the ability to transform visual images.

In their reply, Rauscher and her colleagues criticize the meta study claiming, in particular, that it included studies testing other aspects of an intelligence test for which they never claimed a "Mozart effect". They claim that none of the three experiments by Steele and his colleagues exactly replicates their experiments and report that four other studies that do replicate their results have manuscripts in preparation for publication.

Click here, you can hear Keene, Chabris, and Rauscher giving their positions on the Mozart effect on NPR.

The Nature correspondence and the Psychological Science paper by Steele and his colleagues are available from Steele's web site so you can read this discussion for yourself and decide if you agree with the final remark in the Steele correspondence

A requiem may therefore be in order

or the final remark in the Raucher correspondence

Because some people cannot get bread to rise does not negate the existence of a 'yeast effect'.

or that in the Washington Post article:

If nothing else, it seems, the rise and fall of the Mozart effect may teach the public a lesson about the tentativenes of all scientific discovery. If that happens, then the incomparable composer will have made people wiser after all, if not actually smarter.

DISCUSSION QUESTIONS:

(1) In the Rauscher experiments the students were given a preliminary spacial-temporal test, used to divide the students into treatment and control groups in such a way as to make the groups comparable at this ability before the experiment. Steele and his colleagues randomized the groups to accomplish this. Rauscher writes

spacial task abilities vary widely between individuals, making randomization an inefficient way to ensure uniform before-treatment performance groups.

Does this seem to you to be a significant difference between the two methodologies?

(2) It is natural for experimenters studying the Mozart effect to try to identify other aspects of an I.Q. test for which listening to Mozart improves a student's performance. Do you think a meta study should include such studies? Is this a general problem with meta-studies?

(3) Raucher and others report that they have also shown that music training can help brain development. You can here Raucher discussing these experiments at NPR's April 4, 1997 Talk of the Nation: Science Friday

Does this seem a more plausible conclusion than the Mozart effect?

(4) It has been said that mathematicians tend to enjoy music and
mountain climbing -- the 'three M effect'. How would you design a
study to test this theory?

<<<========<<

>>>>>==============>

Ask Marilyn.

Parade Magazine, 29 August 1999 pp. 20-21

Marilyn vos Savant

A reader asks: "What are your views on the 'Mozart effect'? That is, the belief (or fact) that classical music enhances the capacity of the human mind. I have also read about a study demonstrating the 'Mozart effect' on laboratory mice."

Marilyn says that she wouldn't be surprised if mice exposed to heavy metal music were slower in mazes but adds that she does not believe that classical music increases human intelligence. She points out the research reported in Nature magazine in 1993 had a much more limited scope than popular accounts would have us believe. Specifically, a sample of college students was found to perform better on tests after listening to a Mozart sonata than after listening either to relaxation tapes or not listening to any tapes.

DISCUSSION QUESTIONS:

(1) Is Marilyn's first comment made in jest? Do you think it makes sense to talk about a Mozart effect in mice?

(2) What exactly is Marilyn's objection? Would you be happy if
listening to Mozart could improve your exam scores, even if we
couldn't agree whether it really increased your overall
"intelligence"?

<<<========<<

>>>>>==============>

How we learned to cheat in online poker

Arkin, Hill, Marks, Schmid, Walls and McGraw

Reliable Software Technologies

The authors are software professionals who help companies with security problems and some of whom also enjoy playing poker. They combined their interests to ask themselves how secure are poker games on the internet. They found that Planet Poker Internet Cardroom offers real-time Texas Hold'em games in which you play against other players on the Web for real money. Recall that Texas Hold'em is the game played at the World Series of Poker. See Chance News 8.05 for a description of how Texas Hold'em is played.

This is the story of how they found that for this casino and three other similar on-line casinos they could play Texas Hold'em and after seeing the first five cards (the two dealt to them and the three common cards) they could see into the future and tell exactly what cards each player has from that time on.

They started by looking at the FAQs at PlanetPoker where they found a description of the shuffling algorithm to show how fair the game was.

They first found errors in this algorithm. This algorithm shuffled a deck of cards which was: for each for i equal 1 to 52, swap the number at position i with that at a position randomly chosen from 1 to 52. So, for example, for a three card deck this process might go 123, 213,231. The authors point out that this shuffling method does not make all shuffles equally likely. The easiest way to see this for a three card deck is to observe that there are 6 possible orderings of 3 cards and 3^3 = 27 possible outcomes for the shuffle under this method of shuffling. Since 27 is not divisible by 6 it is not possible that the number of ways that each can occur is the same. For the case of a 52 card deck the same argument applies since 52^52 is not divisible by 52!

If the shuffling method does not produce random shuffles it is conceivable that a card shark could take advantage of this information. The authors point out that a correct method of shuffling would be; for i = 1 to 52 swap the number at position i with that at a position chosen random from i to 52.

A pseudo-random generator is used to do the swapping. Such generators typically produce an integer between 0 and N and return that number divided by N to get a random number between 0 and 1. Subsequent calls take the previous integer and pass it through a function to return another integer which is then again divided by N. In most random number generators N = 2^32 which is the largest value that will fit into a 32-bit number. Thus such a generator can produce only about 4 billion numbers. A number called the seed is chosen to start the algorithm and from here on the outcome is completely determined by the algorithm. Thus there can be at most 4 billion possible shuffles which is a far cry from the 52! or about 2^226 possible shuffles. The algorithm used chose the seed by the number of milliseconds since midnight. There are 86,400,000 milliseconds in a day. Thus there are, in fact, only about 86 million possible shuffles.

By synchronizing their computer with the system clock on the server generating the pseudo-random number, our sleuths were able to reduce the number of possible shuffles to about 200,000. An ordinary PC can easily run through this many shuffles in real time.

As we remarked earlier, in Texas Hold-em each player is dealt two cards and then three cards are dealt face up which are common cards to be used by any of the players. Thus our sleuths just ran through the 200,000 possible shuffles until they found one that produced these five cards and then knowing the shuffle, from that time on they knew exactly what cards every player had and what cards they or other players would have next.

Alas, instead of making a lot of money from this knowledge they arranged to show the folks at CNN that their method really worked by using the casino where Texas Hold'em was played but not for real money. CNN then put an account of this feat on their nightly news program. You can view this program.

DISCUSSION QUESTIONS:

(1) The sleuths said that it would be against the law to use their system in a real internet game. Why would this be illegal?

(2) Of course, the software companies involved have improved their shuffling algorithm and method for choosing random seeds. How would you have done this? (See for example HotBits: Genuine random numbers for a method of creating random numbers from radioactive decay processes)

(3) Why else might you worry that internet poker might not be a
completely fair game? For example, would collusion be possible
and effective for internet poker?

<<<========<<

>>>>>==============>

Colleges by the numbers.

The Washington Post, 29 August 1999, H1

Albert B. Crenshaw

The annual US News college rankings have appeared (US News & World Report, 30 August 1999). You can see the results online at 2000 college rankings.

What has Caltech done to move into the #1 slot as the best university in the country (the first time that an engineering school has occupied that position)? The change is due at least in part to a change in the ranking methodology, as US News itself acknowledges. In past years, critics of the rankings have charged that this annual fiddling with the formulas is just a ploy to sell magazines. Their point is that institutions don't change that much in a year, but US News wouldn't generate much excitement at the newsstand by printing the same rankings over and over.

Now the Post is reporting a more serious concern. James Monks of the Consortium on Financing Higher Education and Ronald Ehrenberg of the Cornell Higher Education Research Institute have published a study of relationship between college application rates and fluctuations in the rankings. They found that moving up in the rankings results in more applicants and higher admissions yield; dips in the rankings have the opposite effect. In fact, Monks and Ehrenberg were able to statistically predict the effect that an observed move up or down in the rankings would have on a school's ability to be selective the following year.

This kind of effect creates an incentive for college administrators to pay extra attention to the ranks, perhaps at the expense of focusing on their basic mission. Indeed, Ehrenberg claims that two years ago Cornell successfully jockeyed for position in the rankings by instituting changes that actually had "no effect on...academic quality." The result was a move from 14th in 1997 to 6th in 1998. But this year the US News formula was again altered, and Cornell has now dropped to 11th place. Ehrenberg promises to tell more of the Cornell story in a forthcoming book.

DISCUSSION QUESTION:

After reading this article, you might get the idea of looking at
the ranks over several years. Do you think an institution that
consistently ranks near a certain level is in some way "better"
than on whose rank fluctuates more widely around that level?

<<<========<<

>>>>>==============>

A statistician reads the sports pages.

Chance Magazine, Spring 1999

Scott Berry

As we reported in Chance News 8.05, Scott Berry in his Chance Magazine column developed a model for home-run hits by major league players and used it to predict the top 25 home-run hitters for the 1999 season. He also predicted the number of at bats for these 25 players. Well, the 1999 season is over and we can see how his predictions came out. Here are the predicted and the observed outcomes for the number of home runs and at bats. We have added the home run rate derived which is the number of at bats per home run. P means predicted and O means outcome.

Rank Player Team HR At bats At bats per HR P O P O P O 1 1 Mark McGwire StL 59 65 515 521 8.7 8.0 2 3 Ken Griffey Sea 51 48 602 606 11.8 12.6 3 2 Sammy Sosa ChC 50 63 618 625 12.4 9.9 4 14-15 Juan Gonzalez Tex 44 39 603 562 13.7 14.4 5 Vinny Castilla Col 39 33 604 615 15.5 18.6 6 19-20 Albert Belle Bal 39 37 592 610 15.2 16.5 7 Jose Canseco Atl 39 34 566 430 14.5 12.6 8 Andres Galarraga Atl 39 0 550 0 14.1 9 4 Rafael Palmeiro Tex 37 47 595 565 16.1 12.0 10 5-6 Greg Vaughn Cin 37 45 515 550 13.9 12.2 11 Mo Vaughn Ana 37 33 585 524 15.8 15.9 12 7-8 Manny Ramirez Cle 35 44 568 522 16.2 11.9 13 Barry Bonds SF 35 34 528 355 15.1 10.4 14 9-12 Jeff Bagwell Hou 34 42 552 562 16.2 13.4 15 Jim Thome Cle 35 33 518 494 14.8 15.0 16 9-12 Alex Rodriguez Sea 34 42 650 502 19.1 12.0 17 9-12 Vladimir Guerrero Mon 34 42 602 610 17.7 14.5 18 Jay Buhner Sea 34 0 524 80 15.4 19 13 Mike Piazza NYM 34 40 558 534 16.4 13.4 20 Nomar Garciaparra Bos 33 27 626 532 19.0 19.7 21 19-20 Larry Walker Col 33 37 548 438 16.6 11.8 22 7-8 Carlos Delgado Tor 33 44 239 573 7.2 13.0 23 Tino Martinez NYY 32 28 575 589 18.0 21.0 24 Henry Rodriguez ChC 32 26 506 447 15.8 17.2 25 Frank Thomas CWS 31 15 561 486 18.1 32.4

Scott based his predictions on the expected number of home runs given by his model. The standard deviation for the number of home runs varied from player to player and ranged from 7.3 for Frank Thomas to 10.2 for Mark McGuire.

Two players Galarraga and Buhner were out of the race because of an illness and an injury. Of the remaining players, most were within one standard deviation of the mean and all but one were within two standard deviations of the mean--Thomas being slightly more than two standard deviations from the mean. We note that Scott predicted the top three home-run hitters correctly. Considering that only about one standard deviation separates the 5th position from the 25th, it is impressive that only 4 are out of order among the 14 that made the top 25.

DISCUSSION QUESTIONS:

(1) Paste this data into your favorite statistical package and check the distributions of the home runs and at bats, make a scatter plot of the predicted and observed variable, etc. and see what conclusions you can make about the success of these predictions.

(2) As Scott remarks in his article, the simple model for estimating the number of home runs would be to assume that a particular player has a probability p of getting a home run each time at bat and that he will come to bat n times. Then p and n could be estimated from previous years' experience. (Scott's model assumes that n and p are themselves random variables whose distribution is estimated from previous years' data). What effect would using this simpler model have on the standard deviation of your estimates? Do you think the results would have been as good as those of Scott? Try it out and see if you are right.

(3) Here is an even simpler model to try out. Weather predictions are often checked against the simple method of prediction: the weather tomorrow will be the same as yesterday. Here are last year home run results for 1998 and 1999 for the players Scott considered:

Rank Player Team P 99 98 1 1 Mark McGwire StL 59 65 70 2 3 Ken Griffey Sea 51 48 56 3 2 Sammy Sosa ChC 50 63 66 4 14-15 Juan Gonzalez Tex 44 39 45 5 Vinny Castilla Col 39 33 46 6 19-20 Albert Belle Bal 39 37 49 7 Jose Canseco Atl 39 34 46 8 Andres Galarraga Atl(OL)39 0 9 4 Rafael Palmeiro Tex 37 47 43 10 5-6 Greg Vaughn Cin 37 45 50 11 Mo Vaughn Ana 37 33 40 12 7-8 Manny Ramirez Cle 35 44 45 13 Barry Bonds SF 35 34 37 14 9-12 Jeff Bagwell Hou 34 42 34 15 Jim Thome Cle 35 33 30 16 9-12 Alex Rodriguez Sea 34 42 42 17 9-12 Vladimir Guerrero Mon 34 42 38 18 Jay Buhner Sea 34 0 19 13 Mike Piazza NYM 34 40 32 20 Nomar Garciaparra Bos 33 27 35 21 19-20 Larry Walker Col 33 37 23 22 7-8 Carlos Delgado Tor 33 44 38 23 Tino Martinez NYY 32 28 28 24 Henry Rodriguez ChC 32 26 42 25 Frank Thomas CWS 31 15 29

How would you compare the prediction method "the weather will be the same as yesterday" with Scott's method? Under your comparison method, how do they compare? Do you see a "regression to the mean" effect in comparing 1998 and 1999?

(4) Laurie was betting that Sosa would win the home-run race. On
a critical day at the end of the race when the Cardinals were
playing a double header with Chicago, Laurie bet that McGuire
would not get a home run in either game. Was this a sensible bet?

<<<========<<

>>>>>==============>

The shaky science of predicting tremors.

The Irish Times, 20 August 1999, p26

Brendan McWilliams

Last month's events in Turkey again highlight our limited ability to anticipate the timing of major earthquakes. This article draws a distinction between earthquake forecasting and earthquake prediction. Forecasting involves estimating probabilities of quakes of certain magnitudes in a given region over time scales on the order of decades or centuries. Such forecasts are based on historical records and on satellite or ground-based observations of the rate of movement in the earth near known faults. Prediction, by contrast, means giving advance notice of a quake within of hours, days or perhaps months of the event. While forecasting techniques have gained general acceptance, prediction is now widely viewed as infeasible. The author provided the Richter quote to punctuate these comments. (And of course, since this article appeared we have seen a major quake in Taiwan.)

The importance of the distinction between a forecast and a prediction was illustrated by recent New York Times coverage of the Turkish quake. A 21 August article (Earthquake in Turkey: The overview, by Stephen Kinzer, p.A1) quoted Princeton seismology expert Ahmet Cakmak: "The area where this recent quake was centered will not be hit again any time soon, because the seismic pressure there has dissipated. But the fault that runs closer to Istanbul is still very dangerous, and a major quake here is likely." He added that "There's plenty of time to prepare for that quake, but usually the memory of devastation lasts five years at most...What we should do is learn from this one, expect a bigger one and be prepared." Given this last comment, Cakmak appears to be speaking about a long-range forecast. But readers apparently interpreted his earlier comments as predicting an imminent quake in Istanbul. On 23 August, the Times printed the following correction: "Because of an editing error, an article on Saturday about the earthquake in Turkey referred incompletely to a prediction...that a large earthquake would hit Istanbul. He did not predict such a quake anytime soon; he said there was a 60 percent chance of a major one in Istanbul in 30 to 80 years."

You can find a discussion of previous forecasts regarding the area of the recent quake in Science News online (8/28/99).

As described there, researchers from the U.S. Geological Survey discovered that 9 out of 10 large recorded earthquakes on Turkey's North Anatolian fault had occurred in areas where previous seismic events had increased pressure. One of the two high-risk areas they identified was the Izmit section, the site of the recent quake. Still, they had estimated only a 12 percent probability that this section would experience a major quake by 2026.

DISCUSSION QUESTION:

What is the analog of prediction/forecasting distinction in the
context of weather? How do the success rates there compare with
the situation described in the article?

<<<========<<

>>>>>==============>

Diameter of the World-Wide Web

Reka Albert, Hawong Jeong, and Albert-Laszlo Barabasi

Nature, Sept. 9, 1999, Vol. 401, p. 130

Growth Dynamics of the World-Wide Web

Nature, Sept. 9, 1999, Vol 401, p. 131

Bernardo A. Huberman and Lada A. Adamic

The authors of the first of these articles consider the World-Wide Web as a directed graph, with the vertices representing documents and the edges representing links that point from one document to the other. By looking at a small sub-domain of the Web, they estimate the distributions for P_out(k) and P_in(k), the probabilities that a document has k outgoing or incoming links, respectively. They find that these distributions both follow power laws over several orders of magnitude for k, i.e. they find that P(k) is approximated by k^(-c), where c is approximately 2.45 for outgoing links and 2.1 for incoming links. (The reader is invited to try to reconcile the fact that these two numbers are significantly different with the graph theoretical theorem that the average out-degree and the average in-degree are equal in any directed graph.)

Assuming that these power laws hold for the whole Web, one can simulate the Web in an attempt to estimate the average number of links that must be followed to get from one document to another, where the average is computed over all pairs of documents. The author finds that d is about 0.35 + 2.06(log(N)), where N is the number of documents in the Web (this number is currently estimated to be 8 x 10^8.) Thus, at present, the average distance between any two documents is about 19. Even a doubling in the size of the Web would increase this average distance just slightly. Whether this diameter is relevant to a human or a robot who is negotiating the Web is less than clear to this reviewer; the authors claim that `an intelligent agent, who can interpret the links and follow only the relevant one, can find the desired information quickly by navigating the Web.' Based upon his own experience, this reviewer can conclude that he is not an 'intelligent agent.'

The second article considers the distribution of the number of pages of information available at each web site on the Web. They find that there is a power law, i.e. the probability that a given site has k pages is proportional to k^(-c). They then attempt to account for this behavior by considering a growth process in which the short-term fluctuations of the site size is proportional to the size of the site. It turns out that this model is inadequate, as, over a long period of time, the distributions of sizes under this assumption will be log-normal (not a power law).

The authors claim that, if two additional assumptions are made
about the model, then the distribution of the site sizes will obey
a power law. These two assumptions are that new sites appear at
various times, and that different sites grow at different rates.

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Cloudbusting.

The Economist, 21 August 1999, pp. 69-70.

Cloud-seeding to induce rainfall was first attempted in the 1940s. The goal of seeding is to boost rainfall amounts from a cloud that has already begun raining. Thus any objective measure of success involves estimating how much rain would have fallen without seeding. This turns out not to be easy, and the article notes that despite 50 years of anecdotal evidence, there has been no solid proof that seeding actually works.

Now American researchers at the National Center for Atmospheric Research (NCAR) have spent three years on a rigorous investigation of a new technique called "hygroscopic flare seeding." A hygroscopic compound is one that attracts moisture. Salt is an example, and in clouds over the ocean rain droplets are known to form around salt particles. It seems reasonable to conjecture that seeding other clouds with hygroscopic particles will increase rainfall. Evidence for this was found in South Africa in 1988 by rain researcher Graeme Mather, who noticed that emissions from a paper mill seemed to promote raindrop formation in the clouds overhead. Chemical analysis showed that the emissions were effectively seeding the clouds with two hygroscopic salts: potassium chloride and sodium chloride. Mather followed up with a series of experiments using flares fired from small planes to seed clouds with hygroscopic compounds. The current NCAR efforts in Mexico now aim at reproducing and testing his results.

As observed earlier, conclusive research would require a careful experimental design to compare similar seeded and unseeded clouds. The following "double blind" protocol was used. As a storm develops, a plane seeks out clouds for seeding. When one is found, the pilot notifies a controller on the ground, who opens a randomly chosen envelope to reveal an instruction card. The card is equally likely to say "seed" or "don't seed". The controller accordingly tells the pilot whether or not to ignite the hygroscopic flare. But the pilot then chooses a similar envelope of his own, whose instruction card tells him whether or not to obey the controller's instruction. In either case, he continues along the flight path that would be used for seeding, so the controller is unaware of his action. So far, 99 such flights have been completed, 48 of which resulted in seeding. The results support Mather's finding that seeding increases rainfall by 30- 40%. But Dr. Brant Foote, who oversees the program, points out that at least 150 flights will be needed to provide solid statistical evidence.

As for the practical implications, Foote adds that economic questions will need to be addressed before the technique could be widely applied. In particular, the cost of the flights needs to be assessed in light of the amount of rain produced and the resulting agricultural benefits.

DISCUSSION QUESTIONS:

(1) Why do you think it is important to blind the controller as to whether his instructions are followed?

(2) A 30-40% increase in rainfall sounds impressive, but Dr.
Foote's comments suggest that 99 flights have not been enough to
get statistical significance. What, if anything, does this tell
you about the measurements?

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Group puts L.A. 17th among big cities for children's safety.

The Los Angeles Times, 25 August 1999, pB1

Peter Y. Hong

The Zero Population Growth (ZPG) organization has released its annual report on the "kid-friendliness" of US cities. The LA Times headline naturally highlights Los Angeles, which was ranked 17th among 25 major cities. In a previous Chance News 6.09 we discussed Money Magazine's ratings of the Best Places to Live. But as the article states, the ZPG study differs from such business-oriented ratings, focusing instead on "social, economic and environmental justice." Among the criteria used were teenage pregnancy rates, infant mortality, school dropout rates, air quality and population. Large swings in population--up or down-- hurt cities in the rankings. The logic, according to ZPG's Peter Kostmayer, is that public services are not likely to keep pace with rapid growth, while shrinking population erodes the tax base that funds such services.

You can read more detail on the ranking procedure at the ZPG web site Kid Friendly Cities.

Here is their description of the basic formula used to assign grades in each category:

"In order to determine, for example, the dropout rate score for Worcester, MA (drop-out rate is 15.6%), the first step is to find the suburb with the lowest dropout rate. This was Irvine, CA at 2.1%. Next find the highest dropout rate: Santa Ana, CA at 36.2%. We subtract to get the range (36.2 - 2.1 = 34.1). Then we followed our equation:

(15.6 - 2.1) / 34.1 = 0.61

"That score indicates that the Worcester dropout rate was a little better than average for all of the suburbs that we surveyed. And, in fact, Worcester ended up with an education grade of C and ranked 58th on the list of suburbs, almost right in the middle."

Cities that did not score well were naturally motivated to question the methodology. Los Angeles felt it was a victim of outdated Census figures; for example, 1990 data on home ownership and children living in poverty reflect the national recession at that time, but do not give a fair picture of the city in 1999. Other cities felt that certain figures were just plain wrong. San Bernadino's unemployment rate was given as 10.4% in the study, but the city's own estimate is 5.7%.

The article quotes a media critic named James Fallows as saying "Just as politicians know that gimmicks make the evening news, these groups know that lists make the evening news. The question is, why are we in the press suckers for these?" Ironically, Fallows is a former editor of US News & World Report, whose controversial college rankings are discussed in this Chance News!

DISCUSSION QUESTIONS:

(1) What assumptions are being made in the scoring method? Are you convinced that Worcester is just above average in dropout rate?

(2) Seeking to deflect criticism about the methodology, Kostmayer
stated that "It's really not the rankings, but the information
that's important. It tells cities how to improve the quality of
lives for children." Do you agree?

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Here are two items related to school bus safety.

Bus safer than driving kids to school.

USA Today, 2 September 1999, p1A

Jayne O'Donnell

Bus seat belts: Help or hazard?

USA Today, 2 September 1999, p3B

Jayne O'Donnell

The National Highway Traffic Safety Administration (NHTSA) says that an average of 11 children die each year in school bus crashes, and about 10,000 are injured. The National Coalition for School Bus Safety has put the problem in another perspective. By examining newspaper reports, it identified 50,000 school bus crashes in the last year. This represents an alarming fraction of the 400,000 total buses on the road.

In light of these data, parents might elect to drive children to school rather than letting them ride the bus The first article cautions that this may not really be a safer alternative. It reports that between 1987 and 1997, an average of 600 school-age children died each year in auto accidents during school hours. According to Ricardo Martinez, who heads the NHTSA, the 60-to-one odds when compared to bus riding should "make [parents] think twice before driving their children to school or allowing teen- agers to drive to school unnecessarily."

The second article treats the question of requiring seat belts in school buses, which has been debated on and off for some 20 years. Since 1977, school buses have been required to have high-back padded seats, and government officials have maintained that these provide sufficient safety. Nevertheless, stories of rollovers or side-impact collisions where children were tossed around the bus-- even thrown from windows--have spurred demands for seat belts. Unfortunately, today's buses can accommodate only seat belts which are problematic in their own right because they can cause abdominal injuries. Taking all of this into account, the NHTSA is now studying improvements including arm rests, side pads and shoulder-seat belt combinations.

DISCUSSION QUESTIONS:

(1) Is the number of children who died in auto accidents during school hours the right figure to compare to with the number killed in school bus accidents? Would this argument convince you not to drive your children to school? What else would you like to know?

(2) Referring to the seat-belt question, Marilyn Bull of the
American Academy of Pediatrics says "We can fix the spleens, but
the spine injuries that occur in serious side impacts and rollover
crashes have much more serious and long-term implications." Do
you think the public would be willing to accept increased risk of
abdominal injuries in order to prevent spine injuries?

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Ask Marilyn.

Parade Magazine, 19 September 1999 p8

Marilyn vos Savant

A reader poses the following famous question about bombs on airplanes: "Say that a man who travels a lot is concerned about the possibility of a bomb on his plane. He has determined the probability of this and found it to be low, but not low enough for him. So now he always travels with a bomb in his suitcase. He reasons that the probability of two bombs being on board is infinitesimal."

Marilyn calls the argument "specious," and gives a numerical example to explain. For sake of argument, she uses 1 in one million as the chance that a bomb is on a given plane, noting that this overstates the true chance. Squaring this, she gives the chance of two bombs as 1 in a trillion. But this, she points out, is "only for planes other than the one on which our man travels. This is because he always carries a bomb. So for his plane the probability of a bomb being on board jumps to 1 in 1." It follows that the chance of two bombs on his plane is 1 times (1 in a million), giving "the same likelihood as if he brought nothing at all."

DISCUSSION QUESTION:

What assumptions underlie Marilyn's analysis? Do you find her
argument convincing?

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Fathom: Dynamic Statistics Software

Key Curriculum Press

Fathom is a new statistical software package designed for analyzing data and for teaching the art of analyzing data especially to middle grade and college students. The design of this software was suggested by Key Curriculum's very successful geometry sketchpad software. It is easiest to describe this design in terms of a particular data set.

The Chance course at Dartmouth just started and, as usual, the instructors gave the students a small survey to get some information about them and as a data set for the students to get their first experience at data exploration. The survey asked 18 questions such as: number of hours per week watching television, their pulse, their grade point average, their Math and Verbal SAT scores, and the number of miles Dartmouth is from their home.

The data is kept on the web and so we begin our analysis with Fathom by choosing import from the web and put in the appropriate URL. The data appears, represented by a window full of small gold balls called a "collection". Fathom calls the questions "attributes" of the data and the values of these attributes appear in a table in a separate window when we choose "insert case table" from the insert menu. This table looks like the traditional spread sheet form for data with the labels at the topic indicating the questions asked and columns representing the students answers.

To look at a histogram of the MathSAT scores we choose "graph" from the "insert" menu. A new window appears having only a pair of axes and a menu giving a choice between different types of graphs. We select the MathSAT label from the case table and drag it over the x-axis and we obtain a dot graph of the MathSAT scores. We then choose histogram from the graph menu and a histogram appears.

If we want to compare the scores of men and women we need only drag the sex variable from the case table and over the y-axis. Then we obtain two histograms, one for women and one for men.

The women appear to have slightly higher scores but to check this we can choose "summary table" from the "insert" menu. An empty table appears with two arrows indicating the top and the side of the table. We drag the MathSAT label over the top arrow, the sex label over the side arrow and we obtain the number of cases and the average MathSAT scores for men and for women. We see from this that there are 15 men with an average MathSAT score of 723 and 10 women with an average MathSAT score of 737.

If we want additional information, such as the standard deviations, we can choose add formula from the "summary" menu and type in stdDev for the new formula. We see that the standard deviation for women is 55 and for men is 67. In Chance classes, the MathSAT scores for men and women have usually been about the same; but the standard deviation for women has been consistently lower than that for men.

To see if MathSAT scores are correlated with grade point averages, we go again to the case table and drag the gpa label over the y- axis leaving MathSAT on the x-axis. We then obtain a scatter plot for these two variables. We can add the least squares best fit line and find from this that the correlation is .4. This is a typical correlation for SAT scores and first-year grade-point average. A moveable line is also available providing the sums of the squares of the distances. Students can move this line to find the line the minimizes the sums of the squares.

This is one of a number of situations where Fathom provides both the fast lane for those who just want the answer and a slow lane for understanding the process involved.

You might think that our desk-top is beginning to get crowded with all these windows. However the windows are moveable and sizable. In fact they can be reduced to an icon size when details are not needed, or even removed from the desk-top. Even when removed from the desk-top the continue to be associated with the original collection and can be called back when needed. If at any time we change the value of an attribute or add new cases etc. these changes affect all the windows associated with the collection.

Fathom provides numerous functions to allow the user to define new attributes such a log(salary) and sum of SAT scores. It also has mathematical and logical operations to allow the user to define new functions including those needed for simulation. Thus you can form new collections that represent the outcomes of a sequence of chance experiments such as tossing a coin, rolling a die, taking a sample etc.

Fathom also introduces the notion of a measure as an attribute of a collection. For example, for a collection of numbers the mean and standard deviation are measures. Using this concept you can take samples from a population and have Fathom put them in a new collection consisting of the sample means. This allows you to study the variability of sample means by looking at histograms of the resulting sample mean collection. This makes it easy to illustrate the central limit theorem. The same method makes it easy to carry out the bootstrap method for estimating the variability of a statistics such as the median where the central limit theorem does not apply.

Fathom was clearly designed with the priority on understanding statistical methods rather than just getting the answers. We can say that it is a lot of fun to use. Time alone will tell how it fares in the classroom.

You can try Fathom yourself by downloading a demo version from the Key Curriculum web site Fathom Software.

For more information about Fathom and pricing go to
Fathom Dynamic Statistics Software.

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Chance Rules.

Brian S. Everitt

Springer-Verlag (Copernicus)1999

Paper back version $20.80, hard cover $59.95 at www.Amazon.com

The sub-title of this book is "An Informal Guide to Probability, Risk, and Statistics" and that is exactly what it is. Brian Everitt is Head of the Biostatistics and Computing Department of the Institute of Psychiatry, Kings College, London. He has written extensively on the proper use of statistics in medical experiments.

In this book Everitt writes simply and beautifully to tell his readers what chance is. You will find topics such as lotteries, football pools, horse racing and alternative therapies that may be new to you as well as more familiar topics such as coincidences, statistics in medicine and puzzling probabilities. But even in the familiar topics Everitt always find something new to tell us. For example he starts with a brief history of chance. In tracing the history of the use of throwing dice and drawing lots for divine intervention Everitt remarks

As late as the eighteenth century, John Wesley sought guidance in this manner, giving in his journal, on 4 March 1737, the following charming account of his use of chance to decide whether or not he should marry.

And we guarantee that you will find it charming and bet it is new to you!

The chapter "tossing coins and having babies" begins with an elegant photo of the toss of a coin on the occasion of determining who gets to decide which team bats first in the 1938 Oval Test Match. Everitt apologizes, to those who have been deprived for his brief excursion into cricket. However it is a game where the toss of the coin can have a truly significant effect on the outcome of the game, unlike the toss of a coin in football. After his apology he remarks

On the other hand, perhaps they will be in my debt for exposing them, however briefly to a pursuit that is to baseball or soccer, say, as Pink Floyd is to the Spice Girls.

Everitt's account inspired us not only to find out why the toss of a coin can be so important in cricket but also what Pink Floyd is to the Spice Girls. Incidentally, you will see that the author really tossed coins and roll dice. Instead of the usual acknowledgment of the suffering of the wife and children during the ordeal of writing a book, we learn that they participated in the coin tossing and other chance activities described in the book.

This is the book to give your Uncle George who always wanted to really understand what you teach. It is also the perfect book for students in a reform statistics course who learn only informal probability. They can sneak off to learn when you multiply probabilities and when you add them or what conditional probability is and why it is important. They can learn these important concepts both informally and formally since the author includes the mathematics boxed off for those who want to see the formalities. The author again keeps these mathematical excursions simple and informative and encourages his readers to use them.

After a final chapter on chance, chaos and chromosomes he concludes his book with "An my old uncle used to say, "Be lucky." But then in an epilogue he remarks:

On reflection I thought someone a little grander than my old uncle should have the last word so...

The fabric of the world is woven of necessity and chance. Man's reason takes up its position between them and knows how to control them, treating necessity as the basis of its existence, contriving to steer and direct chance to its own ends.

J. W. Goethe, Whilhelm Meister's Apprenticeship

Of course, Chance students should also read this book!

DISCUSSION QUESTIONS:

(1) Why is the toss of the coin so important in cricket?

(2) What does Everitt mean by saying that cricket is to football
or soccer as Pink Floyd is to the Spice Girls?

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What is Random?

Edward Beltrami

Springer-Verlag 1999

Hard cover $15.40 at www.Amazon.com

This book is an attempt to understand the concept of randomness and its relation to concepts like entropy, thermodynamics, and chaos.

When we toss a coin a number of times we are accustomed to thinking of the resulting sequence of heads and tails as a random sequence. Thus we would consider any sequence that is the result of tossing a coin a sequence of times to be random. This approach makes the sequence HHHHH no less random than the sequence HTHHTH. However, most people would consider the second sequence a random sequence but not the first.

This leads some to believe that people do not understand what random means. However, it led the founder of modern probability Kolmogorov, and others (Chaitin, Solomonov, and Martin Lof) to try to say what it should mean to say that a specific sequence is random.

Martin Lof took the approach that a sequence of heads and tails should be considered random if it would pass a set of statistical tests for randomness such as: the proportion of heads should be near 1/2, there should not be too many or too few runs of heads or tails etc. That is a sequence is random if it is a typical sequence in the sense that it would not be rejected by standard tests of randomness. Kolmogorov, Chaitin and Solomonov took an apparently different approach. They say a sequence of heads and tails is random if it is "complex", meaning that the shortest computer program you can write to produce the sequence is about as long as the sequence itself. The Martin Lof approach was shown to be equivalent to the Komogorov approach, and Beltrami restricts himself to the Kolmogorov approach.

Here is a connection between complexity and entropy. Entropy as defined by Shannon, is a measure of the uncertainly in a chance experiment. It is defined as -sum p(i)log(p(i) where the sum is over all possible outcomes i of the chance experiment and the log is to the base 2. For a single toss of a biased coin with probability p for heads and q for tails the entropy is -plogp -qlogq = -log(pq). This entropy is maximum when p = 1/2 when it has the value 1. For a fair coin tossed n times, the entropy is n and, for a biased coin tossed n times, the entropy is less than n.

Shannon was interested in the problem of encoding sequences for more efficient transmission. He showed that the expected length of the encoded sequence could be at most the entropy, and there is an encoding that achieves this. Writing a computer program to produce sequences of H's and T's of length n is a way to encode a sequence. Thus Shannon's coding theorem is consistent with the fact that most sequences produced by tossing a fair coin are random but this is not true of the tosses of a biased coin.

This is a book that you cannot sit down and read like a novel. You will constantly find intriguing discussions that require you to stop and ponder what is actually going on.

Beltami finds a simple example helpful in discussing the relations between randomness, chaos and entropy. This example was suggested by similar examples presented by M. Bartlett in his paper Chance or Chaos? J. Royal Statistical Soc. A. 153, 321-347, 1990, written in his 80th year. Beltrami named this example the Janus sequence, after the Roman god who was depicted with two heads looking in opposite directions, at once peering into the future and scanning the past.

To construct this example we start with a number u(0) expressed in binary representation. For example u(0) = .1011001101...... meaning that u(0) = 1/2 + 1/8 + 1/16 + 1/128 + 1/256 + 1/1024 +... Starting with u(0) we determine a sequence of numbers u(0),u(1),u(3). We do this by going from u(k) to u(k+1) by shifting the digits of u(k) one to the right and adding a new first digit determined by the toss of a fair coin. We make this first digit 1 if heads turns up and 0 if tails turned up. If the first five tosses are HTHTHH we obtain

u(0) = .1011001101......u(1) = .01011001101.....

u(2) = .101011001101....

u(3) = .0101011001101...

u(4) = .10101011001101..

u(5) = .110101011001101.

We have described how we go from u(0) to u(5). Suppose that we are given u(5) and we want to determine the previous values. To do this we simply start with u(5) and successively chop off the first digit of the sequence and shift the sequence one digit to the left.

These transformations appear to be very different. One seems to introduce a lot of uncertainty as it computes successive values and the other does not. But this is deceptive. They each require an initial value to determine the successive values. Suppose that this initial value can only be measured to a certain accuracy say to 99 digits. What happens if you have the 100th digit wrong and it should be a 1 rather than a 0? For the forward process, such an error makes at most an error of 1/2^100 in any of the successive values. But suppose you make this same error in the initial value for the backward transformation. Then the value we end up with after 100 interactions would have a 0 as first digit rather than a 1 and this would be a serious error. The backward transformation is a chaotic transformation, small errors in the initial value can cause larger errors in future values. Thus in the forward process uncertainty in the future values is caused by the coin tosses. In the reverse process it is caused by the chaotic nature of the transformation.

Here is another way Beltrami uses this example to illustrate relations between randomness, entropy, and chaos. Assume that we start with all the digits of u(0) known to be zero. Then for n iterations the forward transformation is just the result of tossing a coin n times. Thus we start with uncertainty as measured by entropy of n. After k iterations of the forward process the uncertainty has decreased to n-k and the observed sequence of length k should have complexity of about k. When we reverse the process we start with a sequence of complexity n. After we have removed the first k digits we have reduced the complexity to n-k and the entropy is k. Thus entropy plus complexity remains the same in both directions with entropy and complexity playing dual roles for these two transformations.

Beltrami uses this example to help explain interesting physical problems such as the famous Maxwell demon as interpreted by Szilard. A single molecule is put into a rectangular box and moves randomly between two halves of the box. Entropy increases as the second law of thermodynamics predicts, because we become more an more uncertain where the molecule is. The entropy is maximum when the system is in equilibrium and then by the second law we cannot get work out of the system without changing it.

Now the demon blocks the ends of the box by pistons and records on which side the molecule is at each time unit of time. He then moves the piston from the end without the molecule to the center of the box and lets the molecule push it back. The demon thus apparently produces work, increasing entropy without changing the system. One explanation is that the demon, by obtaining information where the molecule is at a sequence of times, has changed the system. Removing this information decreases uncertainty, that is entropy, by the same amount that the demon has increased it. We don't expect you to understand or be convinced by it any more than we were. But the discussion will inspire you to look further. For example, a more complete discussion of the Szilard version of Maxwell's demon can be found.

If this fails try the book "Maxwell's Demon" by Hans Christian von
Baeyer, Random House, 1998.

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Copyright © 1998 Laurie Snell

This work is freely redistributable under the terms of the GNU General Public License as published by the Free Software Foundation. This work comes with ABSOLUTELY NO WARRANTY.

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