MATH 1000 PROJECT IDEAS (1) Birthday Paradox (TAKEN): This question was briefly mentioned in Chapter 13: How many people must be in a room before there is a greater than 50% chance that some pair of people in the room share a birthday? This is a classically counterintuitive probability puzzle, to which most people provide wildly inaccurate guesses. You could carefully answer this question, explain the solution in your own words, provide some history, and maybe discuss a generalization. In particular, what if you replace pair of people with group of three people? (I m not totally sure how to do it, it might be cool to try or estimate.) (2) Party Problem: How many people must be in a room before there is guaranteed to be a group of three mutual acquaintances or a group of three mutual strangers? Four is not enough, because there could be two pairs of friends where each person has never met the two people in the opposite pair, so any group of three has at least one acquainted pair and one unacquainted pair. Is five enough? Is it ever possible to guarantee at least one such group of three people? This question has a definitive answer with an elegant explanation, and turns out to be a special case of an important result known as Ramsey s Theorem in the mathematical research area of graph theory. You could give the answer, and the proof, and discuss some things related to the history of the problem, generalizations (how about four instead of three?), or whatever else you find. (3) The Monty Hall Problem (TAKEN): This is the game with the three doors, one with a car and two with gag prizes, that we discussed and played in class. It is loosely based on the game show Let s Make a Deal and named after its original host. In addition to recalling the details of the game, and carefully explaining in your own words why switching doors is a beneficial strategy, you could delve into the history of the problem, specifically how it sparked a great deal of controversy when the correct explanation was published in Parade magazine. 1
2 MATH 1000 PROJECT IDEAS (4) Pascal s Wager (TAKEN): This one would likely have a bit less mathematical discussion and a bit more history, etc., depending on your approach. The content, and the underlying flaws in logic (think Cat in the Hat t-shirts), of Pascal s Wager were discussed in Chapter 12. To summarize the assertion: if you follow the Christian religion, and the Christian religion is accurate, then the payoff is infinite, so any positive probability that the Christian religion is accurate, no matter how small, mathematically justifies any cost in following the religion. In addition to including the details we have already discussed in your own words, I think it would be cool to hear more about the history of this line of reasoning, and how it has popped up over the centuries in literature, pop culture, etc. Also, if you had some personal thoughts or opinions on the matter, you could totally share those as well. (5) The Prosecutor s Fallacy: This is something we discussed in a fair amount of detail: the misinterpretation that conditional probabilities are the same when the two events switch roles. For example, the chance you ll test positive for a disease given that you re healthy is different from the chance you re healthy given that you tested positive. While we dove into this pretty well, there is still plenty more to be said and plenty of real-world examples (not silly things made up by me) to explore. In particular, Jordan mentioned in the footnotes that there is a book titled Math On Trial by Corlie Colmez and Leila Schneps, that treats several such examples in detail. I think it would be cool if you could recall the core concept in your own words and discuss some examples, history, et cetera, from that book or any other source you uncover. (6) Simpson s Paradox (TAKEN): You and I flip coins, some on Saturday, some on Sunday. On Saturday, a higher percentage of your flips land heads than mine. The same is true on Sunday. Surely, then, for the weekend as a whole, a higher percentage of your flips landed heads than mine, right? Not necessarily! The key issue is that I never told you that we each flipped the same number of coins each day. This counterintuitive reversal phenomenon has been well-studied, and there are lots of realworld examples to delve into. A careful explanation, with some history and several examples, could make for a nice project.
MATH 1000 PROJECT IDEAS 3 (7) Logistic Population Model: Very early in the semester, when we were discussing linearity and nonlinearity, we had some gentle discussions about how populations can grow over time. The most nuanced picture we drew involved rapid, exponential growth at the beginning, followed by a leveling off based on availability of space, resources, etc. These are the basics of what is known as a logistic model, which is in particular a good model for global human population, and as your final project you could delve into this in more detail. This would involve some gentle discussions relating to calculus (no harder than what we discussed in class during some of those days) that I could guide you through. (8) The Statistical Revolution in Sports (TAKEN): Over the last twenty years or so, the kinds of mathematical logic, probability, expected value calculations, regression, etc., that we have discussed in class, as well as more sophisticated statistical analysis, have been embraced by the world of sports on a remarkably large scale. You may be familiar with the book and movie Moneyball, which chronicled the success of the Oakland Athletics baseball team in using statistical considerations that were novel at the time, but now every professional sports team has a dedicated analytics department. A few small examples: major league baseball teams now value on-base percentage and slugging percentage way more than the more traditionallyrevered RBI and batting average; NBA basketball teams shoot WAY more three-point shots than they used to; NFL teams throw WAY more passes than they used to, all because of mathematical and statistical considerations. Those are just a few scattered thoughts, this a pretty broad topic, you could take any number of approaches here; you could take a very historical angle, you could go heavy on the mathematical content, or anything in between.
4 MATH 1000 PROJECT IDEAS (9) The Law of Large Numbers and the Gambler s Fallacy: We discussed both of these things in class. The law of large numbers says, for example, that if you flip a coin more and more times, then, with greater and greater probability, the proportion of totl flips that are heads will get closer and closer to 50%. However, this does NOT say that if the first 10 flips are heads, that the coin will then favor tails in an effort to balance out the total. This misinterpretation is referred to as the gambler s fallacy. You could take a shot at formulating the law of large numbers a bit more precisely, and maybe discuss some history, psychology, and real-life examples of both of these concepts. (10) The Card Game Set (TAKEN): There is a popular card game called Set which uses a deck consisting of 81 cards varying in four features: number (one, two, or three); symbol (diamond, squiggle, oval); shading (solid, striped, or open); and color (red, green, or purple). A set is any group of three cards that are either ALL the same of ALL different in each of the four categories. For example, one red solid diamond, one red solid squiggle, and one red solid oval would form a set, because the three cards have all the same number, all the same color, all the same shading, and all different shapes. This game gives rise to some interesting mathematical questions. For example, what is the maximum number of cards that can be on the table without having ANY sets? This question is deeply connected with one of our in-class activities, as well as one of the biggest recent results in my research area, which was proven by, among others, Jordan Ellenberg, the author of our book. I would love to see you explain some interesting things about Set in your own words, and we could even play a game together as a class!
MATH 1000 PROJECT IDEAS 5 (11) The Prisoner s Dilemma (TAKEN): Two partners commit a major bank robbery together, get arrested, and are separately interrogated. The district attorney s office does not have sufficient evidence to convict the pair for the robbery, but only for a lesser weapons charge, so the DA hopes to extract a confession, and offers the following deal to each partner: If both partners confess, then each partner receives a slightly reduced 9-year prison sentence for the robbery. If one partner refuses to confess while the other partner confesses, then the confessing partner can testify against the non-confessing partner to avoid prison time, while the non-confessing partner receives the full 10-year prison sentence or the robbery. If neither partner confesses, then the partners each receive the 1-year sentence for the weapons charge. If both partners fully understand all of these potential outcomes, and they act totally rationally in their own self interest, what choices will they make? This might seemed contrived, but the analysis of this type of scenario, and the seeming paradox of decision making that comes with it, have a lot of real-life applications. Research into this question could lead you toward biology, economics, philosophy, who knows! (12) An Infinity of Infinities (TAKEN): Are there more fractions than there are whole numbers? Are there more irrational numbers than there are rational numbers? How can we compare the sizes of two sets if they are both infinite? In the book, Jordan very briefly mentioned this line of questioning, which was made firm by the work of Georg Cantor in the second half of the nineteenth century. Jordan comments that these ideas, once seen as beyond the scope of mathematics, are now standard material for undergraduate math majors, though it does sort of blow their mind. I think I can do him one better and explain these ideas to you in my office in an hour or so, well enough that you can do some research including some history, write things up, and explain them to the class in your own words.