Instructor:	Dr. Jan Pearce		C.P.O.:	1815
Office:	103 Draper 985-3569		Office Hours:	M-F 1:15-2:15 PM,
Home: Email:	986-4057 pearce@berea.edu		Please feel free to drop by other times-- I am in my office a great deal and always happy to help!

INTRODUCTION TO THE MATHEMATICS OF PLAYING GAMES. Ancient game boards and game pieces have been found in nearly every area of the world, indicating that games may have been popular pastimes since the beginning of civilization. Today the abundance of game shows, board games, and lotteries demonstrates the endurance and evolution of various games. People often devise strategies for playing certain games based upon their instinct and/or intuition, but these strategies may or may not be "the best" possible strategies. In this course we will develop mathematical techniques for analyzing games of chance and other types of games, so that we can develop optimum strategies for some simple games. We will discuss topics such as "the law of averages" and what it means for a game to be "fair." Students are encouraged to bring games to class for discussion. Prerequisite: MAT 012 or waiver. This course may be used to fulfill the Quantitative Reasoning requirement. Please see the instructor if you have not yet completed or waived MAT 012.

The Course Goals To apply ideas from probability theory to the analysis of selected games. To apply ideas from game theory to the analysis of selected games. To develop skills in formulating, solving, and interpreting mathematical problems. To discuss and apply the modeling cycle and to gain experience with real world applications of probability concepts. To develop the ability to work in a team.

The Attendance Policy Class lectures, discussions, and in-class work are considered to be a vital key to success in this course. It is the hope of the instructor that class sessions are both informative and useful, therefore attendance is expected at each class session unless a specific exception is made. Quizzes may be announced or occasionally "popped," and because the lowest quiz grade will be dropped, under nearly all circumstances, make-up quizzes will not be given. Likewise, make-up tests will under almost no circumstances be given, since the lowest of the test and/or quiz total will be dropped. Absences from class are noted, and repeated absences will adversely affect the student's grade. The final grade may be lowered by one third of a letter grade for each absence after the third. Thus, it is the responsibility of the student to speak to the instructor about each absence from class. This should be done as soon as possible, and if at all possible before the absence occurs. Students who miss class are held responsible for all of the material covered, assigned, and collected during their absence.

The Text The main text Probability: An Introduction is by Samuel Goldberg. We will cover selected topics from chapters 1-5:
Chapter 1, Sets A set is just a collection of well-defined distinct objects. One example of a set is the collection of all red playing cards from a single full deck. Sets are useful for thinking about games and strategies in precise ways.
Chapter 2, Probability in a Finite Sample Space The probability of an event happening is a mathematical way of defining the likelihood of the event happening. For example, when drawing a single card from a standard full deck of cards, the probability of drawing a red card is 1/2 because 1/2 of the cards are red. The sample space is simply a set of all the possible things that can happen. For example, suppose a coin is tossed twice. Then there are four possible things that can happen: Both can be heads, both can be tails, the first could be a head and the second a tail, or the first could be a tail and the second a head. The sample space for this example is S = {HH, TT, HT, TH}.
Chapter 3, Sophisticated Counting This chapter is dedicated to finding probabilities of given events when the number of possibilities is large. This technique is needed as games become more complicated.
Chapter 4, Random Variables A random variable is a formal way to consider all possible outcomes of a game event. For example, suppose again a coin is tossed twice. Then as discussed above, there are four possible things that can happen: Both can be heads, both can be tails, the first could be a head and the second a tail, or the first could be a tail and the second a head. An example of a random variable is one whose value is the number of heads obtained in these two tosses for a given item in the sample space.
Chapter 5, Binomial Distribution and Some Applications This section is dedicated to certain types of experiments which occur again and again. The word binomial refers to the numbers that occur as coefficients in these experiments.

The System of Evaluation
Evaluated Items Points Grading Percentages Test 1 Test 2 Test 3 Quiz Total Homework Final Project 100 100 100 100 100 100 16.7 % 16.7 % 16.7 % 16.7 % 16.7 % 16.7 %		Maximum 90-100 % 80-89 % 70-79 % 60-69 % 0-59 % Scale A's B's C's D's F

The lowest of the three tests and the quiz total will be dropped before calculating the final grade. Please refer to the GRADING section of the current Berea College Catalog for the College-wide interpretations of these letter grades.

The Grading Policies For the benefit of the students in the class, all course grade computations are continually updated by the instructor, so students may check frequently on their in-progress course grade during the term.

The Tests and Quizzes Tests and frequent short quizzes will be given in this course. In general, the announced quizzes will consist of questions on the assigned text readings or homework-like problems. The most likely dates of the three tests will be:
Test 1: Wednesday, January 10. Test 2: Wednesday, January 17. Test 3: Wednesday, January 21.
Problems that appear on the tests will be more varied in nature, ranging from homework-like problems to problems that require a deeper synthesis of ideas and from true or false questions to short-answer questions.

The Homework Bonus

Homework will be assigned on a near-daily basis, since doing homework thoughtfully and conscientiously is one of the keys to success in this course. Through homework, students get the needed practice of application of the concepts. Because the instructor desires to strongly encourage a diligent effort on homework, students who turn in each of their homework assignments with no more than three assignments submitted late, will be awarded an additional 10% on the homework grade!

On Homework Collection All written work should be neat, organized, and should show sufficiently many steps to demonstrate a clear understanding of the techniques used. Homework is due at the beginning of class on the announced date due. If a student must miss class due to either a sickness or a planned absence, homework is still expected to be submitted on time. Assignments may be requested in advance. Late assignments will be accepted for reduced credit up until the homework is returned, and late work must be labeled as late. Written or printed homework assignments may be turned in before class or at the instructor's office, but should NOT be sent through the CPO, attached in ccMail, or given to a student assistant. A selection of the assigned homework problems will be graded for credit, and assignments not meeting the above standards may receive reduced credit.

The Course Home Page Our course home page is located at http://www.berea.edu/Math/Faculty/Jan/MAT129/. Use this page as a resource to find this syllabus and other course-related information.

For Additional Help The teaching assistant for this course will be assigned later. She or he and most of the other Math Lab Consultants will also be able to answer questions about the mathematical content in the course during consultations in the Math Lab whose hour will be announced later. Best results are obtained trying to solve problems alone or in a group before asking for help, so in either place, students should be prepared to show what they have already tried. Topics in this course build throughout the course, so students should be sure to do their best to keep up with the class, so as to not get behind and possibly forever lost. Remember, no question to which one does not know the answer is ever "dumb" unless it goes unanswered because it remained unasked.

On Teamwork Learning to work in teams effectively is strongly encouraged. Some homework assignments may be specifically designed for teamwork, others for individual work, but on most homework you can choose to work alone or in a team. All homework assignments must clearly include all of the authors' names at the top of each page. On any assignment in which half or more of the work was completed in a team, a single copy of the assignment should be handed in with all of the team's participants listed as authors. Teams can generally consist of one, two, or three members due to the nature of the work in this course. Unless otherwise stated, teams shall not consist of more than three members for most work. On any assignment where less than half of the work was completed in a team, individual assignments should be handed in with the author acknowledging all of the help received for each problem. This includes significant help received from the instructor or in the Math Lab Consultants. Note that the instructor or a Math Lab Consultant may help with homework, and while this help should not be acknowledged as co-authorship, it should still be mentioned. This is meant to be a sharing process; do not "give credit" to other students who have not attempted to contribute to the work or to the team's work, because it is ultimately not a help for the student who did not contribute to the work. Thoughtful practice, not (even mindful) copying, is ultimately the best way to learn. Note that on all team-completed assignments, students must describe the roles played by each author on the assignment. Warning: Please be careful to conform to these standards for teamwork, since they are designed to encourage good learning practices. (Furthermore, copying another's work or otherwise failing to adhere to these standards may even result in a charge of academic dishonesty.)

The Class Atmosphere The members of this class constitute a learning community. Learning in such a community best takes place in an atmosphere in which instructor and the students treat everyone with mutual respect. Students need not always raise their hands in order to ask questions or to make comments, but they should not interrupt the instructor or fellow students in doing so. Students typically find the atmosphere set by the instructor to be a sometimes playful and nearly always relaxed one, but students will still need to work hard and consistently both in and out of class in order to do well. If at anytime you have thoughts, comments, or suggestions about how the class atmosphere could be improved or made into one which is more supportive of your learning, please come by or drop me a note about it. I welcome such suggestions.

The Final Project The final project in this course will be to find the solution to a complex and involved probability problem based upon a game. Students may work in a team of up to three students on this project. Some suggestions for problems are listed below, but teams are invited to propose their own problems. In any case the deadline for final projects proposals is Monday, January 15. Because easier problems are easier to explain, minor errors will be looked upon more leniently in a challenging problem.

Monopoly Problem A player in Monopoly rolls two dice to see how many spaces the player's piece must be moved on the board. If the player gets doubles, then the player moves and rolls again. If the player rolls doubles a second time, the player moves and rolls again. If the player gets doubles a third time, the player goes to jail. The player's piece can be moved from 2 to 35 spaces, or it can go to jail. Find the probability of the player being able to move each of these number of spaces and the probability of the player going to jail.

Price is Right Problem In the game show, "The Price is Right," three contestants face each other to decide who plays for the largest prize. A wheel is marked with 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, and 100. Each player spins the wheel once to get a score. If desired, the player may then spin the wheel a second time and add the result to the result from the first spin. If the player gets a score over 100, then she is out of the game. After the first person spins to get a score, the remaining players attempt to beat that initial score. After all three players have gone, the person with the highest score goes on to compete for big prizes. The second and third players know exactly what score they must beat to stay in the game, but this is not so clear for the first player. Calculate at what point the first person should stop in order to maximize her or his chances to win.

Poker Problem Compute the probability of getting each of the possible hands in a game of poker. Rank these hands according to the relative probabilities of getting each hand where the less likely a hand is, the higher the ranking. Next calculate these probabilities and ranking given that a single joker wild card (which can stand for any hand and any suit) is added to the deck. (Do not allow for 5 of a kind of any suit.) Explain whether or not adding a wild card changes the relative rankings of the hands.

Risk Problem Risk is a board game of world conquest. When two players do battle in risk, the attacker rolls three dice and the defender rolls two dice. If the highest number that appeared on one of the attacker's dice is greater than the highest number on the defender's dice then the defender loses one army. If the attacker's highest roll is equal to or less than the defender's highest roll, then the attacker loses one army. In the same way, the attacker's second highest roll and the defender's second highest roll are compared and either the attacker or defender lose an army. So ultimately the attacker can lose two armies, the attacker can lose one army, or the attacker can lose no armies. Find the probability of each, and the average number of armies that the attacker will lose each time the dice are rolled.