CHANCE News 4.01
              (11 Dec 1994  to 10 Jan 1995)

Prepared by J. Laurie Snell, with help from 
William Peterson, Fuxing Hou 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

Back issues of Chance News and other materials for
teaching a CHANCE course are available from the
Chance Web Data Base

        Data, Data, Data! He cried impatiently.  
        I can't make bricks without clay.

                                  Sherlock Holmes

Note:  We have included, at the end of the newsletter, 
announcements for two workshops this summer; one is a 
quantitative literacy workshop for science teachers in 
grades 6-12 from the Baltimore/Washington Metropolitan 
area, and the other is a workshop on the CHANCE course 
for college teachers interested in learning about the 
course, to be held at Dartmouth. 

John Paulos (author of "Innumeracy") has two forthcoming 
publications that will be helpful to those teaching a 
CHANCE course and to future newsletters. Starting in 
April he will be writing a column for "The Economist" on 
the mathematical aspects of stories in the news, and in 
March his new book, "A Mathematician Reads the 
Newspaper", will be published by Basic Books.

>>>>>==========>> Peter Doyle and Bob Norman both pointed out that we missed a golden opportunity to comment that, like the FBI, Marilyn vos Savant is too quick to assume independence. We repeat this item together with discussion questions they suggested. Ask Marilyn. Parade Magazine, 27 November, 1994, p.13. Marilyn vos Savant "Suppose a person was having two surgeries performed at the same time. If the chances of success for surgery A are 85%, and the chances of success for surgery B are 90%, what are the chances that both would fail?" In her answer Marilyn gave the answwer assuming independence. DISCUSSION QUESTIONS: (1) Explain why it is ridiculous to assume independence in this case. (2) Try to do your own analysis by making a plausible guess as to the dependence between these two events. (3) Do you think that Marilyn is trying to land a job with the FBI? <<<========<<

>>>>>==========>> Raphael Falk suggested the following article. The earth is round (p < .05) American Psychologist, December 1994, pp. 997 to 1003 Jacob Cohen The author attacks, in many interesting ways, the ritual of null hypothesis significance testing. He starts by explaining why so many people think that rejection of the null hypothesis, given the data, means that it is unlikely the null hypothesis is true, given the data. He employs amusing examples from simple deductive syllogistic reasoning and colorful terminology such as "the illusion of attaining improbability" or "the Bayesian ID's wishful thinking error." The wishful thinking error leads to the belief that, after a successful rejection of H0, it is very probable that a replication of the research will again result in rejecting H0. In fact, Tversky and Kahneman showed that most members of audiences at psychology meetings even believed that replication will occur even with a small sample. The author goes on to point out that even when interpreted correctly, rejection of the null hypothesis is not very informative. He urges his colleague to use newer methods of exploratory data analysis to get more information from the data and to routinely report confidence intervals. DISCUSSION QUESTION 1. Your test of significance shows that it is unlikely that the data would have occurred if the hypothesis were true. What is needed to allow you to say that it is unlikely that the hyphothesis is true given the data? 2. You design an experiment to determine if your friend is right when he claims he can tell the difference between pepsi and coke. You give him a series of ten trials. On each trial, you toss a fair coin to determine whether he gets pepsi or coke. You require him to get 7 or more correct to establish his claim. He gets 8 correct. Assuming that he can get it right 80% of the time, what is the probability that he will establish his claim if you repeat the experiment? FROM THE INTERNET David Hemmer sent an interesting discussion from the newsgroup rec.games.bridge posted by Evan Bailey. Bailey remarks: Larry Cohen, in reporting the 1994 Spingold finals in the Granovetter's magazine "Bridge Today", states: "Almost one half of the boards in the final contained voids, and more than one quarter of them featured bid slams!" One can calculate the probability that a deal contains one or more voids, and it is .183766, and so you expect 11.76 deals in a 64 board final to contain a void. The probability that one half or more of the boards contains a void is 1.01e-8. If we interpret "almost half" as only 24 boards, then Larry's observation would be more common -- once every 4,000. Spingold finals. Duplicate players often suspect wild deals in ACBL events. Never do the players comment on tame deals. My feeling is that wildness tends to occur toward the end of a tournament. If the ACBL is selecting deals, then we should be told about the selection criteria. The ACBL, of course, denies this, and the only conclusion is that their deal generation program is flawed and should be replaced. Bailey goes on to describe his analysis of data from WBF Open Pairs recently held in Albuquerque. He comments: There are many analyses one can do with these data. The hands of the first four days satisfied my statistics checks. The last three days, the finals of the event, were a different story. Here are the problems I found: 1. The average length of longest suit in hand was higher than expected. 2. The number of hands with six or more cards in longest suit was high. 3. The number of North-South (or East-West) deals with 18 to 22 high card points was low. 4. The number of hands with 0-3 points was high. DISCUSSION QUESTION Would poor shuffling lead to the problems that Baily found in the data he analyzed? <<<========<<

>>>>>==========>> Here is an amusing probability problem discussed on sci.math: N+1 people labeled 0,1,2,..,N are sitting at a round table drinking beer. Person 0 gets a very large pitcher of beer. He takes a drink and passes it with equal probability to one of his neighbors. This continues until everyone has had a drink. Show that the last drinker is equally likely to be any of the drinkers other than 0. This is not a new result, but S. C. Wan Sung and C. W. Sung came up with the following elegant proof that involves no calculation. Let E be the event that i-1 gets a drink before i+1 and F that i+1 gets a drink before i-1. Note that E and F are complimentary events. Then P(i wins) = P(E)(P(i wins|E) + P(F)P(i wins|F). By symmetry P(i wins|E) and P(i wins|F) are equal and do not depend on i. Call the common value q. Summing p(i wins) for all i gives 1 = Nq(P(E) + P(F)) = Nq. Thus q = 1/N. Putting this back in the first equation gives also that p(i wins) = 1/N. DISCUSSION QUESTIONS: (1) What happens if each person passes the beer to the right with probability p and to the left with probaility 1-p? (2) Consider a simple random walk on the integers 0,1,2,...,N that goes from j to j+1 or j-1 with equal probabilities. Show that the above argument proves that the probability starting at 1 of reaching N before 0 is 1/N. <<<========<<

>>>>>==========>> ARTICLES ABSTRACTED


>>>>>==========>> Statistics can throw us a curve: Controversy over a book linking race and IQ shows the potential pitfalls in analyzing data. Most people think numbers don't lie, but mathematicians know better. Los Angeles Times, 4 Jan 1995, A1 K. C. Cole

As the title suggests, this long article is clearly meant to be an attack on the "Bell Curve". The idea is to list a lot of errors and pitfalls that can be made in interpreting statistical concepts, with the hope that these were all made in the Bell Curve. Special attention is paid to the difference between correlation and causation. The article starts with one of our favorite examples: There is a direct correlation, mathematicians have found, between children's achievement on math tests and shoe size. A clear signal that big feet make you smarter?" (A letter to the editor commented: "no no no no no. It's big shoes that make you smarter."). Less frivolous examples are given, such as the discovery that Japan's low-fat diet was correlated with a high incidence of stomach cancer -- later explained by an excessive use of soy sauce. While the examples are interesting in their own right, they often have little to do with the book. An exception is the following example: 'Genetics may not be the main reason that identical twins raised apart seem to share so many tastes and habits', said Richard Rose, a professor of medical genetics at Indiana University. 'You're comparing individuals who grew up in the same epoch, whether they're related or not. If you asked strangers born on the same day about their political views, food preferences, athletic heroes, clothing choices, you'd find lots of similarities. It has nothing to do with genetics.'" The article quotes a number of experts on the dangers of using the mean, when the median is more appropriate, and of assuming a perfectly normal curve when the date is not completely normal. Examples are given where meta- analyses have reversed conventional wisdom. This is an interesting article, but it is hard to see how it is more damning to the research in the "Bell Curve" than to most research commented on in the daily press. DISCUSSION QUESTION: In the Bell Curve, it is shown that having a high IQ is correlated with going to an elite college. As one expert points out, it is probably also true that have parents with a high income is probably also correlated with going to an elite college. How would you try to determine which is the more important variable in predicting who goes to elite colleges? <<<========<<

>>>>>==========>> Promises of miracles: news releases go where journals fear to tread. The New York Times, 10 January 1995, C3, Lawrence K. Altman

Altman remarks that scientists rarely make exaggerated claims in their writing but are often willing to make them in public pronouncements, especially in cooperation with their institution's press office. He discusses in detail the recent announcement from the Scripps Research Institute in La Jolla that appeared to promise a cure for cancer: a single injection of either of two proteins was found to cut off the blood supply to many kinds of tumors, causing them to shrink away leaving normal tissue to thrive. In their paper the authors said that they were not sure that what they saw in the laboratory would occur in humans. Testing with humans had not taken place and would not begin until mid-1996. The press release had no such note of caution, commenting that "this approach not only is expected to eliminate primary tumors but also will likely prevent the metastatic spread of tumor cells by eliminating their access to the blood supply." Altman gives many reasons why scientists and laboratories make bolder statements to the public than can be justified. Most of the reasons have to do with money, but at least one doctor remarked, more charitably, that "the human mind has a need for some level of optimism." DISCUSSION QUESTIONS 1. Give three reasons why researchers make bolder statements to the public than they do to their colleagues. 2. Do you think that there are cases where the researchers should be bolder in public than in their research articles? <<<========<<

>>>>>==========>> Odds you just can't grasp.fear to tread. The Times, 19 December 1994, Features Jon Turney

We mentioned last time that Richard Lewontin, in a letter to "Nature", argued that one of the major problems with the use of DNA in the courts is that juries lack knowledge of simple probability concepts, and it is not possible to explain these concepts in the time allotted during a trial. The author of this article sought expert opinions to see if they agreed with Lewontin. Stuart Sutherland has written a book "Irrationality" that discusses how our intuitions get us into trouble. He remarks: "Since people make consistent mistakes, they have not developed a good intuition for approximately where the correct answer lies. Our best guesses are even less reliable when very large numbers are involved, something both DNA profiling and the National Lottery have in common." Bruce Weir believes that the relevant probability concepts can successfully be explained to jurors and he has had occasion to do so. He states that even such a simple statement as "'This type is estimated to occur in one in a million people' conveys the correct impression that the evidence of a match is powerful evidence." Nicholas Maxwell agrees that "there is something screwy about how people work with probabilistic information" but says "that doesn't mean that there is not some way to present it so it is useful. After all, we do succeed in teaching it to our students." It is agreed that the famous "prosecutor's fallacy" is still commonly made and pretty difficult for a jury or a judge to understand. DISCUSSION QUESTIONS: 1. In his letter to "Nature" Lewontin remarked that "it is common for people to believe that a 1 in 4 chance means that the event is bound to happen on the fourth trial." What did he mean? Do you believe that it is as bad as all that? 2. What example would you give if you wanted to show a simple probability statement that is typically misinterpreted? 3. Imagine the following scenario: Simpson's DNA is found to match that found at the scene of the crime. You are the prosecutor and claim that the chance of such a match is 1 in a million. Simpson's lawyer says you have made the classic ``prosecutor's fallacy" and, considering the 50 million people in the area, this evidence only shows a 1 in 50 chance that Simpson is guilty. How would you reply? 4. Do you agree with Maxwell's remark: "After all we do teach it (probability) to our students." <<<========<<

>>>>>==========>> Homeopathic medicine for asthma said to best a placebo. The Boston Globe, 9 December 1994, p19. Richard A. Knox

According to the first paragraph in the article: "In a new study of patients comparing the effects of pure water with pure water, the pure water labeled 'homeopathic medicine' consistently came out ahead." The homeopathic preparation actually started as a weak solution of an asthma-causing substance such as household dust, but was then repeatedly diluted until it was unlikely that a patient would receive a single molecule of the original substance. The study was published in "The Lancet". When neither patients nor doctors knew which preparation they got, the homeopathic medicine reduced symptoms by 20%, and the treated patients did better on tests of breathing function. When these results are combined with two earlier studies of hay fever patients, homeopathically treated patients had a 33% reduction in symptoms, compared to a 10% reduction for placebo patients. Dr. Thomas Delbanco of Boston's Beth Israel Hospital is still skeptical, and notes that the study failed to measure patients' expectations of homeopathic medicine. Supposing that both treatments are placebos, he reasons: "Let's say by chance that 8 patients in the homeopathic group thought they would get better from the medication and only two in the placebo group thought that. We know that expectation has a tremendous effect." DISCUSSION QUESTION: (1) What do you think about Dr. Delbanco's argument? (2) What is your explanation for the success of homeopathic medicine? <<<========<<

>>>>>==========>> Computer admissions test to be given less often. The New York Times, 4 January 1995, A16 William H. Honan

In response to criticism that the computerized version of the Graduate Record Examination, or G.R.E., is easy to cheat on, the Educational Testing Service, responsible for the test, has reduced the amount of times the tests will be offered via computer. The security problem became an issue when 20 investigators from Kaplan Educational Centers, a test preparation firm, took the exam and were able to reconstruct "a significant portion" of the exam. Jose Ferreira, director of G.R.E. programs at Kaplan, said that the solution depends on whether ETS is willing to spend money to make improvements, such as adding four to five thousand questions, instead of just recycling questions that can be memorized and passed along. Nancy S. Cole, president of ETS, countered by accusing Kaplan of having a vested interest in exposing computer-testing flaws. She said that students who take computerized tests "tend to prepare on their own rather than en masse." ETS filed suit to keep Kaplan from sending investigators. Kaplan agreed to keep investigators out while its officials meet with the testing service to try to settle the lawsuit. <<<========<<

>>>>>==========>> Ask Marilyn. Parade Magazine, 1 January 1995, p10. Marilyn vos Savant

An "anonymous" reader asks: "Let's say we decide to dispense with men entirely and boost the number of women in the country. All women would get together and agree to the following: As long as a woman gives birth to a boy, she would have no more children. But as long as she gives birth to a girl she can have another child. This way, no family would have more than one boy, but plenty of families would have several girls. Do you see anything wrong with this?" The familiar martingale (fair game) argument shows that we should expect no difference between the average number of boys and girls. Marilyn's solution argues with averages, starting with 64 women giving birth. Half have boys, completing their families, the other half have girls and continue. Of the thirty two who had girls, 16 have a boy as a second child, 16 have a girl, and so on. At each stage Marilyn notes that the total number of male offspring equals the total number of female offspring. Though no finite stopping rule is given in the reader's question, Marilyn's "solution" ends with the one woman in 64 who has six girls entering a convent! DISCUSSION QUESTIONS: (1) When the problem is listed as an anonymous reader are you suspicious that this reader is Marilyn? (2) The above argument assumes that the sex of a child is like the outcome of a coin toss. Assume instead that the probability that the first child is a girl is 1/2 but that the probability of a girl when the previous child was a girl is p > 1/2 and, similarly, the probability a boy when the previous child was a boy is also p. Would there now be more boys or more girls in the population? <<<========<<

>>>>>==========>> Abortion seekers come from many backgrounds. The Boston Globe, 9 January 1995, p35. Lynda Gorov

This article reports on conflicting interpretations of statistics related to abortion. For example, in 1992, the abortion rate dropped to its lowest level in 16 years. Reasons cited for the decline ranged from decreased demand for abortions, since it has become more acceptable for unmarried women to have children, to the drop in the number of abortion providers. Explanations for the latter included doctors' fears of anti-abortion groups, younger doctors' not having been exposed to the traumas of illegal abortions, and doctors' realizing that abortion doesn't square with their roles as healers. Apparently, which explanation one finds most compelling tends to square with one's own political or moral position on abortion. <<<========<<

>>>>>==========>> Life-span forecasts are full of 'ifs'. The Boston Globe, 9 January 1995, p25. Judy Foreman

The author discusses the uncertainties involved in forecasts of life expectancies, and the implications of these uncertaintities for funding Social Security and Medicare. The basic components of a demographic model are identified as: live birth rate, net immigration and total mortality rate. The article notes that even straightforward models based on these factors already have large uncertainties associated with projections for older ages, primarily due to the difficulty in projecting the death rate. For example, any attempt to take into account potential breakthroughs in treatments for leading causes of death such as heart disease and cancer leads to wide fluctuations. To illustrate how dramatically projections may vary, a data graphic compares estimates for the US population aged 85 and older for the year 2050 (today there are 3.3 million). The Census Bureau gives low, mid-range and high values of 9.9, 18.9 and 27.3 million, respectively. The low estimate assumes that death rates will not decline from their current levels. The high estimate assumes that the 18% improvement in death rate achieved in the 1970s will continue through 2050. But demographer Kenneth Manton of Duke University comes up with a much higher figure of 48.7 million, under the assumption that the entire population makes the life-style changes necessary to reduce their risk factors. This would include quitting smoking, keeping cholesterol levels to 200, remaining close to "ideal weight", limiting alcohol consumption to 2 or fewer drinks per day, and meeting recommended fitness routines. While admitting that not everyone may respond in the same way to reducing risk factors, Manton still maintains that life expectancy could rise to 95-100 without assuming cancer cures or changes to the genetics of aging. DISCUSSION QUESTION: University of Chicago demographer Jay Olshansky believes that life expectancies of 100 are not achievable without curing or postponing both cancer and heart disease. He is quoted as saying that as a matter of sheer mathematics, to get to an expectancy of 100, 18% of babies would have to live to be 120. Why do you think he says this? Do you agree? <<<========<<

>>>>>==========>> US breast cancer death rate said to fall. The Boston Globe, 11 January 1995, p1. Alison Bass

The National Cancer Institute has reported that the death rate for cancer for white women has declined 6% overall since 1989, which is the largest short-term decline. The decline is attributed to early mammogram screening and advances in treatment. Unfortunately, there has not been a comparable decline for black women. Although the reasons for this are not entirely clear, it is suggested in the article, that due to differential access to care and/or cultural reluctance to seek treatment, blacks are diagnosed at later stages of the disease. Among white women, the declines were more dramatic in some age groups: 8.7% for ages 30-39, 8.2% for ages 40- 49, 9.3% in ages 50-59, 4.8% in ages 60-69, and 3.4% in ages 70-79. Officials at the American Cancer Society say that the new data provide compelling evidence of the benefits of regular mammograms for all women over age 40. However, Dr. Edward Sodnik of the National Cancer Institute said that, while there is clear evidence in favor of screening for women over 50, it is not yet clear that screening saves lives for women under 50. The Institute does not plan to change its current guidelines, which were changed amid some controversy, a little more than a year ago, to recommend screening for women over 50 only, rather than for all women over 40. DISCUSSION QUESTION: In what sense could the above data be viewed as providing evidence for the benefit of screening for women under 50? What else would you like to know? <<<========<<

>>>>>==========>> The Bell Curve - continued. Hernstein and Murray The Free Press 1994 Part III The National Context

Part III discusses difference in performance on intelligence tests within ethnic groups and between ethnic groups. The authors start by pointing out they have already shown that differences in cognitive abilities within a group (in particular within the white group analyzed in Part 2) can be very large and that this fact has political repercussions. They remark that the differences within ethnic classes are much larger than between classes, so any problems that these differences cause would not go away in a homogenous population. In Chapter 13 they describe the differences between groups as they see it. They say that studies suggest that Asians have a slightly higher IQ on average than whites but these results are not conclusive. On the other hand, studies have consistently shown that blacks have an average IQ of about one standard deviation less than for whites. The difference in the NLSY data (that the authors used throughout the book) was 1.21 standard deviations. The authors remark that a difference of one standard deviation allows for a lot of overlap in the distributions of IQ scores between blacks and whites. In particular, there should be about 100,000 blacks with IQ scores 125 or above. On the other hand since there are six times as many whites as blacks in the United States the disproportion's between whites and blacks at the higher levels become very large. The authors ask if these differences are authentic? They ask first if they could be due to cultural bias or other artifacts of the test. Studies that conclude that this is not the case are discussed briefly here and in detail in an appendix. They next ask if the differences are due to socioeconomic status. Looking at the NLSY data and controlling for socioeconomic difference they find that socioeconomic status explains 37 percent of the difference. They remark that "controlled" is hard to interpret here since socioeconomic status can also be a result of IQ. They suggest that if the differences were socioeconomic then the gap should decrease as their measure of socioeconomic status increases. They present a graph for their data showing this is not the case. They remark that the difference between black and white IQ does appear to be decreasing in time and attribute to this to environmental changes. They next turn to the question of whether the difference is do to genetics or environment or both. They point out that there is no consensus on this question. They present the arguments for and against a genetic explanation. The arguments presented make it clear why it is difficult to claim a solution to this problem. For all the obvious explanations they present studies to show that these explanations don't hold up. They conclude this discussion with an uncharacteristically bold statement: "In sum: If tomorrow you knew beyond a shadow of a doubt that all cognitive differences between races were 100 percent genetic in origin, nothing of any significance should change...The impulse to think that environmental sources of difference are less threatening than genetic ones is natural but illusory." The arguments presented in this chapter are well know and I feel better presented in the book "Intelligence" by Nathan Brody, Academic Press 1992. Chapter 14 looks at what happens when you control for IQ. Here the authors find that this removes the difference for some variables and not for others. For example, after controlling for IQ, the probability of graduating from college is higher for blacks as is the probability of being in a high-IQ occupation and wage differentials shrink to a few hundred dollars. Controlling for IQ does not change significantly the difference in black-white marriage rates, or welfare recipiency. It does reduce significantly the difference for the proportion of children living in poverty and for those who are incarcerated. Chapter 15 is entitled "The Demography of Intelligence". The fact that women with low IQ have more children than those with high IQ and changes in the immigrant population suggest to the authors that demographic trends are exerting downward pressure on the distribution of cognitive ability in the United States. They point out that this has been a difficult matter to settle. The "Flynn effect" says that in some sense IQ scores increase worldwide with time. However, the authors conclude that there are worrisome trends in the demographic effects even though there may well be improvements in the cognitive abilities by improved health and education. Chapter 16 considers the relation between low cognitive ability and social problems. The authors confess that causal relations are complex and hard to establish definitely. This leads them to simply ask if persons with serious social problems tend to be in the lower IQ groups. Looking at the NLSY data they present graphs with the x-axis the ten IQ deciles and the y-axis the proportion of the people with a specific problem. They start with poverty. The bar chart starts with about 30 percent poverty among the lowest IQ decile and decrease to about 7% in the highest IQ decile. When the dependent variable is high school dropouts, men interviewed in jail, or women who had receive welfare, the result is the same. After these and many more indications that low IQ is associated with trouble, the authors conclude with their Middle Class Values Index. To qualify for "yes" answer an NLSY person had to be married to his or her first spouse, in the labor force (if man), bearing children within wedlock (if a woman), and never have been interviewed in jail. Here there are respectable proportions of those saying "yes" in all IQ deciles which the authors remark should remind us that most people in the lower half of the cognitive distribution are behaving themselves. DISCUSTION QUESTIONS: (1) In the Bell Curve we find the following statement: “The most modern study of identical twins reared in separate homes suggests a heritatbility for general intelligence between .75 and .80”. This apparently accounts for their upper bound when they say say later that the heritability of IQ falls in the range of .4 to .8. On the other hand, the .75 and .8 is actually a correlation. What seems wrong here? (2) What do you think of the "Middle Class Values Index" as defined by Hernstein and Murray? <<<========<<

>>>>>==========>> QUANTITATIVE LITERACY WORKSHOP The American Statistical Association's Science Education And Quantitative Literacy Workshop Fellowships ASA's fifth Quantitative Literacy project, Science Education and QL (SEAQL), is developing materials for biology, chemistry, physics, earth science, and general science classes grades 6-12 that will employ QL techniques. Dates: July 10 - August 4, 1995 Location: Baltimore/Washington Metropolitan Area Participants: 45 Science Teachers grades 6-12 in Maryland, Delaware, New Jersey, Pennsylvania, West Virginia, Virginia, or the District of Columbia. Knowledge of statistics is not necessary, and not presumed. Credits: 3 graduate credits will be available. Workshop Fellows Receive: Travel reimbursement up to $200, Meals, and lodging at local site. Stipend of $1,200 paid upon completion of the workshop. For more information, contact Cathy Crocker, Director of Education, or Wayne Jones, Manager of Educational Programs, American Statistical Association, 1429 Duke Street, Alexandria, VA 22314-3402 Phone: 703/684-1221; Fax: 703/684-2037; E-mail: seaql@asa.mhs.compuserve.com <<<========<<

>>>>>==========>> CHANCE WORKSHOP June 21 to June 24 Dartmouth College CHANCE is an introductory mathematics course whose aim is to make students more informed and more critical readers of current issues in the news involving concepts of probability and statistics. The workshop will allow college teachers to experience a shortened version of the CHANCE course, complete with discussions of current issues in the news, computer labs with data analysis software, hands-on-activities, videos, journal writing and the use of the Internet. Representatives of other NSF projects will present results of their projects that are useful in teaching a CHANCE course. Participants will receive food, housing and course materials but will be asked to provide their own transportation. An e-mail network will be established to allow participants to share information as they develop their own CHANCE courses, and there will be a follow-up meeting at an annual meeting of the MAA. This workshop is supported by the National Science Foundation. Presenters: J. Laurie Snell, Peter Doyle, Joan Garfield, William Peterson, and Nagambal Shah. Application forms are available by e-mail from jlsnell@dartmouth.edu or by mail from J. Laurie Snell, Dartmouth College, Department of Mathematics, 6188 Bradley Hall, Hanover NH 03755. Phone 603-646-3507. The deadline to apply is March 15, 1995. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! CHANCE News 4.01 (11 Dec 1994 to 10 Jan 1995) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Please send suggestions to: jlsnell@dartmouth.edu >>>==========>>|<<==========<<<