Older links, Matlab programs, etc.
  • Matlab programs: We played with the steppingStones simulation in class. Here is the Javascript version (which runs faster).

  • Midterm solutions: Solutions for the second midterm are here.

  • Matlab programs: We played Monopoly! Here is the verbose version (tells you what's happening at each move), and here is the version that just gives you the frequencies with which you landed on each square after a number of moves.

  • Practice problems: Here are some practice problems from previous years' exams that are relevant for our second midterm. They are in two files: here and here. (These are not an accurate reflection of what our exam will be like.) For more practice problems, see previously posted exams (they had expected value problems that we skipped over), Probability Online, examples from notes and the text, and all the exercises in the book (remember that all of the odd ones have solutions on the book website).

  • Matlab programs: We saw a few Matlab files which illustrated the general version of the central limit theorem. The files are CLT_general, prob_convolve, generate_random_experiment, die_nstar_distribution_compare_nd, and get_distribution_sum. (Download all of them into the same folder on your computer.)

  • Midterm solutions: Solutions to the first midterm are here.

  • Matlab programs: In class we played with the function Craps, which shows the progress of a game of craps. The file craps-2 shows the average winnings after a large number of craps games have been played.

  • Practice problems:: Solutions to another previous midterm someone found online are here.

  • Practice problems: The first midterm from Summer 2009 is here. Note that some of the questions ask about expected value, which will not be tested on our first midterm; after Wednesday's class, problems 1 through 8 should all be doable. Remember that you can use the Probability Online page to check your numerical answers!

  • Matlab programs: We played with Bomb_Dropper in class, and used the function geornd to plot a histogram of geometrically distributed random variables.

  • Matlab programs: The functions we constructed in class are here (broken down into steps). The first hour of Wednesday's office hours (1:30-2:30pm) will be primarily devoted to programming questions, so if you have questions about what we did in class today or about the programming assignments to be turned in Friday, please make sure to come! (Non-programming questions are also welcome.)

  • Matlab programs: We looked at the function DiceRolls, and constructed the function cointosses together (right click and save). If you're confused about any of that, make sure to look up all of the functions in the documentation center on the MathWorks website, and go through some of the tutorials (links in prior announcements). As much as I would love to teach you all there is to know about Matlab (I really would -- it's an amazing piece of software!) we just don't have that kind of time in class.

  • Practice problems: Probability Online is now up and running! This is the "online homework" referred to in the grading scheme for this course, worth 5% of your grade. Log in and take a look around. Some of the questions that appear on weekly written homework will also appear online -- use the online portion to test your numerical answers. Also note that there are achievements to unlock, some of which will give prizes (both individually and for the class as a whole).

    Your login for Project Probability is your first name (as it is listed on the roster) followed by the first letter of your last name (all lowercase), and your password is your student ID. If you have trouble logging in, send me an e-mail ASAP.

    Note: You do not have to do all of the questions all the time. Some will be assigned as required problems in the future, and others will not. However, you should try to keep up with the online problems as much as possible -- I will occasionally post minimum point goals which must be achieved in order to keep up with this portion of the homework. These will just be bare minimums -- you should still be working to answer more questions than that, in order to unlock achievements that will benefit you and everyone else in the class!
  • External links: You can find the book algorithms I mentioned here, and the Java Applet versions are here. The Plinko simulation we played with is here. You may also be interested in doing some practice problems -- solutions to odd-numbered exercises are available here.

  • Matlab programs: We used DiceRolls.m in class (right click - save as). Read through the comments for some Matlab tips. For more Matlab tips, check out the Documentation Center -- enter the name of any function you'd like to learn more about in the search bar. For yet more, here is an excellent Matlab tutorial.
"Think about it" assignments
  • For 7/25:
    • Show that if X and Y are independent then V(X+Y) = V(X) + V(Y)

    • Find two random variables X and Y that are not independent where this equality does not hold.

    • Find two random variables X and Y that are not independent where this equality does hold, or prove no such pair of random variables exists.

  • For 7/11: Find a combinatorial explanation for why interchanging k and m in P(X=s) for a hypergeometric random variable X does not change the value of P(X=s).

  • For 7/8: Prove the statement on page 189:
    b(n,p,k)/b(n,p,k-1) = (λ - (k-1)p)/(k(1-p)).

  • For 7/1:
    • Find the number of ways of being dealt each of the 5-card poker hands (i.e. royal flush, straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, one pair). You don't have to write these out, but be prepared to wing it at the board if your number comes up. :)

    • What experiment is being modeled by the grids on the sides of this webpage? What are the random variable, sample space, and distribution?

  • For 6/29: How many valid choices for New Hampshire phone numbers are there?

  • For 6/28: Prove that if E1, E2, ..., En are disjoint events (you can define the entire sample space as Ω=E1∪E2∪...∪En) and A is another event in Ω, then
    P(A) = P(A∩E1) + P(A∩E2) + ... + P(A∩En).

  • For 6/27: Can you have a uniform distribution on a countably infinite space?