MALS 178 - CHANCE, Fall 1996

Do you believe in streaks?
What do you mean by streaks?
How would you recognize streaky behavior?

More Discussion:

What would it take to believe in streaks?
What would it take to not believe in them?

Chapter 26, Review ex. 2, 5.
Chapter 28, Review ex. 2, 3.
Chapter 29, Review ex. 1, 2, 4.

Journal Assignment:

Read the Discover article "Decisions, Decisions" and comment on it.

1. ESP Experiment

• What is a null hypothesis? What is an alternative hypothesis?
• What is a p-value? What is meant by statistical significance?
• How much evidence would it take to convince you that ESP exists?

2. More on Streaks

What is a streak? How would you recognize streaky behavior?
Computer simulations.

Journal Assignment:

Read the article on ESP "They Laughed at Galileo Too", NYT, August 11, 1996 and comment on it in your journal.

Part I-- Guest Speaker.

Charlie Lewis from ETS will talk about security problems on SAT exams.

Part II-- Chi Squared Test

• What are they for?
• How do you use them?
• Activity on contingency tables.

	    Left Handed 		Right Handed		Total
	________________________________________________________
	Men

	Women

	Total

No more homeworks from the text! Time to really focus on your projects. We will collect your journals at the end of the term instead of next week.

Discussion Questions:

a) Is it irresponsible for a woman to refuse prenatal testing?
b) Is she morally responsible to access all information or only that which is inexpensive'? Or only that which poses no lisk to the fetus?

a) How is "increased risk" for fetal anomaly like or different from "increased risk" for having some inherited trait such as breast cancer, depression, pattern baldness.
b) What is the relevance of variation determined by diagnostic tests'? What are the advantages and disadvantages of gaining information about the fetus'?

Part II -- Card Shuffling

I ) Read the NYT article, 1/9/90 , In Shuffling Cards, 7 Is Winning Number.

2) ShufNing activity.

A boring game of solitaire, which I call Yin/Yang, shows that 7 ordinary riffle shuffles, followed by a cut, of a 52-card deck are not enough to make every permutation equally likely.

Hearts and Clubs are called the Yin suits, and Diamonds and Spades are called the Yang suits. We shuffle the deck of cards 7 times, then cut it, and then start removing and revealing each card from the top of the deck, making a new pile of them face-up (so if this were all we did, we'd just have the deck unchanged after going through it once, except that the deck would be lying face-up on the table).

We start the pile for each suit when we discover its ace, and add cards of the same suit to each of these 4 piles, according to the rule that we must add the cards of each suit in order.

Thus a single pass through the deck is not going to accomplish much in the way of completing the 4 piles, so having made this pass, we turn the remaining deck back over, and make another pass.

We continue this until we complete either the two Yin piles (hearts & clubs), or the two Yang piles (diamonds & spades). If the Yin piles get completed first, we call the game a win; it's a loss if the Yang piles get completed first.

If the deck has been thoroughly permuted (by having put the cards through a clothes dryer, say), then the Yin and Yangs will be equally likely to be first to get completed. Thus our expected proportion of wins will be 1/2.

Discuss your reactions to the video, and respond to some of the discussion questions in today's handout.

Notes on second journal assignments.

Everyone seemed to appreciate Tami Burh's talk and learned a lot from it. You had mixed feelings about Steven Jay Gould's talk. Some felt he was unnecessarily rude and egotistical. Most thought his ideas were interesting and it was worthwhile hearing his talk.

You raised some interesting questions about coincidences and you all had had plenty of incidents that you think of as coincidences. Much has been written about whether more "coincidences" have happened they should by chance alone and we are not going to settle this issue easily. However, it is important to bear in mind our earlier discussion about the different between the probability of a specific event and the probability of this event or a similar event sometime during a longer period, for example, during your lifetime. For example, one of you mentioned that you and your mother both had dreams the same night that involved animals with weird colors. This, by itself, would seem very unlikely but perhaps to have in a lifetime to a dream very similar in some weird way to someone else on the same night is not so strange.

It is possible to make models to show more concretely how the an event on a particular day can have very small probability but "once in a lifetime" not so small. The famous psychiatrist Jung was one of those who believed that coincidences occulTed more often than they could be expected to by chance. He mentioned once that he was struck by hearing references to fish 6 times during a 24 hour period. We can make a model to see how unlikely this would be in a lifetime. We have to specify how often on average you hear a story involving fish. The time between fish stories is random . It might be 2 hours or 2 weeks. Let's assume the average time between fish stories is one week. Then you can write a program to simulate this process where the time between stories is random with average time between stories one week. Then run this program for the equivalent of 40 years and record whether or not there is a 24 hour period that contains 6 or more fish stories. Finally repeat this many times to estimate the chance that 6 or more fish stories in a single day occurs during a 40 year period.

Amy asked if you could make a map of acquaintances in a small town and then try to see who knew who. This kind of a problem is a favorite of mathematicians called "graph theorists" They draw a line between each two people who are know each other. Then there is a "path" between two people a and b if you can go from a to b following these lines. It is natural to ask for the smallest number k such that there is a path between x and y of length less than k for every pair of people x and y in the town. If it is a small town you might guess that there is a path of length at most 3 between any two people. In the movie "Six Degrees of Separation" it was suggested that everyone in the world is connected by a path of length at most 6. Just to show you how berserk mathematicians can get over such problems, here are some remarks from my friend David Gliffeath's home page.

Apparently, sometime within the past few years, MTV talk show host Jon Stewart devised a parlor game in which contestants are to link any movie star to Kevin Bacon by a chain of films that share performers. Thus we imagine actors as vertices in a large network, with edges between any two who have been in a movie together. The goal is to find the path of minimal length connecting x to Kevin Bacon, that length then being the Bacon number, which we denote here as B(x). Quite a cult has grown up around this pastime, as witnessed by more than a dozen web pages now devoted to the game. My favorite link to Bacon fanatics is The Center of the Hollywood Universe.

Now Brett Tjaden and Glenn Wasson at the University of Virginia have automated the calculation of B(x) at their Web site, The Oracle of Bacon. Their program, which makes use of the marvelous Intemet Movie Database (IMDB), will compute the Bacon number of any performer you care to specify within a few seconds. Hard-core cultists seem threatened by the power of the Oracle, but the rest of us welcome this automation of what can be an arduous evaluation. For instance, B(Bara, Theda) = 3, and in fact the Oracle confirmed an outstanding conjecture that for any x from the United States, either B(x) is at most 4 or B(x) is infinite.

Mathematicians have had their own version of this story for many years, centered around the Hungarian number theorist and combinatorist Paul Erdos, where the links are formed by joint authorship of research papers.

Sue wondered about the coincidence of getting an even dollar amount for the total charge of your groceries. This is an interesting problem. You are adding up several chance events that represent the individual costs for your items. You could make some assumption about the probability distribution for each individual item. For example, the probability that it is 17 cents etc. It is reasonable to assume that your total bill will be more than a dollar. I think you could then show that the probabilities that the last two digits are 00,01,02,..,99 are approximately the same so you could conclude that the chance that your total bill is exactly x dollars for some x is about 1 in one hundred. More interesting is the probability that the leading digit of your bill is 1,2,3,...,9. (If you bill is $14.89 then the leading digit is l). It has been found that the distribution of the leading digits in nature are not equally likely but rather follow a logarithmic distribution. Here is an account of how this tact has been used to find people who cheat on their income tax.

Mark Negrini, who teaches accounting at St. Mary's University in Halilax, wrote his PhD thesis on: "The detection of income evasion through an analysis of digital distributions". He has persuaded business and government people to use Benford's law to test suspicious financial records such as bookkeeping checks and tax returns. Benford's law states that the distribution of the leading digit in data sets is typically not equi-distlibuted hut rather given by the distribution p(k) = log(k+l) - log(k) for k = 1,2,...,9. (The leading digit of .0034 is 3, of 243 is 2 etc.). This gives the probabilities .301, .176, .125, .097, .079, .067, .058, .052, .046 for the chance that 1,2,3,4,5,6,7,8, or 9 will be the leading digit.

Numerous explanations for this have been given but perhaps the most persuasive is that Benford's distribution is the unique distribution for the leading digits that is not changed by a change of units, i.e. multiplying the data by a constant c. Negrini's idea is that, if we are honest, the numbers in our tax returns and on our checks should satisfy Benford's law and if they do not there may he some skullduggery.

The article states that "Mr. Negrini has also lent his expertise to federal and state tax authorities, otficials in Denmark and the Netherlands and to several companies. He has even put President Clinton's tax returns to the Benford's Law test. When he analyzed the president's returns for the past 13 years he found that 'the returns by Clinton follow Benford's Law quite closely"'.

Your explanations to your Uncle George as to what the "margin of er or means" indicates that there is still some confusion about what it mean.s The margin of elTor includes only the sampling error and not the other kinds of er ors that Tami talked about such as elTors caused by non-response. You will find that it is typically about l/sqr(n) cor esponding to this estimate of two standard er ors that we discussed. For example, for a sample of 200, l/sqr(200) = .0707 so this would be reported as a 7 percent margin of error. You have also be careful not to tell Uncle George that the error will be no more than 7 percent since even George will realize that it is possible for the poll to really screw up. A couple of you described it in terms of what would happen when you toss a coin, say 100 times and look at the proportion of heads that comes up. We can say with 95% confidence that this number will be between 40 and 60. Thus if our estimate the probability for heads coming is the proportion of heads in the sample we can be quite certain we are not off by more than 10% for the true probability of heads. I think George would be best served by just giving him the "box" that is included at the end of the New York Times poll reports.

How the Poll Was Conducted

The latest New York Times/ CBS News Poll is based on telephone interviews conducted Feb. 22 to 24 with 1,223 adults throughout the United States.

The sample of telephone exchanges called was randomly selected by a computer from a complete list of active residential exchanges in the country. The list of more than 36,000 residential exchanges is maintained by Marketing Systems Group of Philadelphia.

Within each exchange, random digits were added to folm a complete telephone number, thus permitting access to both listed and unlisted numbers. Within each household, one adult was designated by a random procedure to be the respondent for the survey.

The results have been weighted to take account of household size and number of telephone lines into the residence and to adjust for variations in the sample relating to geographic region, race, sex, age, and education.

In theory, in 19 cases out of 20 the results based on such samples will differ by no more than three percentage points in either direction from what would have been obtained by seeking out all American adults.

For smaller subgroups the potential sampling error is larger. For example, it is plus or minus five percentage points for those who say they are likely to vote in a Republican primary or caucus this year.

In addition to sampling error, the practical difficulties of conducting any survey of public opinion may introduce other sources of error into the poll. Variations in question wording or the order of questions, for instance, can lead to somewhat different results.

So you see, as usual, your journals raised as many interesting questions as they solved which is a sign of good journals

Part I -- Card Shuffling

1) Read the NYT article, 1/9/90, "In Shuffling Cards, 7 Is Winning Number."

2) Shuffling activity.

A game of solitaire, which we call Yin/Yang, shows that 7 ordinary riffle shuffles, followed by a cut, of a 52-card deck are not enough to make every permutation equally likely.

Hearts and Clubs are called the Yin suits, and Diamonds and Spades are called the Yang suits. We shuffle the deck of cards 7 times, then cut it, and then start removing and revealing each card from the top of the deck, making a new pile of them face-up (so if this were all we did, we'd just have the deck unchanged after going through it once, except that the deck would be lying face-up on the table).

We start the pile for each suit when we discover its ace, and add cards of the same suit to each of these 4 piles, according to the rule that we must add the cards of each suit in order.

Thus a single pass through the deck is not going to accomplish much in the way of completing the 4 piles, so having made this pass, we turn the remaining deck back over, and make another pass.

We continue this until we complete either the two Yin piles (hearts & clubs), or the two Yang piles (diamonds & spades). If the Yin piles get completed first, we call the game a win; it's a loss if the Yang piles get completed first.

If the deck has been thoroughly permuted (by having put the cards through a clothes dryer, say), then the Yin and Yangs will be equally likely to be first to get completed. Thus our expected proportion of wins will be 1/2.

Part II -- Review of Tests in JMP

• Single variable
  • z-test (known population standard deviation)
  • t-test (unknown population statistics, small samples)
• More than one variable
  • correlation and regression
  • comparing means
  • Chi-square test

Part III- Games of Chance