!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
CHANCE News 4.09
(8 June 1995 to 1 July 1995)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Prepared by J. Laurie Snell, with help from William Peterson, Fuxing Hou, Jeanne Albert and Ma.Katrina Munoz Dy, as part of the CHANCE Course Project supported by the National Science Foundation. 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 Data Base.
http://www.geom.umn.edu/locate/chance
==================================================
Once you have eliminated the impossible, then whatever remains, however improbable, must be the truth.
Sherlock Holmes
==================================================
NOTE: We are setting up an e-mail discussion group for teachers planning or teaching a Chance type course, or other probability or statistics course that uses current events in the news. Our aim is to share experiences, successes and failures, etc. related to teaching such a course. If you would like to join this group please send a note indicating this to jlsnell@dartmouth.edu.
From our Readers:
We had a request for a reference to the "New Yorker" article on "the day the law of large numbers was repealed". This appeared in "The New Yorker" in about 1947 and was reprinted in "The World of Mathematics" by James R. Newman, Vol. 4, p. 2268, Simon and Shuster 1956.
Contents
>>>>>==========>>
Bob Abelson suggested the following article, describing it as "an amazing mangling of a potentially interesting probability problem".
Sharing gold medals and multiple sclerosis.
The New York Times, 25 June 1995, p. 25
Evelyn Kanter
KEY WORDS: Coincidence
The article begins with:
It may be just a coincidence, what scientists call a cluster effect. Three skiers who won medals in the 1964 Olympics at Innsbruck, Austria, now have multiple sclerosis. Since there are an estimated 1 million people with multiple sclerosis, the odds of this are beyond calculation.
The three skiers are Jimmie Heuga on the U. S. team who learned that he had the disease in 1971, Egon Zimmerman of Austria who learned in 1987 that he had the disease and now Josef Stiegler, also from the Austrian team, has made it public that he too has the disease, for which there is no known cure.
The article discusses how these skiers have lived with the illness. The only other remarks related to the apparently rare event are:
The National Sclerosis Society estimates there are about 350,000 diagnosed cases in the United States.
The incidence of the disease is higher among people who live in cold climates, both north and south of the equator. However, neither Heuga nor U. S. Ski Team officials know of any other Olympic skier, or even one from another winter sport, with the disease.
DISCUSSION QUESTIONS:
1. What do you think the author means by saying "the odds of this are beyond calculation"? What is the "this" ?
2. With the information given in the article, could you estimate the odds in question? If not, what additional information would you need? How would you estimate the odds?
<<<========<<
>>>>>==========>>
Milt Eisner suggested the following two articles.
Schools bring math to life by teaching statistics.
The Washington Post, 19 June 1995, p. A1
David Leonhardt
KEY WORDS: teaching statistics
Leonhardt reports that there is a quiet revolution occurring in mathematics teaching. Probability and Statistics lessons at all grade levels are on the rise. The appeal stems from the fact that the subjects are practical and relevant; teachers argue that probability and statistics will help students become skeptical consumers of the deluge of information presented to society.
Allen P. Cook, director of research for the National Council of Teachers of Mathematics, said statistics teaching has grown steadily since 1989, when the council issued a national recommendation urging all schools to teach more hands-on math. Cook said that, with test scores showing the U.S. falling behind its global counterparts and with an economy that is no longer internationally dominant, instructors realize that they cannot afford to continue teaching abstract mathematical concepts.
Educators say that the trend towards teaching Statistics is difficult to gauge because most of the new emphasis on Statistics is occurring within other math studies, either in lower grades or in high school algebra classes. Only 1% of 1990 high school graduates took a statistics course, said Vance Grant, a National Library of Education statistician.
The effect of this revolution is evident throughout the country's school system. In Fairfax County, statistics compose 25% of middle school math curriculum. D.C. public school instructors teach kindergartners about poll taking and second graders about probability. Within the next 2 years, an Advanced Placement test in Statistics will be offered. Educators conjecture that the test will probably cause hundreds of high schools to offer statistics classes.
The article reports that teachers say that the shift toward the practical subject of statistics is unmistakable and has helped convince students that mathematics and real life intersect. The trend also reflects the long-growing demand among parents and business executives that math instruction become more relevant to students and the jobs they will eventually seek.
Perhaps the overall importance of this revolution in mathematics instruction is best articulated by Tom Nuttall, Fairfax County's math coordinator: "For the layperson, the person who is not going to be a mathematician, which is most of us, it's probably the most important strand of math. It helps people make decisions." And in a world inundated by information, such a tool is indeed a powerful one.
DISCUSSION QUESTIONS:
1. The article states that "like any curriculum change, this one causes debate. In some schools, including Oakland Mills, the emphasis on Statistics means that fewer students will progress to Calculus and be ready for upper-level mathematics and science in college." Is this so bad?
2. The existence of the advanced placement test in Computer Science has influenced the teaching of Computer Science in the high schools -- for example, it pretty much dictates the computer language to be used in high school. Similarly, the use of calculators on the College Board exams influences the teaching of Calculus in the high schools. Do you think the advanced placement test in statistics will have a positive or negative effect on the teaching of Probability and Statistics in high school?
<<<========<<
>>>>>==========>>
Statistics: when the numbers are victim.
The Washington Post, 19 June 1995, A2
Kathy Sawyer
KEY WORDS: Bayes theorem, law
In early March, Alan M. Dershowitz, a defense attorney for O.J. Simpson, commented on television that only about a tenth of 1 percent (one in a thousand) of the batterers actually murder their wives.
In the June 15 issue of Nature, I, J. Good suggests that a more appropriate computation for a jury would be to use Bayes theorem to find the probability that a wife was murdered by her husband, given that she had been battered by her husband and had been murdered. Good estimates this probability to be at least 1/3 and possibly as high as 1/2.
Good calls the Dershowitz statement "a perfect example of how statistics can be used to mislead people, even when the statistic in question is true."
EDITORS NOTE: This problem is also discussed in the current issue of "Chance Magazine" in the following article:
Propensity to Abuse---Propensity to Murder?
Chance Magazine, Spring 1995, p 14.
Jon F. Merz and Jonathan P. Caulkins
DISCUSSION QUESTIONS:
(1) What additional information did Good need to find the probability that he thought was more appropriate?
(2) Read both the "Nature" and "Chance" articles and compare their solutions. Why do their answers differ?
<<<========<<
>>>>>==========>>
Joan Garfield told us that she found the following two earlier articles useful in her Chance course. Since they are not in our database we review them here so they will end up there.
A statistical portrait of the 'typical' American.
The New York Times, 26 July 1992, Section 4 p. 5
Anne Cronin
KEY WORDS: descriptive statistics
The Census Bureau used the 1990 data from the long census questionnaires mailed to one in six, or 17.7 million American households, to determine the average woman, called Jane here. Here are a few facts about Jane. The article provides many more.
Jane is a white women who is 32.7 years old.
Jane is married and is a mother.
Jane has some German blood.
Jane graduated from high school.
Jane works for a private company or corporation. She is a clerical worker. The company is in manufacturing.
Jane does not own a gun.
Jane owed $2,317 on her credit cards at the end of 1991.
Jane generates about 3 pounds of garbage every day.
Jane read a newspaper today.
Jane believes in the Devil, but does not believe in ghosts or witches.
Jane hates liver.
Discussion questions:
1. The author remarks that "people who exactly match Jane's description will be rare indeed". How rare do you think such an exact match would be?
2. The article comments that presidential candidates like to refer to the typical American. Would this data help you understand these references?
3. How tall is Jane? How many days did Jane contact a doctor last year? How many hours of television did Jane watch last week?
<<<========<<
>>>>>==========>>
What's wrong with this picture?
Chance Magazine, Vol. 2, No. 1, 1990 p. 34
Albert Shanker
KEY WORDS: graph, SAT scores
Shanker shows a graph that was distributed at government meetings in 1989 to show that, from 1963 to 1988, there was a direct relation between the rise in elementary and secondary school spending and the decline in SAT scores.
Shanker comments that the graph was very convincing. The SAT scores declined steeply at almost exactly the same rate the expenditure on education was increasing. He points out that this was achieved by using current dollars, rather than adjusting by inflation, and by plotting the SAT scores on an 800-1000 scale, rather than on a 400-1600 scale which was the true range. He comments that nothing was said about the changing population taking the exam. He provides interesting additional information about the source of the graph and about its use.
<<<========<<
>>>>>==========>>
Revisionist General's report: Glowing in the dark is good for you.
The Denver Post, 18 June 1995, p. E4
Alston Chase
KEY WORDS: environment, radon
This article addresses a timely and politically charged issue in the nature of scientific studies: different research groups investigating the same environmental problem often come to different conclusions. While some authors have focused on the scientific and political ideologies of research institutions, media organizations, or funding sources to explain these differences, Mr. Chase seems more interested in describing the latest "environmental optimism", an outlook endorsed by Gregg Easterbrook, author of the recent book, "A Moment on the Earth". Easterbrook, who has been criticized by environmental groups for (among other things) getting his statistics wrong, nevertheless maintains, as Chase points out, that the planet is in much better shape than these same environmentalists would have you believe, in areas such as forests, global warming, acid rain, asbestos, and pesticides.
The article really gets interesting, though, when we learn that "plutonium--often dubbed 'the most toxic substance known to man'- - may actually be good for you." Chase writes that, according to an article published in The Atlantic Monthly in April, a study of 7000 workers at the Rocky Flats nuclear plant in Colorado found that "men with plutonium in their urine turned out to be healthier than the American populace overall," and that the mortality rate among 26 men exposed to radiation at Los Alamos in 1944 was "much lower than for white males in general".
Chase also cites two studies on the effects of radon. One story, distributed by the Associated Press from a study published in the "Journal of the National Cancer Institute", reports that "radon may cause thousands of lung cancer deaths", while a second; published in the journal "Health Physics" finds "a strong tendency for lung cancer rates to decrease with increasing radon exposure." Moreover, according to Chase this study "contains an argument that, if valid, could topple the entire superstructure of scientific pessimism" by "disconfirming...the 'no threshold theory' of risk," which presumes that, if large exposures to a substance are hazardous, than so are smaller ones, only less so. Perhaps, the argument goes, radon actually lengthens life because it stimulates "biological defense mechanisms that improve health."
DISCUSSION QUESTIONS:
1. Can both radon studies be "right"?
2. What information would you want to know about the studies mentioned above to reach your own conclusions about their validity?
3. Chase uses the following analogy to illustrate the no threshold theory: "It is like supposing that since consuming 5000 calories a day makes you fat, eating only five calories will do likewise." How does this sound to you?
4. Chase also says "good news" like his examples above is likely to be ignored by the press and politicians. Do you agree? Why is this important?
<<<========<<
>>>>>==========>>
Ideas & Trends -- Medical Fads: Bran, Midwives, and Leeches.
The New York Times, 25 June, 1995, Section 4 p. 16
Sherwin B. Nuland
KEY WORDS: medical studies
The tendency of physicians to flip-flop about cures and remedies has become obvious with the publication of several conflicting medical reports. New medical discoveries are announced with the same assuredness and supported with just as much evidence as those used to support the precisely opposite viewpoint.
The writer, Dr. Nuland (a clinical professor of surgery at the Yale School of Medicine) gives the following examples,
1) Last week, a report was published that testosterone's role in male aggression may be the opposite of what has long been assumed.
2) Thirty years ago, patients with diverticulitis, an inflammation of small oupouchings of the colon, were routinely treated with a diet low in roughage. The value of using roughage was never verified. Then, a few years later, medical opinion reversed: decreased roughage was a cause, not a cure, of the disease.
He remarks that such swings in medical findings are much more rampant than anyone wants to admit. Such a dilemma is the topic of the lead article in the June 15 issue of The New England Journal of Medicine. The article describes the increased risk of breast cancer in postmenopausal women who are given hormone replacement therapy. This is well within memory of the teaching that hormone treatment doesn't affect the likelihood of cancer at all. The data that supported the old finding seemed just as unequivocal as today's contradicting data.
The author gives a number of other examples. He remarks that clinical theory is a mix of science, experience, contemporary culture, authoritarianism, and emotion. Each time a factor changes, it becomes possible for medical opinion once again to swing towards the opposite of what is currently accepted.
With increased recognition of the present confusing state of affairs, the Federal Government has established the Agency for Health Care Policy and Research to encourage the investigation of long-term therapeutic outcomes.
DISCUSSION QUESTION:
1. The author remarks that "Perhaps one day the pendulum will stop swinging altogether. But I wouldn't stake my life on it. How do you feel about this?
2. Whenever contradictory studies are reported you are apt to find a general article warning the readers that they should not make a medical decision based on a single study. How does the article discussed above differ from such an article?
<<<========<<
>>>>>==========>>
When the boomster slams the doomster, bet on a new wager.
The Wall Street Journal, 5 June 1995, A1
Charles McCoy
KEY WORDS: wager
Wagering between environmental opponents seems to be "in" these days, but it is unclear if both sides truly want to win. This article, subtitled "Clash of eco-titans rekindles a rancorous core debate over future of the earth," describes the latest encounter between Paul Ehrlich, author of "The Population Bomb", the top-selling environmental book of all time, and Julian L. Simon, a University of Maryland business professor and author of Resampling Stats. The two men have apparently been at it for more than 20 years. A favorite among conservatives, Simon has publicly mocked Ehrlich for his dire predictions about the implications of increased population and other ecological warning signs, most of which have never materialized. In 1990, the article says, Mr. Simon won a $576.07 bet with Ehrlich, made a decade before, after the price of several strategic minerals did not in fact skyrocket as Ehrlich had predicted. And now, Mr. Simon has gone further: he claims that "every measure of material and environmental welfare" is improving.
In response, Ehrlich, with additional support from Stanford environmental scientist Stephen Schneider, has offered Simon his most comprehensive wager to date: 15 separate $1,000 bets, each predicting that one of 15 environmental indicators of ecological health (greenhouse gases, biodiversity, fishery stocks, etc.) will worsen over the next decade. The two scientists want "to shut Julian Simon up for a little while", says Ehrlich, but do they really want to win?
So far Simon has not accepted the challenge but instead has offered his own wager on the effects that Ehrlich and Schneider's environmental health indicators will have on human society. (Simon thinks there will be no significant, or at least detrimental, effects.) While the two sides haggle, environmentalists, politicians, and policy makers line up to make side bets. And some scientists have come up with their own wagers. According to the article "Atomic overreaction: a close look at plutonium's unearned reputation as the pre-eminent poison" in the April, 1995 issue of The Atlantic, the health physicist and pro-nuclear power activist Bernard Cohen has "a long-standing offer to ingest pure plutonium if a nuclear-power critic will eat an equal quantity of pure caffeine. No one has taken him up on his offer."
DISCUSSION QUESTIONS: 1. How would you determine if Ehrlich and Schneider's offer is a good wager? The Harvard entomologist and biodiversity expert E. O. Wilson calls it a "sucker bet". What do you think?
<<<========<<
>>>>>==========>>
Productivity is all, but it doesn't pay well.
The New York Times, 25 June 1995, p E4, Week in Review Section
Keith Bradsher
KEY WORDS: econonomy, index
This article describes some of the interplay between productivity and labor benefits. While wages and benefits of workers (as opposed to the dividends and profits of owners of capital) have together typically comprised about two-thirds of the total economic output of the United States, Mr. Bradsher reports that "over the last six years, compensation for American workers seems to have stagnated even as they have worked ever more efficiently and produced ever more goods." According to the article, the rate of wage and salary increase and that of productivity have oscillated somewhat during the past 30 years, with wages outstripping productivity in the late 1960's and early 1970's, and the reverse during the late 1970's and most of the '80's.
More recently, Labor Department figures indicate that this gap has continued to widen: "after adjusting for inflation, average wages and salaries apparently fell 2.3 percent over the 12 month period that ended in March. Productivity rose 2.1 percent during the same period." This information, along with comparable data for each year since 1987, is displayed in a 3-D bar graph that looked straightforward--the "Productivity" portion of the graph goes up faster than the "Wages and Benefits" portion (which twice actually goes down, from 1990 to 1991, and from 1994 to 1995) -- until I (Jeanne Albert) read the explanatory remarks above the display.
Titled "Less for their Labor", the remarks state that the graph depicts "productivity and worker compensation per hour, as an index where 1987 equals 100." In addition, the wages and benefits figures have been "weighted for annual changes in employment patterns among occupations. All numbers were adjusted for inflation using the gross domestic product implicit price deflator." Unfortunately, we are never told why "1987 equals 100", how the "weighting" is achieved, or the meaning of the GDP "implicit price deflator." In fact, after speaking with several people at the Bureau of Labor Statistics (BLS) wing of the Labor Department, about all I was able to determine with any confidence is that the BLS does not use the GDP implicit price deflator (they use one based on the Consumer Price Index), although possibly someone in Secretary Reich's office does. Perhaps more significantly, while the BLS confirmed the 2.1 percent productivity increase, they said their figure for compensation was 0.0, not -2.3, the discrepancy most likely due to applying a different price deflator.
DISCUSSION QUESTIONS: 1. How important do you think graphical displays are for conveying statistical information?
2. Can you explain the remarks displayed with the graph? Is it meaningful to describe productivity and worker compensation using a single index?
<<<========<<
>>>>>==========>>
Screening mammography and public health policy; the need for perspective.
The Lancet, 1 July 1995, pp. 29-31
C. J. Wright and C. B. Mueller
KEY WORDS: screening, mamograms
The authors review the studies on the effects of screening for breast cancer and conclude that screening is not good public policy.
They observe that there has been a lot of publicity about the early studies that showed a 30% relative reduction in breast cancer for women over 50 but very little attention paid to the newer studies which showed no significant benefit to any group. In addition, they claim that little attention has been paid to the costs and possible harmful effects of screening.
The authors say that the randomized prospective trials showed that the numbers of women screened to achieve one less death per year ranged from 7000 to 63,000. About 5% of screening mammograms are positive or suspicious, and of these 80-93% are false positive causing unnecessary concern and further procedures including
surgery. False reassurance by negative mammography occurs in 10- 15% of women with breast cancer that will manifest clinically within a year. They estimate that the mean annual cost per life saved is around $1.2 million dollars which they say is consistent with other estimates.
The authors consider the outcomes of the 1985 study which showed a significant 30% reduction in breast cancer mortality but no reduction in overall mortality. They present a graph analyzing the outcome of the trial in terms of population benefit rather than of relative mortality reduction, giving a striking example of the effect of different ways of presenting data.
DISCUSSION QUESTION
In a related newspaper article, the director of the Ottawa breast cancer screening program says that she worries some women will be discouraged from mammography screening because of the study, but she also suspects many women won't believe the findings. Do you think this is an accurate assessment of the effect of an article of this type.
<<<========<<
>>>>>==========>>
The Lancet, 24 June 1995, pp 1586-8
J. Cohen, I. Stewar
KEY WORDS: DNA fingerprinting
The quote at the beginning of this Chance News is from this article. The authors feel that Holmes has swept under the rug the issue of "beyond a reasonable doubt" which juries must deal with. They suggest that in the case of DNA there is an apparent paradox: "a jury that would accept without qualms an error probability of one in a thousand, such as an honest mistake by a key eyewitness, is unwilling to accept a probability of one in a trillion once its attention is drawn to the statistical nature of the evidence."
The authors give a detailed discussion of the three concerns about DNA fingerprinting: (a) statistical non-uniformity in human populations, (b) technical error by forensic scientists, and (c) misunderstanding of the meaning of statistical claims. In the discussion, they show how these issues are regarded differently by scientists, expert witnesses, and jurors.
It is, of course, interesting to read what Joel Cohen and Ian Stewart have to say about this very current topic.
DISCUSSION QUESTION:
The authors remark that "the fact that an individual has confessed to a crime may in some circumstances increase the probability of his or her innocence. What do you think they have in mind by this comment? In case this really intrigues you, the authors give the reference (Matthews RAJ. The interrogator's fallacy. Bull. Inst. Math. Appl. 1995; 31: 3-5) for this assertion.
<<<========<<
>>>>>==========>>
Defense cites faulty DNA calculations.
Los Angeles Times, 27 June 1995, B1
Stephanie Simon and Tim Rutten
KEY WORDS: DNA fingerprinting, Simpson trial
The Simpson trial has been plagued recently by what the previous authors call "technical error by forensic scientist". Bruce Weir admitted that bloodstains on O. J. Simpson's Ford Bronco and a crime scene glove containing genetic markers were more common that previously estimated.
This article, as well as all others we read, did not explain what mistake Weir actually made. Some articles referred to a mistake in a computer program, others to an arithmetical error and still others to Weir leaving out an allelle. We will give a prize to the first person who can tell us what the error really was.
The article does state that the original frequencies for the blood stand on the bronco were estimated by Weir in testimony to be 1 in 1400. His corrected value was 1 in 570. Similarly, the original frequency for the glove was 1 in 3900 and the revised estimate was 1 in 1600.
The article also mentions that Simpson's lawyer, Peter Neufeld, questioned the frequencies being based on small data-bases. Neufeld pointed out that the data-bases used do not contain certain subgroups such as Asian-Americans who make up a significant part of the population of Los Angeles County.
DISCUSSION QUESTION:
(1) If you were a juror, would you regard the errors reported here significant? Would the fact that the error occurred at all influence your evaluation of the overall DNA evidence?
(2) How large do you think the reference database should be? Is it important that most racial sub-population be represented in the database. (In fact it appears from the article that the database consisted of several hundred FBI agents and blood donors from Detroit, Miami and Houston).
(3) The article states that Peter Neufeld asked, with heavy sarcasm in his voice: "Do you consider 220 white FBI agents to be in the population of potential perpetrators for the murders of Ronald Goldman and Nicole Brown Simpson?" and Weir answered "Oh, yes it's a lovely collection?" How would you have answered this question?
<<<========<<
>>>>>==========>>
The New York Times, 2 July 1995, E7 (Advertisement)
Albert Shanker
KEY WORDS: education, class size
Shanker discusses an article by Fred Mosteller "The Tennessee Study of Class Size," to appear in the journal "The Future of Children." Shanker says "the study leaves no doubt that children in early elementary school achieve at a significantly higher level in classes of 15 than they do in classes of 25. For poor, minority children, the gains are even greater."
The study Mosteller analyzed included some 7,000 children from 79 elementary schools. Children were selected and randomly placed in one of three types of classes: a small class (13 to 17); a standard-size class (22 to 25), in which there was an aide to assist the teacher, and a standard-size class with a teacher only. The third type served as the control group. The children selected were in kindergarten and were followed through grade three. Students in classes of 25 with aides did slightly better than students in classes with teacher only. However, on standardized tests of reading and math, the children in the smaller classes gained .25 of a standard deviation over children in classes of 25 with a teacher only. Mosteller is quoted as saying that this study is "one of the great experiments in education in United States history."
The advertisement says that you can get a copy of Mosteller's paper by writing to "Initiatives for Children", American Academy of Arts and Sciences, 136 Irving St. Cambridge, MA 02138.
DISCUSSION QUESTION:
Would you worry that the President of the American Federation of Teachers would present a biased account of this study?
<<<========<<
>>>>>==========>>
Fortune Magazine, 10 July 1995, Keeping up; p. 211
Daniel Seligman
KEY WORDS: Mr. Statistics, Shakespeare, Hamlet
Mr. Statistics states that he was asked about the wager the King makes with Laertes about the outcome of his duel with Hamlet. This wager appears in the final act of Shakespeare's Hamlet. The court flunky Osric presents to Hamlet the terms of the proposed match with Laertes:
The King, sir, hath laid, sir, that in a dozen passes, between yourself and him, [Laertes] shall not exceed you three hits; he hath laid on twelve for nine.
Mr. Statistics reports that he consulted ProfNet for expert advice. (ProfNet is an e-mail service run by Dan Forbush out of his home which provides academic experts opinions for journalists).
He says that the usual interpretation is that there will be 12 "passes" or bouts each ending with the first "hit". If Laertes is to win the bet with the King, his total hits must exceed Hamlet's by more than three--for example, 8 to 4.
Mr. Statistics says that he found, from computer simulation of 100,000 duels, that for a three point spread to be fair, Hamlet would have to have only a 38% chance of winning a bout.
Editor's note: For a detailed discussion of this wager and its mathematical aspects see the very interesting article "The Odds on Hamlet" by Evert Sprinchorn. This article originally appeared in the "Columbia Forum", Fall, 1964, Vol. XII, No. 4, and was reprinted in "The American Statistician", Vol. 24, No. 5, December 1970, pp. 14-17. Sprinchorn does not think the above interpretation is very reasonable and argues instead for the interpretation that to win the match a contestant has to get three hits in a row. With this interpretation, and the assumption that the two contestants are equally matched, he shows that the probability that Laeretes wins is .443 with corresponding odds very close to 9 to 12.
DISCUSSION QUESTION:
1. Do you think the interpretation given is correct? If not, what would be your interpretation of the wager?
2. What is the exact probability required for Hamlet winning a bout to make a three point spread fair?
3. If you assume that Laertes and Hamlet are equally matched, what is the probability that the King will win his bet?
4. How did Sprinchor get his probability of .443?
<<<========<<
>>>>>==========>>
Water fluoridation harmful as lead.
Addison [VT] County Independent, 3 July 1995, p3.
Ross E. Conrad (Letter to the Editor)
KEY WORDS: flouridation, medical studies
An intriguing challenge from the anti-fluoridation camp. This letter cites 15 statements, with varying degrees of documentation, each warning against fluoridation of drinking water. According to the author, the sum of $100,000 has been placed into a special "Did You know?" account at Bank One (61 North Sandusky St., Delaware, OH 614-369-5555) by the Safe Water Foundation, and will be paid to the first person who provides proof in writing that any of the statements is false. Here they are:
(1) According to the handbook "Clinical Toxicology of Commercial Products" fluoride is more poisonous than lead and just slightly less poisonous than arsenic.
(2) According to the Physicians' Desk Reference," in hypersensitive individuals , fluorides occasionally cause skin eruptions such as atopic dermatitis, eczema or urticaria...These hypersensitive reactions usually disappear promptly after discontinuation of the fluoride.
(3) The Canadian Dental Association recommends "fluoride supplements should not be recommended for children less than 3 years old.
(4) From 1990 to 1992, the Journal of the American Medical Association published three separate articles linking increased hip fracture rates to fluoride in the water.
(5) In the March 22, 1990 issue of the New England Journal of Medicine, Mayo Clinic researchers reported that fluoride treatment of osteoporosis increased bone fracture rate and bone fragility.
(6) A study by Procter and Gamble showed that as little as half the amount of fluoride used to fluoridate public water supplies resulted in a sizable and significant increase in genetic damage.
(7) In 1993, researchers form the National Institute of Environmental Health Sciences admitted that "in cultured human and rodent cells, the weight of the evidence leads to the conclusion that fluoride exposure results in increased chromosome aberrations (genetic change)".
(8) In 1988, the ability of fluoride to transform normal cells into cancer cells was confirmed by the Argonne National Laboratory.
(9) The research of Dr. Dean Burk, former chief chemist of the national Cancer Institute, showed that 10,00 or more fluoridation- linked cancer deaths occur yearly in the US.
(10) Research from Battelle Research Institute showed that fluoride was linked to a rare form of liver cancer in mice, oral tumors and cancers in rats, and bone cancer in male rats.
(11) Since 1990, the national Cancer Institute, the New Jersey Dept. of Health, and the Safe Water Foundation all found that the incidence of osteosarcoma, a type of bone cancer, was substantially higher in young men exposed to fluoridated water as compared to those who were not.
(12) In the latest US study on fluoridation and tooth decay (as of 1993, according to the US National Institute of Dental Research), US Public Health Service dental records of more than 39,000 schoolchildren, ages 5-17, from 84 areas around the US, showed that the number of decayed, missing and filled permanent teeth was virtually the same in fluoridated and non fluoridated areas.
(13) Dr. John Colquhoun, former chief dental officer of the Dept. of Health for Auckland, New Zealand, investigated tooth decay statistics from about 60,000 12- to 13-year old children and showed that fluoridation had no significant effect on the decay rate of permanent teeth.
(14) According to the October 1987 issue of the Journal of the Canadian Dental Association, "Survey results in British Columbia with only 11 percent of the population using fluoridated water show lower average DMFT (tooth decay) rates than provinces with 40-70 percent of the population drinking fluoridated water" and "school districts recently reporting the highest cavities-free rates in the province were totally unfluoridated."
(15) In 1993, the Subcommittee on Health Effects of Ingested Fluoride of the National Research Council admitted that 8 to 51 percent and sometimes up to 80 percent of the children living in areas fluoridated with the amount of fluoride recommended by promoters of fluoridation have dental fluorisis (fluoride poisoning).
DISCUSSION QUESTION: Do you think any of the above statements is false? True but potentially misleading? What further information would you like to have? Do you think that anyone will get the $100,000?
<<<========<<
>>>>>==========>>
Stephan Henrikson asked if we have seen any articles giving information about the proportion of boys and girls who use the Internet. There seems to be surprising little information on this subject. The following article in the "Boston Globe" gives some information. Perhaps another reader knows more.
Boston Globe 8 March 1995, Living section, p. 29
Nathan Cobb
KEY WORDS: gender issues, internet
The article discusses gender differences on the Internet. The author states that "statistically speaking, of course, it's still a man's cyberworld out there. Among the major online services, CompuServe estimates that 83 percent of its users are men, while America Online pegs its male subscribers at 84 percent. Prodigy claims a 60/40 male/female ratio among users. Nobody keeps figures for the Internet, but estimates of female participation run from 10 to 35 percent. Indeed, most of the computer culture is male-dominated". Women are less apt to "flame" then men etc. Women often adopt names to disguise their gender for obvious reasons.
DISCUSSION QUESTION
How would you design a survey to estimate the proportion of men and women on the Internet?
<<<========<<
>>>>>==========>>
The New York Times, 17 June 1995, 1-19
Dick Teresi
KEY WORDS: helmets, law
Dick Teresi is editor of VQ, a magazine about Harley-Davidson motorcycles.
Teresi writes on the occasion of the coming of 150,000 motorcycle enthusiasts to the Laconia NH Motorcycle Rally. He remarks that at the 1993 rally, 98 percent of the respondents said they opposed helmet requirements. Forty-seven states have such requirements.
Teresi observes that helmets have a warning label "Some reasonably foreseeable impacts may exceed this helmet's capability to protect against severe injury or death." He claims the Department of Transportation tests the protection by dropping the helmets on an anvil from a height of six feet -- equivalent to an impact of 14.4 miles per hour.
He claims that studies quoted as saying the states with helmet requirements have fewer accidents per million residents than states where helmets are not mandatory are misleading because these latter states have a higher proportion of motorcycle riders.
Terisi claims that while helmets might prevent head injuries at slow speeds there is evidence that they increase the chance of neck injuries at high speeds.
We used this article for discussion at our Chance Workshop. We asked the participants to read this article and the following two articles that present the other side of the issue.
Rocky Mountain News, 3 January 1995, A-24
Roger Simon
Study backs impact of California helmet law.
The San Diego Union-Tribune 16 November, 1994
Stephen Green
We then provided the following discussion questions.
DISCUSSION QUESTIONS:
Read first the Teresi article.
(1) Do you think that 98% is a good estimate for the fraction of bikers at the Laconia rally who oppose helmet laws?
(2) Does the fact that motorcycle helmets can only withstand a 15 mph impact with an anvil mean that motorcyclists should keep their speedometers below 15 mph?
(3) After reading this piece, do you think it is safer to ride a motorcycle without a helmet? What further statistical information would you like to see?
Now read the article from the Rocky Mountain News
(1) Does the 37% drop in fatalities in California represent a significant improvement?
(2) After reading this piece, do you think it is safer to ride a motorcycle without a helmet?
(3) What further statistical information would you like to see?
Now read the article from the San Diego Union-Tribune about the UCLA study.
(1) After reading this piece, do you think it is safer to ride a motorcycle without a helmet?
(2) What further statistical information would you like to see?
Time to philosophize.
(1) Do you think that a motorcyclist of good faith might reasonably conclude that they would be safer riding without a helmet? If so, do you think it is right for the state to prevent them from doing so? How about if they decided it would be safer to wear a lighter helmet, like a bike helmet?
(2) Suppose that everyone agrees that wearing a helmet cuts in half the chance of a fatal or serious or expensive injury when riding a motorcycle. Does the state have the right to require motorcyclists to wear helmets? If so, should it do so?
(3) Suppose that everyone agrees that wearing a helmet cuts in half the chance of a fatal or serious or expensive injury when riding in a car. Does the state have the right to require car passengers to wear helmets? If so, should it do so?
(4) Should motorcyclists who carry catastrophic injury insurance be allowed to ride without a helmet?
(5) According to recent congressional testimony by the head of the CDC, most motorcycle fatalities `occur among young people between the ages of 18--24, resulting in the loss of many years of productive life'. This is one of the costs attributed to motorcycle fatalities. Do you think that increasing fatalities among riders 65 or older might actually be financially beneficial for society? If so, should senior riders be allowed to ride without helmets? Should they be encouraged to ride without helmets? Should the be required to ride without helmets?
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
CHANCE News 4.09
(8 June 1995 to 1 July 1995)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Please send suggestions to: jlsnell@dartmouth.edu