Proving Probability

May 22, 2014 | by Albert J. and B. Michael

Primary Market

Education, Secondary Ed

Character Focus

Cooperation, Teamwork, Perseverance

Items Needed

1 Toobeez set, 1 dry-erase marker, notebook, pen/pencil

The Activity Time

40 - 50 minutes

- Introduce and demonstrate the concept of probability
- Use predicting skills to determine the validity of probability theory
- Conduct an experiment to demonstrate the real world application of a mathematical concept
- Work cooperatively to prepare experimental materials, collect data and draw conclusions

Character Focus

Teamwork, Cooperative Problem Solving & Perseverance

The Challenge

Each group must prove the validity of probability using the ball bucket experiment.

Preparation

**Setup Time: **15 minutes

**Materials**

- 1 Toobeez set
- 1 dry-erase marker
- notebook
- pen/pencil

Activity Plan

**Time:** 40 – 50 minutes

**Instruction:** Whole class, pairs and Individual

**Space:** Medium

1. After dividing the students into groups, each group should use the following steps to build the ball bucket.

2. Insert a 24” Toobeez tube into both ends of all four curtains.

3. Attach spheres to both ends of the 24” tubes in two of the curtains from Step 2.

4. Using four 36” Toobeez tubes, construct two rectangular panels with the two curtains from Step 3.

5. Connect the two panels from Step 4 with the remaining two curtains from Step 2.

6. Stand the resulting box up with one open end facing the floor (see below). This will serve as the ball bucket for the experiment.

7. Using the remaining spheres in the Toobeez set and an erasable marker, students should write one letter (A, B or C) on each sphere or on a label taped to each sphere. Teacher Note:Be sure to have students record the total number of each type of lettered sphere (see below).

*Here are available Training Options!*

8. Toss the lettered spheres into the ball bucket.

**Helpful Hints**

- Be sure to review these tips prior to beginning the activity, and if necessary, share reminders with the group during the activity.

- Probability is an approximation of the likelihood of an event or outcome
- Probability equals the number of ways an event can occur or the number of possible outcomes
- Probability is always a number between 0 and 1, inclusive. The closer that number is to 1, the more likely an event is to occur

**1. After setting up, divide the students into groups.
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**2. Begin with a mini-lesson on probability.
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**3. Read aloud the following Activity Challenge Box to the group.
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**Challenge**: Each group must prove the validity of probability using the ball bucket experiment.

**4. ****As a group, students should define the three possible outcomes (A, B or C) for drawing a ball from the bucket. Teacher Note: This should be a student-led discussion. Allow students to self discover the possible outcomes.
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**5. The students should calculate the probability of each event using the following equations:
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1. P(A)= # of ways to choose “A”/total # of spheres

2. P(B)= # of ways to choose “B”/total # of spheres

3. P(C)= # of ways to choose “C”/total # of spheres

**Teacher Note:** Students should be invited to try and solve the probability of each event using the master probability equation provided during the mini-lesson.

**6. Groups should conduct an experiment by removing one sphere, recording the letter, and replacing the sphere in the bucket. Students should repeat this procedure at least 20 times. Teacher Note: The more attempts conducted, the more accurate the calculated probabilities.
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**7. Using the equations in Step 5, students should calculate the actual frequency of each event.
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**8. Students should compare the predicted probability and actual frequency of each event. Each group should record any interesting observations.
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**9. Each group should compare and share their results with the entire class.
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**10. After the activity, circle up the group and ask them the following question: “Were the actual experimental frequencies similar to the calculated probabilities? What does this information tell you about probability?”
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**11. Finally, move to the “Activity Discussion and Processing” section of the activity.
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- Lead the class in a group discussion as to why the actual frequency and calculated probability may differ slightly
- Explore with the class how the accuracy of the experimental results could be enhanced (
**Hint:**increasing the number of trials from 20 to 40)

To close the lesson, end with a group discussion about what was learned during the activity. Circle up the group and work through the following questions. If possible, record the group’s responses on flip chart paper so all comments are displayed.

- Why is probability an important concept?
- How would increasing the number of attempts in the experiment affect the relationship between the actual frequency and the calculated probability of an event?
- Why/how can probability be useful in everyday life?
- Why are cooperation and perseverance crucial elements in successfully completing an experiment?

Here are some additional topics for discussion:

- The concept of probability as a predictive tool
- The use of experimental techniques to prove a mathematical theory
- The team effort used in conducting the experiment and collecting data

**1. Increasing accuracy.
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Have the students conduct the same experiment, but with 40 trials rather than 20. Let the students compare this result to the original experimental result and the calculated probabilities.

**2. Additional challenge.
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Have students repeat the activity with four types of lettered spheres (A, B, C & D).

**3. Extension/Follow up
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Have the students conduct the experiment by removing only ten spheres from the ball bucket without replacing them. Determine the effect on the calculated probability.