Studio Aletheia · The Center for Quantitative Studies
DPSR Performance TaskSC 6th Grade Math
Design a Fair Carnival Game
Lesson 25 · Performance Task, Data, Probability & Statistical Reasoning
Today you'll design a carnival-style game using real data and probability, then justify that it's fair using everything you've learned across this entire strand. This task asks you to make sense of a real problem and connect ideas across statistics and probability, exactly the way mathematicians do.
Learning Targets and Success Criteria
Today's task asks you to connect ideas across this entire strand, and to solve a problem you haven't seen the exact shape of before.
What I will learn
- I can collect and summarize a small data set using measures I've learned this strand.
- I can calculate the probability of outcomes in a game I design.
- I can use the complement rule to check that my game's odds make sense.
- I can justify, with evidence, whether my game is fair or favors one side.
What success looks like
- I can display my game's outcome data using a method from this strand, like a box plot or a data list.
- I can calculate at least two probabilities correctly for my designed game.
- I can calculate at least one complement to verify my game's probabilities are complete.
- I can write a clear, evidence-based justification of my game's fairness.
Apply data, probability, and statistical reasoning skills to design and justify a fair game.
The words we'll use in today's lesson.
These terms will carry through today's mini-lesson, Data Lab, hands-on activity, journal, and exit challenge, and they'll keep coming back all year.
Sample Size
The number of data values collected from a population to represent it in a data set.
Median
The middle value of a data set when the values are placed in order from least to greatest.
Box Plot
A graph that displays a data set's five key values — lower extreme, lower quartile, median, upper quartile, and upper extreme — using a box and two whiskers.
Outlier
A data value that is much greater or much less than the rest of the data set.
Probability
A number from 0 to 1 (or 0% to 100%) that tells how likely an event is to happen, found as favorable outcomes over total outcomes.
Simple Event
An event made up of a single outcome, or a small set of favorable outcomes, from all the possible outcomes in a situation.
Complementary Event
The event that everything that is NOT the original event happens instead; P(not A) = 1 − P(A).
Where this shows up in the real world.
Thinking like a mathematician is not just a school skill. It's what people get paid to do every day, in jobs you may not have heard of yet.
Testing Whether a Game Is Fair
Data analysts at game companies collect outcome data from thousands of plays and check whether the results match the probability the game was designed for. If the data drifts too far from the expected probability, they flag it, exactly the kind of check you'll run on your own game today.
Pricing the Prize
Before a carnival or arcade sets a prize value, an actuary-style calculation figures out the probability of winning, so the average payout doesn't bankrupt the game. Your fairness justification today uses the very same logic: know the odds before you set the stakes.
One Task, the Whole Strand
Today's task asks you to connect ideas across this entire strand, and to solve a problem you haven't seen the exact shape of before.
This performance task leans hard on two of the five ways mathematicians think. It calls on Connections (MPS.C.1), because you'll notice that a carnival game's fairness question is really the same probability idea you've been using all along, just wearing a new costume. And it calls on Problem Solving (MPS.PS.1), because no one is going to hand you a template — you'll need to make sense of the design problem yourself and keep working even when your first version of the game doesn't turn out fair.
A fair game is one where the probabilityA number from 0 to 1 (or 0% to 100%) that tells how likely an event is to happen, found as favorable outcomes over total outcomes. of winning matches what players are told to expect, and where the prizes make sense given the odds. To design one, you'll choose a simple eventAn event made up of a single outcome, or a small set of favorable outcomes, from all the possible outcomes in a situation., like drawing a colored marble or spinning a section, calculate its probability, and use the complementary eventThe event that everything that is NOT the original event happens instead; P(not A) = 1 − P(A). rule to make sure you've accounted for every outcome. If your data comes from a real trial run, you might even display it with a tool from earlier in this strand, like a box plotA graph that displays a data set's five key values — lower extreme, lower quartile, median, upper quartile, and upper extreme — using a box and two whiskers., checking its medianThe middle value of a data set when the values are placed in order from least to greatest. and watching for any outlierA data value that is much greater or much less than the rest of the data set., to show how consistent your results were. Before you trust any of those results, check your sample sizeThe number of data values collected from a population to represent it in a data set. — a game tested on only two spins doesn't tell you much about whether it's really fair.
Adapted from Studio Aletheia's The Center for Quantitative Studies curriculum library, drawing on mathematical resources and the SC CCR Mathematics Standards.
Materials for the Data Lab.
- A. A spinner or bag of marbles to build your game
- B. Fraction/decimal/percent conversion chart
- C. Grid paper or your Data Journal for a box plot
- D. A calculator
- E. Your Task Packet
Before declaring a game fair, calculate the probability of winning, calculate its complement, and compare both to the prize value.
Task Lab
The task moves through four phases — work through each one in order, using tools from across this strand.
Build and Test Your Game
You'll build your designed game, test it, and present a fairness justification.
Data Journal Entry
Accountability Checklist
Is Your Game Fair?
In two or three sentences, explain whether your carnival game is fair. Use the term probability and reference the complementary event you calculated as evidence.
The Center for Quantitative Studies · Course Navigation
Move between lessons or return to the larger Studio Aletheia ecosystem.