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.
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.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
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 one example of use tree diagrams and simulations for compound events in the Math Box.
- Change one input, predict the effect, and test the prediction.
- State the relationship or invariant that explains what stayed mathematically consistent.
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.
A bag has 3 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/7 × 1/2 = 0.2143For independent stages, multiply probabilities along the branch.
A bag has 4 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/9 × 1/2 = 0.2222For independent stages, multiply probabilities along the branch.
A bag has 5 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/11 × 1/2 = 0.2273For independent stages, multiply probabilities along the branch.
A bag has 2 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
2/9 × 1/2 = 0.1111For independent stages, multiply probabilities along the branch.
A bag has 3 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/6 × 1/2 = 0.25For independent stages, multiply probabilities along the branch.
A bag has 4 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/8 × 1/2 = 0.25For independent stages, multiply probabilities along the branch.
A bag has 5 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/10 × 1/2 = 0.25For 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.
A bag has 3 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/10 × 1/2 = 0.15For independent stages, multiply probabilities along the branch.
A bag has 4 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/7 × 1/2 = 0.2857For independent stages, multiply probabilities along the branch.
A bag has 5 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/9 × 1/2 = 0.2778For independent stages, multiply probabilities along the branch.
A bag has 2 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
2/7 × 1/2 = 0.1429For independent stages, multiply probabilities along the branch.
A bag has 3 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/9 × 1/2 = 0.1667For independent stages, multiply probabilities along the branch.
A bag has 4 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/11 × 1/2 = 0.1818For independent stages, multiply probabilities along the branch.
A bag has 5 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/8 × 1/2 = 0.3125For 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.
A bag has 3 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
3/8 × 1/2 = 0.1875For independent stages, multiply probabilities along the branch.
A bag has 4 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
4/10 × 1/2 = 0.2For independent stages, multiply probabilities along the branch.
A bag has 5 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
5/12 × 1/2 = 0.2083For independent stages, multiply probabilities along the branch.
Practice
Guided practice with hints
01A bag has 2 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 2/5 × 1/2 = 0.2
02A bag has 3 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 3/7 × 1/2 = 0.2143
03A bag has 4 red and 5 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 4/9 × 1/2 = 0.2222
04A bag has 5 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 5/11 × 1/2 = 0.2273
05A bag has 2 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 2/9 × 1/2 = 0.1111
06A bag has 3 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 3/6 × 1/2 = 0.25
Practice
Independent practice
01A 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
02A 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
03A 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
04A 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
05A bag has 4 red and 3 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 4/7 × 1/2 = 0.2857
06A bag has 5 red and 4 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 5/9 × 1/2 = 0.2778
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
08A 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
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
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
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
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 Reasoning Sprint
Solve use tree diagrams and simulations for compound events problems accurately and justify each result with a valid check.
Practice
Mastery check
01A bag has 4 red and 6 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 4/10 × 1/2 = 0.2
02A bag has 5 red and 7 blue tokens, then a fair coin is flipped. Find P(red and heads).
Answer: 5/12 × 1/2 = 0.2083
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
04A 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
05A 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
06A 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
07A 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
08A 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
California Mathematics Framework (2023)
Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.Common Core State Standards for Mathematics — Grade 7
Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.