Mastering Mathematical Reasoning · 6th-class · Mathematical Investigations and Projects · Summer Term

Probability Fair Games

Students will design and analyze simple probability games, determining fairness and calculating expected outcomes.

• NCCA Curriculum Specifications
• NCCA: Primary - Chance
• NCCA: Primary - Problem Solving

About This Topic

Probability Fair Games introduces students to designing and analyzing games of chance, like spinners, dice, or card draws. They calculate probabilities for each outcome, determine fairness by checking if the win chance equals 1/2, and compute expected values to predict long-term results. This connects fractions and ratios from earlier terms to practical applications, such as evaluating carnival stalls or lotteries.

Aligned with NCCA Primary Chance and Problem Solving strands, the unit addresses key questions: explaining why a game is fair or unfair, adjusting rules to balance probabilities, and forecasting outcomes over many plays. Students collect data from trials, graph results, and discuss discrepancies between short-term luck and theoretical expectations. These investigations build skills in prediction, data analysis, and justification vital for summer term projects.

Active learning shines here because students physically create and test games. When they build spinners, run hundreds of trials collaboratively, and tweak rules based on evidence, abstract probabilities turn into observable patterns. This approach deepens understanding, encourages perseverance through variability, and makes maths feel like playful problem solving.

Key Questions

1. Apply probability concepts to explain whether a game of chance is fair or unfair.
2. Analyze how to make an unfair game fair by adjusting its rules or outcomes.
3. Predict the long-term results of playing a probability game multiple times.

Learning Objectives

• Design a simple probability game with clear rules and outcomes.
• Calculate the probability of specific outcomes in a designed game using fractions or decimals.
• Analyze a game to determine if it is fair or unfair based on calculated probabilities.
• Propose specific rule adjustments to transform an unfair game into a fair one.
• Predict the expected number of wins or losses over a set number of trials for a given game.

Before You Start

Introduction to Fractions and Ratios

Why: Students need to understand how to represent parts of a whole and compare quantities to calculate probabilities.

Data Collection and Representation

Why: Students will collect data from game trials, so prior experience with recording and organizing data is beneficial.

Key Vocabulary

Probability	The measure of how likely an event is to occur, often expressed as a fraction or decimal between 0 and 1.
Fair Game	A game where each player has an equal chance of winning. In a two-player game, this means each player has a 1/2 probability of winning.
Outcome	A possible result of a probability experiment or game, such as rolling a 3 on a die or landing on 'red' on a spinner.
Expected Value	The average outcome of an event if it were repeated many times. It helps predict long-term results.
Theoretical Probability	The probability of an event occurring based on mathematical calculation, not on experimental results.

Watch Out for These Misconceptions

Common Misconception

A game is fair if you win about half in 10 plays.

What to Teach Instead

Fairness depends on equal theoretical probabilities, not short-term results which vary due to chance. Group trials of 100+ plays show convergence to expectations; peer sharing of data charts corrects over-reliance on small samples.

Common Misconception

All outcomes in a game have equal chance unless stated.

What to Teach Instead

Custom games often have unequal sections or weights. Students designing their own spinners measure and calculate actual probabilities, revealing hidden biases through hands-on testing and adjustment.

Common Misconception

Expected value is the result of every single play.

What to Teach Instead

Expected value is a long-term average. Class-wide simulations with hundreds of trials demonstrate this; discussions connect individual streaks to the bigger picture, building statistical intuition.

Active Learning Ideas

Game Design Workshop: Custom Spinner Challenge

Pairs draw spinners on paper plates with 4-8 sections, assigning colours for wins and losses. They calculate exact win probability as a fraction and expected value per spin. Pairs test 50 spins, tally results, and graph actual versus predicted wins.

45 min·Pairs

Fairness Testing Circuit: Game Rotations

Prepare three games: biased coin, uneven die, balanced cards. Small groups rotate every 10 minutes, run 20 trials per game, record win rates, and vote on fairness. Debrief as a class compares group data to reveal patterns.

50 min·Small Groups

Unfair to Fair Makeover: Rule Tweaks

Provide an unfair game, like a 3:1 spinner. Groups propose and test rule changes, such as adding sections or combining outcomes, to achieve 1/2 probability. They justify adjustments with calculations and trial data.

35 min·Small Groups

Long-Term Prediction Relay: Trial Marathon

Whole class simulates 200 plays of one game by passing a die or spinner. Tally cumulative wins on a shared chart. Predict and verify if results approach expected value over time.

30 min·Whole Class

Real-World Connections

• Carnival game operators design games of chance. They use probability to ensure their games are profitable over many plays, meaning they are often unfair to the player in the long run.
• Board game designers carefully balance the probabilities of different events, like drawing cards or rolling dice, to create engaging and fair gameplay for all participants.
• Casinos use complex probability calculations to set the odds for games like roulette or slot machines, ensuring a statistical advantage for the house over time.

Assessment Ideas

Quick Check

Provide students with a spinner divided into 4 unequal sections (e.g., 2 red, 1 blue, 1 green). Ask: 'What is the theoretical probability of landing on red? Is this spinner fair for a two-player game where one player wins on red and the other wins on blue or green?'

Exit Ticket

Students are given a simple dice game: Player A wins if they roll a 1 or 2, Player B wins if they roll a 3, 4, 5, or 6. Ask them to write: 1. The probability of Player A winning. 2. The probability of Player B winning. 3. Is the game fair? Explain why or why not.

Discussion Prompt

Present students with a game where Player A wins if they draw an even number from a bag of 10 numbered balls (1-10), and Player B wins if they draw an odd number. Ask: 'How could we change the rules of this game to make it fair for both players? Describe at least two different ways.'

Frequently Asked Questions

How do you teach 6th class students to identify fair probability games?

Start with simple visuals like spinners divided equally. Guide students to list all outcomes, assign probabilities as fractions, and check if win chance equals 1/2. Use real trials to contrast theory with results, reinforcing that fairness means balanced long-term expectations, not every play.

What is expected value in probability fair games?

Expected value is the average outcome over many plays, calculated as sum of each outcome times its probability. For a game with 0.4 win (1 point) and 0.6 loss (0 points), it is 0.4. Students compute this before testing, then verify with class data to see predictions hold long-term.

How can active learning help students understand probability fair games?

Active approaches like building and testing custom games make probabilities concrete. Students run trials, graph variability, and refine rules collaboratively, experiencing how short-term luck evens out. This hands-on cycle boosts engagement, reveals misconceptions through data, and develops problem-solving skills central to NCCA Chance strand.

How to adjust an unfair game to make it fair for 6th class?

Identify unequal probabilities, then tweak: add equal sections to spinners, balance die faces, or group card outcomes. Students calculate before and after to confirm 1/2 win chance. Testing iterations ensures understanding, turning trial-and-error into evidence-based reasoning aligned with problem-solving standards.