Probability Experiments and Likelihood

Active learning engages students directly in probability experiments, making abstract concepts like fractions and decimals tangible. When students physically flip coins, roll dice, or spin spinners, they connect theoretical probability to real-world data, building both number sense and critical thinking skills.

• 6th-class
• Mastering Mathematical Reasoning
• 4 activities
• 20 min–40 min

Learning Objectives

1. Calculate the theoretical probability of simple events using fractions and decimals.
2. Compare experimental results from probability trials to theoretical probabilities, identifying discrepancies.
3. Analyze how the number of possible outcomes affects the likelihood of an event.
4. Explain the application of probability in real-world scenarios like game design or insurance.
5. Design and conduct a simple probability experiment, recording and interpreting the data.

Inquiry Circle

⏱ 25 min·Pairs

Pairs Trial: Coin Flip Tracker

Pairs predict theoretical probability of heads (1/2), then flip a coin 100 times, tally results as fractions and decimals. Graph outcomes and compare to theory. Discuss why results vary and retry with more flips.

Prepare & details

Differentiate between theoretical probability and experimental results.

Facilitation Tip: During the Coin Flip Tracker, circulate and remind pairs to record each flip immediately to prevent tally errors.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 35 min·Small Groups

Small Groups: Dice Sum Challenge

Groups roll two dice 50 times, record sums, and calculate experimental probability of sum 7 (about 1/6). Convert to decimals, compare to theory. Adjust for three dice to see outcome effects.

Prepare & details

Analyze how the probability of an event changes with the number of possible outcomes.

Facilitation Tip: In the Dice Sum Challenge, ask groups to compare their results with another group before calculating probability to highlight variability.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 40 min·Whole Class

Whole Class: Spinner Prediction Board

Class creates spinners with 4-8 sections, predicts fractions for each colour. Everyone spins 20 times, logs on shared board. Analyze class data versus individual trials for patterns.

Prepare & details

Explain how insurance companies or game designers utilize probability in decision-making.

Facilitation Tip: For the Spinner Prediction Board, encourage students to predict outcomes before spinning to make the comparison between theory and practice explicit.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 20 min·Individual

Individual: Card Draw Journal

Each student draws from a deck without replacement 20 times, notes suit probabilities as decimals. Journal reflections on changes with fewer cards left. Share key insights in plenary.

Prepare & details

Differentiate between theoretical probability and experimental results.

Facilitation Tip: When students complete the Card Draw Journal, ask them to write about how their initial guess changed after seeing results.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Teaching This Topic

Start with simple experiments like coin flips to establish foundational ideas, then gradually increase complexity with dice or spinners to show how sample size affects results. Avoid rushing to conclusions; instead, guide students to notice patterns and discrepancies themselves. Research shows that when students articulate their reasoning before seeing outcomes, they engage more deeply with the material.

What to Expect

By the end of these activities, students will confidently express likelihood using fractions and decimals, compare experimental and theoretical probabilities, and explain why repeated trials matter. They will also justify their predictions with evidence from their trials and class discussions.

Watch Out for These Misconceptions

Common Misconception

During the Dice Sum Challenge, watch for students who assume every outcome is equally likely, even when dice have six faces.

What to Teach Instead

Use the Dice Sum Challenge to have students list all possible sums (2 through 12) and count the ways each can occur, showing that some outcomes have more combinations than others.

Common Misconception

During the Coin Flip Tracker, watch for students who believe a few trials accurately represent probability.

What to Teach Instead

In the Coin Flip Tracker, have students pool their results to create a class graph of 100+ flips, showing how the experimental probability approaches the theoretical 1/2 as trials increase.

Common Misconception

During the Spinner Prediction Board, watch for students who think past spins influence future spins.

What to Teach Instead

Use the Spinner Prediction Board to track results over 50 spins and discuss how each spin is independent, using the data to counter the idea of 'streaks' affecting likelihood.

Assessment Ideas

Exit Ticket

After the Coin Flip Tracker, give each student a coin and ask them to flip it 10 times, recording heads or tails. On the ticket, they should write the experimental probability of heads as a fraction and compare it to the theoretical 1/2, explaining any difference.

Discussion Prompt

During the Spinner Prediction Board, pose this question: 'Imagine a spinner with 3 equal sections: red, blue, green. If you spin it 100 times, would you expect exactly 33.3 red spins? Why or why not? What if the spinner had 100 sections instead?'

Quick Check

After the Dice Sum Challenge, show students a bag with 5 red marbles and 5 blue marbles. Ask: 'What is the probability of picking a red marble? What if we added 5 more red marbles? How does that change the probability?'

Extensions & Scaffolding

• Challenge: Ask students to design their own spinner with unequal sections and predict the probability of landing on each color. Have them test it 50 times and compare their predictions to the actual results.
• Scaffolding: Provide a template for recording coin flip trials with pre-labeled columns for heads, tails, and running totals to support organization.
• Deeper exploration: Introduce compound events by having students predict the probability of rolling two dice and getting a sum of 7, then test their hypothesis with 100 trials.

Key Vocabulary

Probability	The measure of how likely an event is to occur, expressed as a number between 0 and 1.
Theoretical Probability	The probability of an event calculated by dividing the number of favorable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely.
Experimental Probability	The probability of an event determined by conducting an experiment and dividing the number of times the event occurred by the total number of trials.
Outcome	A possible result of a probability experiment.
Likelihood	The chance of something happening or being true; often expressed as unlikely, equally likely, or likely.