### Examples

*Example:*If Gabriella rolls a fair die 300 times, she can predict that she will roll a 3 approximately 50 times since the theoretical probability is .

*Example:* Sandra performs an experiment where she flips a coin three times. She finds the theoretical probability of landing on exactly one head as . If she performs this experiment 50 times (for a total of 150 flips), predict the number of repetitions of the experiment that will result in exactly one of the three flips landing on heads.

### Clarifications

*Clarification 1:*Instruction includes making connections to proportional relationships and representing probability as a fraction, percentage or decimal.

*Clarification 2: *Experiments to be repeated are limited to tossing a fair coin, rolling a fair die, picking a card randomly from a deck with replacement, picking marbles randomly from a bag with replacement and spinning a fair spinner.

*Clarification 3: *Repetition of experiments is limited to two times except for tossing a coin.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**8

**Strand:**Data Analysis and Probability

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Theoretical Probability

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students used a simulation of a simple experiment to find experimental probabilities and compare them to theoretical probabilities. In grade 8, students solve real-world problems involving probabilities related to single or repeated experiments, including making predictions based on theoretical probability. In high school, students will expand their knowledge to include more general experiments.- Students should understand that results from an experiment do not always match the theoretical results, but if they do a large number of trials, they should be close.
- When determining experimental probabilities, students should understand that this may be done by performing a simple experiment more than once and also by performing a repeated experiment more than once.
- For example, to determine the experimental probability of tossing four heads in a row, the repeated experiment of tossing a coin four times will need to be performed many times (see also Benchmark Example 2).

- Instruction includes $P$($e$$v$$e$$n$$t$) notation.
- Instruction includes opportunities for students to run various numbers of trials to discover that the increased repetition of the experiment will bring the experimental probability closer to the theoretical. Use virtual simulations to quickly show higher and higher volumes of repetition that would be difficult to create with physical manipulatives
*(MTR.5.1).* - When comparing theoretical to experimental probability, it is important to not just compare the number of times the event occurs, but the probabilities themselves.

### Common Misconceptions or Errors

- Students may incorrectly assume the theoretical and experimental probabilities of the same experiment will always be the same. To address this misconception, provide multiple opportunities for students to experience simulations of different situations, with physical or virtual manipulatives, in order to find and compare the experimental and theoretical probabilities.
- Students may incorrectly expect to see every possible outcome occur during a simulation. While all may occur in a simulation, it is not certain to happen. Students may inadvertently let their own experience with an experiment affect their response.
- For example, during an experiment if a student never draws an ace from a standard deck of cards, this does not indicate it could never happen.

- Students may incorrectly believe the theoretical probability of an event is the proportion of times that event will actually occur.

### Strategies to Support Tiered Instruction

- Teacher reviews the root words theoretical (theory) and experimental (experiment) and discusses the difference between a theoretical probability and experimental probability. Teacher provides graphic organizer to keep as reference for root words.
- For example, experimental probabilities are from simulations whereas theoretical probabilities are from calculations.

- Teacher provides opportunities to discuss the difference between simulating simple experiments and simulating repeated experiments.
- For example, students could discuss tossing a coin 150 times as a simulation of the simple (single) experiment “tossing a coin” and tossing a coin 150 times as a simulation of the repeated experiment “tossing a coin three times.” In the first case, the simulation has 150 trials whereas the second simulation has 50 trials.

- Teacher sets up a simulation with several trials to work through and discuss what ‘we think’ should happen (Theoretical Probability) and what actually happens when the experiment is completed (Experimental Probability). Then, the teacher models how to find experimental probability, showing how the more trials that are done, the closer the results should get closer to the theoretical probability. Teacher provides instruction focused on color-coding when setting up a proportional relationship to ensure corresponding parts are placed in corresponding positions within the proportion.
- For example, if Jason choses a card from a standard deck of cards 104 times and replaces the card each time, what is a reasonable prediction on how many times he will choose a heart?

- For example, if Jason choses a card from a standard deck of cards 104 times and replaces the card each time, what is a reasonable prediction on how many times he will choose a heart?
- Instruction includes providing multiple opportunities to experience simulations of different situations, with physical or virtual manipulatives, in order to find and compare the experimental and theoretical probabilities.

### Instructional Tasks

*Instructional Task 1*

**(***MTR.5.1*)The bar graph shows the results of spinning the spinner 200 times.

- Part A. If the repeated experiment is to spin the spinner twice, predict how many times the event of landing on the same number twice will occur in 100 simulations of the repeated experiment.
- Part B. Using technology, perform a simulation of the experiment described in Part A.
- Part C. Compare the number of times you actually landed on the same number twice during the simulation to your prediction from Part A.

### Instructional Items

*Instructional Item 1*

If Jackson rolls a fair 6-sided die 200 times, what is a reasonable prediction on how many times he will roll a 4?

*Instructional Item 2*

A spinner is divided into three equal parts 1-3. The repeated experiment of spinning the spinner twice is simulated 300 times. A table of outcomes is shown.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Experts

## Problem-Solving Task

## MFAS Formative Assessments

Students are asked to estimate the frequency of an event given its probability and explain why an expected frequency might differ from an observed frequency.

## Original Student Tutorials Mathematics - Grades 6-8

Learn how to use probability to predict expected outcomes at the Carnival in this interactive tutorial.

## Student Resources

## Original Student Tutorial

Learn how to use probability to predict expected outcomes at the Carnival in this interactive tutorial.

Type: Original Student Tutorial

## Problem-Solving Task

As studies in statistics and probability unfold, students will not yet know the rules of probability for compound events. Thus, simulation is used to find an approximate answer to these questions. In fact, part b would be a challenge to students who do know the rules of probability, further illustrating the power of simulation to provide relatively easy approximate answers to wide-ranging problems.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Task

As studies in statistics and probability unfold, students will not yet know the rules of probability for compound events. Thus, simulation is used to find an approximate answer to these questions. In fact, part b would be a challenge to students who do know the rules of probability, further illustrating the power of simulation to provide relatively easy approximate answers to wide-ranging problems.

Type: Problem-Solving Task