Likelihood Lab Worksheet PROBABILITY ARCADE
Lesson 1: Introduction to Probability Models
What is Probability?
Probability is the likelihood of an event occurring. We measure this on a scale from 0 to 1 (or 0% to 100%).
0
Impossible
Won't happen
1/4
Unlikely
Could happen
1/2
Equal
50/50 Chance
3/4
Likely
Probably will
1
Certain
Will happen
Likelihood Lab
Name: __________________________ Date: ____________
Worksheet
Classify the following events using our probability vocabulary (Impossible, Unlikely, Equal, Likely, Certain).
1. Rolling a 7 on a standard 6-sided die.
2. Flipping a coin and it landing on Heads.
3. The sun rising in the East tomorrow morning.
4. Picking a red marble from a bag with 9 red and 1 blue.
Think Like a Game Designer:
Describe an event that has an unlikely chance of happening in a game of your choice.
Arcade Stations
Test your luck at each station!
01 The Coin Flip
Flip a coin 10 times. Record your results. Does "Equal" chance mean you always get 5 heads and 5 tails?
Student Data Area:
02 Spinner Chaos
Spin the 4-color spinner. Which color is the most likely to win? Why?
Sketch the spinner here:
03 Dice Duel
Roll two dice. Sum the numbers. Is getting a "Sum of 2" equally likely as getting a "Sum of 7"?
Record sums here:
04 Mystery Bag
Pull 5 marbles (replace them after each pull). Based on your pulls, which color is unlikely?
Pulls: [Color] [Color] ...
The Rigged Game Slides Level 01
THE RIGGED GAME
Analyzing Chance & Likelihood
The Teacher's Bet
"I will flip this coin 10 times. Every time it lands on Heads, I win. Every time it lands on Tails, you win."
Spoiler Alert: The teacher is using a double-headed coin.
Is this game FAIR? Why or why not?
Decoding the Odds
01
Theoretical Probability
What should happen based on math.
02
Experimental Probability
What actually happens when we test it.
The Formula
\[ P(Event) = \frac{\text{Successes}}{\text{Total Possibilities}} \]
The Likelihood Scale
0
1/2
1
IMPOSSIBLE
0%
UNLIKELY
25%
EQUALLY LIKELY
50%
LIKELY
75%
CERTAIN
100%
Rigged Game Teacher Guide Teacher Facilitation Guide
Lesson 1
The Rigged Game: Intro to Probability
Learning Objectives
Define probability as a number from 0 to 1.
Correctly use vocabulary: impossible, unlikely, equal, likely, certain.
Identify that games can be manipulated to change odds.
Materials Needed
Double-headed coin (or a coin where you "cheat")
Arcade Station Materials: Dice, Coins, Spinners, Marble Bags
"Likelihood Lab" Worksheets
The Hook: The "Rigged" Coin Bet (10 mins)
Challenge a student to a coin flip contest. You choose Heads, they get Tails. Offer a small reward if they win (like a sticker or bonus point). Teacher Tip: Use a double-headed coin or simply "trick" the flip to ensure heads almost every time.
"When you win 10 times in a row, ask the class: 'What are the chances of that happening?' This leads into the scale of likelihood."
Instruction: The Scale (15 mins)
Walk through the slides. Emphasize that Probability is just a way to put a number on "happen-ability."
0: Literally can't happen. (e.g., rolling a 7 on one die)1: Will always happen. (e.g., the sun rising)0.5: True randomness like a fair coin.
Guided Practice: Arcade Stations (25 mins)
Set up 4 stations around the room. Students rotate every 5-6 minutes.
Station Watch For... 01 Coin Flip Students thinking "Equal" means exactly 5/5. Explain variability. 02 Spinner Uneven spinners where one section is visually larger. 03 Dice Duel Students noticing 7 is common because more combinations make 7. 04 Mystery Bag The "Law of Small Numbers"—5 pulls might give a weird result!
Debrief & Exit (5 mins)
Collect the worksheets. Ask one final question: "If a game is 'Certain' that the house wins, would you play it? Why?"
Possibility Pathways Slides Level 02
POSSIBILITY PATHWAYS
Mapping the Sample Space
The Outfit Challenge
You are packing for a weekend trip. You have:
3 Shirts (Red, Blue, Green)
2 Pairs of Pants (Jeans, Khakis)
How many unique outfits can you create?
?
?
?
?
The Sample Space
The Sample Space is the complete set of all possible outcomes for an event.
We use 3 main tools to find it:
1. Lists
Writing them all down
2. Tables
Grid for two events
3. Tree Diagrams
Branching possibilities
Tool: Tree Diagrams
Tree diagrams show outcomes branching out step-by-step.
Example: Flipping a coin twice.
Total Outcomes = 4
HH, HT, TH, TT
START
Heads
Heads (HH)
Tails (HT)
Tails
Heads (TH)
Tails (TT)
Possibility Pathways Worksheet Possibility Pathways
Constructing Sample Spaces & Calculating Probability
Level 02
Player Name
Date
01
Branching Out (Tree Diagrams)
A frozen yogurt shop offers 2 Sizes (Small, Large) and 3 Toppings (Sprinkles, Berries, Chocolate).
A) Complete the Tree Diagram:
START
Small
Sprinkles
__________
__________
Large
__________
__________
__________
B) List the Sample Space:
Write every combination (e.g., Small, Sprinkles)
02
The Power of the Grid
You roll a red die and a blue die. Complete the table to show the SUM of the outcomes.
<table class="w-full text-center border-collapse"><tbody><tr class="bg-indigo-50"><td class="p-3 border border-slate-300 font-black italic text-indigo-600">RED \ BLUE</td><td class="p-3 border border-slate-300 font-bold">1</td><td class="p-3 border border-slate-300 font-bold">2</td><td class="p-3 border border-slate-300 font-bold">3</td><td class="p-3 border border-slate-300 font-bold">4</td><td class="p-3 border border-slate-300 font-bold">5</td><td class="p-3 border border-slate-300 font-bold">6</td></tr><tr><td class="p-3 border border-slate-300 font-bold bg-indigo-50">1</td><td class="p-3 border border-slate-300">2</td><td class="p-3 border border-slate-300">3</td><td class="p-3 border border-slate-300">4</td><td class="p-3 border border-slate-300">5</td><td class="p-3 border border-slate-300">6</td><td class="p-3 border border-slate-300 text-slate-300 italic">...</td></tr><tr><td class="p-3 border border-slate-300 font-bold bg-indigo-50">2</td><td class="p-3 border border-slate-300">3</td><td class="p-3 border border-slate-300">4</td><td class="p-3 border border-slate-300">5</td><td class="p-3 border border-slate-300 text-slate-300 italic">...</td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td></tr><tr><td class="p-3 border border-slate-300 font-bold bg-indigo-50">3</td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td></tr><tr><td class="p-3 border border-slate-300 font-bold bg-indigo-50">4</td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td></tr><tr><td class="p-3 border border-slate-300 font-bold bg-indigo-50">5</td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td></tr><tr><td class="p-3 border border-slate-300 font-bold bg-indigo-50">6</td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300"></td><td class="p-3 border border-slate-300 font-bold">12</td></tr></tbody></table>
Possibility Pathways Answer Key Teacher Answer Key
Lesson 2
Possibility Pathways: Sample Spaces
Part 1: Branching Out (Yogurt Shop)
A) Tree Diagram Content:
Small → Sprinkles, Berries, ChocolateLarge → Sprinkles, Berries, Chocolate
B) Sample Space List:
{S,Sprinkles}, {S,Berries}, {S,Choc}, {L,Sprinkles}, {L,Berries}, {L,Choc}
Total Outcomes: 6
Part 2: The Dice Grid (Sums)
<table class="w-full max-w-md text-center border-collapse text-xs mb-6"><tbody><tr class="bg-slate-100"><td class="p-1 border border-slate-300 font-bold">R\B</td><td class="p-1 border border-slate-300">1</td><td class="p-1 border border-slate-300">2</td><td class="p-1 border border-slate-300">3</td><td class="p-1 border border-slate-300">4</td><td class="p-1 border border-slate-300">5</td><td class="p-1 border border-slate-300">6</td></tr><tr><td class="p-1 border border-slate-300 font-bold bg-slate-100">1</td><td class="p-1 border border-slate-300 font-bold text-emerald-600">2</td><td class="p-1 border border-slate-300">3</td><td class="p-1 border border-slate-300">4</td><td class="p-1 border border-slate-300">5</td><td class="p-1 border border-slate-300">6</td><td class="p-1 border border-slate-300 font-bold text-red-600">7</td></tr><tr><td class="p-1 border border-slate-300 font-bold bg-slate-100">2</td><td class="p-1 border border-slate-300">3</td><td class="p-1 border border-slate-300">4</td><td class="p-1 border border-slate-300">5</td><td class="p-1 border border-slate-300">6</td><td class="p-1 border border-slate-300 font-bold text-red-600">7</td><td class="p-1 border border-slate-300">8</td></tr><tr><td class="p-1 border border-slate-300 font-bold bg-slate-100">3</td><td class="p-1 border border-slate-300">4</td><td class="p-1 border border-slate-300">5</td><td class="p-1 border border-slate-300">6</td><td class="p-1 border border-slate-300 font-bold text-red-600">7</td><td class="p-1 border border-slate-300">8</td><td class="p-1 border border-slate-300">9</td></tr><tr><td class="p-1 border border-slate-300 font-bold bg-slate-100">4</td><td class="p-1 border border-slate-300">5</td><td class="p-1 border border-slate-300">6</td><td class="p-1 border border-slate-300 font-bold text-red-600">7</td><td class="p-1 border border-slate-300">8</td><td class="p-1 border border-slate-300">9</td><td class="p-1 border border-slate-300">10</td></tr><tr><td class="p-1 border border-slate-300 font-bold bg-slate-100">5</td><td class="p-1 border border-slate-300">6</td><td class="p-1 border border-slate-300 font-bold text-red-600">7</td><td class="p-1 border border-slate-300">8</td><td class="p-1 border border-slate-300">9</td><td class="p-1 border border-slate-300">10</td><td class="p-1 border border-slate-300">11</td></tr><tr><td class="p-1 border border-slate-300 font-bold bg-slate-100">6</td><td class="p-1 border border-slate-300 font-bold text-red-600">7</td><td class="p-1 border border-slate-300">8</td><td class="p-1 border border-slate-300">9</td><td class="p-1 border border-slate-300">10</td><td class="p-1 border border-slate-300">11</td><td class="p-1 border border-slate-300 font-bold text-indigo-600">12</td></tr></tbody></table>
Carnival Crimes Slides Level 03
Carnival Crimes
The Math of "Fairness"
Would You Play?
THE DEAL
"Pay $1 to play. You have a 10% chance to win $2."
YES
NO
Think: If you played 100 times, what would happen?
What is "Fair"?
Fair Game
A game where everyone has an equal chance of winning, or where the expected value is zero.
The House Edge
The mathematical advantage that the person running the game has over the player.
In a fair game:
Case Study: The Duck Pond
There are 50 plastic ducks in a pond.
1 Duck has a Gold Star (Big Prize)
9 Ducks have Blue Dots (Small Prize)
40 Ducks have No Marking (No Prize)
Calculate the P(Win Any Prize):
10 / 50 = 1/5 = 20%
🦆🦆🦆🦆🦆 🦆🦆🦆🦆⭐ 🔵🔵🦆🦆🦆 🦆🦆🦆🦆🦆
Is 20% fair for a $5 ticket?
Carnival Crimes Analysis Worksheet Carnival Crimes
Student Forensic Probability Report
Detective Name
"The local carnival is in town, but the math doesn't add up. Your job is to analyze three popular games and determine if they are mathematically FAIR or if a CRIME against probability is being committed."
Case #01
The Color Wheel
The Rules
The wheel has 8 equal sections. 1 Section is GOLD, 3 are SILVER, and 4 are BLACK. It costs $2 to spin. To win a prize, you must land on GOLD.
P(Win Gold)
P(Lose - Black or Silver)
Verdict & Why?
Case #02
Dice Dash
The Rules
Roll two standard dice. If the sum is 10, 11, or 12, you win a giant teddy bear. Any other sum is a loss.
Total Possible Sums
P(Win Sum ≥ 10)
Verdict & Why?
Case #03
The Marble Mine
The Rules
A bag contains 5 Red, 5 Blue, and 10 White marbles. You pull one. If it is NOT WHITE, you win.
P(Win - Red or Blue)
Is this game "Fair"?
Verdict & Why?
Carnival Crimes Teacher Guide Teacher Discussion Guide
Lesson 3
Carnival Crimes: The Fairness Debate
The Mathematical Verdicts
Case #01 Color Wheel
P(Win) = 1/8 (12.5%)
Highly Unfair. The house wins 87.5% of the time.
Case #02 Dice Dash
P(Win) = 6/36 (16.7%)
Unfair. Only sums of 10 (3), 11 (2), and 12 (1) win.
Case #03 Marble Mine
P(Win) = 10/20 (50%)
Mathematically Fair. Equal chance to win or lose.
Facilitation Questions
Question 1: The "Feel" of Fairness
"Why do people play Game #1 (Color Wheel) if the odds are so bad?"
Look for: Bright colors, visual excitement, low cost ($2), "I feel lucky" fallacy.
Question 2: Ethical Design
"Is it 'wrong' for a carnival to have unfair games?"
Look for: It's entertainment, not an investment; the prizes cost money so the carnival has to make profit; transparency issues.
Question 3: Fixing the Game
"How could we make Game #2 (Dice Dash) perfectly fair?"
Look for: Change the winning sums to include 18 outcomes total (e.g., all even numbers, or sums 7-12, etc.)
Preparation for Lesson 4
"As you debrief, tell students that tomorrow they will be the ones designing the games. They need to decide if they want their game to be 'Fair' or 'House Favored'."
Game Architect Slides Level 04
Game Architect
Building the Probability Engine
Your Mission
The Goal
Design a game of chance using 1-2 manipulatives (dice, coins, cards, or spinners) that looks appealing to play.
The Constraint
You must prove the probability of winning using a sample space model (list, table, or tree).
Choose Your Strategy
THE FAIR PLAY
P(Win) is exactly 50%. A true contest of luck where no one has an advantage.
THE HOUSE WINS
P(Win) is between 20-40%. Harder for the player, better for the carnival profit.
THE "SURE BET"
P(Win) is over 60%. Used to attract players or for "bonus rounds" in an arcade.
Architecture Checklist
1
Name Your Game: Make it sound exciting!
2
Choose Gear: Will you use 2 dice? 1 spinner and 1 coin?
3
Define "Win": What exact outcome leads to a prize?
4
Calculate Probability: Fill out your blueprint before testing!
Game Architect Blueprint Worksheet Game Architect Blueprint
Design & Probability Model Phase
Project Phase 01
Designer(s): ____________________
01 Concept & Story
Arcade Game Name
Manipulatives Used (Dice, Coins, etc.)
How to play (The Rules)
02 The Probability Model (Sample Space)
Show ALL possible outcomes. Use a list, table, or tree diagram in the space below.
03 The Winning Probability
Winning Outcomes
Total Outcomes
P(Win) %
Is your game FAIR, HOUSE-FAVORED, or PLAYER-FAVORED? Explain why based on your math:
Game Design Assessment Rubric Game Design Rubric
Subject: Probability & Decision Making
Criteria Expert (4) Proficient (3) Developing (2) Novice (1) Sample Space Model Model is 100% accurate, clearly organized, and shows all possible outcomes. Model is mostly accurate with only minor formatting errors. Model is incomplete or has significant mathematical errors. No sample space model provided or entirely incorrect. Theoretical Probability Probability of winning is correctly calculated as a fraction, decimal, and %. Probability is correctly calculated in at least two forms. Calculation is attempted but contains errors in division or identification of outcomes. No calculation or incorrect use of formula. Fairness Analysis Provides a deep explanation of fairness using mathematical evidence. Correctly identifies fairness type with basic evidence. Identifies fairness type but explanation lacks mathematical support. Unable to explain if the game is fair or unfair. Game Mechanics Rules are clear, creative, and the game is fully functional. Rules are understandable and the game works as intended. Rules are confusing or game mechanics have "bugs." Game is unplayable or lacks clear rules.
Teacher Feedback
Final Level / 16
The Casino Test Slides Level 05
The Casino Test
Experimental vs. Theoretical Results
Grand Opening
The arcade is now open for business!
The Player's Job:
Test at least 5 different games and record every win/loss.
The Designer's Job:
Manage your station and help players track their data.
The Final Showdown
Theoretical
"The Math Prediction"
Calculated by dividing successful outcomes by total outcomes. This is what SHOULD happen in a perfect world.
Experimental
"The Real Data"
Calculated by dividing actual wins by actual trials. This is what ACTUALLY happened during our test today.
"Do they always match? Why might they be different?"
Analyzing the Night
If we played a game 10,000 times, would the experimental result get closer or further from the theoretical?
Which game was the Luckiest (Experiment > Theory)?
Which game was the Unluckiest (Experiment < Theory)?
Casino Trial Log Worksheet Casino Trial Log
Experimental Data Collection
Testing Phase
Name: ______________________
Game Name Tally of Trials (10 total) Total Wins Experimental P(Win) 1. / 10 2. / 10 3. / 10 4. / 10 5. / 10
Designer Post-Analysis
Compare YOUR game's theoretical probability to the actual data collected from your players:
Theoretical P(Win)
Experimental P(Win)
Based on the data, would you change any rules to make the game fairer or more profitable? Why?
"The experimental probability will approach the theoretical probability as the number of trials increases." (Law of Large Numbers)
Probability Post Mortem Exit Ticket Probability Post-Mortem
Exit Ticket
Designer Name
Score
1. Which of these is a FAIR probability for a coin-flip game where the winner gets a point for heads?
P(Win) = 0.25
P(Win) = 0.50
P(Win) = 1.00
2. If you play a game 1,000 times and your Experimental Probability is 48%, but the Theoretical is 50%, did you have "good luck" or "bad luck"? Explain.
True or False:
TRUE
Experimental results always match theoretical math exactly.
