1. Exploring Probability Through Yahtzee Extensions
Chris Bolognese
Upper Arlington High School

2. Goals of the Talk Motivation with Supportive Research
Data Collection & Analysis via Nspires
Java Simulation
Showcase Sample Student Work
Reflection

3. Motivation
Q.U.A.N.T. [Quantifying Uncertainty and Analyzing Numerical Trends] Summer Workshop, 2010
Goal was to create cognitively-demanding tasks related to probability and statistics.

4. Motivation
Implementing Standards-Based Mathematics Instruction (Stein, et. al, 2009) outlines a Task Analysis Guide to evaluate the demand of a task.
GAISE Report (Franklin, et. al, 2007) outlines levels of probabilistic and statistical reasoning

5. Motivation Memorization Tasks
-Not ambiguous -No connection to concepts -Focus is to reproduce previously learned facts
Procedures Without Connections -Algorithmic
-Little ambiguity about methods
-No explanations required
Procedures Without Connections -Procedures deepen students’ conceptual understanding
-Multiple representations -Solution pathways suggested
Doing Mathematics -Requires complex and non-algorithmic thinking.
-Students examine different solution strategies.
-Pathway is ambiguous

6. Context Honors Precalculus class consisting mainly of juniors.
Previously exposed to basic probability and counting techniques.
Wanted a rich problem that had no clear answer yet was accessible due to real life experiences.

7. The Solution

8. Standard Yahtzee
Roll five dice to get different combinations. The dice can be rolled up to three times in a turn, reserving dice as needed. The goal is to make one of thirteen different combinations (such as three-of-a-kind, or a large straight). The roll worth the most points is a Yahtzee (all five dice have the same value).

9. Some Dicey Questions What’s the probability that we get Yahtzee
on the very first roll?
There are 65 possible rolls, with only 6 outcomes as Yahtzees, one for each of the six face values. Thus, the probability of a Yahtzee is 6/65 ≈ 0.077%

10. Some Dicey Questions
What is the probability a Yahtzee is rolled within three rolls?
This is much harder as it requires a transition matrix. One can show that the probability is roughly %.

11. Essential Question
In Yahtzee, we have to stop rolling after at most three rolls… But that’s no fun!
What if we are determined to get a Yahtzee
and continue to roll. What is the expected
number of rolls to obtain a Yahtzee?

12. Data Collection
With a partner, experimentally determine the number of rolls to get at least one Yahtzee, reserving matching dice as needed. Then share your data.

13. Java Simulation
While we could continually roll to collect data, let’s use technology to simulate this task! How does this simulation compare to our experimental data?

14. Data Collection
This graph (DataGenetics, 2012) shows the chance of getting Yahtzee in n-rolls.

15. Sample Student Work
Data Representations

16. Sample Student Work
Distribution Characteristics

17. Sample Student Work
Combinatorics

18. Sample Student Work
Transition Matrix

19. Sample Student Work
Tree Diagram

20. Sample Student Work
Probability of Yahtzee in at most 2 Rolls

21. Sample Student Work

22. Other Possible Extensions
How does the expected number of rolls to get a Yahtzee depend of the number of dice?
What are the probabilities of other types
of combinations in Yahtzee?
Pick another probabilistic board game
and use probability theory/
combinatorics to analyze the game.

23. Reflection
The Common Core Mathematical Practices urges students to: 1. Make sense of problems 2. Reason abstractly 3. Construct & critique arguments 4. Model with mathematics 5. Use tools appropriately 6. Attend to precision
7./8. Look for patterns and structure

24. Reflection
When designing a cognitively-demanding task, build in connections, open-ended problems, mathematical modeling, and multiple approaches.
I highly encourage you to use, modify, or create your own task involving probability that engages students.
Thanks for attending!