Prepared by J. Laurie Snell, Bill Peterson and Charles Grinstead, with help from Fuxing Hou, and Joan Snell.
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"You will not apply my precept," he said, shaking his head.
"How often have I said to you that when you have eliminated
the impossible, whatever remains, however improbable,
must be the truth?"
Sir Arthur Conan Doyle (The Sign of Four, Chapter 6)
Contents of Chance News 9.04
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In his contribution to Media Highlights in the March 2000 College Mathematics Journal, Norton Starr introduces the readers to the Forsooth column in RSS News by giving some (non random) samples from the past. Here they are:
Due to deforestation the rainfall (in the Peruvian rainforest) is now 120% less than 25 years ago.
Michael Palin, BBC TV, 2 November 1997
The chances for the baby's chromosomes being defective were 250 to 1, which sounds like reasonable odds. Except that all odds are, in reality, 50-50: it may happen and it may not.
The Times Magazine, 2 August 1997
One of Labour's priorities is to find a new definition of poverty to replace the popular one which includes anyone earning less than half of average earnings (an unhelpful definition making it statistically impossible to reduce poverty).
The Guardian 8 May 1997
Only 25% of households consist of the classic couple with 2.4 children...
The Observer (Matthew Fort) 10 November 1996
It is important to note that: many beers and wines are stronger than average.
Drink/drive campaign leaflet,
Here are two complementary views on what determines people's ability to navigate.
The Globe article reports a study in the April issue of the journal Nature Neuroscience, in which German researchers scanned the brains of 12 men and 12 women navigating a virtual reality maze. One observed difference concerned the use of the activity of the hippocampus, an inner brain structure known to be involved in memory. The brains scans showed men using both their right and left hippocampi during the task, whereas women used only their right. On the other hand, women appeared to also use an outer part of the brain called the prefrontal cortex.
Earlier studies have found that women are more inclined to use landmarks for navigation, while men rely on a sense of geometrical direction. The researchers speculated that activity in the cortex was related to storing landmarks, while the hippocampus was related to geometric reasoning. The present study does not shed light on whether the sex differences are genetic or learned. However, similar differences have shown up in rat studies, which suggests at least a genetic component.
The Economist reports related results from the current edition of the Proceedings of the National Academy of Sciences. Researchers at University College London compared brain scans of London taxi drivers to a reference group. Drivers must pass a rigorous exam on routes through the city to obtain a taxi badge, which the researchers took as evidence of superior navigational ability. The sample of drivers were all right-handed males, so a reference standard was compiled by averaging scans from a 50 right-handed "ordinary" men.
The only differences that emerged were in the hippocampi. The back of the hippocampus was larger in the taxi drivers than in the reference group. When the differences were plotted against years of taxi driving experience, it was observed that the back of the hippocampi grow and the front parts shrink over time. This suggests that the brain may be more physiologically adaptable than neuroscientists have traditionally thought.
DISCUSSION QUESTION:The Globe article raises the question: does the German study explain why men never seem to want to ask directions and women do? Do you think it does?
This article discusses the pressure from social scientists, businessmen, religious leaders, congressmen and others to include their own favorite questions on the census long form. The census decides on the questions to ask but Congress has veto power.
The short form asks: Tenure (owned or rented), Name, Sex, Age, Relationship to household, Hispanic Origin, and Race. The long form (sent to an average of one in six households) has has more detailed questions relating to the family structure, economic status, housing etc.
The article mentions wish lists from various sectors. Examples include sexual orientation, religion and the use of the internet.
The government needs the information on the long form to help allocate resources, while others would like to have it for research, marketing, etc. The census bureau has to balance its needs against those of others and also try to keep the form to a reasonable length. It is estimated that it will take about 38 minutes to fill out but evidently this estimate has a large variance.
The only question that was not on the 1990 form relates to grandparents as caregivers. This question was added by Congress. To improve the return of the short form, five questions that were on the short form in 1990 were moved to the long form and five 1990 questions were dropped from the long form.
One of the questions moved from the long form to the short form was marital status, which slipped by Congress's review of the forms. According to the article: in an impassioned speech to the Senate, Jesse Helms said the bureau "obviously no longer regards marriage as having any importance." The Senate voted unanimously to have the question on the short form in future censuses.
(1) Why do so many people want to know information about religion? Why do you think the census does not include this?(2) What considerations does the census bureau have to take into account in preparing the two types of forms?
The Office of Management and Budget (OMB) is responsible for the official federal government methodology for the collection of information on race and ethnicity. In 1997, after extensive hearings and consultations, the OMB set the standards to include five categories for race: (1) American Indian or Alaska Native, (2) Asian, (3) Black or African American, (4) Native Hawaiian or other Pacific Islander, and (5) White. They established two categories for ethnicity:(1) Hispanic or Latino and (2) Not Hispanic or Latino. The standards require that respondents be offered the option of selecting one or more racial designations. These are minimal standards. The census follows these standards but asks for more detailed information on race.
The first question is on ethnicity and asks whether you are or are not "Spanish/Hispanic/Latino". If you are, you can then tick a box for (1) Mexican, Mexican American, Chicano, (2) Puerto Rican, (3) Cuban or (4) other Spanish/ Hispanic/Latino.
The second question asks what your race is. You are asked to mark one or more races to indicate what you consider yourself to be from the categories:
(1) White, (2) Black/African American/Negro, (3)American Indian or Alaskan Native,(4) Asian Indian, (5) Chinese, (6) Filipino, (7) Japanese, (8) Korean, (9) Vietnamese, (10) Native Hawaiian, (11) Guamanian, (12) Chamorro, (13) Samoan, (14) Other Pacific Islander, and (15) Some other race.
Parents of mixed-race children applaud the change while others worry that this will reduce the number officially considered to be of a specific minority. In response to this concern the administration declared that those listing themselves as white and also a member of a minority will be consider to be of that minority.
The guidelines set down by the OMB also clarify the responsibilities of business, government agencies and educational institutions who have to report racial breakdown of the work force, student body etc. Rather than requiring that they provide data on all 63 combinations possible under the minimal requirements, they will be asked to report the numbers in the five races listed in the minimal requirements and four racial combinations: (1) American Indian/Alaskan Native and white, (2) Asian and white, (3) African-American and white, and (4) American Indian/Alaskan Native and African-American.
(1) How many possibilities are there for the way that people can indicate their race on the census form?
(2) In considerations of the OMB, some groups recommended the simpler solution of just having the single races and one other multi-racial category. Why do you think this simpler idea was rejected?(3) Anthropologists say that race is not a meaningful classification. On the other hand it is used in civil-rights legislation, clinical studies etc. Is this a contradiction?
Listen to this program to get a real feeling for what is going on in carrying out census 2000. It is an hour program with principal guest Kenneth Prewitt, Director, Census Bureau. Prewitt has had a lot of practice explaining what the census is doing and why -- and he is good at it. Both in his interview with narrator Williams and in fielding questions from listeners, Prewitt answers tough questions in a diplomatic and informative manner. For example, when callers want to know why the government cares about such things as how many miles they drive to work, Prewitt assured them that the government does not care about how many miles THEY drive but they do need the statistical information for determining funds for highway construction and maintenance in their areas. He was even able to explain why the government cared about how many toilets they have.
Other guests commenting on the census are: Maris Demeo from the Mexican American Legal Defense and Educational fund; Peter Skerry, author of the book "Counting on the Census: Race, Group Identity and the Evasion of Politics"; and Wes Marsh, Arizona State Representative.
From Representative Marsh we learn that Arizona, like Alaska, Kansas, and Colorado, has passed legislation banning the use of sampling in redistricting in its state. In defending his state's decision, Marsh says that he does not believe in scientific polls -- at one point referring to them as "scientific scheming polls, or sampling." Prewitt asked Marsh how he reconciles this with the fact that all of the economic data used by Alan Greenspan and others in making government decisions is based on sampling. Marsh replies "As you know, you can get a poll to come up with any data you want."
(1) Senator Chuck Hagel of Nebraska has suggested that people can simply ignore questions on the long form that they find intrusive. Presidential candidate George W. Bush has said that people should fill out the form; but that if he received a long form, he was not sure he would want to fill it out either. In answer to a caller who asked if it was o.k. to not answer some of the questions, Prewitt said that the law requires that all the questions be answered. Who do you think does not know the law?
(2) If you do not return your census form, you are subject to $100 fine. How many such fines do you think the census bureau has collected?
(3) Arlene Polonsky writes to the New York Times (Letter to the Editor April 4, 2000):
The director of the Census Bureau is alarmed because
as of March 30 only 50 percent of all households that
had received census questionnaires had returned them
(news article, April 2).
Question 1 on the short form asks, "How many people were living or staying in this house, apartment or mobile home on April 1, 2000?" This question could only be answered on or after April 1. I am surprised that 50 percent of respondents considered that they were in a position to answer before that date. They must have received a crystal ball with their census forms.
How do you think Hewitt would answer this?
Calculating the risks of hormone therapy.
Boston Globe, 7 March 2000, E1
Previous Chance News articles have discussed the benefits and risks associated with hormone replacement therapy for post- menopausal women. Major health concerns including cancer of the breast and uterus, osteoporosis, and heart disease are all part of the picture.
One problem is that the bulk of data currently available comes from observational studies. However, a $625 million nationwide study called the Women's Health Initiative is now in progress. It involves 27,000 women who have been randomly assigned to hormone therapy or placebo. When it is completed in 2005, the study will have followed the treatments for an average of nine years.
But how should women be advised in the meantime? Dr. Nananda Col and colleagues at the New England Medical Center have designed a computer model to provide some insight. It uses existing observational data (including the two recent studies described in the last issue of Chance News) to give 10- to 20-year projections of health risks associated with hormone therapy. To illustrate the model for this article, Dr. Col entered data for a hypothetical 50-year-old woman. To create a composite health profile for the woman, representative population data were used as input. For example, she was given a total cholesterol of 239 mg/dl and a systolic blood pressure of 134 mm Hg. Also, she was assigned 44% of the health risks attributable to smoking, a 20% chance of a family history of breast cancer, and was assumed to have had her first child between the ages of 25 and 29.
With these data as input, the model was used to give project health risks after ten years of therapy. The risk of breast cancer was one in 45 under estrogen therapy, but increased to one in 39 under combined estrogen-progestin therapy. However, the risk of uterine cancer was one in 19 with estrogen alone compared to one in 102 with the combination. The benefits of hormone therapy in preventing hip fractures increased over time, becoming even more substantial after 20 or 30 years. Reduced risk of heart disease showed up even earlier in treatment and remained as long as the therapy continued.
(1) The article presents absolute risks for breast and uterine cancer. What are the corresponding relative risks? Which do you think would be more helpful for women to hear?
(2) Typically when such studies are reported, women are advised to consult with their own doctors also. If your doctor had access to this model, you could input your personal health data. Presumably, the model would have to be updated as future studies are completed. How do you think this would be received?Editor's note: The April 5 New York Times has an article by Gina Kolata reporting that a preliminary review of the data from the Women's Health Initiative suggests that hormone replacement therapy actually increases the risk of heart attack rather than decreases it as had been expected. However, the data is not conclusive. This has been reported to women in the study but they have also been strongly encouraged to continue in the study.
Two finance professors, Lubos Pastor, at the University of Chicago, and Robert F. Stambaugh, at the University of Pennsylvania, have published a study that argues against picking index funds exclusively. They assumed the Fama-French model, which says that one stock does better or worse than others because of its greater sensitivities to three factors: the movement of the overall stock market, the performance of value stocks relative to growth issues and the performance of small-capitalization stocks relative to large-cap stocks.
The authors tracked all funds from 1963 to 1998 that had no sales charges and whose managers had at least three years with their fund. There were 505 funds meeting these criteria. They found that index funds do not respond quickly to the Fama-French factors, while a few of the actively managed funds were very sensitive to these factors.
DISCUSSION QUESTION:Suppose that in fact the 505 funds' sensitivity to the Fama-French factors were essentially random, i. e. not due to any skill on the part of the managers. Might one still expect to find a few funds that were sensitive to these factors?
An Internet start-up firm, Sandbox.com, this year put on a contest that asked contestants to pick the winners of each of the 63 games played in the NCAA Men's Basketball Tournament. If any contestant picked all of the winners correctly, a prize of $10,000,000 would be awarded. It is stated in the article that "statisticians put the odds of selecting all 63 winners at something in the neighborhood of eight or nine quintillion (a quintillion has 18 zeros)." (These odds are not stated quite correctly; it should say something like "10^18 to 1 against anyone picking all 63 winners.") The president of Sandbox.com stated that he thought that the "true" odds against winning were about five billion to one, because he felt that basketball fans bring some skill to the contest.
How would one estimate the "true" odds? First of all, if we assume that the probability that the favorite wins the i'th game to be p_i > .5, then clearly one has the largest probability of winning if one picks all of the favorites. In this case, the probability of winning is just the product of the probabilities p_i, assuming that the games are independent events. In the first round, involving 32 games, in each of the four brackets, the best teams play the worst teams, so the first games have high p_i values (ranging, say, from .5 for the "middle" team matchups to .9 for the mismatches). In the last 31 games, the p_i's are much closer to .5. Let us say that a basketball fan can pick the winners in these games with probability .55. Then the probability that all of the favorites win is about 1.9 x 10^(-14). This is a much smaller probability than the president of Sandbox.com came up with.
Given the above probability, this contest strikes this reviewer as a "safe" contest to put on. The article goes on to say that an insurance examiner assessed a fee of about 25 cents per contestant. If we assume that the examiner doubled his or her estimate of the actual probability, to provide a profit margin, then the examiner was assessing the probability to be 1.25 x 10^(-8).
How does Sandbox.com make money on this contest? Each contestant generates about $3.50 in commissions and fees for Sandbox.com, because when they register, the contestants are presented with trial magazine subscriptions and memberships to other web sites. At the writing of the article, 600,000 contestants had registered, generating 2.1 million dollars to Sandbox.com (which paid $150,000 in insurance).
Finally, it is interesting to note that last year, in essentially the same type of contest, not one of the 160,000 contestants picked Weber State to beat North Carolina in the first round (Weber State won), so the contest was over after the first week.
(1) Is it reasonable to assume that the games are independent events?
(2) How might a basketball fan estimate the p_i's?(3) Which estimate of the probability that a given fan wins the contest do you think is more accurate?
A study published last May in the journal Nature found that children who slept with nightlights up to the age of two years had a two to three times greater risk of becoming nearsighted. One possible explanation was that excess exposure to light might over- stimulate eye growth. However, the current issue of Nature contains reports from two groups of researchers who dispute the earlier findings.
One of the new studies was done at the New England College of Optometry. Looking at 213 people with ages from 2 to 24, investigators compared the eyesight of those who had slept with and without nightlights. It turned out that 20% of each group was now myopic.
The other study, conducted at Ohio State University, involved 1220 individuals with an median age of 10. It found that 17% of those who had slept with nightlights were now myopic, compared with 20% of those who had slept without nightlights. The study also identified those who had slept in fully lit rooms. In this group, 22% were now myopic. None of the differences was statistically significant.
Genetic predisposition to nearsightedness may have been a lurking variable in the original study. According to the article, the strongest association that turned up in the Ohio State study was a tendency for nearsighted parents to use nightlights. One of the researchers is quoted as saying "Surprise, surprise: those of us who can't see like to leave the lights on."
(1) The article reports the overall age distribution in each study, but not the ages within the groups of nightlight users and non-users. Why is it important that these be comparable? How would you as an investigator go about ensuring that they are?(2) Dr. Richard Stone, a co-author of the original study, said the publicity it received might have caused parents to deny using nightlights because they felt guilty. (In an item on food surveys in the last Chance news, it was pointed out that people will tend to underreport fat consumption once they are told fat is bad for them.) What do you think of this explanation? What data might help you evaluate it?
A medical study on sexually transmitted diseases has raised some thorny questions concerning the design of experiments that involve human subjects. This study, led by Dr. Thomas Quinn of Johns Hopkins University, was reported in the March 30, 2000 issue of the New England Journal of Medicine.
The project was carried out in 10 clusters of rural villages in Uganda with 15,127 subjects. The aim of the first part of the study was to determine if the presence of other sexually transmitted diseases such as syphilis and gonorrhea increased the risk of being infected with the HIV virus.
The subjects in both clusters were initially counseled on prevention of HIV, offered free condoms, confidential testing, and encouraged to inform their partners if they become infected with the HIV virus.
Then in 5 of the clusters subjects were periodically given antibiotic treatment to reduce the chance of sexually transmitted diseases while the subjects in the other 5 clusters were not given antibiotic treatment. Five community based surveys were conducted in intervals of 10 months. In these follow-up periods interviews and tests were given relevant to the incidence and progress of sexual diseases.
In the main study it was found that antibiotic treatment did not decrease the prevalence of HIV but did decrease the number of other sexually transmitted diseases. These results were reported earlier in an article in Lancet.
This article reports on a second aspect of the study. During the first four follow-up periods the researchers identified 415 couples who had, at the beginning of the study, had one person HIV positive and the other HIV negative. By the end of a 30-month follow up period, 90 of the HIV negative partners had become HIV positive.
The principle finding of this part of the study was that the viral load is the chief predictor of the risk of heterosexual transmission of HIV. (Viral load tests measure what's called HIV RNA. RNA is the part of HIV that knows how to make more of the virus.)
In the editorial that accompanied the article, New England Journal of Medicine editor, Dr. Marcia Angell, remarked that a study in the U.S. that withheld treatment for HIV patients and did not inform the HIV negative subjects their partners were HIV positive would not be approved. She reviewed the arguments for having different guidelines for developed and underdeveloped countries but stated that she continues to believe that ethical standards should not be different in different countries.
Dr. Quinn responded by noting that Ugandan authorities had insisted that only the infected subjects had the right to tell their partners that they were infected. In addition, the subjects were given better care, including antibiotics for such diseases as gonorrhea and syphilis, than they would have obtained had they not been part of the study. He also stated that in rural Uganda, there is no way to monitor the antiviral drugs that are given to H.I.V. patients, and these drugs are highly toxic.
As usual such studies bring to mind the Tuskegee study, in which black men in the United States, who had contracted syphilis, were not given penicillin so that the researchers could track the course of the disease. The difference between these two studies is that in the Uganda, anti-H.I.V. drugs are simply unavailable. However, some people argue that in a clinical research study, all subjects deserve to benefit from the fact that they are subjects.
(1) The study was approved by the AIDS Research Subcommittee of the Uganda National Council for Science and Technology, the human- subjects review boards of Columbia University and John Hopkins University, and the Office for Protection from Research Risk of the National Institutes of Health. Do you find it convincing that this was an acceptable study?(2) An earlier study, also carried out in Africa, designed to determine if low doses of AZT given to a pregnant woman would protect the child, raised similar ethical questions. (See Chance News 6.11.) This trial also could not be carried out in the U.S. because it is known that larger doses can protect the child. One argument for different standards in different countries is based on the special relevance of the results of the study to the country where it is carried out. Do you think this applies to this AZT study? What about the Uganda study?
Why do humans use chance in decision making? Is a decision made by chance fair? Are extremely rare events likely in the long run? Does true randomness exist? Why do even experts disagree about the meanings of randomness? Why is probability so counter-intuitive?
In her book Randomness, Deborah Bennett poses, discusses, and attempts to answer these and other intriguing questions that lie at the heart of all probabilistic ideas.
Specifically, Bennett states that "this book will examine randomness and several other notions that were critical to the historical development of probabilistic thinking--and that also play an important role in any individual's understanding (or misunderstanding) of the laws of the chance".
To illustrate how probability can be counter-intuitive, Bennett begins with several versions of well-known problems. A few of these are quite good, and in fact I discussed one of them-- Kahneman and Tversky's "cab" problem--in my introductory probability and statistics class this semester. The problem (p.2) goes like this:
A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data:
(a) 85% of the cabs are Green and 15% Blue
(b) A witness identified the cab as Blue.
(c) On nights with conditions similar to the accident, the witness can correctly identify Green and Blue 80% of the time.
What is the probability that the cab involved in the accident was Blue?
Bennett states that a "typical" answer is around 80% (the correct answer is about 41%), and argues that this and other examples show that many people (lay persons and experts alike) make errors determining probabilities because they ignore, or at least don't understand how to use, base rate information (here, the fact that only 15% of cabs are Blue.)
(A side note: Not only my students liked this problem; I told it to my mother, and she passed it on to a friend. Then she sent me this email: "My friend then proceeded to email it to all his friends and there was a massive work stoppage in the NY-NJ metro area while his friends and their co-workers sent a flurry of emails back and forth. Since most of them did not get it, they ended up accusing him of sending an English language puzzle, saying he had not worded it correctly! He asked me to ask you to send more puzzles.")
Chapters 2 through 5 trace the history of the use of chance devices in decision-making, games, and gambling, and there is wealth of fascinating information here. In very engaging prose, Bennett touches on such topics as the earliest known dice, dating from around 2750 BC in Mesopotamia and the Indus valley; Vedic texts, written between 400 BC and 400 AC, whose central theme is gaming; and other examples from literature, religion, and philosophy, such as the Iliad, the Old Testament, and the I Ching, where chance mechanisms are used "to give the gods a clear channel through which to express their divine will."
We also learn that through the Middle Ages and into the Renaissance many misconceptions about probability remained, though by the early 1600's Galileo (and perhaps others) had a clear understanding of the meaning of a "fair" die. Here too there are lively historical examples, especially an account of how the distribution for the sum of three fair dice was eventually worked out. At this point Bennett argues that progress in understanding such probabilities depended on viewing events sequentially: "In the evolution of the theory of probability, the inability to recognize the sequential nature of random outcomes has been a major stumbling block, and it continues to plague learners even today." (p. 54) This is indeed an important point; however, I found her examples largely unconvincing.
In chapters 6 and 7 Bennett describes the early development of probability distributions and statistical theory, including such topics as measurement error, the Bell curve, the Galton "quincunx" (board), the Central Limit Theorem, random sampling, correlation, Chi squares, normal and t-distributions, and standard deviation. There are also many thoroughly engaging anecdotes and sketches of the main players, including Bernoulli, Gauss, De Moivre, Galton, Pearson, and Gosset.
Chapters 8 and 9 get down to the business of randomness itself, especially the questions: How (and why) do we generate random numbers? and, What is a random sequence? As in previous chapters Bennett focuses heavily on the story of how these ideas have evolved, and in particular the role that computers began to play on generating random digits (e.g., Monte Carlo methods). While these chapters are likely to be the heaviest going for readers who are mathematically very unsophisticated, they contain several examples that are certainly accessible to many undergraduate and high school students. The ideas presented here are fascinating and left me hungry for more.
The last chapter contains analyses of several well-known "paradoxes" in probability, such as the Monty Hall problem (and variants), the birthday problem, and the "gender" problem (given that a family has two children and at least one is a girl, what is the probability that the family has two girls?) To me, this is the least successful chapter, but perhaps this is because I am already familiar with the examples and have developed my own ways of understanding them.I am sure that Randomness will appeal to a fairly wide range of readers: interested "lay" persons, students, teachers ofmathematics, probability, and/or statistics, and "experts" who want some historical background. There is also a useful and extensive reference section.
Readers of Chance News need no introduction to the Bible Codes controversy. Six years after he accepted the paper that started it all, Robert E. Kass wrote an introduction to the paper "Solving the Bible Code Puzzle", by Brendan McKay, Dror Bar-Natan, Maya Bar-Hillel, and Gil Kalai (Statistical Science, May 1999) which ended with: "It indeed appears, as they conclude, that the puzzle has been solved."
We all learned a lot from this controversy and it will remain a wonderful case study for understanding statistical tests and the study of coincidences. Reading this article, you will see why Maya and Laurie have both tried to get their theatrical daughters interested in producing a serious play about the Bible Codes -- the characters are great, it is a real life mystery, and the topic has already shown it can sell millions of books.
Of course the Conant Doyle quote at the beginning of this Chance News is perfect for this article. It was also used by the authors to introduce their article -- a coincidence? We could equally well have used the following quote from Maya and Avishai's fascinating article:
|Suppose you see a magician cutting the pretty lady in two, separating her smiling face, upper torso and all, from her twitching toes, long-booted legs and all, and then putting them together again. It is wrong to ask: "How can the lady be cut in half and glued back together again?", because the obvious answer to that question is: She cannot! But you are on your way if you ask: "How can one appear to be cutting the lady in half and then putting her together again?"|
The radio station WXKS in Boston has started a contest in which the grand prize is $2 million (actually less than this, because this prize would be paid out in a 40-year annuity). The contest is being played 51 times during the period of March 16 to April 21 of this year. The contest is quite simple to describe: The 20'th telephone caller after the contest starts will be asked his or her birthdate. Then the radio station will announce a predetermined birthdate. If the months match, the caller wins $5,000. If the months and days match, the caller wins $10,000. If the birthdates match exactly, the caller wins $1,000,000. In each case, the caller has been asked to name a friend in advance, and the friend wins the same prize as the caller. (This reviewer can imagine that the caller's OTHER friends might feel slighted, were the caller to win a prize.)
The legal language in the description of this contest on the Web is somewhat humorous in some places, although it is probably quite standard. We quote:
|Grand prize winning contestants are subject to verification of their identity by WXKS-FM, American Media and Special Promotions and American Hole in One prior to their grand prize being awarded. Grand prize winning contestants may be required to submit to a polygraph test. A polygraph test, which reveals any fraudulent activity or non-compliance with any of the rules and regulations outlined herein, will result in a grand prize claim denial. Grand prize winning contestants may also be required to submit to a voice anal- ysis to compare their voice to a grand prize winning contest call audio tape to verify an audio match. A voice analysis that does not produce an audio match will disqualify the contestant and result in a grand prize claim denial.|
The American Hole in One company is presumably a company of the same sort as SCA Promotions, which was discussed in a recent Chance News. This company insures a contest promoter against the winning of a grand prize. What are the chances that the grand prize is won?
To estimate this probability, one needs to know two distributions, the distribution of the likely callers' birthdates, and the distribution which governs the birthdates chosen by the radio station. It is likely that both distributions are fairly uniform over the days of the year, as is the case in the general population, but the first distribution is probably not the same with respect to the year as is the general population. This is because most radio stations target their audiences, so the variance of the age of the target audience is much less than the variance in the general population.
Since the radio station presumably knows, to a certain degree, the age distribution of its target audience, it can do one of four things; it can match its distribution to the target audience distribution, it can match its distribution to the general population distribution, it can use a uniform distribution on the set of all dates over a certain period of time, or it can pick a distribution that is unrelated to either the target audience or general population distribution. The last of these options seems dishonest, but any of the other three are perfectly reasonable. However, the choice among these three options clearly affects the probability of an exact match.
If we assume that the radio station makes the third choice, and uses the 20th century as a sample space, then there are 36525 birthdates. Under this assumption, the target audience birthdate distribution becomes irrelevant (as long as all of the audience was born in the 20th century, a fairly reasonable assumption). The probability of the grand prize being won is approximately
1 - (1 - 1/36525)^(51),
which is about .0014.
DISCUSSION QUESTION:Suppose that the target audience is concentrated in a twenty-year age group, and suppose that the radio station uses that age group from which to pick the birthdates (uniformly). What is the probability that the grand prize will be won?
David Savitz is an epidemiologist at the University of North Carolina who has studied possible links between power lines and cancer for the past 20 years. His 1987 study was one of four studies that suggested that power lines were a significant risk factor for childhood leukemia. A large eight-year study coordinated by the National Cancer Institute reported in the New England Journal of Medicine in 1997 concluded that power lines were not a serious risk factor for cancer and this led to a similar assessment by a committee of the American Acadey of Sciences.
In this study Savitz and his colleagues reviewed the health records of 139,000 electricians and other technicians employed in five large electric power companies in the U.S. between 1950 and 1980. 6000 of these workers were selected for a detailed study. The researchers found that there were 536 suicides among the 6000 workers and the suicide deaths were twice as high among those who were regularly exposed to extremely low frequency electromagnetic fields.
The researchers found a "dose response" in that the greater the amount of exposure to EMF's, the greater the likelihood of suicide.
The authors suggest that electromagnetic fields may reduce the production of melatonin, a hormone that regulates mood and sleep patterns. Reduced levels of melatonin have been associated with depression.
This study appears in the April, 2000 issue of the Journal of Epidemiology and Community Health.
Milloy maintains a web site called Junk Science. He adds new examples daily. On August 16 Milloy commented on this study:
|The statistical associations reported in this study are weak. Most are statistically insignificant. All the comparisons involve small groups. Statistical comparisons were only adjusted for work status, social class, location in the U.S. (east or west), exposure to solvents, and exposure to sunlight. I'm no psychiatrist, but I know there are many more risk factors for suicide than were considered by these researchers.|
Milloy also notes that research published at the same time in the Journal of Epidemiology and Community Health shows that divorce and separation doubles the risk of suicide among men. The Savitz study did not include marital status in the screens it applied.
Milloy suggest the results are not very significant. Is this consistent with the claim that, of the 536 suicides, the suicide deaths were twice as high among those who were regularly exposed to extremely low frequency electromagnetic fields?It is often suggested that a study paid for by a company that stands to gain by the results of the study is slightly suspect. This study was funded by the Electric Power Research Institute. Should that make it more believable?
There's good news for chocolate lovers! Chocolate beans contain compounds called flavonoids, which have been correlated with a reduced risk of cardiovascular disease. These compounds work in the body in several ways. One way is to reduce the effects of free radicals on the walls of blood vessels. Another way is to "relax" the inner walls of blood vessels. People in whom such relaxation is absent often suffer from things such as high blood pressure and atherosclerosis.
Another study has found that diets enriched with dark chocolate or cocoa powder raised the subjects' high-density lipoprotein cholesterol. This is the so-called "good" cholesterol, which is associated with a decreased risk of heart disease.
The article points out that these studies have been funded by the chocolate industry, so one might be tempted to dismiss them. On the other hand, this situation has occurred in other industries; once industry-funded studies claim that benefits accrue from eating a certain type of food, the government sometimes steps in and begins funding more studies.While the possibility of health claims on chocolate products is a happy thought, there are those (including this reviewer) who do not need any such inducements to indulge in Nature's most perfect food!
In the last Chance News we described the recent work of physicist Carlton Caves arguing that there are serious flaws in Gott's method for estimating the future life of a phenomenon. Caves' work was described in PhysicsWeb and included a reply by Gott. We promised in this Chance News to try to explain what the argument was really all about.
In this issue we will concentrate on straightening out what Gott really claims. In the next issue we will attempt to explain Caves' approach.
Here is what we believe is the model for Gott's approach. We assume a random time interval I with endpoints B and E representing the beginning and end of a phenomenon X. Then L = E - B is the "lifetime" of X. Let A = L x rnd where rnd is a random variable uniformly distributed between 0 and 1 and independent of U and V. Then A has a uniform distribution between 0 and L and represents the "age" of X when it is observed. Gott feels the assumption that X is observed at a random time is reasonable if there is nothing "special" about the time we observe X. He calls this the "Copernican assumption."
Then R = L-A is the remaining lifetime of X when we observe X. Since A is uniformly distributed on 0 to L it follows that for u and v with 0 < u < v < 1:
(1) Pr(uL < A < vL) = (vL - uL)/L = v-u.
Using the fact that L = A + R this is seen to be equivalent to
(2) Pr[((1-v)/v)A < R < ((1-u)/u)A] = v-u.
Choosing the particular values u = 1/40, v = 39/40 Gott obtains:
(3) Pr(A/39 < R < 39A) = 0.95.
At this point we have a statement about random variables. It sets up a recipe for making predictions about R once we have observed a value a for the random variable A. Given the age a of X Gott obtains a 95% confidence interval for the remaining lifetime R of X as:
a/39 < R < 39a.
The frequency concept of probability says that if makes such estimates for a large number of X's these estimates will be correct about 95% of the time.
It does NOT follow from the (3):
(4) Pr(a/39 < R < 39a | A = a) = 0.95
This is the classical mistake that our students, and for that matter we ourselves (see Chance News 9.03), often make when discussing confidence intervals.
Taking v = 1 and u = 1/(1+c) in (2) gives for any c > 0
Pr(R > cA) = 1/(1+c).
Again this does not say that
(5) Pr(R > ca | A = a) = 1/(1+c)
Choosing c = 2, (5)their life and find that they are 75 then this person has a 33% c would say that if you meet a person at a random time during hance of living at least to 150 which is clearly absurd.
The correct statements:
Pr(A/39 < R < 39*A) = 0.95
Pr(R > cA) = 1/(1+c)
are probabilities relating to the joint distribution of the two random variables A and R. You cannot expect them to imply corresponding probability statements when involving only the random variable R.
In his original paper (Nature, 27 May 1993) Gott was careful to describe his predictions in terms of confidence intervals and not probabilities.
For example, when he applies his results to the lifetime of our species he observes that we have been around for 200,000 years and writes:
The estimate for the future lifetime of our species is
5,100 years < R < 7.8 x 10^6 years (95% confidence limits)
This is similar to reporting in a poll that 53 percent of the voters prefer Gore with a 5 percent margin of error. The New York Times and other newspapers are careful to explain this as saying:
|In theory, in 19 cases out of 20, the results from such polls should differ by no more than plus or minus four to five percentage points from what would have been obtained by polling the entire population of voters.|
In the poll example the unknown true proportion of voters is assumed to be a constant so it is easier to understand why you cannot talk about the probability that the true proportion lies in an interval. In our situation R is a random variable so it is easier to make erroneous statements of the from (4) and (5).
In his recent paper (Predicting future during from present age: a critical assessment, to appear in Contemporary Physics) Caves appears to be convinced that Gott believes that probabilities of the form (4) and (5) are consequences of his theory. This leads him to state that there is no probability model that is consistent with Gott's theory. Another indication of this belief is the following bet that Caves proposes to Gott:
|Caves obtained the ages of 24 dogs owned by his colleagues and chose the six dogs who were over 10 years in age. He then offered Gott a 1000 dollar bet on each of these six dogs against their living to twice their present age. He offers 2 to 1 odds.|
|In accordance with his practice for other phenomena, Gott would make a prediction for each dog based on its present age. In particular, he would predict that each dog will survive beyond twice its present age with probability one-half.|
We do not believe that Gott would make the claim in the second sentance. It is a mistake of the form (4). Gott would simply say that if he did predict that each of the dogs will live beyond twice its present age he would expect to be correct (95% confident) for about 1/3 of the dogs.
Caves also remarks:
|If he (Gott) believed his own predictions, the probability that he would be a net loser on these 6 bets is 7/64 = .109.|
Again this is based on using the incorrect form (5) for Gott's confidence intervals.
Choosing the 6 oldest dogs also poses problems about the assumption that the dogs were observed at a random time during their lifetime.
Thus we feel that Gott's theory does make sense. Of course the mere fact that it makes sense mathematically does not assure its usefulness. The key assumption that you observe the phenomena at a random time during its lifetime is a strong assumption and certainly not too often satisfied.
For example, the people most interested in lifetimes are the insurance companies. For them to set premium prices by Gott's confidence intervals would require them to assume that people purchase insurance at a random time during their lifetime. This is clearly not the case. Even if it were true the insurance company would find it hard to set premium prices from Gott's confidence intervals. For example based on A = 75, Gott's 95% confidence interval for Laurie's future life is:
1.92 < R < 2925
and for his one year old grandchild it is
1/39 < R < 39.
Hopefully Laurie does not need insurance to age 2925 and his one- year- old grandchild will need it for more than 39 years. Thus Laurie might be charged too little and his granddaughter too much.
Of course the Insurance companies has estimates for the distribution of the lifetimes of people and uses them. Gott's method does not assume a knowledge of the distribution of the phenomena being considered.
We consider next an example where we can make use of the distribution of the lifetime. This example will also provide a good introduction to the work of Caves.
Suppose you are in New York for the first time and stop at a bus stop. Then it seems reasonable to assume that you have arrived at a random time between visits of buses arriving at this bus stop. You are naturally concerned about how long you will have to wait for your bus. Then taking X to be the arrival of your bus, A the time since the last bus, and R the time you will have to wait for your bus, Gott tells you that
Pr(R > cA) = 1/(1+c).
For example, you should have 50% confidence that you will have to wait about as long as the time since the last bus.
Waiting times of this form have been studied extensively in the probability literature under the name "renewal theory." They include a variety of applications; waiting times at queues, the lifetimes of light bulbs, the lifetime of machines etc. In these settings, Feller (Probabilility and its Applications Vol 2, p 11) describes "the waiting time paradox" which he illustrates in terms of our bus problem.
Assume that the times between arrivals (interarrival times) of the buses at a bus stop are independent with distribution F(t) having density f(t). Assume also that the buses have been running for a long time so that the effect of the first bus has worn off. Then if you arrive at the bus stop at time t we can ask how long do you have to wait for the next bus and how long was it since the last bus? In renewal theory it is shown that the density for the distribution for both of these times are the same and this common distribution is:
g(t) = (1-F(t))/m
where m is the mean interarrival time.
For example, assume that the interarrival time has an exponential distribution F(t) with mean 30 minutes. Then if you arrive at a random time you might expect you would have to wait on the average about 15 minutes. However, alas, it turns out that you have to wait on average 30 minutes. To see this we note that F(t) = 1-e^(-t/30) so g(t) = 1/30e^(-t/30). Thus in this case both the waiting time and the time since the last bus have the same distribution and the interarrival times. Thus the time between buses for our bus is 60 minutes rather than the mean interarrival time of 30 minutes.
The explanation of this apparent paradox is that the knowledge that you arrived at a random time makes it more likely that you arrived in one of longer possible interarrival times for the buses.
Thus we have two different lifetimes: The lifetime L for a bus observed starting from the time it leaves the bus stop (i.e., a typical interarrival time) and the lifetime L' for your bus when you arrive at a random time. Typically,
Pr(L' > t) > P(L > t)
If we denote by f and f' the density for L and L' then it is shown in renewal theory that:
f'(t) = tf(t)/m
The intuitive story here is that the probability for the lifetime at a random time ought to be proportional to the underlying density f. But the chance of an interval covering t also ought to be proportional to its length (the longer the interval, the better its chance of covering). So we want a density proportional to xf(x). To make this integrate to 1, you need to divide by m.
Assume now that when you arrive at the bus stop you ask when the last bus came. You are told that it arrived a minutes ago. This corresponds to knowing the age A = a in Gott's model. This tells us that our bus started out from the bus stop at time t-a. Since we know this starting time, future predictions should use the interarrival time L with density f(t)and distribution F(t). Thus for c > 1:
(6) Pr(R > ca | A = a) = P(L - A > ca | A = a) = P(L > ca + a | A = a) = [1 - F(ca +a)]/[1-F(a)]Therefore, unlike Gott's analysis that does not assume a knowledge of the distribution of the lifetime L, when we know this distribution we can calculate, for example, the conditional probability that we have to wait twice as long as we have been waiting. Such probabilities will not, in general, agree with those obtained incorrectly from Gott's analysis:
(5) Pr(R > ca | A = a) = 1/(1+c)
In his treatment of Gott's problem, in addition to assuming Gott's copernicus assumption, Caves assumes an apriori distribution F(x) with density f(x) for the lifetime L of X. This F plays the role of our interarrival distribution. Caves establishes (6) for the future lifetime R of X given the age A. He observes that equation (5) will be correct only if the apriori distribution for the lifetime L is given by the density f(t) = 1/t^2. This is a density with infinite mass at 0 but Pr(L > t) = 1/t makes sense for t > 0. This apriori distribution was considered by Harold Jeffreys (See his probability book) as appropriate in certain situations where the distribution is determined by a parameter about which nothing is known. Gott uses this for his examples of particle emissions and the lifetime of the human species.. We will discuss these examples in a later issue of Chance News.
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