Exploring Probability

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Lesson Plan 1: Experimental Probability

Curriculum Area

7th Grade Mathematics
Aligned to Common Core State Standards:

• CCSS.MATH.CONTENT.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.
• CCSS.MATH.CONTENT.7.SP.C.6: Approximate the probability of a chance event by collecting data and using that data to predict frequencies of future events.

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Objective

Students will:

1. Understand experimental probability as the ratio of the number of successful outcomes to the total number of trials.
2. Conduct hands-on experiments to collect data and calculate probabilities.
3. Compare experimental probabilities to theoretical probabilities.

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Materials Needed

• 2 Dice per pair of students
• A small bag of multicolored candies (e.g., Skittles or M&Ms) per group
• Probability recording sheets (teacher-prepared, with pre-marked columns)
• A coin for each student pair
• A whiteboard or an interactive display
• Markers, pencils, scratch paper

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Lesson Breakdown

1. Warm-Up (5 minutes)

• Write the following on the board:
  "If you roll a six-sided die, what is the probability of rolling a 4?"
  "How can we know if a die is fair?"
• Begin by discussing theoretical probability (e.g., 1 out of 6).
• Link to the concept of experimental probability, emphasizing how it can differ due to chance or faulty equipment.

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2. Mini-Lesson (10 minutes)

• Write the formula for experimental probability on the board:
  Experimental Probability = (Number of Successful Outcomes) ÷ (Total Number of Trials)
• Walk through a simple example using coin flips: If you flip a coin 10 times and land on heads 4 of those times, what is the probability of heads? (Answer: 4 ÷ 10 = 0.4)
• Discuss how increasing the number of trials often leads experimental probability closer to theoretical probability (law of large numbers).

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3. Activity 1: Candy Color Experiment (15 minutes)

1. Divide the class into small groups (2-3 students per group).
2. Give each group a small, opaque bag of multicolored candies.
3. Instruct students to record the number of candies of each color inside their bag.
4. Have students randomly draw a single candy (without looking), record its color, then place it back into the bag. Repeat this process 20 times.
5. Have each group calculate the experimental probability of drawing each candy color based on their data.
6. As a class, compare group results and discuss discrepancies. Why are some probabilities from different groups slightly different?

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4. Activity 2: Dice Rolling (10 minutes)

1. Pair students and give each pair a pair of dice.
2. Instruct pairs to roll their dice 30 times and record the outcomes. For each roll, record whether the sum of the dice is:
   • Less than 7
   • Exactly 7
   • Greater than 7
3. Have students calculate experimental probabilities for each category and compare their results to the theoretical probabilities. Discuss any notable differences.

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5. Wrap-Up & Exit Ticket (5 minutes)

• Ask students:
  1. How does experimental probability differ from theoretical probability?
  2. How does increasing the number of trials affect experimental probability?
• Exit Ticket Question:
  "You flip a coin 10 times and it lands on tails 7 times. What is the experimental probability of tails? Why might this number differ from the theoretical probability?"
• Collect exit tickets as students leave.

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Lesson Plan 2: Comparing Probabilities

Curriculum Area

7th Grade Mathematics
Aligned to Common Core State Standards:

• CCSS.MATH.CONTENT.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.
• CCSS.MATH.CONTENT.7.SP.C.7: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies.

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Objective

Students will:

1. Use probability models to predict outcomes.
2. Compare probabilities derived theoretically with experimental probabilities.
3. Analyze and justify discrepancies between theoretical and experimental outcomes.

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Materials Needed

• Spinner templates and paper clips (1 per student)
• Dice (2 per pair of students)
• Deck of cards (1 per group of 4 students)
• Probability chart handouts
• Chart paper
• Markers, pencils

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Lesson Breakdown

1. Warm-Up (5 minutes)

• Write the following probabilities on the board:
  The chance of rain: 0.7
  The chance of sunny weather: 0.2
  The chance of cloudy weather: 0.3
• Ask students to identify the issue (probabilities don’t add to 1).
• Briefly revisit the concept that total probabilities for events in a sample space must equal 1.

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2. Mini-Lesson (10 minutes)

• Introduce probability models (e.g., spinner or dice probabilities).
• Use a spinner with 4 colors (red, blue, green, yellow) and ask:
  What is the probability of landing on each color?
  • Discuss that all spins are equally likely (theoretical model).
• Compare with a real-life event, e.g., flipping a biased coin where the chance of heads = 0.4 and tails = 0.6. Discuss how experimental results can deviate because of uneven weight, user error, etc.

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3. Activity: Spinner Experiment (15 minutes)

1. Distribute spinner templates and paper clips.
2. Instruct students to spin their spinner 30 times and record the results for each color. Provide a simple table for recording.
3. Have each student calculate experimental probabilities for their spins and compare them to the theoretical probabilities (should be equal for all colors at 0.25).
4. Students discuss with a partner:
   • Why are their numbers slightly different?
   • What might happen if they spun the spinner 50 more times?

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4. Activity: Comparing Models (10 minutes)

1. Introduce a deck of cards and ask groups to calculate the theoretical probability of drawing:
   • A red card
   • A face card
   • The Ace of Spades
2. Have each group shuffle their deck, draw 20 cards without replacement, and record their results.
3. Groups compare theoretical probabilities to their experimental outcomes and write observations on chart paper.

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5. Wrap-Up & Exit Ticket (5 minutes)

• Reflect as a group:
  • Why might experimental results differ even if the process is fair?
  • How can we decrease the difference between theoretical and experimental probabilities?
• Exit Ticket Question:
  "A spinner is divided equally into 3 colors: red, blue, and green. If you spin it 50 times and land on green 20 times, what is the experimental probability of green? How does it compare to the theoretical probability of 0.33?"