Jackpot Journey Slides JACKPOT JOURNEY
Mastering Expected Value
Which Game Would You Play?
A
The Sure Thing
Guaranteed gain
B
The Gamble
10% chance to win $100
90% chance to win $0
High risk, high reward
How do we compare these mathematically?
The Probability Distribution
Random Variable (\(X\))
The numerical outcome of a random process (like "Profit").
Probability (\(P(X)\))
The likelihood of each outcome occurring.
Game B Distribution
Outcome (\(x\)) Prob. \(P(x)\) $100 0.10 $0 0.90
Note: Total probability must equal 1.00!
The Expected Value Formula
\[ E(X) = \sum [x \cdot P(x)] \]
"Multiply each outcome by its probability, then add them all up."
This tells us the average outcome if we played the game many times.
Let's Calculate: The Dice Duel
Practice #1
The Rules:
Roll a 6: Win $12
Roll a 1-5: Lose $3
Is this a "fair" game? Will you make money in the long run?
Step 1
List Outcomes (\(x\))
Step 2
Find Probs (\(P(x)\))
Step 3
Multiply & Sum
Outcome (\(x\)) Prob. (\(P(x)\)) \(x \cdot P(x)\) $12 1/6 ? -$3 5/6 ?
Dice Duel: The Math
Win:
\( 12 \cdot \frac{1}{6} = +2 \)
Loss:
\( -3 \cdot \frac{5}{6} = -2.50 \)
Total \(E(X)\):
-$0.50
What does this mean?
"On average, for every game played, you will lose 50 cents."
The casino wins this game in the long run!
Theory vs. Reality
Theoretical
What we expect to happen based on the math.
"The Law of Large Numbers says the more we play, the closer we get to this number."
Experimental
What actually happens in our trials.
"In 10 games, anything can happen! Luck plays a part in the short run."
Ready to test the math? Let's play!
Odds Overcome Guide ODDS OVERCOME GUIDE
Teacher Facilitation & Monitoring
Target Standard
CO HS.S-MD.A.5.a
Lesson Objectives
Define a random variable as the numerical outcome of a chance process.
Construct a probability distribution table for a game of chance.
Calculate expected value (E[X]) using the summation formula.
Contrast theoretical expected value with experimental mean payoff across repeated trials.
Common Misconceptions
Gambler's Fallacy
Students may believe that if they "lose" several times, they are "due" for a win. Emphasize that each trial is independent.
Summing without Weighting
Students often just average the outcomes (e.g., $10 and $0 = $5 expected) without accounting for the probabilities (weights) of those outcomes.
Materials Needed
6-sided dice (1 per student)
Calculators
Jackpot Journey Slides
Payoff Path Worksheets
Trial Tracker Sheets
Slide-by-Slide Facilitation
1-2
The Hook & Intuition
"Ask: If you could play Game A or B 100 times, which would leave you with more money?"
Most students choose B for the 'big win'. Note their reasoning. Transition to: "How can we prove which is better over time?"
3-4
Vocabulary & The Formula
Focus on Random Variables . In gambling, the variable is nearly always 'net profit'. Ensure students see that probabilities must sum to 1.
Key Prompt
"The Sigma (\(\Sigma\)) just means 'total'. We are adding up the 'contribution' of each possible outcome."
5-6
Guided Calculation
Model the table setup. Let students calculate \(12 \cdot (1/6)\) and \(-3 \cdot (5/6)\).Scaffold: If they struggle with fractions, use decimals (0.17 and 0.83), but emphasize the fraction represents the 'exact' chance.
7
Transition to Experiment
Hand out the Trial Tracker . Every student rolls 12 times. This gives the group a significant sample size (e.g., 5 students = 60 trials) to see the mean converge toward -$0.50.
PROGRESS MONITORING TRACKER
Use this table to record student performance during the intervention. Score on a 1-3 scale (1: Needs significant support, 2: Prompting required, 3: Independent mastery).
Payoff Path Worksheet Payoff Path Worksheet
Name: ____________________________________
Date: ____________________________________
Goal: Calculate the "Expected Value" (the average outcome over many games) for different games of chance.
Task 1 The High-Stakes Flip
You flip a fair coin. If it's Heads , you win $10 . If it's Tails , you win $0 .
1. Build the Distribution
Outcome (\(x\)) Probability (\(P(x)\)) $10 $0
2. Calculate E(X)
\( (10 \cdot \text{____}) + (0 \cdot \text{____}) \)
\( E(X) = \) $__________
Task 2 The Lucky Deck
In a small deck of 10 cards, there is 1 Ace and 9 Number Cards.$50 . If you pull a Number Card, you lose $5 .
Outcome (\(x\)) Prob. (\(P(x)\)) \(x \cdot P(x)\) +$50 1/10 (0.10) -$5 Expected Value: $__________
If you play this game 100 times, would you expect to have more money or less money than when you started? Why?
Task 3 The Dice Duel (Trial Setup)
This is the game we will play in class! Calculate the Theoretical Expected Value first.
Roll a 6
+$12
Roll a 1-5
-$3
Step 1: The Math Table
Outcome (\(x\))
Prob. (\(P(x)\))
Product (\(x \cdot P\))
Theoretical E(X):
$
Trial Tracker Sheet TRIAL TRACKER
Record your results for the Dice Duel experiment.
Student: ____________________________
Trial Dice Roll (1-6) Payoff ($) 1 2 3 4 5 6 7 8 9 10 11 12
Total Profit/Loss
Sum all your payoffs from the table on the left.
$
Experimental Mean
Divide your Total by the number of trials (12).
____
/ 12 =
$
The Great Comparison
Theoretical E(X)
-$0.50
Your Result
$________
Was your result higher or lower than the theoretical average? Explain why it might be different.
Reminder:
6 = +$12
1-5 = -$3
Winning Hand Exit Ticket WINNING HAND
Exit Ticket: Expected Value
Name: ________________________
Date: _________
The Challenge: The 4-Card Draw
A special deck has 4 cards. You draw 1 card at random.$20 .$30 .
1. Create the Probability Distribution
Outcome (\(x\))
Probability (\(P(x)\))
+$20
-$30
2. Calculate Expected Value \(E(X)\)
Total E(X): $ ________
3. Interpretation
In the "long run" (if you played this game 1,000 times), would you expect to be winning money or losing money? Explain.
