Fair Play Lab Slides Fair Play Lab
Designing Probability-Based Decisions
HS Statistics Tier 2 Intervention
What is "Fair"?
In statistics, a decision is fair if every person or group involved has the exact same probability of being chosen.
P(Outcome) = \( \frac{1}{n} \)
Where \( n \) = total number of participants
If 4 students want the last laptop, each must have a probability of...
25% or \( \frac{1}{4} \)
Our Lab Tools
Drawing Lots
Physical items (slips of paper, sticks) in a container. One is pulled out.
RNGs
Random Number Generators. Using technology to pick a number within a range.
Standard Cubes
Dice with a set number of sides. Assigning numbers to people.
The Fairness Audit
Scenario: The 5-Player Spin
A spinner has 4 equal sections. Five students want to play. One section says "Spin Again."
Is this fair? Why or why not?
Does everyone have the same P(win)?
Is the process transparent?
Can it be repeated?
Lab Analysis:
While the sections are "equal," the "Spin Again" section changes the probabilities mid-game.
We need a method where every participant is represented identically from the start.
How to Design a Fair Decision
1
Identify
Count the total number of participants (n).
2
Assign
Give each person a unique identifier (ID).
3
Select Tool
Choose a tool that matches the count (n).
4
Verify
Check if P(each) = 1/n.
Ready to start our Fair Play Lab ?
Decision Design Lab Worksheet Fair Play Lab: Decision Design
Student Lab Report | HS Statistics
Name:
Date:
Fundamental Rule of Fairness:
"A decision is fair if every participant has the same probability of being chosen."
P(Chosen) = 1 / Total Participants
Part 1: Identifying Unfairness
Read the scenario below. Why is this method unfair?
Scenario: The Lunch Line Leader
There are 10 students in the class. The teacher rolls a standard 6-sided die. If the number is 1-5, the student with that number is the leader. If the number is 6, the teacher picks one of the remaining 5 students (numbers 6, 7, 8, 9, 10).
What is the probability for Student #1 to be chosen?
What is the probability for Student #10 to be chosen?
Explain why this system does not provide equal opportunity:
Part 2: Design Challenge
The Task: There are 7 students who want to win the "Class DJ" spot for Friday. You must design a fair selection procedure using only a Random Number Generator (RNG) .
Fairness Design Checklist
Total participants (n) identified?
Unique ID assigned to each?
Range of numbers matches students?
What happens if a "bad" number is rolled?
Step 1: The Assignments
Assign each student (S1-S7) a number or range of numbers.
Step 2: The Procedure
Describe exactly how the RNG will be used (e.g., "Generate a number from 1 to 10...").
Step 3: The Proof
Why is this fair? Use probability (fractions) to justify.
Part 3: Tool Evaluation
Tool Pros (Benefits) Cons (Challenges) Drawing Lots
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| RNG (Tech) |
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Final Lab Question:
If you have 1,000 participants for a raffle, which tool would be the most practical and least biased? Explain why.
Fair Play Facilitator Guide Facilitator Guide
Fair Play Lab | Tier 2 Intervention
Standard
S-MD.B.6
Learning Objectives
Define mathematical fairness as equal probability for all participants.
Identify flaws in biased decision-making procedures.
Design and justify a fair procedure using tools like RNGs and drawing lots.
Pacing & Structure (30-40 min)
0-5m
The Hook:
Ask: "If there's one slice of pizza left and 3 friends want it, how do you decide who gets it so NO ONE feels cheated?" Use Slide 2 to formalize this.
5-15m
Direct Instruction:
Review Slides 3-4. Model the "Fairness Audit." Focus on why "Spin Again" or "Teacher's Choice" breaks the 1/n rule.
15-30m
Guided Design:
Students work through Part 2 of the Lab Worksheet. Circulate and ask: "If I'm Student #4, what is my exact chance of winning?"
30-40m
Debrief & Exit Ticket:
Discuss the raffle question (Part 3). Transition to the Progress Monitoring check.
Misconceptions
Students often think "random" means "fair," but a biased RNG (e.g., picking 1-10 for 3 people) isn't fair unless ranges are equal.
Scaffolding
For students struggling with n=7, start with n=2 (coin flip) or n=4 (suits in a deck) to visualize equal 1/n partitions.
Checklist for Success
- Did student identify n?
- Did student assign IDs?
- Is P(each) identical?
- Is the tool valid?
Facilitator Prompt Cards
Probing Questions
"If we use a 6-sided die for 4 people, what happens when we roll a 5 or a 6? Is it still fair? Why?"
Error Correction
"You assigned numbers 1-3 to Person A and 4-6 to Person B. What about Person C? How do we adjust the ranges?"
Real-World Connection
"In a sports draft, teams sometimes get more 'lots' in the bin. Is that a fair decision, or a weighted one? What is the goal?"
Mathematical Justification
"Don't just say 'it feels fair.' Prove it to me using the fraction 1/n. What is your 'n' here?"
Fair Play Exit Ticket Fair Play Exit Ticket
Progress Monitoring | HS Statistics
Name
Score
/ 5
1 Define Fairness
In your own words, what must be true about the probability of every outcome for a decision to be mathematically "fair"?
2 Analyze the Model
"To pick a winner among 3 students (A, B, and C), the teacher flips a coin twice. HH = A wins. HT = B wins. TH = C wins. TT = Flip again."
Is this model fair? Circle one: FAIR | UNFAIR
Justify your answer with probability (fractions):
3 Design a Solution
You have a group of 8 students. You have a standard 6-sided die. Briefly describe a procedure that would be unfair, and then one that would be fair.
Unfair Method:
Fair Method:
How confident do you feel about designing fair decisions?
Lost
Getting There
Pro
Fair Play Answer Key Answer Key & Rubric
Fair Play Lab | HS Statistics Intervention
Lab Worksheet Solutions
Part 1: Identifying Unfairness
P(Student #1): \( 1/6 \) (Directly rolled on the die).P(Student #10): \( (1/6) \times (1/5) = 1/30 \) (Must roll a 6, then be picked from 5).Explanation: The system is unfair because students 1-5 have a 16.7% chance, while students 6-10 only have a 3.3% chance. Fairness requires equal probability for all.
Part 2: Design Challenge (Sample Answer)
Step 1: Assign IDs 1-7 to the seven students.Step 2: Use an RNG to generate a number from 1 to 7. (Alternatively: 1-10 and "reroll" if 8, 9, or 10 is hit).Step 3: Since there are 7 outcomes and each has exactly 1 number assigned, P(each) = \( 1/7 \).
Part 3: Tool Comparison
Drawing Lots: Pros: Visual/Transparent. Cons: Hard to manage for large groups; physical materials needed.RNG: Pros: Fast, supports large numbers, perfectly random. Cons: Requires tech; less "visual" for some students.Raffle Question: RNG is the only practical tool for n=1000. Physical lots would be difficult to mix/draw fairly.
Exit Ticket Solutions
1. Definition of Fairness:
Every possible outcome or participant must have the exact same probability of being chosen (\( P = 1/n \)).
2. Analyze Model:
Answer: FAIR
Sample Justification: There are 4 possible outcomes for 2 flips (HH, HT, TH, TT). Each has a 25% chance. Students A, B, and C each get 25%. By rerolling TT, we maintain the 1:1:1 ratio between the remaining options. \( P(A) = P(B) = P(C) = 1/3 \).
3. Design Solution (n=8, 6-sided die):
Unfair: Rolling once. If 1-6, those students win. Students 7-8 have 0% chance.
Fair: Roll die twice. Assign a unique combination (e.g., (1,1), (1,2), etc.) to each of the 8 students. Reroll combinations not assigned. Every student has \( 1/36 \) chance per roll attempt, resulting in equal final probability.
Mastery Rubric
Level Criteria Mastery (5/5) Clearly explains P=1/n. Correctly identifies unfair scenarios with math. Designs valid procedures.
