Math 180
Instructor: Chris Hays Office: DRL 4N34 Office Hours: T 2:00pm -- 4:00pm Lecture: MWF 1:00pm -- 1:50pm, DRL A2 Midterm 1: Friday, September 27 Midterm 2: Friday, November 22
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[09-Dec-13] Here are the slides for Lecture 37. Homework #11 is available, and is due Firday, December 13. Since we finished all of the material for the course, class is cancelled today. It only remains for you to hand in homeworks 10, 11, and the final paper.
[04-Dec-13] Here are the slides for Lecture 36. We covered interest rates (compounding on different intervals, and present and future value), and began looking at pricing options. Note that due dates have been moved around:   Homework #10 is due Monday, December 9   The final paper is due Wednesday, December 11   Homework #11 is due Friday, December 13 Please submit the paper as a pdf via email. Homework #11 can either be handed in to me at my office, or in my mailbox
[02-Dec-13] Here are the slides for Lecture 35. In class, we finished examining weighted voting systems by studying the Banzhaf Power Index. Homework #10 is available, and is due Firday, December 6.
[25-Nov-13] Here are the slides for Lecture 33. Here are the slides for Lecture 34. In class, we discussed the Chair's Paradox. We then looked into weighted voting, and the Shapley-Shupik Power Index. The next homework will be posted within the next day. It will not be due until Friday, December 6.
[20-Nov-13] As promised, I have found old exams:   Old midterm: look at problems 1,2a,3ab,4   Old final exam: look at problems 5,6,7,10 Lastly, I have attached solutions to homework 9
[15-Nov-13] Here are the slides for Lecture 32. Here is Homework #9. It is not due for class, and is only here to provide sample problems for voting theory. Here is a review list of topics that we have covered since exam #1. To study for the upcoming exam, I recommend looking at these topics, as well as going over the homework, the class handouts, and the problems covered during lecture. In class, we continued look at voting theory. We discussed two new fairness criteria: monotonicity and independence of irrelevant alternatives. We discussed a new method of voting: pairwise comparisons. Lastly we considered Arrow's Impossibility Theorem, which notes that no voting method can satisfy all voting criteria that have been mentioned in class.
[13-Nov-13] Here are the slides for Lecture 31. We continued look at voting theory. We considered various criteria for fairness: Condorcet, majority, public enemy, and monotonicity.
[11-Nov-13] Here are the slides for Lecture 30. In class, we began talking about voting theory. We discussed plurality voting, runoffs, approval voting, instant runoffs, the Borda method, and lastly the Condorcet criterion.
[08-Nov-13] Here are the slides for Lecture 29. Here is Handout #7. In class, we continued talking about the fair division of goods. In particular, we discussed using the method of sealed bids in order to fairly divide indivisible goods. Lastly, Homework #8 is available, and is due Firday, november 15.
[06-Nov-13] Here are the slides for Lecture 28. In class, we continued talking about the fair division of goods. In particular, we talked about a few methods for cutting cake.
[04-Nov-13] Here are the slides for Lecture 27. In class, we talked about the fair division of goods. We discussed different notions of fairness, different properties that the goods may take, and lastly the rule for inheritance that is discussed in the Talmud. Here is a paper outlining this rule.
[01-Nov-13] Here are the slides for Lecture 26. Note that Lectures 24 and 25 incorrectly mention Handout #7. It should be Handout #6, which is available here. For this week's homework, I am only wanting to see the topic, and a quick summary, of what you plan on writing the final essay on.
[30-Oct-13] Here are the slides for Lecture 25. In class, we developed another definition for evolutionary stability. We also looked at mixed strategies that are evolutionarily stable. Lastly, we considered the mating habits of the common side-blotched lizard. Here is the original Nature article that discusses the game theory behind the lizards' mating.
[28-Oct-13] Here are the slides for Lecture 24. In class, we discussed the concept of evolutionarily strategic strategies. Note that the due date for selecting a topic for the paper has been postponed to Friday, November 8. If you have any questions about the paper, feel free to contact me.
[26-Oct-13] Here are the slides for Lecture 23. In class, we covered finding Nash equilibria that consist of mixed strategies, as well as evolutionarily strategic strategies. If this is of interest to you, you can look at the original article on the subject: The Logic of Animal Conflict, by Maynard Smith and Price. Lastly, Homework #7 is available, and is due this Firday (November 1st).
[24-Oct-13] Note that the homework was postponed to this Friday. Also, the link to the slides for Lecture 20 is now correct.
[23-Oct-13] I realized that I forgot to post the previous slides. Here are slides for Lecture 21. Here are slides for Lecture 22. For question 4 of the homework, you may assume that there are only two players. Lastly, in class we began looking at mixed strategies. In particular, we discussed how the best general strategy for Rock Paper Scissors is to play each choice randomly with probability 1/3. Just to show that that is not easy as it seems, you can compete against this Rock Paper Scissors program. If you can truly play randomly, you should at least tie the computer.
[18-Oct-13] Here are slides for Lecture 20. Here is Handout #5. In class, we began studying Nash equilibria. Lastly, Homework #6 is available, and is due on Wednesday, October 23.
[16-Oct-13] Here are slides for Lecture 19. In class, we contined looking at game theory. In particular, we went through ``iterative deletion of dominated strategies'', and the Median Voter Theorem. For the homework, do not use the fact that ``1028'' people were polled, but rather use their given confidence interval. The reason for this difference is that Gallup weights the people they poll (see also the end of Lecture 16).
[14-Oct-13] Here are slides for Lecture 18. I believe that I have corrected all the errors for the lecture 18 slides (sorry about that). If there are any other errors, or any errors in future slides, please let me know so that I can correct them. In class, we began to cover game theory. In particular, we discussed dominating strategies. Lastly, Homework #5 is available, and is due this Firday (October 18th).
[07-Oct-13] Here are slides for Lecture 17. We finished dealing with statistics. In class, we covered confidence intervals are margins of error when polling. The next homework will not be posted until later this week.
[07-Oct-13] Here are slides for Lecture 16. In class, we covered polling, and sampling error.
[04-Oct-13] Here are slides for Lecture 15. In class we discussed two other types of averages: the median and the interquartile range. These are statistics that are not sensitive to outlying data. Additionally, we looked at Simpson's Paradox. This paradox states that depending on how one splits up data, opposite trends can be seen from the same set of raw data. Here is a paper outling Simpson's paradox relating to the graduate admissions at Berkeley. For those of you who are nephrotically inclined, here is the original article specifying differences in kidney stone treatment. Here is a paper discussing the previous results, and how it relates to Simpson's Paradox. You may need to access the internet through Penn to access these three papers.
[03-Oct-13] Here is a pdf of a z-score table. This will be useful for the most recent homework.
[02-Oct-13] Here are slides for Lecture 14. In class, we continued looking at Bernoulli trials, and discussed standard deviation and normal distributions. Lastly, Homework #4 is available, and is due on Wednesday, October 9.
[01-Oct-13] Here are slides for Lecture 13. In class, we covered Bernoulli trials and normal distributions. As an example of Bernoulli trials, we examined the Drake Equation. Below are a few places providing estimates for probabilities used in the Drake Equation. These articles are really just here to prove that I didn't make the numbers in class up.   This BBC article is discussing how most star systems have planets.   This article is arguing that if a planet is capable of sustaining life, it has a .13 probability of developing life (published in Astronomy & Astrophysics 476 (3)).   This article reviews a more complicated form of the Drake Equation, and argues that there is a .2 probability that a planet with life will develop a civilization (published in International Journal of Astrobiology 11 (2)).
[25-Sep-13] Exam 1 is next class. In class, we discussed The Sally Clark case, as well as variance/standard deviation. Here is a reference discussing the misuse of probability in the case, in case you are interested. Here are slides for Lecture 12.
[25-Sep-13] I've attached copies of the handouts:  Handout 1  Handout 2  Handout 3  Handout 4
[23-Sep-13] Here are slides for Lecture 11. In class, we studied the Monty Hall Problem. A good discussion of the Monty Hall Problem is in this article. We also discussed the use of probability in the O.J. Simpson trial. The issues surrounding the relevance of the history of physical abuse were brought up in a letter to the NY Times editor here with a reponse here. Lastly, here is a list of all of the topics that we've covered. To study for the final exam, I would recommend looking at these topics, as well as going over the homework, the class handouts, and the problems covered during lecture.
[20-Sep-13] We continued to cover conditional probability and Bayes' Formula. Here are slides for Lecture 10. If you have previously downloaded the lecture 9 slides, note that Bayes' Second Formula was initially stated incorrectly. The formula in the current slides - as well as the lecture 10 slides - is now correct.
[18-Sep-13] Here are slides for Lecture 9. In class, we continued looking at conditional probability. In particular, we covered Bayes' formula. Lastly, Homework #3 is available, and is due on Wednesday, September 25.
[16-Sep-13] Note that office hours have changed. For the rest of the term, they will be from 2 to 4 on Tuesdays. We studied conditional probability. Here are slides for Lecture 8.
[13-Sep-13] We continued studying probability. In particular, we talked more about the Friendship paradox, and started talking about independent events. Here are slides for Lecture 7. Here is a discussion about the Friendship Paradox and Facebook. Here is a paper discussing the Friendship Paradox and Twitter.
[11-Sep-13] Slides for Lecture 6. We continued studying rules of probability. In particular, we looked at the ``Birthday paradox'', and briefly talked about the paper ``Why your friends have more friends than you do''. The paper is available here (you will have to be accessing the internet through Penn). For an example of the ``fifty-fifty fallacy'', you can look at the Daily Show link. Lastly, Homework #2 is available, and is due on Wednesday, September 19.
[09-Sep-13] Slides for Lecture 5. We continued studying rules of probability.
[06-Sep-13] Slides for Lecture 4. We have started looking at probability.
[04-Sep-13] Slides for Lecture 3. Additionally, Homework #1 is available here, and is due on Wednesday, September 11.
[01-Sep-13] Slides for Lecture 1. Slides for Lecture 2.
[01-Sep-13] This will serve as the class blog. I will post all announcements here, as well as homework and due dates. Additionally, I will provide the slides of each class.