Chapter 5 · Lesson 7 of 8 · Grade 7 · Statistics & Probability · MAT-07-SP-007
Use Tree Diagrams and Simulations for Compound Events
Represent and solve use tree diagrams and simulations for compound events problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 5: Sampling, Probability, and Statistical Inference
Statistics & Probability
Essential question: How can samples and probability models support careful claims about uncertainty and populations?
Where this lesson fits
Lesson 7 of 8. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create and defend a multi-step model using the central ideas of sampling, probability, and statistical inference, including a representation, calculation, interpretation, and independent check.
Learning objectives
What you should be able to do
- Represent and solve use tree diagrams and simulations for compound events problems using precise mathematical notation, models, and reasoning.
- Explain why a method for use tree diagrams and simulations for compound events works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
Grade 7 uses prerequisite knowledge as active reasoning.
Learn
Build the idea from meaning, not memorization.
Chapter 5: Sampling, Probability, and Statistical Inference
Statistics & Probability. Lesson 7 of 8 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 7 chapters, Grade 6 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier chapter representation.
Mathematical meaning
Use Tree Diagrams and Simulations for Compound Events describes data or chance while accounting for variability. Calculations must be interpreted in the context of data collection or a probability model.
Represent the relationship
The Math Box uses probability simulator to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify calculations from the raw values, then decide whether the sample, display, center, spread, or probability model actually supports the claim.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Probability Lab
- Model use tree diagrams and simulations for compound events in the Math Box.
- Change one input and predict the effect before testing it.
- Connect the visual, numerical, and symbolic representations.
- State what remains invariant and why.
Worked examples
Twenty different ways to see the concept work.
A bag has 2 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
2/5 × 1/2 = 0.2For independent stages, multiply probabilities along the branch.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: A bag has 3 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/7 × 1/2 = 0.2143The representation should show the same mathematical relationship as the calculation. For independent stages, multiply probabilities along the branch.
Explain why a valid method works, then solve: A bag has 4 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/9 × 1/2 = 0.2222A complete explanation names the relationship or property being preserved. For independent stages, multiply probabilities along the branch.
Estimate or predict first, then calculate and decide whether the result is reasonable: A bag has 5 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/11 × 1/2 = 0.2273The estimate is a reasonableness check, not a replacement for the exact result. For independent stages, multiply probabilities along the branch.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A bag has 2 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
2/9 × 1/2 = 0.1111Verification should independently support the result. For independent stages, multiply probabilities along the branch.
Interpret the answer in context after solving. What does the result mean here? A bag has 3 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/6 × 1/2 = 0.25State the result with its meaning, units, direction, or comparison—not only a number. For independent stages, multiply probabilities along the branch.
Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: A bag has 4 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
Interpret center with spread, shape, and sampling method. Correct solution: 4/8 × 1/2 = 0.25Error analysis requires identifying the broken idea, not merely replacing the final answer. For independent stages, multiply probabilities along the branch.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A bag has 5 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads). Case B: A bag has 2 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
Case A: 5/10 × 1/2 = 0.25 Case B: 2/6 × 1/2 = 0.1667Compare the structure, representation, units, and result rather than only the numbers. For independent stages, multiply probabilities along the branch.
A bag has 2 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
2/8 × 1/2 = 0.125For independent stages, multiply probabilities along the branch.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: A bag has 3 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/10 × 1/2 = 0.15The representation should show the same mathematical relationship as the calculation. For independent stages, multiply probabilities along the branch.
Explain why a valid method works, then solve: A bag has 4 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/7 × 1/2 = 0.2857A complete explanation names the relationship or property being preserved. For independent stages, multiply probabilities along the branch.
Estimate or predict first, then calculate and decide whether the result is reasonable: A bag has 5 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/9 × 1/2 = 0.2778The estimate is a reasonableness check, not a replacement for the exact result. For independent stages, multiply probabilities along the branch.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A bag has 2 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
2/7 × 1/2 = 0.1429Verification should independently support the result. For independent stages, multiply probabilities along the branch.
Interpret the answer in context after solving. What does the result mean here? A bag has 3 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/9 × 1/2 = 0.1667State the result with its meaning, units, direction, or comparison—not only a number. For independent stages, multiply probabilities along the branch.
Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: A bag has 4 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
Interpret center with spread, shape, and sampling method. Correct solution: 4/11 × 1/2 = 0.1818Error analysis requires identifying the broken idea, not merely replacing the final answer. For independent stages, multiply probabilities along the branch.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A bag has 5 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads). Case B: A bag has 2 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
Case A: 5/8 × 1/2 = 0.3125 Case B: 2/9 × 1/2 = 0.1111Compare the structure, representation, units, and result rather than only the numbers. For independent stages, multiply probabilities along the branch.
A bag has 2 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
2/6 × 1/2 = 0.1667For independent stages, multiply probabilities along the branch.
Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: A bag has 3 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/8 × 1/2 = 0.1875The representation should show the same mathematical relationship as the calculation. For independent stages, multiply probabilities along the branch.
Explain why a valid method works, then solve: A bag has 4 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/10 × 1/2 = 0.2A complete explanation names the relationship or property being preserved. For independent stages, multiply probabilities along the branch.
Estimate or predict first, then calculate and decide whether the result is reasonable: A bag has 5 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/12 × 1/2 = 0.2083The estimate is a reasonableness check, not a replacement for the exact result. For independent stages, multiply probabilities along the branch.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: A bag has 2 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Choose a representation before computing.
Answer: 2/5 × 1/2 = 0.2
02Interpret the answer in context after solving. What does the result mean here? A bag has 3 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Explain why the method works and name a verification strategy.
Answer: 3/7 × 1/2 = 0.2143
03Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: A bag has 4 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Choose a representation before computing.
Answer: Interpret center with spread, shape, and sampling method. Correct solution: 4/9 × 1/2 = 0.2222
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A bag has 5 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads). Case B: A bag has 2 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: 5/11 × 1/2 = 0.2273 Case B: 2/7 × 1/2 = 0.1429
05A bag has 2 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Choose a representation before computing.
Answer: 2/9 × 1/2 = 0.1111
06Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: A bag has 3 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Explain why the method works and name a verification strategy.
Answer: 3/6 × 1/2 = 0.25
Practice
Independent practice
01Explain why a valid method works, then solve: A bag has 4 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 4/8 × 1/2 = 0.25
02Estimate or predict first, then calculate and decide whether the result is reasonable: A bag has 5 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 5/10 × 1/2 = 0.25
03Solve and verify the result with a second method, inverse operation, or equivalent representation: A bag has 2 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 2/8 × 1/2 = 0.125
04Interpret the answer in context after solving. What does the result mean here? A bag has 3 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 3/10 × 1/2 = 0.15
05Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: A bag has 4 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: Interpret center with spread, shape, and sampling method. Correct solution: 4/7 × 1/2 = 0.2857
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A bag has 5 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads). Case B: A bag has 2 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: Case A: 5/9 × 1/2 = 0.2778 Case B: 2/5 × 1/2 = 0.2
07A bag has 2 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 2/7 × 1/2 = 0.1429
08Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: A bag has 3 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 3/9 × 1/2 = 0.1667
Common mistakes
Learn to catch the error, not just the answer.
Interpret center with spread, shape, and sampling method.
Finite experimental frequencies vary even when a model is correct.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Summarize data, compare groups, evaluate survey claims, and describe uncertainty.
- Build probability models and simulations for repeated random processes.
Challenge problems
Explain why a valid method works, then solve: A bag has 4 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/11 × 1/2 = 0.1818
Estimate or predict first, then calculate and decide whether the result is reasonable: A bag has 5 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/8 × 1/2 = 0.3125
Solve and verify the result with a second method, inverse operation, or equivalent representation: A bag has 2 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
2/6 × 1/2 = 0.1667
Interpret the answer in context after solving. What does the result mean here? A bag has 3 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/8 × 1/2 = 0.1875
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 5 Reasoning Lab
Solve use tree diagrams and simulations for compound events tasks using calculation, representation, explanation, interpretation, verification, error analysis, and comparison.
Practice
Mastery check
01Error analysis: a student says, "Reporting one statistic without considering variability." Explain the mistake, then solve this related task correctly: A bag has 4 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: Interpret center with spread, shape, and sampling method. Correct solution: 4/10 × 1/2 = 0.2
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A bag has 5 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads). Case B: A bag has 2 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: Case A: 5/12 × 1/2 = 0.2083 Case B: 2/8 × 1/2 = 0.125
03A bag has 2 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 2/5 × 1/2 = 0.2
04Represent before calculating. Use a data display, ordered data set, summary measure, or written statistical claim for this task, then solve: A bag has 3 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 3/7 × 1/2 = 0.2143
05Explain why a valid method works, then solve: A bag has 4 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 4/9 × 1/2 = 0.2222
06Estimate or predict first, then calculate and decide whether the result is reasonable: A bag has 5 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 5/11 × 1/2 = 0.2273
07Solve and verify the result with a second method, inverse operation, or equivalent representation: A bag has 2 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 2/9 × 1/2 = 0.1111
08Interpret the answer in context after solving. What does the result mean here? A bag has 3 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 3/6 × 1/2 = 0.25
Terminology
Words to know
- distribution
- The pattern of values including center, spread, shape, and unusual features.
- variability
- The degree to which observations differ.
- sample
- A subset of a population used to collect information.
- probability
- A number from 0 to 1 describing likelihood.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 7 scope, sequence, and public task structure
Coherent sequencing, representations, and reasoning-rich tasks.Eureka Math² Grade 7 program structure
Module coherence, cumulative practice, and fluency.enVision Mathematics Grade 7 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 7
Grade-level content and mathematical-practice expectations.