Prepared by J. Laurie Snell, Bill Peterson and Charles Grinstead, with help from Fuxing Hou, and Joan Snell.
Please send comments and suggestions for articles to
Back issues of Chance News and other materials for teaching a Chance course are available from the Chance web site:
Chance News is distributed under the GNU General Public License (so-called 'copyleft'). See the end of the newsletter for details.
Chance News is best read using Courier 12pt font.
The best way to get rich quickly from probability theory is to find someone who knows less about it than you do.
Contents of Chance News
Note: We are working on a revision of our web site and traveling a lot this summer, so Chance News will come at even more random times and be shorter than usual. This is a good time to send us suggestions for Chance News and ways to improve the Chance Web site. In particular what should we name it? We are looking for something as inspired as "The Wizard of Odds", chosen by Michael Shakelford for his homepage on odds in gambling.
Norton Starr sent us the howlers from the Forsooth column in the May 1999 issue of RSS News. The two we liked the best were:
Does it matter that women are 550% (five and a half times) less likely than men to be appointed to a professional grade?
(Association of University Teachers)
Issue 46, Spring 1999
Women born in the 1960's are twice as likely as their mothers never to have children.
6 March 1999
It is not clear to us what the point of this next howler is or even why it is a howler, but this is good for a discussion question.
Each year about 570,000 people die in the UK - 22 every 20 minutes, the time the National Lottery draw is on TV on Saturday. The latest viewing figures are 7.28 million, 13 percent of the population ... which means that, on average, three viewers die for every winner. And remember, sometimes nobody wins.
13 August 1998
DISCUSSION QUESTION:What is the Daily Express trying to say about the lottery?
Tufte has made available serarately Chapter 2 of his most recent book: "Visual Explanations" (see Chance News 6.04). This chapter provides beautiful expositions of two classic case studies in statistics: "John Snow and the cholera epidemic" and "The decision to launch the space shuttle Challenger." It is hard to think of anything your students could read that would give them a better appreciation of statistics in the real world. And the price ($5) is right. You can order this booklet from:Graphics Press
The Wizard of Odds is the web site Michael Shackleford maintains for people who want to "learn the odds of all the major casino games and the best way to play." Michael was a math/economics major in college, became an actuary, and works at the Social Security headquarters in Baltimore.
We particularly enjoyed Michael's discussion of blackjack. After discussing how blackjack is played and the art of card counting, Michael invites us to read the diary of his annual trip to Las Vegas to play blackjack, this year from June 8 to June 11. Michael describes himself as a card counter with modest skill. In his diary you will find a detailed discussion of his fluctuating fortune as he wandered from Casino to Casino playing blackjack and trying to avoid attracting the attention of the pit boss and an invitation to leave the casino. Michael had a pretty good four days this year ending with a net win of $1035, up a bit from last year's $640.
Under "slot machines" you will find a discussion of how modern slot machines work, the payoffs for the various popular machines, and a list of myths about slot machines.
The classic slot machines had three wheels which could stop at any one of 22 positions. At each of the 22 possible stopping positions there was a picture of a bell, an orange, a 7 etc. and at one position the word "Jackpot". You pull the lever, the wheels spin, and one of the 22^3 = 10,648 possible stopping triples occurs-you hope the one with all three wheels "Jackpot" but you might get a smaller payoff if the wheel stops at, for example, (7,7,orange). The nature of the machine makes it reasonable to assume that all 10,648 stopping positions are equally likely. Thus if you can observe what symbols are on each reel you can easily compute the expected payoff of the machine.
How does a modern slot machine with 22 position wheels work? It looks very similar to the classical slot machine-you pull a lever, wheels spin, it stops at one of the 10,648 possible stopping positions showing bells, oranges etc. But now the stopping positions for the wheels have been determined by the computer before you even pull the lever. Whether anyone is playing or not, every second the computer chooses three numbers at random from 0 to (2^32)-1. The machine then divides those numbers by the number of stops on the virtual wheels, often 64, and the remainder determines where the virtual wheel will stop.
If the virtual wheel has 64 stops then the 3^64 = 262,144 possible triples for the computer's choices are equally likely. However the programmer can map these to the 10,648 possible stopping triples on the real wheel to make the payoffs fit the casino owner's wishes. For example, in the classical machine the probability for a jackpot cannot be less than 1/10,648, but in this virtual machine it could be made as small as 1/262,144 to permit larger jackpot prizes. It could be made even smaller by having the virtual reel have more stops, say 128.
One of the popular myths about slot machines is to assume that, for the computer driven machines, the 10,648 possible stopping positions are still equally likely. To show that this is not the case, Michael found a slot machine at the Atlantic City Tropicana with three wheels and 22 stopping positions. He played until he was able to determine the symbols and their frequencies on each wheel. Then assuming that all the 3^22 stopping positions are equally likely, he was able to calculate the expected payoff for a $1 coin. He found this be $4.70! He rejected the hypothesis that all possible stops were equally likely.
Michael often appears in the news because of his annual report on the most popular names for children in the United States. Thanks to Michael's efforts, you can find the most popular names given to babies born between 1880 to 1998 at the Social Security web site: Name Distributions in the Social Security Area, August 1997. You can also find the most popular names by states. Let's see what difference a century makes by looking at the top ten in the U.S. for 1900 and 1998.
Note that the only name that has survived on the top ten list over the century is Elizabeth.
Newspaper accounts of Michael's research stress the demographic changes which lead to changes in popular names. For example, in 1998 in California and Texas the most popular name for a baby boy was Jose.
(1) Joan and Laurie Snell chose John and Mary for the names of their two children. Estimate Laurie's age.
(2) The Wizard of Odds indicates that most slot machines in Las
Vegas have a 90-95% return. It seems to us that more people brag
about coming away from roulette a winner than coming away from a
slot machine a winner. Can you explain this?
Study links welfare paydays to rise in drug deaths
The Los Angeles Times, 8 July, 1999 A3
Sylvia Pagan Westphal
An increase in the number of deaths in the United States in the first week of the month.
The New England Journal of Medicine, 8 July 1999, p 93
David P. Phillips, Nicholas Christenfeld, Natalie M. Ryan
David Phillips is familiar to readers of Chance News as the sociologist at UCSD who likes to find patterns in dates of deaths. For example, he showed that women die more often right after their birthdays and men more often right before their birthdays than can be accounted for by chance. (See Chance News 1.04, 2.19, 3.01.)
Using computerized data from death certificates in the United States between 1973 and 1988, the authors compared the number of deaths in the first week of the month with those in the last week of the previous month. They found that, on average, there were 4,320 more deaths in the first week of the month than in the last week of the previous month.
The authors call this increase the "boundary effect". The boundary effect is clearly shown by a graph in the original article giving the average number of deaths per day for the 14 days before the end of the month and the 14 days at the beginning of the month. The number of deaths is unusually low in the last week of the month and then increases abruptly (by more than 15 standard deviations) on the first day of the month.
The authors define the boundary effect for a specific cause of death to be the ratio of the number of deaths by this cause in the first week of the month to the number in the last week of the preceding month multiplied by 100. Looking at those causes of death for which the boundary effect is significant, the largest boundary effect is for substance abuse, followed in order by homicide, suicide and motor vehicle accidents. Drug abuse includes both alcohol abuse and other drug abuse. The boundary effect for both are about the same.
Federal benefits such as social security, welfare, and military benefits, are usually provided at the beginning of the month. This suggests that those on welfare may be spending some of this money for dangerous drugs leading to the increased mortality at the beginning of the month. Previous research has linked payments for federal benefits with changes in health status.
If this explanation is valid, the boundary effect would be expected to be greater for poor people than for those not poor. Since income is not indicated on the death certificate, this cannot be directly tested. Because blacks are poorer on average than whites and race is indicated on the death certificates, the authors compared these two groups and did find a significant higher boundary effect for blacks than whites.
In the LA Times article, Phillips is quoted as saying: If the federal government were to deliver support in the same way that private agencies do in the form of food, clothing and shelter, there would be less deaths.
(1) Can you suggest a reason that women should tend to die just beyond their birthdays and men just before than should happen by chance?
(2) Can you think of other explanations for the boundary effect?
(See the original paper for some provided by the authors.)
How to predict everything
The New Yorker, July 12, 1999,p 35
Readers of Chance News will recall that Princeton physicist J. Richard Gott III published a paper in Nature (See Chance News 2.11) describing his Copernicus method of estimating confidence intervals for the length of just about anything. He called it the Coperincus method because Copernicus pointed out that there is nothing special about the earth. Not being special plays a key role in Gott's method. In particular, Gott used this Copernicus method to estimate that, with 95% confidence, humans are going to be around at least fifty-one hundred years but less than 7.8 million years.
In this interview Gott explains his method, how he thought of it, and some of its many other applications.
Gott says that he thought of his method for estimating lifetimes in 1959 just after graduating from Harvard. He was traveling in Europe and happened to stop to look at the Wall and wondered how long it would be there. He realized that there was nothing special about his being at the Wall at that time. Thus if you divide up the time from the time the Wall was build to the time it is removed into four equal parts there is a 50% chance that he is in the middle two parts. If he is at the beginning of the middle half then the Wall will last three times as long as it has so far. If he is at the end of the middle half of the Wall it will last 1/3 as long as it has so far. At the time the Wall was eight years old. Thus he concluded that there was a 50% the Wall would last between 2 and 1/3 and 24 years. In fact it lasted 20 more years.
Of course the New Yorker was interested in Gott's application of his theory to estimating how long the various Broadway plays would last. On May 27, 1993 Gott looked up all the plays that were listed in The New Yorker-Broadway and Off Broadway plays and musicals. He called each theater to find when they opened. He then used his theory to predict 95% confidence intervals for the time they would close. Forty-four shows were showing at the time and so far thirty-six have closed, all at times within his confidence intervals. The others are still within the range he predicted, so Gott remarks that he is batting a thousand so far.
Gott gives a number of other amusing applications of the Copernicus theory. For example, the reason that you so often find yourself in the longest line at the supermarket is that there is nothing special about you. You are just a random person waiting to check out. When you choose a random person from those waiting to check out you are more likely to get one from a long line. You can find most of what Gott says in this article also in an article A grim reckoning he wrote for The New Scientist.
(1) Gott remarks that humans have been around about 200,000 years. How does he get his 95% confidence limits from this?
(2) What do you think about Gott's method of estimatomg lifetimes?
Richard Askwith asks Britain's leading expert gambling expert to reveal his secrets.
The Independent 27 June 1998 p 2
John Haigh is a Reader in Mathematics and Statistics at the University of Sussex who does research in probability and has a particular interest in lotteries and other forms of gambling. He has written a book aimed at showing the pervasiveness of probability in everyday life and explaining probability reasoning in the context of real world problems. We will give our impressions of the book but we strongly encourage you to read Askwith's article (available from Lexis Nexis). It is a much better discussion of the book and the author than we can give. Here is a sample:
YOUR BEST strategy here," says Dr. John Haigh, casting a dispassionate eye around Peter Pan's Amusement Arcade on the Brighton seafront, "is to tell that boy over there that if he doesn't give you pounds 5, you'll tell the management he's under 18 and not accompanied by an adult. Either that, or keep your money in your pocket.
In his preface Haigh writes "My greatest mathematical debt is to someone I never met: William Feller, whose books inspired a generation of probabilists." Those of us who had the good fortune to know Feller will agree that Feller would have been pleased with the honest and lively presentation of probability in this book.
Probabilists will find many of their old favorites such as: How long does it take to get a pattern, say HTHH, of H's and T's when you toss a coin a sequence of times? What is the most likely number of times in the lead in a coin tossing game? They will also find much that is new to them. This comes from Haigh's interest in a knowledge of gambling. Despite the fact that he himself is only a "theoretical" gambler, according the account in the Independent, his book has made him a kind of folk hero to those who do gamble. They appreciate the fact that he is using his knowledge of probability to give meaningful advice to real gambling problems.
Haigh explains carefully what a person's chances of winning the lottery are. Knowing that people will still play the lottery, he explains what choices of numbers will give the best chance of not having to share the jackpot if they do end up winning it.
Those who play games such as squash or tennis will find how probability reasoning can help them in real strategic decisions. For example, our photographer Bob Drake is one of our areas top squash players. Bob was pleased to find that some of his ideas about squash were supported by probability theory. Here is an example of this.
For the traditional version of squash in which a player can only score when serving, Haigh considers the problem: if the score reaches 8-8, and you are the receiver, should you exercise your right to have the game played as best of 10 or leave it as first of 9? He first shows how to find the probability of scoring the next point when you have probability p of winning a point. Using this he shows that you should opt to change to 10 only when p is greater than about 38%. He then observes that the fact that this decision only has to be made when the score reaches 8-8 suggests that the normal situation would be that p is near .5 so you should opt for 10. On the other hand if you are close to exhaustion or have just picked up an injury p could well be less than 38% and you should opt to play to 9.
Most tennis players have a fast first serve and a slower, steadier second serve. We might occasionally try our fast serve also on our second serve in hopes of catching our opponent off guard. But is it possible that the best strategy is always to use a fast serve? Haigh shows that this depends on four probabilities: the probability of a successful fast serve, the probability of a successful steady serve, the probability of winning a point given a successful fast serve, and the probability of winning a point given a successful steady serve. Depending on these probabilities, the best strategy can be fast serve both times, fast first serve and steady second serve, or both serves a steady serve. Based on this analysis we have changed our service strategy to using fast serve for both serves and have not lost a match since.
Read this book and your life will change also.
Kent Morrison suggested the next article.
9th grade dip on tests baffles educators.
L.A. Times, 2 July, 1999 A3
For the past two years public school students in California in grades 2 through 11 have taken standardized tests called Stanford tests in reading, math, and language. Students in grades 2 through 8 also take a test in spelling and those in grades 9 through 11 also take tests in Science and Social Science. These test are prepared by Harcourt Educational Measurement. In both 1998 and 1999, there has been what appears to be a significant drop in statewide math scores from 8th to 9th grade. Scores are given in the form of the percent of students who score above the national 50th percentile. Here are the California reading scores for the past two years:
The drop in reading scores from 8th to 9th grade do seem significant but the reports do not say anything about the variation expected in these scores.
According to the article there is no shortage of explanations: transition to high school is difficult, new students coming into the California system at the high school level, a new level of reading expected at the high school level, something is wrong with the test etc.
Some school districts are changing their summer plans to try to improve the reading of students going into high school next year. Of course this takes resources away from other programs and it will be too bad if it turns out just to be something strange about the testing.
You can find more about the Stanford test at www.hbtpc.com. The people at Harcourt Educational Measurement are studying the problem beginning with trying to identify a cohort that would include only the 9th graders who were also 8th graders the year before. They promised to pass on their results. Perhaps we will know more about their results by the next Chance News.
(1) How would you go about trying to understand how significant the drop in reading scores is?
(2) What other explanations can you suggest for the drop in reading scores?
(3) Math scores in 1999 are uniformly better than those for 1998.
Does this mean the students are getting better at math?
False false positive rates
The New England Journal of Medicine, July 8, 1999, p 131
Letter to the Editor from James N. Suojamen
Suojamen is responding to an article by LeLorier et al. (NEJM Aug. 21, 1997)indicating that the results of meta-studies frequently conflict with large randomized controlled experiments.
Sougamen comments that meta-studies often involve estimating false positive rates. This would be true for studies of the reliability of tests such as the PSA test for prostate cancer. He comments:
Errors in the calculation or reporting of false positive rates can adversely affect the results of meta-analyses that uncritically incorporate these data and results in discrepancies with the results of clinical trials.
To determined if false positive rates are reported accurately in tests, Sougamen did his own meta-analysis study to answer this question. Searching Medline between January 1995 and July 1997 he found 63 published studies that reported a false positive rate. Sougamen writes:
For each study, I calculated the false positive rate from the published data and compared it with the reported false positive rate. Thirty of the 63 studies reported the false positive rate incorrectly or miscalculated its value. The reported values ranged from 20 percent to 1135 percent of the actual false positive rates.
Of course calculating a false positive rate just involves a simple calculation from a 2x2 table
|Disease present||Disease absent|
18 of the 30 errors consisted of calculating the false positive rate as b/(a+b). Six studies gave it as b/(a+b+c+d). Six made other errors.
(1) What is the right answer?
(2) What percentage of studies do you think have a serious
epidemiologist as part of the research group?
Studies of leisure time point both up and down
The New York Times, 10 July, 1999, B7
Research on whether Americans have more or less leisure time than they used to gives contradictory results. This article discusses how that happens.
According to the article, one problem is that researches have different methods for trying to answer this question. They use different time periods, different methods-relying on memories, keeping notebooks etc. In addition, their definitions for leisure time and work may differ.
The largest source of public data on the number of hours per week Americans work comes from the Bureau of Labor Statistics. The Bureau regularly surveys 400,000 non-farm businesses for the number of hours worked by production and non-supervisory workers. The Bureau also surveys 50,000 randomly selected households by telephone interviews which ask for the hours worked in the previous week.
Of course, this is also a popular subject for social scientists. In 1991 Harvard economist Juliet Schor reported in her book "The overworked Americans", that time on the job had increased steadily over the past 20 years.
Sociologist John Robinson and Geoffrey Godbey in their 1997 book "Time for Life" reported the opposite. They reported that men and women were working less than they did in 1965 and had gained an hour a day in free time.
Robinson and Godfrey asked their subjects to keep notebooks. Researchers who ask subjects to keep notebooks say that relying on memories causes a bias because people tend to exaggerate the amount that they work. People who rely on the memories of the subjects say that using notebooks biases the results in the other direction, since people who are busy forget to record some of the work periods.
But many researchers feel that the changes in average amount of leisure time is not terribly meaningful anyway because there appear to be two different groups moving in different directions. One group is created by the tendency to take early retirement and for couples to postpone having children or have fewer children, increasing their leisure time. The other group is the baby boomers who seem to be workaholics with little leisure time.
The researchers raise all these questions but make few suggestions for how to obtain more meaningful results.
(1) Do you agree that trends in the average amount of leisure time for the population are not meaningful when there are two subgroups going in opposite directions?
(2) How would you design an experiment to answer the question: Do
Americans have increased leisure time?
The courts vs. scientific certainty.
The New York Times, 27 June, 1999
The Institute of the National Academy of Sciences reported a study which looked at all the studies that have been carried out to test the effect of silicone implants on women's health. They concluded that silicone implants can cause more local problems than have been previously suggested, but there is no convincing evidence that the implants cause more serious ailments such as lupus and rheumatatoid arthritis as has been claimed.
The more serious ailments have been the focus of previous law suits which have awarded $7 billion in verdicts and settlements to women claiming that silicone implants caused them to have serious ailments.
This article focuses on the difference between what the scientists require for evidence and what the court does. Scientists tend to require evidence at the 95% level and continue their experiments until they reach consistent studies at this level. The court has to act at a specific time on a specific case. In civil cases juries are told that a mere preponderance of evidence is enough certainty to favor the plaintiff. The article interprets this as saying there is at least a 51% chance that the charges are valid. In the case of silicone implants we seem to have an example of the courts acting as though there was scientific evidence backing up the plaintiff's claims, when apparently there was no such evidence. In the other direction, until recently it has been hard for those who developed serious illnesses from smoking to successfully sue a tobacco company despite the clear evidence that smoking causes serious illnesses.
According to this article some experts feel that we just have to get used to the fact that the legal system is not perfect and cannot act like science. Others say that the tort system is supposed to be based on the assumption that you only pay when you have done harm.
In their rulings beginning in 1993, the Supreme Court said that judges should act as keepers and be more vigilant in assuring that "scientific evidence is not only relevant but reliable." It would seem that so far these rulings have not had much effect.
(1) Do you think a preponderance of evidence that harm has been done means at least a 51% chance that harm has been done?
(2) The weather bureau does not have trouble with 60 percent chance
of rain; why should the rest of science have trouble with
percentages of less than 95%?
This work is freely redistributable under the terms of the GNU General Public License as published by the Free Software Foundation. This work comes with ABSOLUTELY NO WARRANTY.