The CHANCE Project
Version 0.1
21 September 1994
Welcome to Chance!
Chance is an unconventional math course. The standard elementary math course develops a body of mathematics in a systematic way and gives some highly simplified real-world examples in the hope of suggesting the importance of the subject. In the course Chance, we will choose serious applications of probability and statistics and make these the focus of the course, developing concepts in probability and statistics only to the extent necessary to understand the applications. The goal is to make you better able to come to your own conclusions about news stories involving chance issues.
Topics that might be covered in Chance include:
During the course, we will choose six to ten separate topics to discuss with special emphasis on topics currently in the news. We will start by reading a newspaper account of the topic. In most cases this will be the account in the New York Times. We will read other accounts of the subject as appropriate, including articles in journals like Chance, Science, Nature, and Scientific American, and original journal articles. These articles will be supplemented by readings on the basic probability and statistics concepts relating to the topic. We will use computer simulations and statistical packages to better illustrate the relevant theoretical concepts.
The class will differ from traditional math classes in organization as well as in content: The class meetings will emphasize group discussions, rather than the more traditional lecture format. Students will keep journals to record their thoughts and questions, along with their assignments. There will be a major final project in place of a final exam.
The class meets every during the 10A period (Tuesday and Thursday 10:00 to 11:40) in 13 Silsby. The x-hour will meet every week at 3:00 P.M. in 102 Bradley and will be used for discussion of material in the text, questions about homework, use of the computer, or anything else relating to the course.
We want to enable everyone to be engaged in discussions while at the same time preserving the unity of the course. From time to time, we will break into discussion groups of 3 people.
Every member of each group is expected to take part in the discussion and to make sure that everyone is involved: that everyone is being heard, everyone is listening, that the discussion is not dominated by one person, that everyone understands what is going on, and that the group sticks to the subject.
After a suitable time, we will ask for reports to the entire class. These will not be formal reports. Rather, we will hold a summary discussion between the teachers and reporters from the individual groups.
The required texts for the course are Freedman, Pisani, Purves, and Adhikari, Statistics, 2nd edition and Data Desk by Velleman. They are available at the Dartmouth Bookstore.
Each participant should keep a journal for the course. This journal will include:
A good journal should answer all the questions asked, with evidence that some time has been spent thinking about the questions before answering them. In addition, there should be evidence of original thought: evidence that you have spent some time thinking about things that you weren't specifically asked about. This might take the form of: finding and commenting on news articles about topics relevant to the course; asking us challenging questions; making connections between what went on in class and experiences in your own life; going to a casino and winning a lot of money.
In writing in your journal, exposition is important. If you are presenting the answer to a question, explain what the question is. If you are giving an argument, explain what the point is before you launch into it. What you should aim for is something that could communicate to a friend or a colleague a coherent idea of what you have been thinking and doing in the course.
You are encouraged to cooperate with each other in working on anything in the course, but what you put in your journal should be you. If it is something that has emerged from work with other people, write down who you have worked with. Ideas that come from other people should be given proper attribution. If you have referred to sources other than the texts for the course, cite them.
Your journal should be kept on loose leaf paper. Journals will be collected periodically to be read and commented on. If they are on loose leaf paper, you can hand in those parts which have not yet been read, and continue to work on further entries. Pages should be numbered consecutively and except when otherwise instructed, you should hand in only those pages which have not previously been read. Write your name on each page, and, in the upper right hand corner of the first page you hand in each time, list the pages you have handed in (e.g. [7,12] on page 7 will indicate that you have handed in 6 pages numbered seven to twelve).
Journals will be collected and read as follows:
Thursday 6 October Thursday 20 October Thursday 3 November Thursday 17 November Tuesday 29 November
Homework To supplement the discussion in class and assignments to be written about in your journals, we will assign readings from FPPA, together with accompanying homework. When you write the solutions to these homework problems, you should keep them separate from your journals. Homework assignments will be assigned at each class meeting and should be handed in at the next class meeting.
We will not have a final exam for the course, but in its place, you will undertake a major project. The major project may be a paper investigating more deeply some topic we touch on lightly in class. Alternatively, you could design and carry out your own study. Or you might choose to do a computer-based project. To give you some ideas, a list of possible projects will be circulated. However, you are also encouraged to come up with your own ideas for projects.
At the end of the course we will hold a Chance Fair, where you will have a chance to present your project to the class as a whole, and to demonstrate your mastery of applied probability by playing various games of chance. The Fair will be held from during the final examination time assigned by the registrar.
Enjoy CHANCE!
Read the New York Times article ``When is a coincidence too bad to be true?" by Gina Kolata and answer the following questions.
More people died in airline accidents during the first half of 1994 than in the same period of any other year in the last decade except for the record year of 1985, according to the Flight International Airline Safety Review published this week.
What do you make of that?
Read the article `How numbers can trick you'. See how many examples of these six dealy sins of statistical misrepresentation you can provide.
Read the article DNA fingerprinting; it's a case of probabilities written by Richard Saltus for the Boston Globe and discuss the following questions.
One way to avoid most of the nonsense associated with DNA fingerprinting would be to collect DNA fingerprints of everyone in the country. Then instead of speculating about whether certain genetic markers are independent within subpopulations, and all that hogwash, we can just check a DNA sample against everyone in the population.
Say you're prosecuting a case where there is a DNA match. It occurs to you that you could ask your experts to testify, not that there is only a one-in-a-billion chance of this match, but rather that the chance is `real small', and while opinions differ about exactly how small that might be, nobody will contest that the chance is bigger than one in a hundred.
Read, on Mosaic, DNA Typing: Statistical Basis for Interpretation Chapter 3 from the report of the National Reseach Council.
Divide up into groups and think about information you would like to find out about students in the class. Your objective is to generate 5 good questions that contain at least one question asking for a yes/no answer, one asking for qualitative information (word or phrase) and one asking for numerical information. Each group should write the 5 questions on a card provided.
We will select "the best" of these questions and add a few of our own to end up with a questionnaire of about 25 questions. On Tuesday we will ask you to answer this questionnaire. We will then provide you with the data to explore using Data Desk.
The journals are to be handed in, for the first time, Thursday October 6. Several students have asked about what should be in the journals. Here is what we said about this in the course description:
Each participant should keep a journal for the course. This journal will include:
A good journal should answer all the questions asked, with evidence that some time has been spent thinking about the questions before answering them. In addition, there should be evidence of original thought: evidence that you have spent some time thinking about things that you weren't specifically asked about. This might take the form of: finding and commenting on news articles about topics relevant to the course; asking us challenging questions; making connections between what went on in class and experiences in your own life; going to a casino and winning a lot of money.
In writing in your journal, exposition is important. If you are presenting the answer to a question, explain what the question is. If you are giving an argument, explain what the point is before you launch into it. What you should aim for is something that could communicate to a friend or a colleague a coherent idea of what you have been thinking and doing in the course.
Read Chapters 3 and 4 of FPPA. Do the following review exercises: for Chapter 3, Page 47: 1, 9, 10 and for Chapter 4, Page 70: 1, 2, 4
Read the article in the September 22 Valley News on Bronson v. Hitchcock Clinic
No new assignment from FPPA, but be sure that all the first three homework assignments from FPPA have been handed in by Thursday's class.
We will mail you the results of the questionnaire. Explore this data using Data Desk. Record any interesting findings in your journal. You can get additional help with Data Desk in the x-hour on Wednesday. Hand your complete journal in on Thursday.
Read the materials on weather prediction.
Read Chapter 8 in FPPA. Do review exercises on page 129 problems 1, 2, 4, 9
How has your faith in the reliability of weather forecasting been effected by Harold Brooks discussion of weather predicting.
You are discussing the health risks of cigarette smoking with Doris Puffer, a married, slender, 52 year old post-menopausal female patient of yours who is a current smoker. She has had three children, and has no exciting medical problems in the past. Both of her parents lived to be over 80 years old without serious medical trouble. A recent cholesterol level was in the third quintile for her age, and she ``exercises regularly" (by doing her own shopping, it turns out). Under your care, she has received regular Pap smears and blood pressure checks, all of which have been normal. However, she has refused your advice, and has not taken menopausal estrogens.
You estimate that the 10-year risk of lung cancer in a patient like her who does not smoke is about 0.1%, and you guess that the 10-year risk of lung cancer for her (as a current smoker) is about 1%. You estimate that the 10-year risk of coronary heart disease (CHD) in a non-smoker otherwise similar to her is about 8% and you think that her 10-year CHD risk is around 20%.
Mrs. Puffer reminds you that cigarette smoking is protective against endometrial cancer, a problem a friend of hers was just hospitalized for. Since Mrs. Puffer is slender and not taking estrogens, you estimate the 10-year endometrial cancer risk of a non-smoking women otherwise similar to her is about 1%, but she - as a smoker - has a risk of about 0.5%.
Read Chapter 3 Risks, Risk Factors, and their measurements from Dr. Baron's notes.
Comment in your journal on how you might be able to use Dr. Baron's discussion of evaluating health risks in your own decisions relating to health risks.
When you have a choice between two brands of the same food is it reasonable to expect that the more expensive brand tastes better? Discuss in your groups how you would design an experiment to test if this is the case for different brands of chocolate chip cookies.
Read Chapters 9 and 10 in FPPA. Do the following review exercises from Chapter 10 on page 167: # 2, 4, 6, 8
Do a correlation analysis of the data relating to your own cookie rankings using Data Desk and include the results and your conclusions in your journal.
Read Chapters 14 and 15 in FPPA. Do review exercises in Chapter 15 on page 243 problems 1, 2, 7, 8.
Record in your journal the results of the class experiments and your conclusions from these experiments.
Carry out the following ESP experiments within your group. Choose one member of your group to be a ``sender". The sender will have four cards each of a different suit. He or she will shuffle these four cards and then look at the top card being careful not to let anyone else see it. The sender will then concentrate on the suit of this card: hearts, diamonds, clubs, or spades. Each member of the group will try to ``receive" the message and will write down what he or she thinks the suit chosen was.
Repeat the shuffling and sending ten times with the sender writing down each time what the correct suit was. At the end of the ten trials the sender should reveal the choices and each person will determine his or her score as the number of correct answers. The sender should then make a list of these scores and record these on the black board.
Read Chapters 16 and 17 in FPPA. Do the following review exercises from these chapters. On page 259: # 6,7 and on page 278 #1,2,6,10.
Read the current issue of Chance News and record in your journal your answers to the discussion questions for an article of your choice.
Read Chapter 19 in FPPA and do review exercises on page 323 problems # 1, 3, 4, 5
Hand in next Tuesday (November 1) a brief description of the project that you propose to do for the course. You can find project suggestions on Mosaic. (Click on ``Chance at Dartmouth, 1994 Fall".) Suggestions can also be found from ``Chance Magazine" that we have put on reserve in Kresge Library. You can also find ideas from recent issues of ``Chance News". You can search on these for articles on a given topic.
Read the article ``Induced abortion hikes breast cancer risk, study says" from the October 27 Los Angeles Times.
Read Chapter 20 in FPPA and do review exercises on page 340 # 2, 4, 7,11
Record in your journals your answers to the following questions.
What odds would you give that Cuomo wins in New York?
Here is how New York Times describes how the survey was conducted:
HOW THE SURVEY WAS CONDUCTED
The latest New York Times/ WCBS-TV News Poll is based on telephone interviews conducted Wednesday through Saturday with 1,411 adults throughout New York State. Of these, 1,012 said they were registered to vote.
The random sample of telephone numbers was provided by Survey Sampling of Fairfield, Conn.
Within each household one adult was chosen at random to be the respondent. The results have been weighted to account for household size and number of telephone lines, as well as for variations relating to region, race, sex, age and education.
Results about respondents' intended votes have also been weighted to reflect the statewide distribution of ballots in recent elections from New York City (31 percent), its suburbs (24 percent), and the rest of the state (45 percent).
According to statistical theory, in 19 out of 20 cases the results based on such samples will differ by no more than 3 percentage points in either direction from a survey of all registered voters in New York State.
Do you think the average New York Times reader will understand what all this means? Do you? See if you can explain what they are saying in a way that your Uncle George might understand.
Read the article ``Jack or Jill" from Lancet March 20, 1993.
Suppose that a certain society values sons more than daughters. In this society, a couple will continue bearing children until they produce a son, at which point they will retire from the child- bearing business.
Suppose we assume that 5% of the people are drug-users. A test is 95% accurate, which we'll say means that if a person is a user, the result is positive 95% of the time; and if she or he isn't, it's negative 95% of the time. A randomly chosen person tests positive. Is the individual highly likely to be a drug-user?
Marilyn's answer was:
Given your conditions, once the person has tested positive, you may as well flip a coin to determine whether she or he is a drug-user. The chances are only 50-50.
How can Marilyn's answer be correct?
The standard test for the HIV virus is the Elias test that tests for the presence of HIV antibodies. It is estimated that this test has a 99.8% sensitivity and a 99.8% specificity. 99.8% specificity means that, in a large scale screening test, for every 1000 people tested who do not have the virus we can expect 998 people to have a negative test and 2 to have a false positive test. 99.8% sensitivity means that for every 1000 people tested who have the virus we can expect 998 to test positive and 2 to have a false negative test.
The Times article remarks that it is estimated that about 2 in every 1000 college students have the HIV virus. Assume that a large group of randomly chosen college students, say 100,000, are tested by the Elias test. If a student tests positive, what is the chance this student has the HIV virus? What would this probability be for a population at high risk where 5% of the population has the HIV virus?
If a person tests positive on an Elias test, then two more Elias tests are carried out. If either is positive then one more confirmatory test, called the Western blot test, is carried out. If this is positive the person is assumed to have the HIV virus. In calculating the probability that a person who tests positive on the set of four tests has the disease, is it reasonable to assume that these four tests are independent chance experiments?
Read Chapter 21 in FPPA. Do review exercises on page 357 problems 4,5, 8,11.
Read the article New York Times ``Stark data on missing women in China". What do you think is the best explanation for the missing women in China? Record also your own answers to questions 3 and 4 in the Jack and Jill disscussion.
Read Chapter 28 in FPPA. Do review exercises on page 490 # 1, 3, 6, 7
Read Chapter 26 in FPPA. Do review exercises on page 449 # 1,2,5,6
Here is the infamous Monte Hall problem, as it appeared in the Parade Magazine of 9 September 1990:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He then says to you, ``Do you want to pick door number 2?'' Is it to your advantage to switch your choice?
What assumptions does your answer depend on?
Here is another paradox having to do with switching from one choice to another.
Two envelopes each contain an IOU for a specified amount of gold. One envelope is given to Ali and the other to Baba and they are told that the IOU in one envelope is worth twice as much as the other. However, neither knows who has the larger prize. Before anyone has opened their envelope, Ali is asked if she would like to trade her envelope with Baba. She reasons as follows. With 50 percent probability Baba's envelope contains half as much as mine and with 50 percent probability it contains twice as much. Hence, its expected value is
1/2 (1/2) + 1/2 (2) = 1.25,
which is 25 percent greater than what I already have and so yes, it would be good to switch. Of course, Baba is presented with the same opportunity and reasons in the same way to conclude that he too would like to switch. So they switch and each thinks that his/her net worth just went up by 25 percent. Of course, since neither has yet opened any envelope, this process can be repeated and so again they switch. Now they are back with their original envelopes and yet they think that their fortune has increased 25 percent twice. They could continue this process ad infinitum and watch their expected worth zoom off to infinity.
John Finn has suggested starting with a simpler problem. Suppose Ali and Baba know that I am going to give them either an envelope with 5$ or one with 10$ and I am going to toss a coin to decide which to give to Ali, and then give the other to Baba.
Here is another paradox closely related to the previous one. Ali and Baba are again given two envelopes with an IOU for a specified amount of gold in each envelope. This time they know nothing about the amounts other than that they are non-negative numbers. After opening her envelope, Ali is offered the chance to switch her envelope with that of Baba. Can Ali find a strategy for deciding whether to switch which will make her chance of getting the envelope with the larger of the two numbers greater than one half? At first blush, this would appear to be impossible.
But consider the following strategy: Ali does an auxiliary experiment of choosing a number U by some chance device that makes all non-negative numbers possible. For example, choose U to be the number of tosses of a fair coin until the first head turns up. If the IOU given Ali is greater than U she keeps this envelope, if it is less than U she switches to the other envelope.
We have given you a page of sequences of H's and M's (hits and misses). They represent three different kinds of behavior
We are not giving any new homework from FTTP in order to permit you to finish your projects before going off on your Thanksgiving vacation. This coming Tuesday we will explain and practice the games of chance that you will need to know to succeed at the Chance Fair which will be the following Tuesday when you return.
Chance at Dartmouth Fall 1994
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