Exploring Probability Through Yahtzee Extensions Chris Bolognese Upper Arlington High School cbolognese@uaschools.org

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

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.

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

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

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.

The Solution

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).

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%

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 4.6029%.

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?

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.

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?

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

Sample Student Work Data Representations

Sample Student Work Distribution Characteristics

Sample Student Work Combinatorics

Sample Student Work Transition Matrix

Sample Student Work Tree Diagram

Sample Student Work Probability of Yahtzee in at most 2 Rolls

Sample Student Work

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.

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

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!