Learn on PengiIllustrative Mathematics, Grade 7Chapter 8: Probability and Sampling

Lesson 1: Probabilities of Single Step Events

In this Grade 7 Illustrative Mathematics lesson from Chapter 8: Probability and Sampling, students explore the probability of single-step events by using experimental data to make predictions about future outcomes. Through hands-on activities like drawing colored blocks from a bag and analyzing real-world scenarios, students learn to distinguish between situations where probability can be determined from known conditions versus situations where past observations must guide estimates. The lesson builds foundational understanding of experimental probability and how repeated trials produce data that can predict the likelihood of future events.

Section 1

Understanding Equally Likely Outcomes and Fairness

Property

Outcomes are equally likely when each outcome has the same chance of occurring. The basic probability formula P(event)=number of favorable outcomestotal number of possible outcomesP(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} only applies when all outcomes are equally likely.

Examples

Section 2

Defining Probability and Experiments

Property

An experiment is an activity whose results can be observed and recorded. Each of the possible results of an experiment is an outcome. The set of all possible outcomes for an experiment is a sample space. The sample space SS for rolling a fair die is S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}. An event is a collection of outcomes, a set in the sample space. The set of all even-numbered rolls {2,4,6}\{2, 4, 6\} is a subset of all possible rolls of a die {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\} and is an event.

Examples

  • Flipping a coin is an experiment. The sample space is {H,T}\{H, T\}. The event of getting heads has one outcome.
  • Picking a random letter from "MATH" is an experiment. The sample space is {M,A,T,H}\{M, A, T, H\}. The event of picking a vowel has one outcome, AA.
  • Spinning a spinner with sections Red, Blue, and Green is an experiment. The sample space is {Red, Blue, Green}. The event of landing on a primary color includes two outcomes: Red and Blue.

Explanation

Think of probability as a game. An "experiment" is the action, like rolling a die. An "outcome" is one possible result, like rolling a 4. The "sample space" is all possible results, and an "event" is any group of outcomes.

Section 3

Calculating Experimental Probability from Trial Data

Property

Experimental probability is calculated from actual trial results using the formula:

Pexperimental(event)=number of times event occurstotal number of trialsP_{experimental}(event) = \frac{\text{number of times event occurs}}{\text{total number of trials}}

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 8: Probability and Sampling

  1. Lesson 1Current

    Lesson 1: Probabilities of Single Step Events

    LearnPractice
  2. Lesson 2

    Lesson 2: Probabilities of Multi-step Events

    LearnPractice
  3. Lesson 3

    Lesson 3: Sampling

    LearnPractice
  4. Lesson 4

    Lesson 4: Using Samples

    LearnPractice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Understanding Equally Likely Outcomes and Fairness

Property

Outcomes are equally likely when each outcome has the same chance of occurring. The basic probability formula P(event)=number of favorable outcomestotal number of possible outcomesP(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} only applies when all outcomes are equally likely.

Examples

Section 2

Defining Probability and Experiments

Property

An experiment is an activity whose results can be observed and recorded. Each of the possible results of an experiment is an outcome. The set of all possible outcomes for an experiment is a sample space. The sample space SS for rolling a fair die is S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}. An event is a collection of outcomes, a set in the sample space. The set of all even-numbered rolls {2,4,6}\{2, 4, 6\} is a subset of all possible rolls of a die {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\} and is an event.

Examples

  • Flipping a coin is an experiment. The sample space is {H,T}\{H, T\}. The event of getting heads has one outcome.
  • Picking a random letter from "MATH" is an experiment. The sample space is {M,A,T,H}\{M, A, T, H\}. The event of picking a vowel has one outcome, AA.
  • Spinning a spinner with sections Red, Blue, and Green is an experiment. The sample space is {Red, Blue, Green}. The event of landing on a primary color includes two outcomes: Red and Blue.

Explanation

Think of probability as a game. An "experiment" is the action, like rolling a die. An "outcome" is one possible result, like rolling a 4. The "sample space" is all possible results, and an "event" is any group of outcomes.

Section 3

Calculating Experimental Probability from Trial Data

Property

Experimental probability is calculated from actual trial results using the formula:

Pexperimental(event)=number of times event occurstotal number of trialsP_{experimental}(event) = \frac{\text{number of times event occurs}}{\text{total number of trials}}

Examples

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 8: Probability and Sampling

  1. Lesson 1Current

    Lesson 1: Probabilities of Single Step Events

    LearnPractice
  2. Lesson 2

    Lesson 2: Probabilities of Multi-step Events

    LearnPractice
  3. Lesson 3

    Lesson 3: Sampling

    LearnPractice
  4. Lesson 4

    Lesson 4: Using Samples

    LearnPractice