## Goals

In this PBI unit, students will use principles of probability, game theory, and statistics to develop a fair and engaging family/group game for commercial consumption, and build a prototype of their game. Students will also use their writing skills to write an instruction booklet with the purpose of teaching a person who has never heard of their game before how to play their game. At the end of the unit, students will prepare a group oral presentation to formally present the rules and prototype of their game to the instructor and other students, who will evaluate their presentation and idea and vote for the best idea. Students will utilize peer review and statistical analysis to edit their ideas and their writing throughout the unit.

## Objectives

Students will be able to:

- Calculate the probability of an event
- Distinguish between permutation and combinations
- Calculate the probability of an event with replacement and an event without replacement
- Calculate the expected value of a random variable
- Explain the concept of interdependent decision making or game theory and identify it in everyday life
- Distinguish between observational studies, surveys, and experiments
- Design an experiment to test their game
- Write a multi-step process
- Write a set of instructions
- Review and edit other students’ writing
- Make edits of their own writing based on feedback
- Distinguish between different sampling methods
- Analyze data by interpreting results
- Communicate effectively in a presentation
- Evaluate the presentations of other students
- Work as a team towards a common goal
- Use critical thinking skills to argue for an optimal strategy
- Apply independence in contextual problems to calculate probability of a given event
- Use probabilities to justify decisions about risk
- Calculate expected value to analyze mathematical fairness, risk, and payoff
- Communicate arguments using precise mathematical language via oral communication or written communication
- Explain the difference between theoretical and empirical probability
- Use the Law of Large Numbers to describe the relationship between theoretical and empirical probabilities
- Apply the proportional relationship between the measure of the area of a sector of a circle and the area of a circle to solve problems
- Justify the mathematical concept of probability based on area

## Common Alternative Conceptions

- All games are ultimately chance-based rather than skill-based.
- Permutations and combinations are the same thing.
- The probability of independent events are determined by previous independent events.
- You can choose a number by true randomness from just your thoughts.
- In the Monty Hall problem, there’s a 50-50 chance of getting the car regardless if you switch or stay.
- In the prisoner’s dilemma, the prisoners will never choose to betray each other.
- All experiments must use the scientific method, surveys are not experiments.

## Alignment with Texas Essential Knowledge and Skills (TEKS)

**Geometry**(13) Probability. The student uses the process skills to understand probability in real-world situations and how to apply independence and dependence of events. The student is expected to:

- (A) develop strategies to use permutations and combinations to solve contextual problems;
- (B) determine probabilities based on area to solve contextual problems;
- (C) identify whether two events are independent and compute the probability of the two events occurring together with or without replacement;
- (D) apply conditional probability in contextual problems; and
- (E) apply independence in contextual problems.

**Discrete Mathematics**(6) Game (or competition) theory. The student uses knowledge of basic game theory concepts to calculate optimal strategies. The student analyzes situations and identifies the use of gaming strategies. The student is expected to:

- (A) recognize competitive game situations;
- (C) identify basic game theory concepts and vocabulary;
- (E) explain the concept of and need for a mixed strategy;
- (F) compute the optimal mixed strategy and the expected value for a player in a game who has only two pure strategies;
- (H) identify the nature and implications of the game called "Prisoners' Dilemma";
- (I) explain the game known as "chicken";
- (J) identify examples that illustrate the prevalence of Prisoners' Dilemma and chicken in our society; and

**Statistics**(5) Probability and random variables. The student applies the mathematical process standards to connect probability and statistics. The student is expected to:

- (A) determine probabilities, including the use of a two-way table;
- (B) describe the relationship between theoretical and empirical probabilities using the Law of Large Numbers;
- (C) construct a distribution based on a technology-generated simulation or collected samples for a discrete random variable; and
- (D) compare statistical measures such as sample mean and standard deviation from a technology-simulated sampling distribution to the theoretical sampling distribution.

(2) Statistical process sampling and experimentation. The student applies mathematical processes to apply understandings about statistical studies, surveys, and experiments to design and conduct a study and use graphical, numerical, and analytical techniques to communicate the results of the study. The student is expected to:

- (A) compare and contrast the benefits of different sampling techniques, including random sampling and convenience sampling methods;
- (B) distinguish among observational studies, surveys, and experiments;
- (C) analyze generalizations made from observational studies, surveys, and experiments;
- (D) distinguish between sample statistics and population parameters;
- (E) formulate a meaningful question, determine the data needed to answer the question, gather the appropriate data, analyze the data, and draw reasonable conclusions;
- (F) communicate methods used, analyses conducted, and conclusions drawn for a data-analysis project through the use of one or more of the following: a written report, a visual display, an oral report, or a multi-media presentation; and

**Advanced Quantitative Reasoning**(4) Probabilistic and statistical reasoning. The student uses the process standards in mathematics to generate new understandings of probability and statistics. The student analyzes statistical information and evaluates risk and return to connect mathematical ideas and make informed decisions. The student applies a problem-solving model and statistical methods to design and conduct a study that addresses one or more particular question(s). The student uses multiple representations to communicate effectively the results of student-generated statistical studies and the critical analysis of published statistical studies. The student is expected to:

- (E) use probabilities to make and justify decisions about risks in everyday life;
- (F) calculate expected value to analyze mathematical fairness, payoff, and risk;

**Biology**(12) Science concepts. The student knows that interdependence and interactions occur within an environmental system. The student is expected to:

- (A) interpret relationships, including predation, parasitism, commensalism, mutualism, and competition among organisms;
- (B) compare variations and adaptations of organisms in different ecosystems;

**English Language Arts and Reading**(13) Writing/Writing Process. Students use elements of the writing process (planning, drafting, revising, editing, and publishing) to compose text. Students are expected to:

- (A) plan a first draft by selecting the correct genre for conveying the intended meaning to multiple audiences, determining appropriate topics through a range of strategies (e.g., discussion, background reading, personal interests, interviews), and developing a thesis or controlling idea;
- (B) structure ideas in a sustained and persuasive way (e.g., using outlines, note taking, graphic organizers, lists) and develop drafts in timed and open-ended situations that include transitions and the rhetorical devices used to convey meaning;
- (C) revise drafts to improve style, word choice, figurative language, sentence variety, and subtlety of meaning after rethinking how well questions of purpose, audience, and genre have been addressed;
- (D) edit drafts for grammar, mechanics, and spelling; and
- (E) revise final draft in response to feedback from peers and teacher and publish written work for appropriate audiences.

(24) Listening and Speaking/Listening. Students will use comprehension skills to listen attentively to others in formal and informal settings. Students will continue to apply earlier standards with greater complexity. Students are expected to:

- (A) listen responsively to a speaker by taking notes that summarize, synthesize, or highlight the speaker's ideas for critical reflection and by asking questions related to the content for clarification and elaboration;
- (B) follow and give complex oral instructions to perform specific tasks, answer questions, solve problems, and complete processes; and
- (C) evaluate the effectiveness of a speaker's main and supporting ideas.

(25) Listening and Speaking/Speaking. Students speak clearly and to the point, using the conventions of language. Students will continue to apply earlier standards with greater complexity. Students are expected to give presentations using informal, formal, and technical language effectively to meet the needs of audience, purpose, and occasion, employing eye contact, speaking rate (e.g., pauses for effect), volume, enunciation, purposeful gestures, and conventions of language to communicate ideas effectively.

(26) Listening and Speaking/Teamwork. Students work productively with others in teams. Students will continue to apply earlier standards with greater complexity. Students are expected to participate productively in teams, building on the ideas of others, contributing relevant information, developing a plan for consensus-building, and setting ground rules for decision-making.