Probability Fair Games

Active learning lets students see probability in action rather than just calculate it on paper. When they design spinners or test games, they confront real probabilities and fairness, making abstract fractions concrete.

• 6th-class
• Mastering Mathematical Reasoning
• 4 activities
• 30 min–50 min

Learning Objectives

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

Experiential Learning

⏱ 45 min·Pairs

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.

Prepare & details

Apply probability concepts to explain whether a game of chance is fair or unfair.

Facilitation Tip: During Game Design Workshop, circulate and ask each group to justify their spinner’s probability distribution before they cut and assemble it.

Setup: Varies; may include outdoor space, lab, or community setting

Materials: Experience setup materials, Reflection journal with prompts, Observation worksheet, Connection-to-content framework

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Experiential Learning

⏱ 50 min·Small Groups

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.

Prepare & details

Analyze how to make an unfair game fair by adjusting its rules or outcomes.

Facilitation Tip: During Fairness Testing Circuit, set a timer for each station so students rotate quickly and collect enough data to see patterns in the results.

Setup: Varies; may include outdoor space, lab, or community setting

Materials: Experience setup materials, Reflection journal with prompts, Observation worksheet, Connection-to-content framework

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Experiential Learning

⏱ 35 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.

Prepare & details

Predict the long-term results of playing a probability game multiple times.

Facilitation Tip: During Unfair to Fair Makeover, require students to present their rule changes with both probability calculations and a revised game board or card set.

Setup: Varies; may include outdoor space, lab, or community setting

Materials: Experience setup materials, Reflection journal with prompts, Observation worksheet, Connection-to-content framework

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Experiential Learning

⏱ 30 min·Whole Class

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.

Prepare & details

Apply probability concepts to explain whether a game of chance is fair or unfair.

Facilitation Tip: During Long-Term Prediction Relay, have students graph class results as they accumulate; this visual helps them see the law of large numbers in action.

Setup: Varies; may include outdoor space, lab, or community setting

Materials: Experience setup materials, Reflection journal with prompts, Observation worksheet, Connection-to-content framework

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Teaching This Topic

Start with hands-on design to build intuition, then move to systematic testing to challenge misconceptions. Avoid rushing to formulas—instead, let students discover why expected value predicts long-term results through repeated trials. Research shows that students grasp fairness better when they see how small sample sizes mislead, so emphasize data collection and class sharing over single attempts.

What to Expect

Students will explain why probabilities matter in game design, adjust rules to create fair games, and use expected values to predict long-term outcomes. They will also justify their decisions using calculations and data from trials.

Watch Out for These Misconceptions

Common Misconception

During Game Design Workshop, watch for students who claim a game is fair because 'it feels balanced' or 'we won about half in 10 spins.'

What to Teach Instead

Ask them to calculate the theoretical probability first, then collect 100 spins to compare. Have them present their data on a class chart to see how short-term results differ from long-term expectations.

Common Misconception

During Fairness Testing Circuit, watch for students who assume all outcomes in a custom spinner are equally likely unless told otherwise.

What to Teach Instead

Direct them to measure the angle of each section and calculate the actual probability before testing. Ask them to adjust their spinner if the sections are unequal and recalculate.

Common Misconception

During Long-Term Prediction Relay, watch for students who think the expected value must match every single trial outcome.

What to Teach Instead

Have them simulate 500 trials as a class and graph the results. Ask them to explain why streaks occur but the average approaches the expected value over time.

Assessment Ideas

Quick Check

After Game Design Workshop, provide a spinner with 3 unequal sections (e.g., 3/8, 3/8, 2/8). Ask students to calculate the probability of landing on each section and decide if it is fair for a two-player game where one wins on the first section and the other wins on the other two.

Exit Ticket

After Fairness Testing Circuit, give students a dice game where Player A wins on 1, 2, or 3 and Player B wins on 4, 5, or 6. Ask them to write the probability for each player, determine if the game is fair, and explain their answer using calculations.

Discussion Prompt

During Unfair to Fair Makeover, present a game where Player A wins on red cards and Player B wins on black cards from a standard deck. Ask students to suggest two different ways to change the rules or materials to make the game fair, and justify each with probability calculations.

Extensions & Scaffolding

• Challenge: Ask students to design a game with three players where each has a 1/3 chance of winning, using spinners or dice they create.
• Scaffolding: Provide pre-labeled spinners with unequal sections and ask students to calculate the actual probabilities before testing.
• Deeper exploration: Have students research real carnival games, collect probability data from photos or videos, and redesign one to be fair while keeping it fun.

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.