Understanding Probability

Grade Level and Standards

Grade: 7
Curriculum Area: Mathematical Practices – Probability and Mathematical Reasoning
Aligned Standards:

Common Core Math Standards:
- 7.SP.C.5: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.
- 7.SP.C.6: Approximate the probability of a chance event by collecting data on repeated trials of a random process.
- 7.SP.C.7: Develop a probability model and use it to find probabilities of events. Compare theoretical and experimental probabilities.

By the end of the lesson, students will be able to:

Explain terms such as likelihood of events, theoretical probability, and experimental probability.
Use a probability scale from 0 to 1 to describe the likelihood of outcomes.
Identify and distinguish between theoretical and experimental probability.
Conduct a hands-on activity to calculate experimental probabilities and compare them with theoretical probabilities.

Whiteboard and markers
Graph paper
A die (one per pair of students)
Coins (one per student)
A spinner (can be created on paper or digitally)
Probability worksheets with pre-drawn tables for data collection
A poster with a visual probability scale (from 0 – “impossible” to 1 – “certain”)
Calculator (optional, for quick calculations)

Objective: Set the stage by engaging students with familiar scenarios before diving into theoretical concepts.

Key Question Discussion (3 minutes):
Write on the board:
- If you flip a fair coin, what is the chance it lands on heads?
- If you roll a die, what are the chances of rolling a number greater than 4?
Prompt students for answers and encourage them to justify their reasoning. Write these probabilities as fractions and decimals.
Visual Representation of Likelihood (7 minutes):
- Show a large probability scale on the board (0 = Impossible, 1 = Certain).
- Use relatable, real-world examples to label points on the scale:
  - “The chance it will snow tomorrow in July.” (Impossible = near 0)
  - “The chance the sun will rise tomorrow.” (Certain = near 1)
  - “The chance it rains tomorrow.” (Likely = somewhere between 0.5 and 0.99).
- Involve students by asking them to come up with examples and place them on the scale.

Objective: Develop a strong conceptual understanding of theoretical and experimental probabilities.

Theoretical Probability (5 minutes):
Write the formula on the board:
[ \text{Theoretical Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} ]

Example: Rolling a die.
- Discuss: “What is the probability of rolling a 6?” (1 favorable outcome, total 6 outcomes → Probability = 1/6).
- Explain this prediction is theoretical because it is based on logic, not experimentation.
Experimental Probability (5 minutes):
Discuss how experimental probability compares:
[ \text{Experimental Probability} = \frac{\text{Number of Favorable Outcomes Observed in Trials}}{\text{Total Trials Conducted}} ]
Example:
- Imagine rolling the die 12 times and counting how often a 6 appears. If it appears 4 times, the experimental probability is 4/12 = 1/3.
- Highlight differences: “Notice how this result differs from the theoretical 1/6. Why might this happen?”

Objective: Allow students to conduct experiments, collect data, and compare theoretical vs. experimental probabilities.

Task 1: Coin Flip Probability (10 minutes)
- Students work individually.
- Each student flips a coin 20 times and records the outcome (Heads/Tails) in a table.
- Ask students to calculate:
  - Theoretical Probability: Heads = 50% or 1/2, Tails = 50% or 1/2
  - Experimental Probability: Based on their collected data.
- Group Discussion:
  - “Do your experimental probabilities match the theoretical ones? Why or why not?”
Task 2: Dice Rolls in Pairs (10 minutes)
- Students pair up and roll a die 30 times together (alternating rolls).
- Each pair records how often they roll a "4.”
- Pairs calculate:
  - Theoretical Probability of rolling a “4” = 1/6
  - Experimental Probability = Based on how often they rolled a “4” in 30 trials.
- Open Discussion: Each pair shares their results and recognizes variability in outcomes.

Objective: Solidify understanding with reflection and discussion.

Class Sharing (2 minutes):
- Ask questions to spark reflection:
  - “Why don’t experimental probabilities always match theoretical probabilities?”
  - “If we did 10,000 trials, would experimental results get closer to theoretical probabilities? Why?”
Exit Question Cards (3 minutes):
Students write brief answers to one of the following to turn in on their way out:
- “Describe one difference between theoretical and experimental probability.”
- “Write an example of an event with a probability close to 0, and one close to 1.”

For Advanced Learners:
- Challenge these students to create spinner diagrams or dice with modified outcomes (e.g., weighted sides) to explore biased probability.
- Compare results with unbiased spinners/dice and reflect on fairness.
For Struggling Learners:
- Pair these learners with more confident peers during the hands-on tasks.
- Give additional visual aids, such as completed data tables or probability wheels, for reference during the lesson.
Whole Class Engagement:
- Incorporate movement by creating a “human probability scale”: students physically position themselves between 0 and 1 based on how likely they think a given event will occur.

Formative Assessment:
- Observation during class discussions, table completion, and group work.
- Quick “Thumbs Up/Thumbs Down” checks for understanding of likelihood placement on the scale.
Exit Question Cards:
- Review responses for clarity on theoretical vs. experimental differences.
Follow-Up Task:
- Assign a short homework problem set where students calculate probabilities for various scenarios and compare results when outcomes are simulated.

Real-Life Application:
- Ask students to research and report one real-world event (e.g., probability of rain, sports outcomes). Determine if the prediction was based on theoretical or experimental probability.
Online Simulations (if technology is available):
- Encourage students to use simple probability simulator apps to test outcomes for dice rolls or coin flips with larger numbers of trials.