Carnival Cash-In Slides Step Right Up!
Carnival
Cash-In
Mastering the Math of Expected Payoffs
Would You Play?
The Coin Flip Challenge
1 It costs $5 to play.
2 If it's Heads, you win $12.
3 If it's Tails, you win $0.
Think about it...
If you played 100 times, would you come out ahead?
YES
NO
Expected Value (EV)
The "average" amount you win or lose per play in the long run.
Outcome
×
Probability
=
Product
Sum of all (Outcome × Probability) = Expected Value
Step 1
List every possible outcome.
Step 2
Find the probability of each.
Step 3
Multiply & Add them up!
Game 1: The Golden Duck
Pick a rubber duck from the pond!
Cost to Play: $2
🦆
1/10 Ducks
Win $10 (Golden Duck!)
🦆
9/10 Ducks
Win $0 (Sad Quack!)
The Math Table
Outcome Prob. Product $10 0.1 $1.00 $0 0.9 $0.00 Total Expected Payoff $1.00
Wait... It costs $2 to play, but you only "expect" to get $1 back?
Net Gain
Expected Payoff
$1.00
-
Cost to Play
$2.00
NET GAIN = -$1.00
You lose $1 per game on average!
Is this a "fair" game? Why or why not?
Game 2: Bottle Ring Toss
Aim carefully! Three tiers of prizes.
Cost to Play: $5
🏆
Gold Tier
$20 Prize
Prob = 0.05 (5%)
🥈
Silver Tier
$10 Prize
Prob = 0.15 (15%)
🎪
No Prize
$0 Prize
Prob = 0.80 (80%)
Calculate the Expected Payoff:
Gold
($20 × 0.05)
Silver
($10 × 0.15)
None
($0 × 0.80)
=
$2.50
Why does the Carnival win?
The carnival is a business. They design games so that the Cost to Play is almost always HIGHER than the Expected Payoff.
The Player
Negative Net Gain
The House
Positive Profit
Mission: Carnival Auditor
Grab your Prize Picker Worksheet. You've been hired to find which games are worth the tickets and which ones are daylight robbery!
Calculate EV
Determine Gain
Explain Why
Prize Picker Worksheet Prize Picker Worksheet
Carnival Auditor Division • Department of Probability
Name:
Date:
The Auditor's Mission: Your job is to analyze the "Lucky Spinner" game. You will simulate playing the game, record your results, and then use math to find the "True Expected Value" of the game.
1
The Simulation: "The Lucky Spinner"
Game Rules: Each spin costs $3 . There are 4 sections on the spinner: Gold ($10), Blue ($2), and Gray ($0).
Tally Your Results (Spin 20 Times)
Gold ($10)
Blue ($2)
Gray ($0)
Total Tickets Spent (20 spins × $3):
Total Prize Money Won (Total $):
Did you "break even"? Based on your 20 spins, would you play this game again? Explain why.
2
The Math: Calculating Expected Payoff
Simulations are fun, but the Expected Value (EV) tells us what happens in the long run . Use the theoretical probabilities provided below to fill in the auditor's table.
Prob. of Gold
0.10
Prob. of Blue
0.30
Prob. of Gray
0.60
Prize Outcome (\(x\)) Probability (\(P(x)\)) Product (\(x \cdot P(x)\)) $10 (Gold) 0.10 $2 (Blue) 0.30
|
| $0 (Gray) | 0.60 |
|
| Expected Payoff (Total Sum) | $ |
Auditor's Final Report
Cost to Play:
$3.00
Expected Payoff:
Net Gain/Loss:
Official Recommendation:
Should the carnival keep this game exactly as it is? Why or why not? Use your calculations to support your answer.
Carnival Cash-In Teacher Guide Carnival Manager's Manual
Instructional Facilitation Guide • Tier 2 Intervention
Lesson: Carnival Cash-In
Objective
Students will calculate the expected payoff of a game of chance using a probability distribution and use that value to determine the game's "net gain" and overall fairness.
Standard: CO HS.S-MD.A.5.a Duration: 45-60 min
Materials
Carnival Cash-In Slides
Prize Picker Worksheet
10-sided dice or Spinners
Basic Calculators
The Opening Act (10 min)
1
The Coin Hook
Use Slide 2 to present the $5 Coin Flip game. Ask students to vote: "Would you play this 100 times? Why or why not?"
Facilitator Script: "Think about the average. Half the time you win $12, half the time you win $0. If you played twice, you'd spend $10 but win $12. That sounds like a good deal, right?"
The Main Event (30 min)
2
The "Lucky Spinner" Simulation
Hand out the Prize Picker Worksheet . Have students simulate the Lucky Spinner 20 times using a 10-sided die (1=Gold, 2-4=Blue, 5-10=Gray) or a physical spinner.
Monitor for accurate tallying and multiplication (Spins × Cost).
Encourage students to share if they are "winning" or "losing" mid-simulation.
3
Formalizing the Math
Transition to Part 2 of the worksheet. Use Slide 3 to introduce the formula: Outcome × Probability.
Scaffolding Tip
Compare the simulation results to the math. Ask: "Our math says we expect $2.60 back. Was your simulation close?"
Common Pitfall
Students often forget to subtract the "Cost to Play." Remind them: "Payoff is what the game gives you; Net Gain is what you actually keep."
Auditor's Checklist (Progress Monitoring)
Skill / Checkpoint Evidence of Mastery Intervention Strategy Identifying Outcomes Student correctly lists $10, $2, and $0 in the table. Ask: "What are all the different amounts of money you could walk away with?" Multiplying Decimals Student calculates \(10 \times 0.10 = 1.00\) correctly. Relate decimals to money (0.10 is 10 cents, 10 groups of 10 cents is $1). Concept of EV Student identifies that an EV of $2.60 means losing $0.40 per game.
Fair Share Exit Ticket ADMIT ONE
Fair Share Exit Ticket
Audit Completion Certificate • Grade 12 Statistics
Auditor:
Date:
Final Audit: "The Big Top Ball Toss"
The carnival is launching a new game. As the lead auditor, you must determine the Expected Payoff and Net Gain to see if it's fair for the players.
Cost to Play
$4.00
Rules
Throw a ball into a bucket. 20% chance of winning $10 , 80% chance of winning $0 .
Outcome (\(x\)) Prob (\(P(x)\)) Product (\(x \cdot P(x)\)) $10 0.20 $0 0.80
|
| Total Expected Payoff | $ _________ |
1. What is the Net Gain for the player?
(Expected Payoff - Cost to Play)
2. Is this game fair? Why or why not?
(Support your answer with math!)
