Week | Day | Date | Section in text |
---|---|---|---|
Week 1 | Day 1 | March 28 | Introduction and basic concepts |
Day 2 | March 30 | 1.2 | |
x-hour | March 31 | Optional: Proof-writing tips | |
Day 3 | April 1 | 1.2 | |
Week 2 | Day 4 | April 4 | 3.1 |
Day 5 | April 6 | 3.1 | |
x-hour | April 7 | 3.2 | |
Day 6 | April 8 | 3.2 | |
Week 3 | Day 7 | April 11 | More combinatorics! |
Day 8 | April 13 | 4.1 | |
x-hour | April 14 | Quiz #1 | |
Day 9 | April 15 | 4.1 | |
Week 4 | Day 10 | April 18 | 4.1 |
Day 11 | April 20 | 6.1 | |
x-hour | April 21 | 6.1 | |
Day 12 | April 22 | 6.1 | |
Week 5 | Day 13 | April 25 | no class |
Day 14 | April 27 | no class | |
x-hour | April 28 | no x-hour | |
Day 15 | April 29 | 6.2 | |
Week 6 | Day 16 | May 2 | 5.1 |
Day 17 | May 4 | 5.1 | |
x-hour | May 5 | Quiz #2 | |
Day 18 | May 6 | 5.1 | |
Week 7 | Day 19 | May 9 | 8.1 |
Day 20 | May 11 | 8.1 | |
x-hour | May 12 | no x-hour | |
Day 21 | May 13 | 9.1 | |
Week 8 | Day 22 | May 16 | 9.1 |
Day 23 | May 18 | 9.2 | |
x-hour | May 19 | Quiz #3 | |
Day 24 | May 20 | Confidence intervals and embarrassing questions | |
Week 9 | Day 25 | May 23 | 11.1 |
Day 26 | May 25 | 11.2 | |
x-hour | May 26 | no x-hour | |
Day 27 | May 27 | 11.3 | |
Week 10 | Day 28 | May 30 | no class: Memorial Day! |
Day 29 | June 1 | Review |
I also told you that it's useful for us all to share a mathematical language. I suggest looking at this handout to understand mathematical statements a little better.
Day 2 (Wednesday, March 30)
Remember that when defining an experiment, it's very important to define your sample space correctly. In class today, I set up an experiment where we chose a sequence of two numbers at random from the set {1,2,3}. In this case, the outcomes were the possible sequences. What if I had asked for the outcomes to be the number of 2s in the resulting sequences? The number of odd numbers in the resulting sequences?
Day 3 (Friday, April 1)
Remember that the experiment I talked about at the end of class was flipping a fair coin until tails came up and counting the number of flips it took. We agreed that any number of coin flips was possible. I have a question for you: since the coin is fair, the probability of getting a tails on one flip is the same as the probability of getting a tails on any other flip. Why is it that the probabilities of needing different numbers of flips to get a tails are different?
Day 4 (Monday, April 4)
Here's the problem I promised you: Student parking is difficult to find at College, so one student parks in the faculty lots every day. He has just noticed that none of the last ten tickets he's gotten has been issued on a Monday or Friday. He is wondering if the campus police actually patrol the faculty lots on those days. If the campus police at College don't give tickets on weekends, is it reasonable to assume that the campus police don't actually patrol the faculty lots on Mondays and Fridays? Why?
Day 5 (Wednesday, April 6)
I mentioned in class that you might want to show that if < a_n > ~ < b_n >, then < b_n > ~ < a_n >. I'll also leave you with a calculation: Suppose that three math books, two physics books, and three government books are shelved randomly (and that all the books are different). What is the probability that all of the books in each subject are shelved together? (Think carefully about what needs to be ordered.)
x-hour, Week 2 (Thursday, April 7)
Here's a problem for you: If you draw 8 cards from a standard deck without replacement, what is the probability that you will draw at least one king? (Remember to be careful about not inducing order in your hand!)
Day 6 (Friday, April 8)
The equation we talked about at the end of class today is called the Inclusion-Exclusion Principle in your book. I prefer the Principle of Inclusion and Exclusion because the acronym is nicer.
The problem I recommended to you about the professors grading multiple-choice tests is #18 in section 3.2.
Day 7 (Monday, April 11)
Here's another problem: Suppose that you've put all the possible nonnegative integer solutions of x_1 + x_2 + x_3 = 7 on slips of paper, put them into a container, and shaken it so well that you have equal probability of drawing out any slip. If you draw a slip, what is the probability that the solution it contains will have x_1=3?
We aren't going over section 3.3 in this course, but it's a rather interesting piece on card shuffling. You might want to have a look at it (after this week's quiz, of course).
Day 8 (Wednesday, April 13)
First of all, the Monty Hall simulation I was using is here.
Second, several of you came up with nice variations on this problem. I'll write them here and leave them to you to think about. Analyzing them involves exactly the same strategy I used at the board today.
The limit of the probability that a sample average differs from the expected value by more than epsilon is zero for every positive epsilon.
The Strong Law of Large Numbers is as follows:
The probability that the limit of the sample average differs from the expected value by more than epsilon is zero for every positive epsilon.
The difference between these two is subtle but important. The first says that as you take a larger and larger sample, the probability that your sample average will be outside some particular margin of error shrinks. The second says that as you take a larger and larger sample, your sample average will almost certainly get closer and closer to the actual expected value. The second implies the first, but not vice versa (see #16 in Section 8.1).
Day 21 (Friday, May 13)
If you're reading this, you know we're working through section 9.1 right now. Everything we've done today is on pp. 326-27. It's dense stuff, but I assure you that pp. 328-29 will fly right by!
Day 22 (Monday, May 16)
The moral of the Central Limit Theorem is that if you normalize a binomial distribution correctly, you can approximate it by a normal distribution with expected value 0 and variance 1. I showed you some tables of probabilities for this normal distribution. Please understand how to use them!
Day 23 (Wednesday, May 18)
Today we talked about confidence intervals and polling. Towards the end, questions came up about how to estimate the p and q to plug into the calculation of the standard deviation. One possibility is just to use the observed p and q from another poll. Another possibility (and a very good one!) is just to note that pq is never more than .25, so we could use .25 in place of pq. That might result in a larger n, but at least we can be sure it will be big enough!
You might want to take the poll at Fallacy Watch, and you might be interested in this list of questions for understanding how much data from a given poll are worth.
Day 24 (Friday, May 20)
First of all, we found that approximately 0% of you have gone longer than a month without washing your sheets and that approximately 22% of you sleep with a stuffed animal or baby blanket.
To do confidence intervals for this sort of question, we consider the fraction of the "yes" answers that come from the coin flip. This is the only part of the problem for which we need to worry about a confidence interval (think about why!). Now we can just calculate the confidence interval as for a standard binomial distribution: we have (1-p)k coin tosses that matter, and each has probability of success .5.
Day 25 (Monday, May 23)
Ian Stewart wrote two wonderful articles about Markov chains and Monopoly for Scientific American in April and October 1996. You should be able to download the .pdfs from the Scientific American archives as long as you're online via Dartmouth's secure network.
If you want to review matrix multiplication and finding inverses of matrices, the Dartmouth library has several e-books. Here's how to find them: