Cube Chaos Slides
Cube Chaos
Exploring the Math of "Maybe"
The Lab Kit
Sample Space
The Sample Space is the list of all possible outcomes.
Our Cubes:
- 3 Purple Cubes
- 2 Red Cubes
- 2 Green Cubes
- 1 Blue Cube
- 1 Yellow Cube
- 1 Orange Cube
Quick Check: Count it!
What is the Total Number of outcomes in our sample space?
?
Add them ALL up from the kit!
P
3
R
2
G
2
B
1
Y
1
O
1
The Scale of "Maybe"
Likelihood
How likely is an event to happen? We use a scale from Impossible to Certain.
Impossible
Unlikely
Equally Likely
Likely
Certain
0% / 0
< 50%
50% / 1/2
> 50%
100% / 1
Quick Check: Predict it!
Picking a Purple Cube
You have 3 out of 10.
Unlikely
Picking a Pink Cube
There are NO pink cubes!
Impossible
Picking a Green, Red, Purple, Blue, Yellow, or Orange Cube
Certain
Theoretical Probability
The Math
What should happen based on math.
\( P(\text{event}) = \)
Ways it CAN happen
Total outcomes
Hint: Write it as a fraction, decimal, or percent!
Quick Check: Calculate it!
What is \( P(\text{Red}) \)?
- How many Red? 2
- How many Total? 10
\( \frac{2}{10} \)
or 20%
\( P(\text{Purple}) \)
\( \frac{3}{10} \)
\( P(\text{Blue}) \)
\( \frac{1}{10} \)
\( P(\text{Orange}) \)
\( \frac{1}{10} \)
The Complement
Opposites
The Complement is the probability of the event NOT happening.
Event
Pick Red
Complement
NOT Red
\( P(\text{Event}) + P(\text{Complement}) = 100\% \)
Quick Check: Find the Other Half!
If \( P(\text{Purple}) = \frac{3}{10} \)...
What is \( P(\text{NOT Purple}) \)?
\( \frac{7}{10} \)
If \( P(\text{Orange}) = 10\% \)...
What is \( P(\text{NOT Orange}) \)?
\( 90\% \)
Pro Tip!
Subtract from the total (1 or 100%) to find the complement fast!
Experimental Probability
The Result
What actually happens during an experiment.
\( P(\text{event}) = \)
Times it DID happen
Total number of trials
Example: I picked 5 cubes, and 2 were green. My experimental probability is \( \frac{2}{5} \).
Quick Check: Tally it Up!
Class Experiment: 20 Trials
Result
5
Blue Picked
Exp. Prob.
\( \frac{5}{20} = 25\% \)
Wait! Theoretical said \( \frac{1}{10} = 10\% \). Why is it different?
Theory vs. Experiment
Theoretical
It's what should happen on paper.
Experimental
It's what did happen during the trials.
Big Secret:
The more trials you do, the closer your experimental results will get to the theoretical probability!
Cube Chaos Activity Worksheet
Cube Chaos Activity
Name: ________________________
Date: _________________________
3 Purple
2 Red
2 Green
1 Blue
1 Yellow
1 Orange
1. Sample Space
Total outcomes in the bag:
Total = _____
2. Likelihood
Predict: Impossible, Unlikely, Likely, or Certain.
Picking a Purple: _________________
Picking a Black: _________________
3. Theory Zone
Calculate the theoretical probability for each color (Fraction form).
P(Purple)
P(NOT Purple)
P(Blue)
P(NOT Blue)
P(Red)
P(NOT Red)
P(Yellow)
P(NOT Yellow)
P(Green)
P(NOT Green)
P(Orange)
P(NOT Orange)
4. Experimental Zone
Conduct 20 trials. Pick, tally, replace, and shake!
| Color | Tally Marks | Count | Experimental Prob. |
|---|
| Purple | | | ____ / 20 |
| Red | | | ____ / 20 |
| Green | | | ____ / 20 |
| Blue | | | ____ / 20 |
| Yellow | | | ____ / 20 |
| Orange | | | ____ / 20 |
5. Compare & Reflect
1. Which color had the highest experimental probability? Was it the same as your theoretical prediction?
2. Why might your results be different from another group's results, even though you used the same cubes?
Cube Chaos Teacher Guide
Teacher Guide: Cube Chaos
7th Grade Probability Exploration Facilitator Guide
Objectives
- Identify sample space (Total outcomes = 10).
- Calculate theoretical probability as a ratio.
- Determine the complement of an event.
- Collect data to find experimental probability.
- Compare outcomes and explain variance in data.
Materials Needed
- 10 Colored Cubes
- • 3 Purple, 2 Red, 2 Green
- • 1 Blue, 1 Yellow, 1 Orange
- Opaque Bag
- Worksheets
Instructional Flow
1
Introduction (10 mins)
Introduce the 10-cube kit using the slides. Define Sample Space as the set of all possible outcomes. Model the likelihood scale using physical cubes.
2
Theory Phase (15 mins)
Model theoretical probability (e.g., P(Purple) = 3/10). Explain the Complement: the probability of NOT picking a color. Students complete Section 3 of the worksheet.
3
The Experiment (20 mins)
Students conduct 20 trials. Key Rule: Replace the cube after every pick to ensure the probability doesn't change for the next trial.
4
Debrief (10 mins)
Discuss why experimental results vary from theory. Explain that more trials usually leads to results closer to the math.
Answer Key
Theories
- P(Purple) 3/10 (30%)
- P(Red) 2/10 (20%)
- P(Green) 2/10 (20%)
- P(Blue) 1/10 (10%)
- P(Yellow) 1/10 (10%)
- P(Orange) 1/10 (10%)
Complements
- P(NOT Purple) 7/10 (70%)
- P(NOT Red) 8/10 (80%)
- P(NOT Green) 8/10 (80%)
- P(NOT Blue) 9/10 (90%)
- P(NOT Yellow) 9/10 (90%)
- P(NOT Orange) 9/10 (90%)
Teaching Tip
Remind students that "Theoretical Probability" is what should happen, while "Experimental Probability" is what actually happens. Differences in results are natural in small sample sizes (like 20 trials). To demonstrate the "Law of Large Numbers," combine all groups' results on the whiteboard!
