Experimental Probability

Active learning lets students physically roll dice, spin spinners, and record outcomes, which builds concrete experience before abstract reasoning. When students collect their own data, they feel the pull of chance variability and see why large samples matter more than quick guesses.

• Grade 7
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Calculate the experimental probability of an event based on collected trial data.
2. Compare experimental probabilities to theoretical probabilities for simple chance events.
3. Analyze the effect of increasing the number of trials on the reliability of experimental probability.
4. Explain why short-term experimental results may deviate from theoretical predictions.
5. Design a simple simulation to model a real-world event using probability concepts.

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Probability Stations

Prepare four stations with spinners divided into unequal sections, dice rolls for sums, coin flips, and marble draws from bags. Students predict theoretical probabilities, conduct 20 trials at each, record frequencies on charts, and compare to predictions. Rotate groups every 10 minutes.

Prepare & details

Justify why experimental results might differ significantly from theoretical predictions in the short term.

Facilitation Tip: During Probability Stations, circulate with a quick checklist to note which students still add outcomes instead of replacing them in the sample space.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Inquiry Circle

⏱ 30 min·Pairs

Pairs Challenge: Increasing Trials

Partners select a chance device like a six-sided die. They predict outcomes, run 10, 30, and 50 trials cumulatively, plotting relative frequencies on a class graph after each set. Discuss how the line approaches theoretical values.

Prepare & details

Analyze how increasing the number of trials affects the reliability of experimental probability.

Facilitation Tip: In the Increasing Trials challenge, have pairs start with 10 trials, then add 10 more each round so everyone sees the cumulative graph grow.

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·Whole Class

Whole Class Simulation: Election Poll

Simulate an election with coloured beads in a bag representing votes. Class predicts winner based on theoretical probability, draws with replacement 50 times, tallies results, and revises predictions. Graph class data to show convergence.

Prepare & details

Explain in what ways a simulation can accurately model a complex real-world event.

Facilitation Tip: For the Election Poll simulation, assign roles like pollster, recorder, and data analyst so every student contributes to the shared data set.

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

⏱ 25 min·Individual

Individual: Custom Spinner Design

Each student draws a spinner with 3-6 sections of varying sizes, calculates theoretical probabilities, runs 100 trials, and creates a bar graph comparing results. Share one insight in a quick gallery walk.

Prepare & details

Justify why experimental results might differ significantly from theoretical predictions in the short term.

Facilitation Tip: When students design custom spinners, provide protractors and color-coded templates so geometric accuracy supports probability accuracy.

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

Teach experimental probability by letting variance happen first, then guiding students to quantify it with relative frequency. Avoid rushing to the ‘correct’ answer; instead, let repeated trials demonstrate convergence. Research shows students grasp the law of large numbers best when they plot their own messy data before smoothing it into a trend.

What to Expect

Students will explain why experimental results differ from theory and how more trials shrink that gap. They will justify short-term deviations and connect relative frequency to theoretical probability with clear calculations and graphs.

Watch Out for These Misconceptions

Common Misconception

During Probability Stations, watch for students who expect every 10-flip set to land exactly 5 heads and 5 tails.

What to Teach Instead

Pause the station rotation after the first set and ask groups to plot their results on a shared class graph, then prompt them to predict what might happen if they did 50 flips instead of 10.

Common Misconception

During Increasing Trials, listen for pairs who claim a fair coin is unfair after five consecutive heads.

What to Teach Instead

In that moment, have them add another 10 trials live and recalculate, then ask whether the new data changes their view before moving on.

Common Misconception

During the Election Poll simulation, watch for comments that the simulated poll should match the actual outcome exactly.

What to Teach Instead

Point to the shared results table and ask students to estimate the error margin by comparing different sample groups within the same class poll.

Assessment Ideas

Quick Check

After Probability Stations, give students a table with 20 recorded die rolls (e.g., 4 sixes) and ask them to calculate the experimental probability of rolling a six and compare it to the theoretical 1/6. Collect responses to identify who still confuses sample size with theoretical probability.

Discussion Prompt

During Increasing Trials, pose the scenario: ‘A class gets 7 heads in 10 flips—does this mean the coin is biased?’ Have students share their thoughts in pairs, then call on volunteers to explain their reasoning using the law of large numbers and their own cumulative graphs from the activity.

Exit Ticket

After the Custom Spinner Design, hand out a scenario: ‘A spinner with three equal sections lands on red 12 times in 20 spins.’ Ask students to write the experimental probability of red and explain one reason it might differ from the theoretical 1/3, then collect tickets to gauge understanding of variability and sample size.

Extensions & Scaffolding

• Challenge: Ask students to design a spinner that yields exactly 60% red over 100 trials and defend their sector angles with theoretical calculations.
• Scaffolding: Provide a pre-labeled graph with axes for trials and relative frequency to focus students on plotting rather than setup.
• Deeper exploration: Have students research real-world polls, compare sample sizes, and discuss how margin of error relates to the number of trials in your simulation.

Key Vocabulary

Experimental Probability	The probability of an event occurring based on the results of an experiment or observed trials. It is calculated as the number of times an event occurs divided by the total number of trials.
Theoretical Probability	The probability of an event occurring based on mathematical reasoning and the possible outcomes, assuming all outcomes are equally likely. It is calculated as the number of favorable outcomes divided by the total number of possible outcomes.
Trial	A single instance of an experiment or chance event being performed, such as flipping a coin once or rolling a die once.
Relative Frequency	The ratio of the number of times an event occurs to the total number of trials conducted; this is how experimental probability is calculated.
Law of Large Numbers	A principle stating that as the number of trials in a probability experiment increases, the experimental probability tends to approach the theoretical probability.