Exploring Theoretical vs. Experimental Probability

Grade Levels

7th Grade

Course, Subject

Mathematics

Related Academic Standards

Determine the probability of a chance event given relative frequency. Predict the approximate relative frequency given the probability. Example: When rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times but probably not exactly 200 times.

M07.D-S.3.2.2 M07.D-S.3.2.2

Find the probability of a simple event, including the probability of a simple event not occurring. Example: What is the probability of not rolling a 1 on a number cube?

M07.D-S.3.2.3 M07.D-S.3.2.3

Find probabilities of independent compound events using organized lists, tables, tree diagrams, and simulation.

Expand

Big Ideas

Data can be modeled and used to make inferences.

Mathematical relations and functions can be modeled through multiple representations and analyzed to raise and answer questions.

Measurement attributes can be quantified, and estimated using customary and non-customary units of measure.

Numerical quantities, calculations, and measurements can be estimated or analyzed by using appropriate strategies and tools.
Concepts

Probability
Competencies

Find probabilities of independent compound events.

Find the probability of a simple event, including the probability of a simple event not occurring.

Predict the approximate relative frequency given the probability.

Rationale

Vocabulary

Theoretical Probability: the likeliness of an event happening based on all possible outcomes

Experimental Probability: the ratio of the number of times that an event occurs to the total number of trials

Frequency: the number of times that an event occurs

Relative Frequency: the ratio of the number of times that an event occurs to the total number of possible outcomes

Histogram: a graphical display of data using bars of different heights that are touching.

Objectives

The student will be able to find the relative frequencies for each M&M color and create a relative frequency histogram based on this data.

The student will be able to determine the theoretical probability for each M&M color based off of the relative frequency.

The student will be able to calculate the experimental probability for each M&M color based off of data collected during 20 trials.

The student will be able to display the experimental probability for each M&M color by creating a histogram.

The student will be able to explain the difference between theoretical and experimental probability using specific evidence from their result sheet.

The students will be able to provide an example of how theoretical and experimental probability is used in a real-world scenario.

Lesson Essential Question(s)

How can data be organized and represented to provide insight into the relationship between quantities?

How can probability and data analysis be used to make predictions?

In what ways are the mathematical attributes of objects or processes measured, calculated and/or interpreted?

What makes a tool and/or strategy appropriate for a given task?

What is the difference between theoretical probability and experimental probability?

How are theoretical and experimental probability used in real-world scenarios?

Duration

45 minutes

Materials

M&M’s (20 for each student) or beads/chips

Brown paper bags

Result sheet

Pencils

Ticket Out the Door

Suggested Instructional Strategies

Active Engagement,

Simulation,

Kinesthetic/Tactile,

Visual/Spatial,

Higher Order Thinking,

Application,

Analysis,

Evaluation

Instructional Procedures

Instructional Procedure:

The teacher will begin the lesson by introducing the terms experimental and theoretical probability. The students will be asked to raise their hand if they think that these terms mean the same thing, then to raise their hand if they think that these terms have different meanings (correct answer).
The teacher will tell the students that they will be doing an activity today that will help them determine how theoretical and experimental probabilities are different.
The teacher will provide each student with a brown paper bag containing 20 M&M’s and a result sheet. The teacher will tell the students not to eat any of their candies until designated to by the teacher at the end of the lesson.
The teacher will ask the students to carefully open their bags of M&M’s and sort them into groups based on color (Brown, Yellow, Green, Red, Blue, Orange).
The students will then count how many M&M’s they have of each color, and record the frequency for each on their result sheet.
Next, the students will find the relative frequencies of each color by dividing each frequency by the total number of M&M’s (fraction). They will then be asked to convert each relative frequency into a decimal. This should all be recorded in the frequency table on their result sheet.
The students will construct a relative frequency histogram for their M&M colors on the result sheet.
Next, the teacher will ask students to put their 20 M&M’s back into their paper bags.
The teacher will then ask students to find the experimental probabilities of pulling each colored M&M out of the bag for 20 trials (with replacement). The students can use their frequency tables/histograms to find the theoretical probability for each color. The students will record this on their result sheets.
The students will then carefully shake up their bags and pull out one M&M without looking in the bag. They will record the color that they chose for the first trial on their result sheet, and then put the M&M back into the bag. The students will repeat this step for 19 more trials.
Once they have completed all 20 trials, the students will add up how many times each colored M&M was pulled from the bag out of the 20, and then find the experimental probabilities on their result sheet.
Once students have calculated the experimental probabilities for their trials, the teacher will ask them to create a histogram to display the experimental probabilities for each M&M color.
The students will then compare their experimental probabilities and theoretical probabilities using the histograms that they created. Are they the same? Are they different? They will explain on their result sheets.
To complete the activity, the students will be asked to find both the theoretical probability and the experimental probability of picking either a brown or yellow M&M out of the bag based on the information that they have gathered through their trials. They must explain how they got their answers on their result sheet.
The teacher will ask the students to compare their results sheet with a partner. As every bag of M&M’s is different, the student’s results should differ as well.
Finally, the teacher will prompt the students to eat their M&M candies while they go over their results.
The teacher will ask the students to raise their hand if their experimental probabilities ended up being different than their theoretical probabilities (most of the students should raise their hand).
The students will be given a moment to think about why that is and discuss their ideas with an elbow buddy.
The teacher will then wrap up the lesson by giving students a ticket out the door to complete. On the ticket, they must answer the lesson essential questions: What is the difference between experimental and theoretical probability? Provide an example of how they can be used in the real-world.
The teacher will collect each student’s result sheet and ticket-out-the-door.

Formative Assessment

The teacher will informally assess students’ progress by circulating around the room during the activity, observing and taking notes, and offering support and clarification when necessary.

The teacher will informally assess student’s understanding of the essential questions by reviewing their answers to the ticket-out-the-door question.

The teacher will determine whether or not the lesson objectives were met by looking over each student’s result worksheet.

Exploring Theoretical vs. Experimental Probability