Freshman Seminar: Probability in Science, Sports and Life (Fall 2003)

Instructor: David Aldous

This is a "debriefing" version edited after the course finished.

Courses in mathematical probability teach you to do certain mathematical calculations, but are often far removed from broader questions about the the role of randomness in the ``real world" of science or of human affairs. This Freshman Seminar course is intended as an introduction to such questions using minimal mathematics. I will talk about topics such as

Where does randomness come from? -- the fine-grain principle.
Coincidence and Noise: the background randomness of Life
What math says about the Stock Market
Game theory: from rock-paper-scissors to the Battle of the Sexes
Chance in sports: why this year's top teams will probably do less well next year
The psychology of risk
Card shuffling and computer sorting algorithms
The tipping point: when small changes can make a big difference
Chance models and the phylogeny of species
Coding: for secrecy or for efficient communication.

Text

There is no text, but I suggest students look at one or both of the two following books.

Bernstein, Peter L. Against the Gods: The Remarkable Story of Risk. Wiley, 1996.
Haigh, John, Taking Chances Oxford University Press, 1999.

The first is primarily historical and verbal. The second gives simple mathematical study of various real-world questions about games of chance and sports. I will talk about Chapter 16 of Bernstein and Chapter 6 of Haigh.

Student responsibilities

The student's responsibility is to carry out a small project, write it up and present it in class (15 minutes). In choosing a project you are encouraged to pursue your own interests. Here are some suggestions for types of project, from least to most imaginative.

1. A topic report. Choose a topic from one the the books above, or below, or a book of your own choice. Best to seek one interesting idea to discuss; don't try to summarize a whole book.

2. A project involving data-collection or experiment to confirm or refute some general idea in probability. We'll see examples in the first two classes. Suggestions below.

Books (sources for topic reports)

Popular Science

Peterson, Ivars. The Jungles of Randomness. Wiley, 1998.
Bennett, Deborah J. Randomness . Harvard University Press, 1999.
Everitt, Brian S. Chance Rules . Copernicus, 1999.

Sports, Games of Chance

Albert, Jim and Bennett, Jay. Curve Ball: Baseball, Statistics and the Role of Chance in the Game Copernicus Books, 2001.
Packel, Edward W. The Mathematics of Games and Gambling Mathematical Assoc. America, 1981.

Evolution

Dawkins, Richard. Climbing Mount Improbable. Norton, 1997.

Stock Market

Taleb, Nassim Nicholas Fooled by Randomness: The Hidden Role of Chance in the Markets and in Life . Texere, 2001.
Shiller, Robert J. Irrational Exuberance. Broadway 2001.
Malkiel, Burton Gordon A Random Walk Down Wall Street. Norton, 1999.

Everyday Life

Ropeik, David and Gray, George. Risk (a practical guide for deciding what's really safe and what's really dangerous in the world around you) Houghton Mifflin 2002.

More mathematical books

Morris, S. Brent. Magic Tricks, Card Shuffling and Dynamic Computer memories. Mathematical Association of America 1998.

..... and my particular favorite

Shiller, Robert J. The New Financial Order. Princeton University Press 2003.

Some possible projects

Talk with me before starting any project.

Little real-world experiments

Biased pennies. If you spin a penny fast on a flat surface, it eventually stops "heads" or "tails", but (surprise!) it's more often tails than heads; and the chance (p, say) of tails varies from one penny to another. What are the largest and smallest values of p you can find?
Dropping thumbtacks. OK, this is pretty boring, but a natural example. Drop 100 thumbtacks on a surface, and count how many are "point up". Repeat. Theory says the two counts (maybe 68 and 62) are likely to be within 10 of each other. Does this prediction work?

Looking for data (library or Internet)

Risky plays in sport. Theory says: if you're winning then play conservatively (that is, non-risky), but if you're losing then you need to take risks. Look for data to show if this really happens. For instance, in football one can use interceptions as an indicator of risky play. Theory says that a more-than-usual number of interceptions will be thrown in the last quarter by the team that's behind. Football fans say "of course this happens", but can you get actual data? What about soccer? Suppose than, 5 minutes before the end, one team is ahead by 1 goal. Theory says (can you think why?) that during the final 5 minutes, the "ahead" team is more likely to score than the "behind" team. Can you find data?
Separating skill and luck in sports. If all teams in a league were equally good, then we can think of the winner of each game as being random; at season end the teams would have different winning percentages, just by chance, and theory predicts the s.d. (size of spread) of these winning percentages. In fact, some teams are better than others, and the actual s.d. is larger. So by comparing actual and theoretical s.d. one can measure relative effects of skill and luck. Within this topic there are many possible projects
- Comparing different sports -- does luck play a larger role in football or baseball?
- Sometimes a team may seem to have a slump or a surge in the final month, say. Does this in fact happen more often than "just by chance" would predict?
- Hot streaks. Individual athletes often believe in "hot streaks" where they are unusually successful (basketball shots, golf, baseball pitcher's no-hit innings, etc). Do these actually happen more often than "just by chance" would predict?
- The regression effect predicts that, if you look at (say) baseball teams with an overall winning record one season, then the majority of them will have worse records next season than this season. Does this actually happen?
Zipf's law. This concerns data-sets which show relative frequencies of different alternatives. For instance, if you ask a sample of people for their favorite flavor of ice cream, you might get data like (chocolate 42%; vanilla 26%; strawberry 15%; etc). Or if you look at the Statistics Dept web site data on files downloaded each month, you see (December 2000) that the most-often-downloaded files have frequencies (1.91%, 1.58%, 1.57%, 1.46%, 1.45%,.....). Or take a long book and find the frequencies of different words ("the" 3.1%, "a" 1.7%, .....). Zipf's law predicts that if f_i is the i'th largest frequency then (at least for i in the middle of the list) there should be a power law approximation f_i = c i^{-a} for some c and a. Find some actual data and see if this is really true.
Risks in everyday life It's often said that people are usually inaccurate in assessing relative risks of unlikely events -- being killed in a auto accident, in a plane crash, being struck by lightning, winning the lottery, etc. Find data on actual chances, and give the class a quiz to see how well they can guess which are more or less likely.

More complex projects (for groups of students)

Media scares caused by coincidences. One of the general themes of probability is "coincidences happen more often than you would imagine". In particular, statisticians believe that when some kind of dramatic event (plane crash, train crash, school shooting, fatal fire, kidnapping) happens twice within a short time, it attracts excessive media attention, even if there has been no change in the actual frequency of such events. So the project is: look through (say) Time magazine for the year 2000 to find articles like this; then find actual data about frequencies of these events for (say) the last ten years, to see whether anything unusual really was happening in 2000.
Insurance deductibles. By increasing the "deductible" (the amount of a loss that you must pay yourself) on auto or fire or theft insurance, you can reduce the cost of your insurance policy. Theory suggests that on average this is a "gamble" which is in your favor (because the insurance company must change enough to cover its payouts, plus expenses and profit). Can you find data?