Fair Play Slides Fair Play Lab
Using Probability to Make Decisions
High School Statistics | Tier 2 Intervention
The Logic of Fairness
In statistics, a decision method is FAIR if every person or outcome has an equal chance of being selected.
Think about it:
"If we use a coin flip to decide who goes first, does anyone have an advantage?"
Fairness = Equal Probability
If there are n people...
1
/
n
...every person must have a probability of exactly 1 out of n.
2 People
1/2 (50%)
4 People
1/4 (25%)
10 People
1/10 (10%)
Method 1: Drawing Lots
How it works:
Identical slips of paper are placed in a container.
Each person's name is on exactly one slip.
A slip is drawn without looking.
Potential Problems:
• Slips are different sizes or textures.
• Names are written multiple times.
• Some slips are stuck together.
Is every slip physically the same?
Method 2: Random Number Gen
RNG System v2.0 STATUS: READY
RANGE
1 - 30
RESULT
17
The Statistics:
The computer uses a complex algorithm to pick a number where every digit in the range is equally likely.
Why use this?
✅ Removes human bias
✅ Fast for large groups
✅ Verifiable results
Spot the "Rigged" Method
Which of these methods is UNFAIR for choosing 1 out of 5 people?
Method A
Roll a standard 6-sided die. If it lands on 1-5, that person is picked. If it lands on 6, roll again.
PROBABILITY:
1/5 (20%) per person
Method B
Assign each person a number 1-5. Flip two coins. If both are heads, pick person #1. Otherwise, pick person #2.
PROBABILITY:
Does everyone have 1/5?
Ready for the Lab?
Today we will test these methods ourselves. Your goal is to find which one is the most reliable way to make a fair decision.
Collect Data
Analyze Probabilities
Justify Choices
Fair Play Lab Sheet Fair Play Lab Report
Topic: Probability & Decision Making (S-MD.B.6)
Scientist:
Date:
Investigation Goal:
Determine if selection methods are statistically fair by testing outcomes and calculating probabilities.
Phase 1: Defining Fairness
In statistics, a decision is FAIR if every person has an EQUAL chance of being chosen.
Scenario:
There are 4 friends choosing who gets the last slice of pizza. To be fair, what should the probability be for each friend?
\(P(\text{Friend}) = \)
(Write as a fraction)
Experiment 1: Drawing Lots
Procedure: Label 3 slips of paper (A, B, C). Fold them and put them in a cup. Draw one, record it, and put it back. Repeat 15 times.
Outcome Tally (15 draws) Total Count Paper A Paper B Paper C
Reflection:
Did one paper seem to get picked much more than the others? If yes, why might that happen physically (e.g., size, folding)?
Experiment 2: Random Number Generator
Procedure: Use the RNG tool provided. Set the range from 1 to 5. Run it 20 times and record how often each number appears.
1
2
3
4
5
Create a bar chart of your results above
Statistical Analysis:
The Theoretical Probability for each number is 1/5 (20%). How do your actual results compare to the theoretical probability? Why aren't they exactly the same?
Final Recommendation
Imagine you are the principal of the school. You need to pick one student out of 500 to win a free laptop. Which method would you use to ensure it is COMPLETELY FAIR ?
Drawing Lots (Names in a hat)
Random Number Generator
Rolling a Die
Justify your choice using the word "PROBABILITY":
Fair Play Facilitator Guide Facilitator Guide
Teacher Resource
Lesson 1: Fair Play Lab | Probability & Fairness
Instructional Goal
Students will understand that a "fair" decision method requires all outcomes to have an equal probability (\(1/n\)). They will transition from intuitive fairness to mathematical justification using experimental data.
Common Misconceptions
Physical Bias: Students may assume drawing names is always fair, ignoring physical variables (paper size, texture).Law of Small Numbers: Students might think a method is "rigged" if they see a slight imbalance in 10-20 trials.Subjective Fairness: Thinking a method is fair just because "everyone was there" rather than calculating \(1/n\).
Quick Specs
Standard
CO HS.S-MD.B.6
Group Size
3-5 Students
Materials Needed
• Slips of paper
Instructional Routine
10m
Launch & Hook (Slides 1-3)
Discuss the concept of "fairness" in a game or contest. Introduce the mathematical formula for fairness: \(P = 1/n\).
"If I have 5 people and I pick a number between 1 and 10, is that fair? Why not?"
20m
Guided Exploration (Lab Sheet Section 2)
Students perform the "Drawing Lots" experiment. Pro-tip: Intentionally make one slip of paper slightly larger or thicker to see if students catch the physical bias during their reflection.
15m
Independent Testing (Lab Sheet Section 3)
Students use an RNG tool. Monitor their bar charts. Ensure they understand that variability is normal, but the long-run probability should be equal.
Progress Monitoring
Look-Fors
Can the student define fairness as "equal probability"?
Can they calculate \(1/n\) for a given group size?
Do they distinguish between physical bias and random chance?
Scaffolding Move
If students struggle with fractions, use visual area models (pie charts). Compare a "fair" pie (equal slices) to an "unfair" pie (different size slices) to represent probability.
Differentiation Strategies
Student Need Support Strategy Visual Learners
Fair Play Lab Key Answer Key
Fair Play Lab Sheet | Lesson 1
Teacher Reference
Phase 1: Defining Fairness
Scenario: 4 friends choosing pizza slice.
\(P(\text{Friend}) = \)
1/4
(or 25% or 0.25)
Experiment 1: Drawing Lots
Sample Reflection Responses:
"The results for Paper A were much higher because the paper was slightly larger and easier to grab."
"Physical differences like how the paper is folded or the texture of the paper can create BIAS, making the method unfair even if it looks random."
*Note: Student data will vary. Look for them identifying physical bias if you intentionally tampered with the slips.
Experiment 2: RNG
Statistical Analysis Key Points:
Theoretical vs. Experimental: "The theoretical probability is exactly 1/5 for each, but in 20 trials, chance variation (luck) means some numbers will show up more. This is normal in short experiments."
Sample Size: "If we ran it 10,000 times, the counts would be much closer to exactly 20% each."
Final Recommendation
Random Number Generator
Ideal Justification:
"The RNG is the best choice because it uses an algorithm to ensure that every student (1 to 500) has an equal probability of exactly 1/500. Unlike drawing names, it cannot be biased by the size of the paper or how well the container is shaken."
