Math 408 (Prof. Hildebrand), Spring 2008
Honors Credit Projects for James Scholars

General Guidelines

Eligibility/application

The projects are open to students in the James Scholar program who have submitted an appropriate online application, called Honors Credit Learning Agreement (HCLA), earlier in the semester. (The deadline for HCLA applications has long passed, so if you haven't applied yet, it's too late to do it now.)

General guidelines

According to the general guidelines of James Scholar program, the work required to earn James Scholar Honors Credit must be substantial (it should take at least 15 hours to complete); it must go well beyond homework type assignments; and it must be related to the subject of the course (which in our case is statistics/probability, in a rather broad sense). A list of suitable projects is given below.

What you need to do

Deadlines

Other notes

The projects

Below is the list of projects. Take a look at the descriptions and links provided, and pick the one that suits you best. I have selected these projects since they are at a level and length appropriate for a James Scholar project, and I have had students in years past successfully complete the project as part of a James Scholar or EC assignment. I might consider other project ideas, but you should first consider the projects below. Project ideas that might sound great at first glance may turn out to be either way above the scope and level of a James Scholar project, or too little or too shallow for such a project. The projects below are all tried and proven, and at just the right level. If you have questions about any of these, talk to me!

Lotteries in the real world

In its simplest form, a standard lottery is easy to analyze, Also, if the prizes to be won are fixed and not shared (e.g., if everyone who gets all 6 numbers right in a 6 number lottery wins a million dollars), then the expected win at a single drawing, and hence the fair price of a lottery ticket, can easily be computed. However, things get more complicated if the prizes may have to be shared, or are variable, as in the case of a "jackpot" that accumulates if there are no winners at a given drawing. Occasionally, such jackpots reach almost astronomical numbers and then make headlines and cause a big rush at convenience stores by people wanting to buy tickets. One can ask if it is advantageous to buy lottery tickets during such periods of high jackpots. (The answer is not clear: On the one hand, a high jackpot means that the potential winnings are larger, but on the other hand, a high jackpot attracts a greater number of lottery players, so there is also a greater chance of having to share a prize.) The above article analyzes this situation for the Powerball lottery.

Probability and the Law: The People vs. Collins case

This project deals with a famous court case from the 1960s in which a suspect was convicted based on flimsy probabilistic reasoning, and the conviction was overturned in appeal, after the probabilistic argument was shown to be seriously flawed. This project is less mathematical than the others, and involves more old-fashioned literature search, legwork (e.g., a trip to the law library), and digging out and reading legal documents. (Since the case occurred nearly 40 years ago, well before the internet age, most of the relevant literature is only available in print form.)

Doing such research in itself can be fun, interesting, and educating (have you even seen an official court opinion?), especially for those who have an interest in public policy, politics, and the legal system. This is a case where you should definitely avail yourself of the vast resources of the University Library. A good starting point for a literature search is the Reference Library, located at the second floor of the Main Library. Don't be shy asking one of the reference librarians there; they are there to help!

Below are some sources that I am familiar with. You should get all of these; if you find others, great, but make sure you cite all sources you use, and copies of the source documents to your paper. I have intentionally been vague in the references, so you'll have to find exact citations, and library locations, etc., yourself (or, better, ask a librarian to help).

Card shuffling

How many times do you have to shuffle a standard deck of card in order to mix them reasonably well? The correct answer is seven, according to Persi Diaconis, a well-known Professor of Statistics and Mathematics at Stanford and former professional magician. Diaconis' work, which appeared in the early 1990s, made headlines at the time, and was written up in the New York Times and other main stream media. The above article by Brad Mann gives an exposition of Diaconis' work and the underlying mathematics. Compared to the other projects, this project involves deeper (and probably more interesting) mathematics (but still well within the level of a basic undergraduate probability course). On the other hand, it requires little or no literature search, trips to the library, etc., as it can be based almost exclusively on the Mann article. (You might still want to look at the NY Times article referenced in the Mann paper, or other, secondary, sources, if only to get a different, nontechnical, perspective.)

Programming projects

For those who have strong programming skills, I could provide suitable projects. If you are interested, talk to me. (In my Math 461 classes, the programming projects tend to be by far the most popular. This is no surprise, since those classes are populated by students in Engineering and Computer Science for whom programming is second nature. By contrast, in all of my Math 408 classes I have had only one student take on a programming project. Superficial familiarity with a programming language is not enough; you should have a very solid knowledge in at least one of the major programming languages (e.g., C, C++, Java) as well as hands-on experience in writing programs before considering a programming project.)


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