# Exploring Probability Through Yahtzee Extensions - PowerPoint PPT Presentation

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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.

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Exploring Probability Through Yahtzee Extensions

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## 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

### 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.

### 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

### Java Simulation

While we could continually roll to collect

data, let’s use technology to simulate this

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

Combinatorics

### Sample Student Work

Transition Matrix

Tree Diagram

### Sample Student Work

Probability of Yahtzee in at most 2 Rolls

### 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 problems2. Reason abstractly3. Construct & critique arguments4. Model with mathematics5. Use tools appropriately6. 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!