Math 331 Fall 2009
“The fear of the LORD is the beginning of knowledge;
Fools despise wisdom and instruction.” Proverbs 1:7
Objective :
To learn to apply a variety of aspects of probability theory by ‘doing it’ in a fun context
Directions : You will be assigned to a group. Play the game with your group a few times to get a feel for it. As the semester unfolds, you will learn more of the tools necessary for solving different aspects of the guided analysis. In particular, I will demonstrate some YAHTZEE calculations during class. As such, it is recommended to work on successive sections as the semester progresses. The write-up guidelines are provided at the end.
1.
See the reverse of this page for the YAHTZEE categories. For all probability calculations, assume you begin a particular hand (sequence of three rolls) with the goal of filling a pre-specified category. Note: You will need to determine rules for which decisions will be made when selecting which dice to keep between rolls. Make such rules explicit in your write-ups.
www.yahtzee.org.uk/
2.
(Ch. 1) Select three of the following YAHTZEE categories: “3 of a kind”, “4 of a kind”, “Full house”, “Small straight”, and “Large straight”. Calculate the probability of obtaining each category in a single hand.
3.
(Ch. 2, 3) Propose a discrete random variable related to YAHTZEE. Analyze this random variable as best as you can. For example, calculate its expected value and variance. If you cannot think of a good random variable, then try the following: model the “Chance” category with the sum of discrete Uniform(1,6) random variables.
4.
(Ch. 4) Compute the expected value and variance of the score of one of the following categories:
“Aces”, “Twos”, “Threes”, “Fours”, “Fives”, or “Sixes”.
5.
(Ch. 5) Select one of the following tasks and write an R script that will accomplish it. ( Note: You need only do one of the three. You may do more than one, if you like.
) a.
Write a script that will simulate one of the three categories selected in part (2). Run the simulation at least 10,000 times and record the proportion of times the category occurred.
Compare this simulated proportion with your theoretical calculation. b.
Write a script that will simulate a hand for the “Chance” category. Run the simulation at least 10,000 times and compare your results to part (3). c.
Write a script that will simulate a hand for the selection you made in part (4). Run the simulation at least 10,000 times and compute the mean and the variance. Compare with your theoretical results from part (4).
Write-up
Play the game with your group a few times before beginning the project, and after completing the calculations.
1.
Write up the calculations from your project in such a way that it is clear how the final results were obtained. This may be done by hand. (A typed version is preferable, although if such has a significant cost in communication of results or time, please do not do it. See me to discuss LaTeX, the better choice than Word’s Equation Editor).
2.
If, as the semester unfolds, you would like to add an additional component, or possibly substitute a component, please feel free to discuss this with me.

Math 331 Fall 2009
3.
Write a sketch of how the work went. That is, how was the work divided, how long did the parts take, and the major benchmarks during the process.
4.
There shall be one copy of the above three sections, with each person from the group’s name on it. For the conclusion, each individual member shall write a response to the following items
(typed). a.
Comment on the difficulty of the project. b.
Would you do anything differently if you had to do it over again? c.
If you were to pursue the project further, what additional questions, or avenues of exploration, would you like to address? d.
When you played the game after doing the calculations, do you feel you were able to perform better as a result of your study? e.
What did you learn from this project?
5.
On a designated day at the end of the semester, we will discuss the projects as a class. There is no need to prepare a formal presentation, but all group members should be prepared to intelligently discuss their own projects on that day.