Probability Models

Active learning works for probability models because students need to see probabilities as dynamic relationships, not fixed numbers. When they spin, record, and recalculate, they experience how experimental data shapes a model and how theoretical predictions align with real results.

• 7th Grade
• Mathematics
• 3 activities
• 25 min–35 min

Learning Objectives

1. Calculate the theoretical probability of simple and compound events using fractions, decimals, and percents.
2. Compare theoretical probabilities with experimental probabilities derived from data sets.
3. Construct tree diagrams and tables to represent sample spaces for compound events.
4. Explain the relationship between the number of trials and the accuracy of experimental probability estimates.
5. Evaluate the validity of a probability model based on given data.

Inquiry Circle

⏱ 35 min·Small Groups

Spinner Investigation: Building a Frequency Model

Groups spin a non-uniform spinner 40 times and record results. They calculate experimental probabilities and compare to what a uniform model would predict. Each group adjusts their model based on data and then tests their model with 20 more spins to see how well it predicts.

Prepare & details

What is the difference between theoretical probability and experimental probability?

Facilitation Tip: During Spinner Investigation, circulate and ask students to explain how their frequency data connects to their probability model.

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

Think-Pair-Share

⏱ 25 min·Pairs

Think-Pair-Share: Tree Diagram or Table?

Present a compound event (e.g., choosing a shirt color then a pants color from given options). Students individually draw a tree diagram OR a table to find all outcomes and calculate a specified probability. Partners compare their methods and discuss which they found easier for this specific problem.

Prepare & details

How do tree diagrams and tables help us find the probability of compound events?

Facilitation Tip: During Think-Pair-Share, listen for students to justify why one representation (table vs. tree diagram) better captures the probabilities in their scenario.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Gallery Walk

⏱ 30 min·Small Groups

Gallery Walk: Probability Model Critique

Post four probability model setups around the room, each with either a correctly or incorrectly constructed tree diagram or table. Groups rotate and identify whether each model is valid, marking errors and explaining the correction. Final whole-class discussion addresses common errors found.

Prepare & details

Why does the experimental probability get closer to the theoretical probability as we perform more trials?

Facilitation Tip: During Gallery Walk, provide a simple checklist so students focus on whether models sum to 1 and whether labels are clear.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Teaching This Topic

Experienced teachers approach this topic by first letting students generate data, then introducing formal models as tools for prediction. Avoid starting with formulas; instead, build intuition through repeated trials. Research shows that students retain concepts better when they construct models from their own data before comparing to theoretical values.

What to Expect

Students will move from guessing probabilities to justifying them with data and structured tools. They will explain why models differ, choose appropriate representations, and critique models based on evidence rather than assumptions.

Watch Out for These Misconceptions

Common Misconception

During Spinner Investigation, watch for students who assume every outcome must be equally likely and adjust the spinner to make outcomes equal.

What to Teach Instead

Redirect their attention to their frequency data. Ask them to calculate the experimental probability for each color and justify their model based on the recorded outcomes.

Common Misconception

During Think-Pair-Share, watch for students who say tree diagrams only work for coin flips or dice rolls.

What to Teach Instead

Have them sketch a tree diagram for a scenario with unequal probabilities, such as spinning a spinner with different sized sections, and label each branch with its probability.

Assessment Ideas

Quick Check

After Spinner Investigation, give each pair a scenario with two independent events, such as flipping a coin twice. Ask them to calculate the theoretical probability of two heads and then simulate the event 20 times, recording experimental results. Collect one sample per pair to compare theoretical and experimental probabilities.

Exit Ticket

During Gallery Walk, provide an exit ticket with a table showing the results of rolling a die 50 times. Ask students to calculate the experimental probability for rolling a 3 and write one sentence explaining why this might differ from the theoretical probability of 1/6.

Discussion Prompt

After Think-Pair-Share, pose the question: 'Why does the experimental probability get closer to the theoretical probability as we perform more trials?' Facilitate a class discussion where students reference their own Spinner Investigation data and connect it to the law of large numbers.

Extensions & Scaffolding

• Challenge: Ask students to design a non-uniform spinner with exactly three outcomes whose probabilities are not equal, then calculate the theoretical probability of landing on red.
• Scaffolding: Provide a partially completed table for the Spinner Investigation so students focus on recording and summing frequencies.
• Deeper exploration: Have students research how weather forecasts use probability models, then compare to their spinner model in a brief written reflection.

Key Vocabulary

Theoretical Probability	The probability of an event occurring based on mathematical reasoning, calculated as the ratio of favorable outcomes to total possible outcomes.
Experimental Probability	The probability of an event occurring based on the results of an experiment or observation, calculated as the ratio of observed frequencies to the total number of trials.
Sample Space	The set of all possible outcomes of a probability experiment or event.
Compound Event	An event that consists of two or more simple events occurring in sequence or simultaneously.
Frequency	The number of times a specific outcome or event occurs in a set of data or trials.