Unit Plan: Probability in Action – The Mathematics of Digital Reward Systems

I. Unit Overview and Teacher’s Guide

This section provides a comprehensive framework for the educator, establishing the unit’s pedagogical rationale, learning objectives, curriculum alignment, and instructional sequence. It is designed to equip the teacher with the necessary context and tools to deliver a rigorous and engaging learning experience. The structure adheres to established best practices for high-quality mathematics lesson planning, which emphasize clear objectives, real-world application, and a coherent progression of concepts.¹

A. Rationale and Context

In an era where digital entertainment is ubiquitous, students interact daily with complex systems governed by probability and statistics, often without realizing it. This unit leverages the inherent student interest in video games and digital applications to demystify the abstract concepts of probability and expected value. By deconstructing the mechanics of modern reward systems—such as “loot boxes,” “prize wheels,” or “gacha” pulls—students will discover how sophisticated mathematical models are used to design engaging user experiences and drive commercial outcomes.³

The central theme directly addresses the call within modern educational standards for students to “apply statistics ideas to real-world situations” and “link classroom mathematics and statistics to everyday life, work, and decision making“.⁵ Instead of focusing on traditional, and potentially sensitive, examples like casino games, this unit reframes the core mathematical inquiry into a critical exploration of game design, consumer awareness, and applied mathematics. Students will move beyond being passive consumers of these systems to become informed analysts who can mathematically evaluate their structure and fairness. This approach fosters not only mathematical proficiency but also digital literacy, empowering students to make more informed decisions in a world increasingly shaped by algorithms and data.⁶

B. Overarching Learning Goals

Upon successful completion of this unit, students will have demonstrated the ability to:

1. Calculate the theoretical probabilities of simple and compound events by determining the size of a sample space, utilizing principles of combinatorics.⁷
2. Define a random variable for a given scenario, and subsequently develop and represent its corresponding probability distribution in tabular and graphical formats.⁵
3. Calculate the expected value (EV) of a discrete random variable and accurately interpret its meaning as the long-run average or mean of the probability distribution.¹⁰
4. Utilize expected value as a primary analytical tool to evaluate the long-term outcomes of a chance-based system and determine its mathematical “fairness”.¹²
5. Synthesize and apply probability concepts to design, analyze, and justify a balanced digital reward system, demonstrating a comprehensive understanding of the intricate relationship between probability, payout values, and long-term profitability.³

C. Curriculum Standards Alignment (Common Core State Standards for Mathematics)

This unit is designed to target specific standards within the High School Statistics and Probability (HSS) conceptual category, making it an ideal component for an Algebra 2, Precalculus, or introductory Statistics course.⁵

• HSS.CP.A: Understand independence and conditional probability and use them to interpret data. Students will work with independent events in the core lessons, with an optional extension exploring conditional probability through “pity systems”.⁶
• HSS.CP.B: Use the rules of probability to compute probabilities of compound events in a uniform probability model. Students will calculate probabilities of compound events when determining the likelihood of specific outcomes on a multi-reel prize wheel.⁶
• HSS.MD.A.1 (+): Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution.⁵
• HSS.MD.A.2 (+): Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.⁵
• HSS.MD.B.5 (+): Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. This is the central skill of the unit, applied in both analysis and design tasks.⁶
• HSS.MD.B.6 (+): Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). The concept of a “fair game” (EV=0) is a key theme.⁶
• HSS.MD.B.7 (+): Analyze decisions and strategies using probability concepts. The final project requires students to analyze the strategic decisions made in the design of their reward system.⁶

D. Unit Progression and Pacing (5-6 Class Periods)

The unit is structured to scaffold learning from foundational concepts to complex application over approximately one school week. The pacing is flexible to accommodate varying class needs.

• Lesson 1: The Anatomy of Chance (1-2 periods): This lesson focuses on the building blocks of probability theory: sample space, events, and the use of combinatorics to count outcomes.
• Lesson 2: The Long Run (2 periods): This is the core lesson that introduces probability distributions and the pivotal concept of expected value, linking it directly to the design of chance-based systems.
• Lesson 3: The Designer’s Seat (2 periods): This is a project-based summative assessment where students apply all learned concepts to design and analyze their own digital reward system.

E. Key Vocabulary

A consistent understanding of mathematical language is crucial. The following terms will be explicitly defined and used throughout the unit.¹⁵

• Probability
• Outcome
• Sample Space
• Event
• Independent Events
• Combinations
• Permutation
• Random Variable
• Probability Distribution
• Expected Value (EV)
• Fair Game
• Return-to-Player (RTP)

F. Required Materials

• Scientific or graphing calculators
• Computers or tablets with spreadsheet software (e.g., Google Sheets, Microsoft Excel) for the final project
• All handouts, worksheets, and project guides provided in the Appendices of this plan.¹⁶

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II. Lesson 1: The Anatomy of Chance – Simple Probability and Combinatorics

This foundational lesson aims to transition students from an intuitive, everyday understanding of “chance” to a formal, mathematical framework. The primary goal is to establish the core principle that probability is derived from systematically counting all possible outcomes of an experiment. The lesson is structured using an “Experience First, Formalize Later” (EFFL) model, which encourages student inquiry before direct instruction.¹⁷

A. Lesson Objectives

• Students will be able to define sample space and event in the context of a chance experiment.
• Students will be able to calculate the total number of possible outcomes in a multi-stage chance experiment using the fundamental counting principle and the concept of combinations.
• Students will be able to calculate the theoretical probability of a single event occurring.

B. Lesson Procedure

1. Introduction/Hook (10 min)

The lesson begins by activating students’ prior knowledge and connecting the topic to their personal experiences.¹⁸

• Opening Question: Pose a broad question to the class: “Think about a video game, board game, or even a social media app you’ve used. What are some things that happen by chance? For example, think about getting an item from a treasure chest, rolling dice, drawing a card, or which video appears next in your feed.”
• Class Discussion: Facilitate a brief, whole-class discussion where students share examples. Guide them to recognize the prevalence of randomness in systems they use every day. This establishes the relevance of the topic before any formal mathematics is introduced.²

2. Core Activity: “Deconstructing the Prize Wheel” (25 min)

This inquiry-based activity challenges students to derive a core mathematical principle through problem-solving.

• Scenario Introduction: Present the class with a hypothetical “Prize Wheel” from a fictional mobile game called ‘Gem Quest’ (see Appendix A.1). Explain that this wheel is more complex than a simple spinner; it’s composed of three independent reels that spin simultaneously.

• Reel 1 has 4 unique symbols (e.g., Ruby, Sapphire, Emerald, Diamond).
• Reel 2 has 5 unique symbols (e.g., Gold, Silver, Bronze, Platinum, Copper).
• Reel 3 has 3 unique symbols (e.g., Sword, Shield, Potion).

• Inquiry Task: In small groups, students must answer the question: “How many different prize combinations can this wheel possibly land on?”
• Mathematical Discovery: This task is designed to lead students to discover or apply the fundamental counting principle.¹⁹ Without prior instruction, they must reason that for each of the 4 possibilities on the first reel, there are 5 possibilities on the second, and for each of those results, there are 3 possibilities on the third. The solution is the product of the possibilities at each stage:
  4×5×3=60 total combinations. This process of counting combinations is a fundamental aspect of gambling mathematics and, by extension, the mathematics of any chance-based system.⁷ This simple calculation is the first step in understanding the “math model” that operates behind the scenes of such games. The “flashy animation” of the spinning wheel is merely a visual representation of a process selecting one outcome from this finite sample space.⁸

3. Formalization & Guided Notes (20 min)

This phase connects the students’ discovery to formal mathematical language and notation.

• Share and Compare: Bring the class back together and have a few groups share their methodology for finding the total number of combinations. Highlight the common strategy of multiplication.
• Introduce Vocabulary: Use the “Prize Wheel” as the central example to formally define key terms:

• Sample Space: The set of all possible outcomes. In this case, the sample space consists of all 60 unique combinations (e.g., {Ruby, Gold, Sword}, {Ruby, Gold, Shield}, etc.).
• Event: A specific outcome or a subset of outcomes from the sample space. For example, “Winning the Grand Prize” or “Winning any prize involving a Sword.”
• Probability: Formalize the classical definition of probability: P(Event)=Total Number of Outcomes in Sample SpaceNumber of Favorable Outcomes​.⁷
• Worked Example: Guide the class through a probability calculation. “The ‘Grand Prize’ is awarded for the single combination of {Diamond, Platinum, Sword}. What is the probability of winning the Grand Prize?”

• Number of Favorable Outcomes = 1.
• Total Number of Outcomes = 60.
• P(Grand Prize)=601​.

4. Independent Practice (15 min)

Students apply the newly formalized concepts to solidify their understanding.

• Worksheet: Distribute the “Probability Problems” worksheet (Appendix A.2). This worksheet presents variations of the prize wheel or similar systems, such as a simple “loot box” containing a known quantity of different items. Students will practice calculating total outcomes and the probabilities of various events. This reinforces the core logic of dividing the number of winning combinations by the total number of possible combinations.⁸

5. Closure/Exit Ticket (5 min)

This serves as a quick formative assessment to gauge understanding of the lesson’s main concept.²

• Final Problem: “A game designer is creating a new, simpler prize wheel with 10 symbols on each of its 3 reels. The jackpot requires one specific combination of symbols. What is the probability of hitting the jackpot?”
• Expected Answer: Students should quickly calculate the sample space as 10×10×10=1000 and state the probability as 10001​. This problem directly mirrors the logic used to analyze simple slot machines and confirms that students can apply the fundamental counting principle to determine a probability.⁸

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III. Lesson 2: The Long Run – Expected Value and Fair Games

This lesson forms the conceptual core of the unit. It introduces students to Expected Value (EV), the mathematical tool used to predict long-term averages in situations involving chance. The lesson directly incorporates the user-requested text to bridge the gap between the visual appeal of games and the underlying mathematical models that govern them. The structure follows a classic “I do, We do, You do” instructional model to provide strong scaffolding for this new, multi-step concept.¹⁸

A. Lesson Objectives

• Students will be able to define a random variable and construct a probability distribution table for a given scenario.
• Students will be able to calculate the expected value of a discrete random variable.
• Students will be able to interpret the meaning of the expected value in the context of a problem and use it to analyze the mathematical “fairness” of a game.

B. Lesson Procedure

1. Introduction/Hook (15 min)

The lesson opens by presenting the provided text, prompting students to think critically about the purpose of mathematics in game design.

• Display the Text: Project the following text for the class to read:“Real Online Slots Statistics. You should know by now that behind all that flashy animation and artwork there’s something that is responsible for how slot payouts are distributed. Well, this “something” is a math model that is designed by mathematicians. Here, we are discussing some of the key slot stats that determine how slots behave.”
• Facilitated Discussion: Initiate a discussion with probing questions: “What do you think this ‘math model’ is for? Why would a company hire mathematicians just to design a game of chance? What does it mean to control how payouts are ‘distributed’?”
• Guiding the Conversation: Students may initially suggest the math is just to ensure randomness.⁸ The teacher’s role is to guide them toward a deeper understanding. The mathematical model is not merely for creating chance; it is for
  managing and predicting the outcomes of that chance over many repetitions.²⁰ The core purpose is to ensure the system is predictable and, from a commercial standpoint, profitable for the operator. This is achieved by designing for a specific long-term average payout, often referred to in the industry as the Return-to-Player (RTP).⁸ The “flashy animation” creates an illusion of pure, unpredictable luck, but the underlying mathematics ensures a predictable financial result for the game’s owner.²⁰ This discussion culminates in the central question of the lesson:
  How can we calculate this long-term average outcome before we even play?

2. “I Do” – Guided Instruction on Expected Value (20 min)

The teacher explicitly introduces Expected Value as the answer to the hook’s central question and models the calculation process.

• Defining Expected Value: Introduce Expected Value (EV) as the mathematical tool for this prediction. Define it formally as “the average gain or loss of an event if the procedure is repeated many times”.¹⁰ Emphasize that it’s a long-term prediction, not a guarantee for any single trial.
• The Key Tool: The Probability Distribution & Expected Value Table: Introduce the structured table that will be used to organize all calculations. This table is the most critical scaffold for the lesson.
• Modeling with a Simple Example: Use a clear, non-threatening example to model the process from start to finish. The carnival game scenario is ideal: “A carnival game consists of drawing one ball from a box containing two yellow balls, five red balls, and eight white balls (15 total). If the ball is yellow, you win $5. If red, you win $2. If white, you lose $3”.¹²

• Model filling out the table step-by-step on the board, explaining each column.
• Outcome (Event): Draw Yellow, Draw Red, Draw White.
• Value (X): The net value. Here, the values are given directly as +$5, +$2, and -$3.
• Probability P(X): Calculate the probability for each event: P(Yellow)=152​, P(Red)=155​, P(White)=158​.
• Weighted Value (X⋅P(X)): Multiply the value by its probability for each row.
• Final Calculation and Interpretation:

• Demonstrate the final step: Sum the “Weighted Value” column to find the EV. The formula is EV=∑[X⋅P(X)].¹⁰
• EV=(5)(152​)+(2)(155​)+(−3)(158​)=1510​+1510​−1524​=−154​≈−$0.27.
• Explicitly interpret the result: “The expected value is approximately negative 27 cents. This means that if you were to play this game many, many times, you would expect to lose, on average, about 27 cents per game. This is not a ‘fair game.’ A mathematically fair game is one where the expected value is exactly 0, meaning neither the player nor the operator has an advantage in the long run”.¹⁰

3. “We Do” – Guided Practice (20 min)

The class applies the new concept to the familiar context from the previous lesson.

• Return to the Prize Wheel: Revisit the ‘Gem Quest’ Prize Wheel from Lesson 1. Add a new layer of information: “It costs 100 gems to spin the wheel.” Assign payout values to the different prize categories (e.g., Grand Prize = 5000 gems, Gold Prize = 500 gems, Silver Prize = 150 gems, No Prize = 0 gems).
• Collaborative Calculation: As a class, construct and complete a new EV table for the Prize Wheel. Students should be able to provide the probabilities (calculated in Lesson 1) and determine the net values for each outcome (Payout – Cost). For example, the net value for the Grand Prize is 5000−100=4900 gems.
• Analysis: Once the final EV is calculated, analyze the result as a class. Is the Prize Wheel a fair game? Based on the EV, is it designed for the player to win or lose gems in the long run? This directly connects the calculated number back to the “math model” from the hook—the EV is the core statistic the game’s mathematicians would have designed.

4. “You Do” – Independent Practice (20 min)

Students work independently to demonstrate their grasp of the procedure and its interpretation.

• Worksheet: Distribute the “Calculating Expected Value” worksheet (Appendix A.3). This worksheet contains 2-3 new scenarios involving digital reward systems, drawing on the mathematics of loot boxes and gacha games.³
• Sample Problem: “A ‘Cosmic Chest’ in a sci-fi game costs $2 to open. It is guaranteed to contain one of three items: a Common Blaster (80% chance, resale value $0.50), a Rare Shield (15% chance, resale value $5), or a Legendary Helmet (5% chance, resale value $20). Calculate the expected value of opening one chest. Based on your calculation, is it a financially sound strategy to buy these chests if your goal is to make a profit?”
• This problem requires students to set up the EV table, correctly calculate net values (resale value – cost), and write a concluding sentence interpreting their result.¹⁰

5. Closure & Reflection (5 min)

A final check for conceptual understanding.

• Reflection Prompt: Ask students to write a one-sentence summary that answers: “What does Expected Value tell you about a game of chance?”
• Desired Response: A successful response should capture the idea that EV predicts the average outcome over many trials, not a single result.¹⁰ This ensures they grasp the concept, not just the calculation procedure.

C. Key Table: Probability Distribution & Expected Value

This table is the central organizational tool for this lesson and the entire unit. Its structure breaks down a complex calculation into a logical, step-by-step process, making it accessible to students and mirroring the analytical approach used by professionals.¹⁰

Outcome (Event)	Value (X) (Net Gain/Loss)	Probability P(X)	Weighted Value (X⋅P(X))
Description of Outcome 1	Value 1 – Cost	P(1)	X_1 \cdot P(X_1)
Description of Outcome 2	Value 2 – Cost	P(2)	X_2 \cdot P(X_2)
…	…	…	…
Description of Outcome n	Value n – Cost	P(n)	X_n \cdot P(X_n)
Total		1 (or 100%)	EV=∑[X⋅P(X)]

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IV. Lesson 3: The Designer’s Seat – Project-Based Summative Assessment

This capstone lesson transitions students from analysis to the higher-order skills of synthesis and creation.¹⁵ By taking on the role of game designers, students must apply their entire toolkit of probabilistic and statistical knowledge to solve a practical, open-ended problem. This project forces them to grapple with the real-world constraints and trade-offs inherent in designing chance-based systems, providing a rich and authentic assessment of their understanding.

A. Lesson Objectives

• Students will apply their understanding of probability, combinatorics, and expected value to a creative design task.
• Students will justify their design choices using quantitative evidence derived from their mathematical models.
• Students will communicate their mathematical thinking and design rationale clearly and coherently to their peers.

B. Project Overview

1. Launch the Project (Day 1, 20 min)

Introduce the final project, setting the stage and clarifying expectations.

• Project Introduction: Announce the final project: “You are a Game Designer!” Distribute the comprehensive project guide (Appendix B.1).
• The Prompt: “Your design team has been tasked by a major game studio to create a new digital reward system (e.g., a ‘loot box,’ ‘prize wheel,’ ‘gacha pull’) for their next blockbuster game. Your system must be exciting and engaging for players, making them want to participate. However, it must also be mathematically designed to be profitable for the company in the long run. You must use the principles of probability and expected value to design, analyze, and justify your system.” This prompt is directly modeled on the real-world design process where developers balance player experience with monetization goals.³

2. Project Work Time (Day 1, 30 min & Day 2, 30 min)

Students work in collaborative groups to design and analyze their systems.

• Project Requirements: The project guide (Appendix B.1) outlines the following deliverables for each group:

1. System Design Concept: A name and theme for their game and reward system. A clear definition of the “cost to play” (e.g., 100 gold coins, $1.99, etc.).
2. Outcomes and Probabilities: A list of at least 5-7 unique outcomes or prize tiers (e.g., Common, Uncommon, Rare, Epic, Legendary). Each outcome must have an assigned probability, and the sum of all probabilities must equal exactly 1 (or 100%).
3. Payout Values: A defined “payout value” for each prize, in the same units as the cost to play.
4. Mathematical Analysis: A complete and clearly formatted Probability Distribution & Expected Value table for their system. This table must show the calculation of the final EV for the player.
5. Design Justification: A well-written paragraph (150-200 words) that explains their design choices. This justification must answer: Why did you choose these specific probabilities and payout values? How does your calculated Expected Value reflect your dual goals of keeping players engaged while ensuring long-term profitability for the company? This written component is essential for assessing higher-order thinking and the ability to communicate mathematical reasoning.¹⁵

3. Group Presentations (Day 2, 20 min)

Students share their work, reinforcing learning across the entire class.

• Each group will give a brief (2-3 minute) presentation of their reward system. They must display their EV table, state their final EV, and summarize their justification for the design. This public sharing component holds students accountable and allows them to see a variety of approaches to the same problem.¹⁵

4. Submission and Assessment

• At the end of the second day, groups submit their final project document, which includes their design concept, EV table, and written justification.
• The project is graded using the provided, transparent rubric (Appendix B.2), which assesses both mathematical correctness and the quality of the analytical reasoning.

C. The Core Challenge: Design and Justification

This project moves students beyond simply finding a pre-determined answer. The central challenge is to design for a target EV. Students will quickly discover that to create a system that is profitable for the developer, the expected value for the player must be negative.¹⁰ This reinforces the primary lesson from the entire unit: these systems are mathematically structured to favor the operator over the long term.

However, they will also face the inherent design tension: if the EV is too negative (achieved through very low probabilities for valuable items or very low payout values), the game will feel unrewarding and will fail to engage players. This forces them to make strategic trade-offs. For example, they might include a “jackpot” item with a very high payout but an extremely low probability to create excitement, while the more common items have values far below the cost to play. This process of balancing the perception of high value with the mathematical reality of a negative EV is the authentic work of a game designer. By engaging in this process, students are not just solving a math problem; they are designing a solution within a set of realistic, conflicting constraints. This embodies the “application” aspect of mathematical rigor called for in leading educational frameworks.¹

D. Key Table: Project Assessment Rubric

A clear, criteria-based rubric is essential for a project-based assessment. It ensures that expectations are transparent to students from the outset and that grading is objective, consistent, and fair.² The rubric (Appendix B.2) is designed to measure all of the unit’s learning goals, from procedural fluency to conceptual application.

Criteria	4 – Exemplary	3 – Proficient	2 – Developing	1 – Beginning
Mathematical Concepts	All concepts (probability, net value, EV) are applied correctly and insightfully.	All key concepts are applied correctly.	Minor conceptual errors are present (e.g., using payout instead of net value).	Major conceptual errors are present.
Mathematical Accuracy	All calculations are correct. Probabilities sum precisely to 1. The EV table is flawless.	Calculations are mostly correct. Probabilities sum to 1. Minor errors may exist.	Multiple calculation errors are present. Probabilities may not sum to 1.	Calculations are largely incorrect.
Analysis & Justification	Justification provides a clear, compelling, and insightful explanation of design choices, explicitly linking the EV to the project’s dual goals.	Justification explains the design choices and correctly connects them to the calculated EV.	Justification is brief or lacks a clear connection between the design and the EV.	Justification is missing or does not address the prompt.
Clarity & Organization	The project is exceptionally well-organized and easy to understand. The EV table is clear and professionally formatted.	The project is well-organized. The EV table is clear and easy to read.	The project lacks clear organization. The EV table may be messy or hard to follow.	The project is disorganized and difficult to understand.
Creativity & Design	The game concept is highly creative, thoughtful, and well-themed.	The game concept is clear and complete.	The game concept is incomplete or lacks detail.	The game concept is missing.

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V. Appendices

A. Student Handouts & Worksheets

A.1: Lesson 1 Handout: Deconstructing the Prize Wheel

Game: Gem Quest

Reward System: The Triple-Reel Prize Wheel

Welcome, adventurer! The ‘Gem Quest’ Prize Wheel offers a chance at fabulous riches. To play, you spin three reels at once. Your prize is determined by the combination of symbols you land on.

• Reel 1 (Gems): Has 4 symbols: Ruby, Sapphire, Emerald, Diamond
• Reel 2 (Metals): Has 5 symbols: Gold, Silver, Bronze, Platinum, Copper
• Reel 3 (Items): Has 3 symbols: Sword, Shield, Potion

Your Task (Part 1): How Many Possibilities?

In your group, determine the total number of unique combinations possible on the Prize Wheel. Show your work and be prepared to explain your reasoning.

Your Task (Part 2): Calculating Probabilities

The game’s prize list is below. Calculate the probability of winning each prize. Remember: P(Event)=Total OutcomesFavorable Outcomes​.

Prize Category	Required Combination(s)	Number of Winning Combinations	Probability (Fraction & Percent)
Grand Prize	{Diamond, Platinum, Sword}
Ruby Sword	{Ruby, Any Metal, Sword}
Any Potion	{Any Gem, Any Metal, Potion}
A Bronze Prize	{Any Gem, Bronze, Any Item}

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A.2: Lesson 1 Practice: Probability Problems

Problem 1: The Dragon’s Hoard Loot Box

A “Dragon’s Hoard” loot box contains a mix of 20 gems. The contents are:

• 10 Rubies (Common)
• 6 Sapphires (Uncommon)
• 3 Emeralds (Rare)
• 1 Diamond (Legendary)

You open the box and receive one gem at random.

a. What is the sample space for this experiment?

b. What is the probability of receiving a Ruby? P(Ruby)=?

c. What is the probability of receiving a Rare gem (an Emerald)? P(Rare)=?

d. What is the probability of receiving the Legendary gem? P(Legendary)=?

e. What is the probability of receiving a gem that is not Common? P(Not Common)=?

Problem 2: The Gacha Summon

A gacha game features 100 unique characters in its general summoning pool.

• 80 characters are 3-star rarity.
• 15 characters are 4-star rarity.
• 5 characters are 5-star rarity (the highest).

You perform one summon.

a. What is the probability of summoning a 5-star character? P(5-star)=?

b. What is the probability of summoning a 4-star character? P(4-star)=?

c. A special event “rate-up” banner is active. On this banner, one specific 5-star character, ‘Arcturus,’ has a 1% chance of appearing. All other 99 characters have an equal chance of appearing within the remaining 99% probability. What is the probability of summoning any character other than Arcturus?

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A.3: Lesson 2 Practice: Calculating Expected Value

For each problem, create a full Probability Distribution & Expected Value table to find the EV. State your conclusion clearly.

Problem 1: The Cosmic Chest

A ‘Cosmic Chest’ in a sci-fi game costs $2.00 to open. It is guaranteed to contain one of three items with the following resale values and probabilities:

• Common Blaster: Resale Value 0.50,P=80%
• Rare Shield: Resale Value 5.00,P=15%
• Legendary Helmet: Resale Value 20.00,P=5%

a. Complete the EV table below. Remember to use the net value (X = Resale Value – Cost).

b. What is the expected value of opening one Cosmic Chest?

c. Based on the EV, is this a profitable activity in the long run? Explain.

Outcome	Net Value (X)	Probability P(X)	Weighted Value (X⋅P(X))
Common Blaster
Rare Shield
Legendary Helmet
Total			EV =

Problem 2: Spinning the Daily Wheel

A free-to-play mobile game lets you spin a “Daily Wheel” once for free. The wheel has 10 equal-sized sections with the following rewards:

• 5 sections award 100 Gold.
• 3 sections award 500 Gold.
• 1 section awards 2000 Gold.
• 1 section awards 50 Gems (a premium currency).

A data analyst determines that 1 Gem is roughly equivalent to 40 Gold in the game’s economy.

a. To calculate the EV in terms of Gold, first convert the Gem prize to its Gold equivalent.

b. Create an EV table for one spin of the Daily Wheel. The “cost” is 0.

c. What is the expected value (in Gold) of a single spin? This represents the average amount of Gold the game gives away to each player daily through this mechanic.

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B. Project Materials

B.1: Lesson 3 Project Guide: “You are a Game Designer!”

The Scenario:

Congratulations! Your team has been hired by “Infinity Games,” a top-tier game development studio. Your first assignment is to design the flagship digital reward system for their upcoming title. This system could be a loot box, a prize wheel, a gacha summon, or anything else you can imagine.

The Dual Mandate:

Your system must achieve two goals:

1. Player Engagement: It must be exciting, feel rewarding, and make players want to spend their in-game currency or real money. High-value “jackpot” items are a must!
2. Studio Profitability: The underlying math must ensure that, over thousands or millions of plays, the system is profitable for the studio.

Your Task:

In your design team of 2-3, you will design, analyze, and prepare a presentation for your reward system.

Project Deliverables:

1. Design Concept (1 page):

• Game Title & Theme: What is the name of your game and its genre (e.g., fantasy, sci-fi, sports)?
• Reward System Name: e.g., “The Astral Forge,” “The Winner’s Wheel,” “Heroic Summon.”
• Cost to Play: How much does it cost for one activation of your system? (e.g., 100 Gems, $1.99, 50 Dragon Scales).

2. Mathematical Model (1 page):

• Prizes and Payouts: A list of at least 5-7 distinct prizes or prize tiers. Each prize must have a defined Payout Value in the same units as your cost.
• Probabilities: Each prize must have an assigned probability. The sum of all probabilities MUST equal 100%.
• Expected Value Analysis: A complete and neatly formatted Probability Distribution & Expected Value Table. This is the core of your analysis and must clearly show:

• Each outcome.
• The Net Value (X) of each outcome (Payout – Cost).
• The probability of each outcome P(X).
• The weighted value of each outcome (X⋅P(X)).
• The final calculated Expected Value (EV) for the player.

3. Design Justification (1 paragraph):

• A well-written paragraph (approx. 150-200 words) that explains the logic behind your design.
• Address these questions: Why did you choose your specific probabilities and payout values? How did you create a sense of excitement for the player? How does your final EV ensure profitability for the studio? What is the long-term financial outcome for a player who uses your system repeatedly?

Presentation:

Be prepared to give a 2-3 minute summary of your design to the class (the “studio executives”). You should display your EV table and explain your justification.

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B.2: Lesson 3 Project Rubric

(See table in Section IV.D of the main report)

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C. Answer Keys

(Note: A full document with worked-out solutions for A.2 and A.3 would be provided to the teacher.)

Quick Solutions:

• A.3, Problem 1 (Cosmic Chest):

• Net Values: Common = -$1.50, Rare = +$3.00, Legendary = +$18.00.
• EV = (-1.50)(0.80) + (3.00)(0.15) + (18.00)(0.05) = -1.20 + 0.45 + 0.90 = +$0.15.
• Conclusion: Surprisingly, this is a profitable chest for the player in the long run, with an average gain of 15 cents per chest.
• A.3, Problem 2 (Daily Wheel):

• Gem Prize = 50 Gems * 40 Gold/Gem = 2000 Gold.
• EV = (100)(0.5) + (500)(0.3) + (2000)(0.1) + (2000)(0.1) = 50 + 150 + 200 + 200 = 600 Gold.
• Conclusion: The expected value of a free daily spin is 600 Gold.

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D. Optional Extension Activity: Conditional Probability and “Pity Systems”

Target: Advanced students or as a follow-up lesson after the main unit.

Concept Focus: Conditional Probability, Dependent Events.6

Introduction:

“In our project, we assumed every play was an independent event—the result of the last play had no effect on the next. But many modern games use more complex systems to keep players from getting too frustrated. One of these is a ‘pity system.’ Let’s analyze one.”.26

The Scenario:

“The gacha game ‘Mythic Heroes’ has a base 1% chance of pulling a 5-star hero. However, it includes a ‘hard pity’ system. If a player performs 89 consecutive pulls without receiving a 5-star hero, the 90th pull is guaranteed to be a 5-star hero.” 27

Discussion Questions:

1. What is the probability of not getting a 5-star hero on a single, normal pull? (P(fail)=1−0.01=0.99).
2. What is the probability of failing 89 times in a row? (P(89 fails)=0.9989).
3. How does the “pity” mechanic change the probability of the 90th pull if you have already failed 89 times? (The probability becomes 1, or 100%. This is an example of conditional probability).
4. Does this system change the overall Expected Value compared to a simple 1% system with no pity? How would a mathematician or game designer begin to calculate the new, more complex EV?

Rationale: This extension introduces the concept that probabilities are not always static. It demonstrates a real-world application of conditional probability and dependent events, showing how designers use more sophisticated mathematical models to precisely manage player experience and retention. This provides a deeper and more nuanced look into the “math model” that governs these systems, directly addressing the goals of rigor and coherence by linking the unit’s core concepts to more advanced topics.¹