Dice Duel Slides Dice Duel
The Math of Independence
Compound Probability | Lesson 1
The "Hot" Hand?
"A roulette wheel has landed on RED ten times in a row."
Player A says:
"Black is 'due' to hit! I'm betting the farm on black."
Player B says:
"Red is on a streak! It's going to hit red again."
Who is mathematically correct?
Independent Events
Definition
Events where the outcome of the first event does not affect the probability of the second event.
Rolling a die
Flipping a coin
Spinning a roulette wheel
Memoryless
The dice don't know what happened last time.
The Multiplication Rule
For independent events A and B:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Probability of a 6
\( \frac{1}{6} \)
Probability of a 6
\( \frac{1}{6} \)
Probability of two 6s in a row: \( \frac{1}{36} \approx 2.7\% \)
The Gambler's Fallacy
The Sequence Probability
Probability of rolling five 7s in a row in Craps:
\( (\frac{1}{6})^5 = \frac{1}{7,776} \)
Very unlikely before you start rolling.
The Next Trial Probability
If you've already rolled four 7s, the probability of the next roll being a 7:
\( \frac{1}{6} \)
The history is irrelevant to the current roll.
The Mistake:
Confusing the probability of a long streak with the probability of the next outcome.
Dice Duel Challenge
?
The Scenario:
In a game of Craps, "Snake Eyes" (rolling a sum of 2) has a probability of \( \frac{1}{36} \).
What is the probability of rolling Snake Eyes twice in a row?
What is the probability of NOT rolling Snake Eyes for 50 rolls straight?
Is it ever "due"?
Work with your partner to calculate these before moving to the worksheet!
Dice Duel Worksheet Dice Duel
The Math of Independent Streaks
Name:
Date:
Core Concept: For independent events, \( P(A \cap B) = P(A) \cdot P(B) \). The outcome of previous trials has zero mathematical impact on the current trial.
1
Calculating the Odds
You are playing a simplified game of Craps where you win if you roll a "Natural" (a sum of 7 or 11). There are 8 ways to roll a 7 or 11 out of 36 possible outcomes.
A. Probability of one "Natural":
B. Probability of three "Naturals" in a row:
C. Probability of rolling "Snake Eyes" (Sum = 2) four times in a row:
2
The Fallacy Investigation
"The roulette wheel has landed on RED 6 times in a row. A gambler bets \$1,000 on BLACK, claiming it is 'overdue' because the law of averages must balance the colors out."
1. Mathematically, what is the probability that the 7th spin is Black? (Assume a European wheel with 18 Red, 18 Black, and 1 Green zero).
2. Contrast the probability of the entire sequence (7 Reds in a row) versus the probability of the single outcome (the 7th spin being Red).
3. Why do humans tend to believe the Gambler's Fallacy? Use the term "Independent Events" in your reasoning.
3
The Long Dry Spell
Casinos profit because players underestimate how long "independent dry spells" can last. If the probability of winning a specific slot machine spin is \( 1/20 \):
A. What is the probability of LOSING a single spin?
B. What is the probability of losing 50 spins in a row?
C. If a player has already lost 49 times, what is the probability they lose on the 50th spin?
D. Reflection: How does the answer to "B" explain why "chasing your losses" is a dangerous mathematical strategy?
"In a game of pure chance, the past has no power over the future."
Dice Duel Teacher Key Dice Duel | Teacher Key
Answer Guide & Instructional Tips
Part 1: Calculating the Odds
A. Probability of one "Natural" (7 or 11):
8/36 = 2/9 ≈ 22.2%
B. Probability of three "Naturals" in a row:
(2/9) × (2/9) × (2/9) = 8/729 ≈ 1.1%
C. Probability of rolling "Snake Eyes" (Sum = 2) four times in a row:
(1/36)^4 = 1/1,679,616 ≈ 0.0000595%
Instructional Tip: Emphasize how quickly compound probability for independent events approaches zero as the streak length increases.
Part 2: The Fallacy Investigation
1. Probability 7th spin is Black:
18/37 ≈ 48.65%
Crucial: It is NOT 50% or 100%. The previous 6 spins don't change the physics of the wheel.
2. Entire Sequence vs. Single Outcome:
The probability of anticipating a streak of 7 Reds before it starts is (18/37)^7 ≈ 0.61%. However, once 6 Reds have already occurred, the probability of the 7th spin being Red is simply the base probability of (18/37) ≈ 48.65%.
3. Human Psychology & Reasoning:
Expected answer: Humans possess a cognitive bias where they believe nature "corrects" streaks to reach an average. However, since the wheel spin is an independent event, it has no memory and no mechanism to compensate for previous results.
Part 3: Deep Dive - Probability of Absence
A. Probability of losing a single spin (1/20 win chance):
19/20 = 0.95 (95%)
B. Probability of losing 50 spins in a row:
(0.95)^50 ≈ 0.0769 (7.69%)
Insight: This is surprisingly high! Over 7% of players will lose 50 times in a row without a single win.
C. Probability of losing 50th spin after 49 losses:
19/20 (95%)
D. Chasing Losses Reflection:
Players think they are "due" for a win, but the math shows the probability of winning the next hand remains low. Chasing losses assumes the odds improve over time, but for independent events, the house edge remains constant and lethal to bankrolls over time.
Discussion Hook for Lesson 2:
"If I draw an Ace from a deck of cards and keep it, is the probability of the next card being an Ace still the same? Why is this different from rolling a die?"
Card Clash Slides Card Clash
The Power of Information
Compound Probability | Lesson 2
Everything Changes
Scenario A:
"I roll a die and get a 6. I put it back and roll again."
Independent
The first 6 doesn't care about the second roll.
Scenario B:
"I draw an Ace from a deck and keep it. I draw again."
Dependent
The missing Ace fundamentally changes the math.
In cards, information is profit.
Dependent Events
Definition
Events where the outcome of the first event changes the probability of subsequent events.
Drawing "without replacement"
Conditional probability
A♥
?
Trial 1: \( P(\text{Ace}) = \frac{4}{52} \)
Trial 2: \( P(\text{Ace}) = \frac{3}{51} \)
The Dependent Rule
For dependent events A and B:
\[ P(A \cap B) = P(A) \cdot P(B|A) \]
"Probability of A, then probability of B given that A already happened."
Trial A
Draw a King from a fresh deck.
\( \frac{4}{52} \)
Trial B | A
Draw another King given 1st was a King.
\( \frac{3}{51} \)
Blackjack: 21
A "Natural Blackjack" is an Ace and a 10-value card (10, J, Q, K).
Probability of Ace first:
\( \frac{4}{52} \)
Probability of 10-val second:
\( \frac{16}{51} \)
Compound Calculation:
\( \frac{4}{52} \cdot \frac{16}{51} \cdot 2 \)
(Why times 2?)
Because order could be Ace then 10, or 10 then Ace!
≈ 4.8%
Card Counting Logic
"If the table has seen ten Small Cards (2-6) and zero High Cards (10-A) dealt, what happens to the probability of the next card being an Ace?"
Denominator
52 → 42
Numerators (Aces)
4 → 4
New Chance
9.5%
(Up from 7.7%)
This is the Conditional Advantage. By tracking removed cards, you update your compound probability in real-time.
Deck Dynamics Activity Deck Dynamics
Analyzing Dependent Compound Events
Card Player:
Date:
The Rules of Removal
Consider a standard 52-card deck. You draw three cards without replacement. Show your setup for each compound probability.
A. Drawing three Hearts in a row:
Calculation: \( P(H_1) \cdot P(H_2 | H_1) \cdot P(H_3 | H_1 \cap H_2) \)
B. Drawing three Aces in a row:
Calculation: \( \frac{4}{52} \cdot \text{___} \cdot \text{___} \)
The Flush Draw Decision
Your Hand
9♠
J♠
The Board (The Flop)
2♠
4♥
A♠
You have 4 spades. There are 2 cards left to be dealt (the Turn and the River). To win, you need at least one of the next two cards to be a spade.
1. How many cards are left in the deck that you haven't seen?
2. How many spades are left in the deck?
3. Calculate the probability that neither of the next two cards is a spade.
\( P(\text{Not Spade}) = \frac{\text{Unknown - Spades}}{\text{Unknown}} \times \dots \)
Updating the Odds
In Blackjack, if you "Hit," you take another card. If your total exceeds 21, you lose (Bust).
You hold a 10 and a 6 (Total 16). You know that cards 6 or higher will make you bust.
Scenario A: Fresh Deck
Probability of drawing a 6, 7, 8, 9, 10, J, Q, K:
Scenario B: You saw 4 Kings and 4 Queens already dealt to other players.
Recalculate your "Bust" probability (assume cards seen are removed from the deck of 52):
Did your chance of busting go up or down? Why?
Dependent Compound Rule:
\( P(A \cap B) = P(A) \cdot P(B|A) \)
Card Clash Teacher Key Deck Dynamics | Teacher Key
Answer Guide & Strategic Insights
Part 1: Rules of Removal
A. Three Hearts in a row:
(13/52) × (12/51) × (11/50)
= 1,716 / 132,600 ≈ 1.29%
B. Three Aces in a row:
(4/52) × (3/51) × (2/50)
= 24 / 132,600 ≈ 0.018%
Part 2: Poker - The Flush Draw
1. Cards left: 47 (52 total - 2 in hand - 3 on board).
2. Spades left: 9 (13 total - 2 in hand - 2 on board).
3. Probability neither of next two is a spade:
(38/47) × (37/46) ≈ 0.651 (65.1%)
Pro-tip: To find the chance of getting at least one spade, subtract from 1. 1 - 0.651 = 34.9%.
Part 3: Blackjack Bust Probability
Scenario A (Fresh Deck):
Bust cards: 6, 7, 8, 9, 10, J, Q, K (8 types).
4 of each = 32 cards.
32/52 ≈ 61.5%
Scenario B (Counting):
Deck size: 52 - 8 (Kings/Queens) = 44.
Bust cards: 32 - 8 (Kings/Queens) = 24.
24/44 ≈ 54.5%
Result: Chance of busting went DOWN because high cards (which cause you to bust) were removed from the deck.
Teaching Insight: The Power of Information
The transition from independent to dependent probability is where students start to see math as a "weapon." In dice (Lesson 1), information is useless. In cards (Lesson 2), information changes the compound outcome. This is the foundation of why casinos kick out card counters—they are using conditional probability to shift the house edge.
Jackpot Odds Slides 7
21
49
Jackpot Odds
The Scale of Large Numbers
Compound Probability | Lesson 3
The Multiplication Principle
If there are n ways to do one thing, and m ways to do another, there are n × m ways to do both.
Simple Lottery (Pick 3)
10 × 10 × 10 = 1,000
Possible combinations for digits 0-9.
7
?
?
Winning Chance: 1 in 1,000
Scaling Up
Pick 4
1 in 10,000
Like picking a specific person in a crowded stadium.
Pick 6
1 in 1,000,000
Like picking a specific person in the city of Phoenix.
Powerball
1 in 292 Million
Like picking a specific person in almost the entire USA.
The Lesson:
When compound events have many stages, the probability collapses at an exponential rate.
Double Your Odds?
A common myth:
The Truth:
0.0000000034
Risk vs. Reward
Mathematically, doubling your odds of a near-impossible event still results in a near-impossible event.
"The change is negligible."
Human Intuition Fail
You are more likely to:
Be struck by lightning (1 in 1.2M)
Become a Saint (1 in 20M)
"The human brain is not evolved to intuitively grasp probabilities this small."
Lottery Logic Worksheet Lottery Logic
The Mathematics of Impossibility
Applicant Name:
Date of Entry:
1
Building the Odds
Use the multiplication principle to calculate the total number of possible combinations for each lottery format.
Scenario A: The "Daily 4"
You must pick 4 digits, each from 0-9. Digits can be repeated.
Total Combinations:
Scenario B: License Plate Lottery
3 letters (A-Z) followed by 3 numbers (0-9). Repeats allowed.
Total Combinations:
Powerball Simplified
To win, you must match 5 white balls (numbered 1-69) and 1 red "Powerball" (numbered 1-26).
Note: For this exercise, assume the white balls are drawn with replacement (even though they aren't in reality) to simplify the compound calculation.
Calculation Path:
\( (69)^5 \times 26 = \dots \)
2
Negligible Gains
If the probability of winning the Mega Millions is exactly \( \frac{1}{302,575,350} \):
Win Probability (1 ticket)
(Express as a decimal)
Win Probability (2 tickets)
(Express as a decimal)
Observation: Look at the decimals above. Does buying a second ticket change your "expected reality" in any meaningful way? Explain using the concept of scale.
3
Case Study: The $2.00 Choice
A person spends \$10 a week on lottery tickets for 40 years. Calculate the total expenditure: \$_________________
Contrast this financial outcome with the probability of winning a major jackpot. If you were a financial advisor, what mathematical argument would you use to discourage playing the lottery?
Jackpot Odds Teacher Key Lottery Logic | Teacher Key
Answer Guide & Large Number Context
1. Building the Odds
Scenario A: The "Daily 4"
10 × 10 × 10 × 10 = 10,000 combinations
Odds of winning: 0.01%
Scenario B: License Plate Lottery
26^3 × 10^3 = 17,576 × 1,000 = 17,576,000 combinations
Teaching Note: Point out how adding just 2 characters (letters) increases the difficulty by 1,700 times compared to Scenario A.
Scenario C: Powerball Simplified
69^5 × 26 ≈ 1,564,031,349 × 26 ≈ 40,664,815,074
Note: Real Powerball uses combinations (without replacement) for the white balls, resulting in 1 in 292,201,338. This simplified version (with replacement) shows an even larger scale, emphasizing the power of the multiplication principle.
2. Negligible Gains
1 Ticket Probability:
0.000000003305
2 Tickets Probability:
0.000000006610
Reflection Insight: While technically "doubled," the outcome for both numbers is statistically zero in any single human lifetime. The difference between 3 billionths and 6 billionths is negligible when the threshold for "likely" is so high. It is mathematically comparable to "almost impossible" vs. "still almost impossible."
3. Case Study: The $2.00 Choice
Total Expenditure Calculation:
\$10/week × 52 weeks × 40 years = \$20,800
The Mathematical Argument:
The "Opportunity Cost" argument: If that \$20,800 were invested in a simple 7% index fund instead, it would grow to nearly \$120,000. The mathematical expectation of the lottery is negative (usually \$0.50 back for every \$1.00 spent), whereas the expectation of compound interest is positive. You are essentially paying \$20k for a 0.00000003 chance of winning, versus a 100% chance of having \$120k.
"The lottery is a tax on people who are bad at math." — Common mathematical adage.
Game Architect Slides Game Architect
Designing for the House Edge
Compound Probability | Lesson 4
The Project
"Create a game of chance that requires two stages and calculates a win probability between 5% and 15%."
Phase 1: Design
Sketch your components (dice, cards, spinners) and define the win condition.
Phase 2: Math
Prove the theoretical probability using the compound rules from Lessons 1 & 2.
Multi-Stage Examples
Example A: Mixed Tools
"Draw a Red Card AND Roll a 6."
\( \frac{1}{2} \times \frac{1}{6} = 8.3\% \)
Example B: Dependency
"Draw a King, then draw another King without replacement."
\( \frac{4}{52} \times \frac{3}{51} = 0.45\% \)
The Complexity Rule
The more stages you add, the lower the probability goes. How do you keep the player engaged while ensuring the "House" wins most of the time?
The Business Side
Win vs. Reward
A game that is too hard to win will be ignored by players. A game that is too easy will bankupt the casino.
House Edge:
The mathematical advantage the game designer keeps.
The Architect's Checklist:
Clear win/loss conditions
Exactly 2+ stages
Show your work for \( P(\text{Win}) \)
Visual appeal & presentation
Prepare for Playtesting
Tomorrow, your classmates will play your game 50 times to see if your "Theoretical Math" matches their "Experimental Reality."
"Will the House always win?"
Casino Blueprint Project Guide Casino Blueprint
Project Design & Probability Proof
Game Architect
Architect(s):
Game Name:
1. Structural Design
Components (Circle all that apply):
Standard Dice Playing Cards Custom Spinner Coin Flip Colored Marbles
Stage 1 Condition:
Stage 2 Condition:
Visual Sketch / Layout
Draw your game board or setup here.
2. The Theoretical Proof
You must prove that your game is mathematically designed to win between 5% and 15% of the time. Show the fraction and the final decimal percentage.
P(Stage 1):
___ / ___
P(Stage 2 | Stage 1):
___ / ___
Compound Probability Calculation:
\( P(\text{Win}) = \)
= __________%
*Double check your math! If your win probability is over 15%, the "House" isn't making enough money. If it's under 5%, nobody will play!
3. Business Strategy
How did you adjust your game to hit the 5-15% target? (e.g., "I added another die," or "I made it without replacement").
Explain why a compound event is better for a "House Edge" than a single-stage event.
Game Evaluation Rubric Project Rubric
Probability Game Design Assessment
Criterion Exceptional (10 pts) Proficient (7 pts) Needs Work (4 pts) Mathematical Proof Probability is calculated flawlessly using correct compound rules. Includes fractions and decimals. Calculation is mostly correct but may contain a minor arithmetic error. Significant errors in compound rule application or missing proof entirely. Two-Stage Design Clear two-stage mechanic. Stages are logically connected and easy to follow. Includes two stages, but instructions are slightly ambiguous. Game is single-stage or stages are unrelated/nonsensical. House Edge Target Theoretical win probability falls exactly between 5% and 15%. Win probability is close (4-20%) but misses the optimal target. Game is either a "guaranteed win" or "statistically impossible." Visual Clarity Game setup is visually professional. All tools/components are clear. Game setup is legible and functional but lacks polish. Setup is messy or difficult to understand/play.
Assessment Summary
TOTAL SCORE:
/ 40
Feedback & Notes:
[Insert teacher observations on mathematical reasoning and creative game mechanics]
Lesson 4: Game Architect Rubric
Applied Probability in Gaming Scenarios Unit
House Edge Slides The House Edge
Theory vs. Experimental Reality
Compound Probability | Lesson 5
The Great Convergence
Law of Large Numbers
As the number of trials increases, the Experimental Probability will get closer and closer to the Theoretical Probability.
After 10 Plays:
"Chaos & Luck"
After 1,000 Plays:
"Mathematical Certainty"
Testing the Architects
Today you are the Audit Team. Your job is to play your classmates' games and gather data.
1
Play 50 trials of the game.
2
Record every Win and Loss.
3
Calculate the Experimental Win Rate.
The Comparison
"Does the architect's calculation survive the reality of the dice?"
Why the House Wins
Win Rate
10%
House Edge
90%
The House Edge is the mathematical certainity that the casino will win more than the player over time. Compound events make this edge predictable and scalable.
Unit Reflection
"If the math is guaranteed, is it still a game of CHANCE for the casino?"
Discuss: Why do casinos prefer high volumes of small bets over low volumes of large bets?
Playtest Log Sheet Playtest Log
Auditing Theoretical Probability
Auditor:
Station:
Game Architect(s):
Theoretical Win Probability (\( P_{th} \)):
__________ %
Auditor's Initial Reaction:
Before playing, look at the game. Does the math seem realistic for the physical components? Why or why not?
Trial Results (50 Rounds)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Total Wins:
____
Total Losses:
____
Experimental % (\( P_{ex} \)):
____ %
Data Comparison & Conclusion
1. Was \( P_{ex} \) higher, lower, or very close to \( P_{th} \)? Use the Law of Large Numbers to explain why this might be the case after only 50 trials.
2. Based on your results, is this game "fair"? (Define fairness as being consistent with its advertised theoretical odds).
House Edge Verification
"Trust the math, but verify the reality."
House Edge Teacher Key House Edge | Teacher Guide
Facilitation & Playtest Debrief
Facilitation Strategy
The Setup:
Divide the class into "Architects" (who stay at their game stations) and "Auditors" (who rotate). Each rotation should last approximately 15 minutes, allowing for 50 rapid-fire trials. Ensure students use tally marks or chips to keep the pace fast.
Data Collection:
Encourage auditors to watch for "hot streaks." If a game has a 10% win rate but someone wins 3 times in 5 trials, use that as a "teachable moment" for the Law of Small Numbers (the tendency to see patterns in small samples) vs. the Law of Large Numbers.
Discussion & Debrief
Q: Did anyone find a game where the experimental result was drastically different from the theory?
A: Likely yes. With only 50 trials, variation is high. This reinforces that casinos need thousands of players to ensure their profit margin becomes stable.
Q: How do casinos handle "lucky" players who beat the odds in the short term?
A: They keep them playing. As long as the player stays at the table, the Law of Large Numbers will eventually pull their results back toward the house edge. The only way to "beat" the house is to quit while you are in a high-variance streak.
Summary of Unit Skills:
Differentiate Independent (Dice) vs. Dependent (Cards).
Apply Multiplication Rule for multi-stage events.
Analyze the impact of scale (Lotteries).
Understand the relationship between Theoretical and Experimental outcomes.
The "House" Mindset:
"The casino does not gamble. The casino manages a portfolio of predictable mathematical outcomes. Only the players gamble."
Applied Probability in Gaming Scenarios © 2026
