SCHC 212: The Mathematics of Game Shows

Frank Thorne - Spring 2018
University of South Carolina

Come on down! Are you a fan of games shows such as The Price is Right, Who Wants to be a Millionaire, Let's Make a Deal, Jeopardy, Deal or No Deal, Wheel of Fortune, and more? Contestants on these shows are asked to make decisions which could win or cost them a lot of money, and mathematics can be used to figure out the best strategies. This course will explore these game shows while explaining the mathematics behind good decision-making.

Specific subjects to be covered include: probability and expected value; conditional probability and Bayesian inference; and classical game theory. The course will also have a heavy modeling component: some games are too complicated to analyze completely, but can be analyzed if you make some simplifying assumptions.

This class will teach real mathematics in a fun context. (Beware: "fun" does not mean easy.)

Instructor : Frank Thorne, LeConte 317O, thorne [at] math.sc.edu.

Office Hours: Wednesday, 3:30-5:00; Thursday, 9:00-10:30. I am also frequently (but not always) available immediately after class.

Exception: There are Faculty Senate Meetings on Wednesdays -- February 7, March 7, and April 4. Wednesday office hours these weeks will be moved to Tuesdays.

Learning outcomes:

"Education is what survives when what has been learned has been forgotten." -- B.F. Skinner

Successful students will:

• Master the mathematical topics described above.
• Better understand how to apply mathematics in the real world. While it may be debated whether or not game shows are the "real world", the real point is that math isn't in the statement of many of the questions to be considered. Half the battle is knowing which math problems to solve.
• Gain comfort with ambiguous questions. For example, you may be familiar with the Monty Hall Problem. (If not, don't worry, I'll explain it.) Should you switch doors? The correct answer is "it depends on what assumptions you make".
• Gain comfort with complicated problems. In courses like calculus we break everything up into bite-sized chunks for you. This course will teach the use of a knife and fork.
• Practice your written and oral presentation skills. There will be a term project (see below).
• Have a lot of fun! I mean, how could you not? This subject is just cool.

Course notes:

I will be using these notes as a basis for this spring's notes, and making revisions as I go along. (Your feedback, as always, will be welcome.)

March 5 version (with some solutions).

Meeting schedule : MWF 9:40-10:30, Honors Residence Hall B111.

Note that there will be some tweaks to the schedule, to be discussed and agreed upon in class.

I also hope to have a special presentation (outside of class hours) by a guest speaker. If this happens, attendance will be expected unless you have a very good excuse.

You will be graded both on correctness and on quality of exposition. The standard is that someone who doesn't know the answer should be able to easily follow your work. Any work that is confusing, ambiguous, or poorly explained will not receive full credit.

You are guaranteed at least the following grades: A for 88%, B+ for 83%, B for 76%, C+ for 70%, C for 64%, and D for 50%.

 % of grade Homework: 30% Weekly Feedback: 10% Participation: 10% Term Project: 20% In-class midterm: 10% Final exam: 20%

Homework : Homework will be assigned and graded up to weekly. You are welcome to work with others but you are responsible for your own written solutions. You are always welcome, and strongly encouraged, to come to office hours or stick around after class, show me your written up solutions to homework problems, and ask whether they are correct.

Weekly Feedback : I am developing this course from scratch; to my knowledge, my Fall 2016 course was the first time a university-level course had been taught based on this.

As such, you are responsible for helping me do a good job. Every week (by Friday at 5:00 pm), an e-mail is due reflecting on the lectures, course notes, and/or homework assignment. What are you struggling with? What do you enjoy, and what is boring? Is there anything in the lecture notes which you don't think is adequately explained?

Suggestions to improve the lecture notes are especially appreciated -- as I have some aspiration of eventually turning these into a book.

Being brief is okay, but you're expected to say something more substantial than "Good job, no comments!" Weeks without lectures on all three days, and in addition you may skip at most two weeks. This is not only for my benefit, but also very much for yours. Reflecting on your own learning will help you cement your study habits and assist with retention of the material for years to come.

Participation : Active participation in the course is also very important. The most obvious (and most highly recommended) way to fulfill this requirement is to actively participate in class discussions and ask questions during class. You may also fulfill this requirement by periodically coming to office hours to ask questions, by e-mailing me with your questions, or by being particularly thorough in your weekly feedback.

Term Project : You are also responsible for a term project. Generally, this will involve analyzing a game (or a part of a game), writing a short paper on it, and making a 10-15 minute presentation to the class. You are welcome and encouraged to work in groups.

I recommend (but do not require) that you use software called LaTeX for writing up your term project. Here are some resources and some Windows downloads. If you use a Mac, this is the software which I use and which you can download.

Here is a sample of how your term project might be formatted if you use LaTeX, together with a description of how to get started using LaTeX. This is the .tex file which was used to produce this document, and here is the same file renamed .txt, so that a web browser will open it immediately.

You can also download the TeX file for the course notes above -- if you see something in the notes you want to imitate, you can see how I did it.

Exam schedule :

• In-Class Midterm: Wednesday, March 7 (rescheduled from March 5), in class. Here is last year's (75 minute) midterm, with solutions, and here is this year's, again with solutions. The drop deadline is March 9.
• Final Exam: Wednesday, May 2 at 9:00 am, in the usual classroom.

Requirements: There are no formal prerequisites for the class. Indeed, the class is designed especially for freshmen and/or non-STEM majors. Two guidelines:

• Your high school math background -- in algebra, precalculus, and overall "mathematical maturity" should be strong. A good guideline is that if you have not taken Honors Calculus I (Math 141H) or its equivalent already, you would feel confident doing so.
• The course is not designed for students with an advanced university-level math background, and it has some redundancies with our courses on probability and discrete math. If you have taken any math course at the 300 level or above, you should double check with me about what your expectations are.

Calculators are permitted but not required, including on the exams. A four-function calculator is more than adequate; graphing functionality won't be useful. Programming functionality is not permitted on the homework or exams. If you bring a programmable calculator to the exams, you are bound by the Honor Code not to use this functionality.

Access to the internet is required. Many of the course videos will refer to clips from game shows which are on Youtube. You must be able to watch these (with sound). Try the links below and make sure you can see and hear everything.

Make-up policy :

If you have a legitimate conflict with any of the exams it is your responsibility to inform me at least a week before the exam. Otherwise, makeups will only be given in case of emergency.

Accommodations :

Please contact the Student Disability Resource Center if you have any sort of disability which requires any sort of accommodations. I am always happy to follow their recommendations; it is your responsibility to inform me at least a week before the exam.

Rough schedule of topics to be covered.

This will be updated with much more detail as the course progresses.

• Probability. You throw two dice. What is the probability they sum to eight? You deal yourself a poker hand. What is the probability you deal yourself a royal flush? You play Ten Chances and guess at random. What is the probability you win the car?

Such computations are at the root of most of what we will do.

• Expectation. The expected value of a game is the average amount of money you expect to win, with each possibility weighted by its probability. You're playing Who Wants to Be a Millionaire. Quit or keep going? Do you press your luck?

We will also look at limitations to what the mathematics says about human behavior. Would you rather have \$900,000, or a one-in-ten chance at \$10,000,000?

• Inference. You pull a coin out of your pocket and flip it twenty times. It comes up heads each time. What is the probability that it's a trick coin, with heads on both sides?

You actually can't answer this without making additional assumptions. We will discuss Bayes' theorem and Bayesian inference and watch old episodes of Let's Make a Deal. We will of course discuss the Monty Hall Problem, which actually did not appear on the show.

• Multi-Step Strategy. You play Cover Up. What's the probability that you win the car? What if you know what the first two digits are, and only have to guess the last three. You should certainly pick the correct two digits on the first round. ..... right?
• Multi-player Strategy. Special considerations are required if you are competing with other players -- especially if you all play simultaneously.

Here is a round of Final Jeopardy. Perhaps an English class would be most helpful here, but the math question here is: how much should you wager?

• Modeling. Try and figure out the optimal strategy for Switcheroo. I dare you. (The rabbit hole runs very deep.) You are going to have to make some simplfiying assumptions if you want to get started at all. Which are useful to make?

• Student Presentations and Conclusion.