lesson 3 problem solving practice probability of compound events

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Course: 7th grade   >   Unit 7

• Sample spaces for compound events
• Die rolling probability

Probability of a compound event

• Probabilities of compound events
• Counting outcomes: flower pots
• Count outcomes using tree diagram
• The counting principle

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Video transcript

Practice problems for probability of compound events: video lesson

In this video lesson, I solve several probability problems that involve compound events by listing all the possible outcomes or by drawing a tree diagram. Either way, we have the complete sample space, and we can figure out the probabilities just by writing the ratio of the favorable outcomes to all the possible outcomes (definition of simple probability).

The situations include:

• choosing two students from among five to clean the classroom
• tossing a coin 2 times (there are four possible outcomes)
• tossing a coin 3 times (there are eight possible outcomes)
• picking two cards randomly from a set of cards.

This lesson suits grades 7-8, and meets the Common Core standard 7.SP.8a and 7.SP.8b:

a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.

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Probability of compound events - video lesson

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Into Math Grade 7 Module 14 Lesson 3 Answer Key Find Experimental Probability of Compound Events

We included  HMH Into Math Grade 7 Answer Key  PDF   Module 14 Lesson 3 Find Experimental Probability of Compound Events to make students experts in learning maths.

HMH Into Math Grade 7 Module 14 Lesson 3 Answer Key Find Experimental Probability of Compound Events

I Can find the experimental probability of a compound event.

Spark Your Learning

Turn and Talk The menu also had a choice of two drinks. When considering that choice, how does that affect the total number of combinations that are available to order? Answer: The menu adds a choice of two drinks, The sample space for the dinner is {appetizer, entree, two drinks}. It affects the total number of combinations that are available to order. Before we have only 2 choices after adding the 2 drinks, we have the 4 choices.

Build Understanding

At Felix’s awards dinner, T-shirts were being handed out to the award winners. The choices for the shirts are shown. The selection of a shirt is a compound event due to the available choices in different categories.

Connect to Vocabulary A coin landing heads up when flipped or rolling a 6 on a number cube are simple events. A compound event is an event that includes two or more simple events.

B. How many sizes are there? What are the choices? Answer: There are 3 sizes and 3 choices.

C. Write all the possible combinations of sizes and colors. Answer: The combination of sizes and colors is (red, small), (red, large), (red, medium). (blue, small), (blue, large), (blue, medium). (green, small), (green, large), (green, medium).

D. There are _____________ possible outcomes for this compound event. Answer: There is 9 possible outcomes for this compound event.

F. Suppose a compound event includes two simple events. Explain how many rows and columns are needed in a table of the sample space. Answer: The compound event includes two simple events. You needed 2 columns and 2 rows in a table of the sample space.

Turn and Talk Does it matter whether sizes or colors are listed in rows or columns? Explain. Answer:

Step It Out

Conducting a survey is a type of experiment. Each time a question is asked of one person counts as one trial. Each answer is an outcome. Compare the number of times one answer is given to the total number of times a question is asked to find the experimental probability that a new, randomly chosen person, if asked, will give this answer.

B. Describe the two choices that together make up each compound event. Answer:

C. How can you use a number cube and a coin to simulate the experiment? Answer:

Check Understanding

Question 1. How many possible outcomes are in the sample space of toast choices? Answer: The possible outcomes are (wheat, butter), (whites, butter), (rye, butter), (wheat, dry), (white, dry), (rye, dry). There are 6 possible outcomes are in the sample space of toast choices.

Question 2. What is the experimental probability that a new customer orders buttered white toast? Write your answer as a fraction and a percent. Answer: The customer orders buttered white toast. The number of possible outcomes is 6 The experimental probability of buttered white toast = number of events occurring/ total number of trials = 1/6. Or 16%

On Your Own

B. How many possible outcomes are in the sample space? Answer: The possible outcomes are (Hibiscus tea, small), (Macha tea, small), (black tea, small), (Hibiscus tea, large), (Macha tea, large), (Black tea, large). There are 6 possible outcomes in the sample space.

B. Find the experimental probability of a new customer ordering a large popcorn without butter. Answer: The large popcorn without butter = 15 The total number of popcorns = 125 The experimental probability of the popcorn without butter = number of events occurring/ total number of trials = 15/125 = 3/25.

Question 5. Construct Arguments When simulating a compound event by rolling a number cube and flipping a coin, does it matter which one is done first? Explain your answer. Answer:

Question 6. swimming in the afternoon _______________ Answer:

Question 7. field sports in the morning _______________ Answer:

Question 8. boating in the morning _______________ Answer:

Question 9. arts & crafts in the afternoon _______________ Answer:

Question 10. Choosing a pet at the animal shelter: a male or female; a cat, a dog, or a rabbit Answer:

Question 11. Choosing colors for a shed: a red, blue, or yellow shed with white, gray, or black trim Answer:

Question 12. Choosing an activity: going hiking or going swimming; traveling by bus or traveling by bicycle Answer:

Question 13. Open-Ended Write a word problem that includes finding the possible outcomes, and the size of the sample space, for a compound event. Then find the size of the sample space described in your problem. Answer:

I’m in a Learning Mindset!

If you want to adjust the level of challenge in a simulation, would changing the order of simple events work? Why or why not? Answer:

Lesson 14.3 More Practice/Homework

Question 1. Math on the Spot A compound event is simulated by flipping a coin (H or T) and rolling a number cube (1—6). A. List all the different possible outcomes. Answer: The possible outcomes is (1, H), (2, H), (3, H), (4, H), (5, H), (6, H). (1, T), (2,T), (3,T), (4, T), (5,T), (6,7).

Model with Mathematics Nico rolled two number cubes 250 times. The table shows his results. Use the table for Problems 2-4.

Question 3. A. Find the experimental probability of rolling a 1 on the second number cube. Answer: The experimental probability = number of events occurring/ total number of trials The experimental probability of rolling a 1 on the second number cube = 1/6.

B. Use the complement to find the experimental probability of NOT rolling a 1 on the second number cube. Answer: The experimental probability = number of events occurring/ total number of trials The experimental probability of rolling a 1 on the second number cube = 1/6.

Question 4. Reason Find the experimental probability of rolling double sixes. Is this experimental probability close to what you would expect? Explain. Answer:

Question 5. Carlotta is doing an experiment by flipping a coin and rollling a number cube. Select all that apply to the sample space of the experiment. (A) The possible outcomes for the simulation can be represented in a table with 2 rows and 6 columns. (B) The possible outcomes for the simulation can be represented in a table with 6 rows and 2 columns. (C) The sample space of the simulation can only be represented by a table. (D The sample space has 36 total possible outcomes. (E) The sample space has 8 total possible outcomes. Answer:

A. Find the experimental probability of the outcome H-4. Answer: H-4 = 14/14+7 = 14/21 = 2/3

B. Use the complement to find the experimental probability of NOT H-4. Answer: The experimental probability of NOT H-4 = 1 – p = 1 – 2/3 = 3-2/3 = 1/3

Spiral Review

Question 7. John is using a number cube to simulate the outcome of rain or no rain with a 50% chance of rain forecast. Describe how he can interpret the outcome from each roll of the number cube to perform his simulation. Answer:

Question 8. Using a number or a number range, describe the probability of rolling a 3 on a number cube. Answer: The cube has 6 sides. The experimental probability = number of events occurring/ total number of trials The probability of rolling 3 on a number cube = 1/6

Question 9. Describe the graph of a proportional relationship on a coordinate plane. Answer:

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• → Resources
• → 7th Grade
• → Statistics & Probability

Probability of Compound Events (Independent & Dependent Events) Lesson Plan

Get the lesson materials.

Probability of Compound Events Guided Notes w/ Doodles | Independent & Dependent

Ever wondered how to teach probability of compound events in an engaging way to your 7th-grade students? In this lesson plan, students will learn about finding probabilities of compound events, including independent, dependent, and mutually exclusive events. Through artistic, interactive guided notes, check for understanding activities, a doodle & color by number worksheet, and a maze worksheet, students will gain a comprehensive understanding of probability of compound events. The lesson ends with a real-life example that explores the real-life applications of this math skill.

• Standards : CCSS 7.SP.C.8 , CCSS 7.SP.C.8.a , CCSS 7.SP.C.8.b , CCSS 7.SP.C.8.c
• Topic : Statistics & Probability
• Grade : 7th Grade
• Type : Lesson Plans

Learning Objectives

After this lesson, students will be able to:

Define compound events in probability

Differentiate between independent and dependent events

Determine the probability of independent events

Determine the probability of dependent events

Solve problems involving compound events using sample spaces, lists, and tree diagrams

Identify and explain real-life situations where probability of compound events is used

Prerequisites

Before this lesson, students should be familiar with:

Basic understanding of probability of simple events

Familiarity with tree diagrams (optional, but helpful)

Colored pencils or markers

Probability of Compound Events Guided Notes

Key Vocabulary

Compound events

Simple events

Independent events

Dependent events

Mutually exclusive events

Sample space

Tree diagrams

Introduction

As a hook, ask students why understanding the probability of compound events is important in real life. You can provide examples such as determining the likelihood of winning a raffle with multiple prizes, predicting the chances of getting heads or tails when flipping multiple coins, or calculating the probability of drawing certain cards from a deck during a card game. Refer to the last page of the guided notes for additional ideas.

Use the first two pages of the guided notes to introduce the topic of compound events and provide an overview of the different types of events: independent events, dependent events, and mutually exclusive events. Walk through the concept of sample space and how it helps us determine the total number of possible outcomes in a given situation. Emphasize the importance of organizing the outcomes using lists, tree diagrams, or the fundamental counting principle. Refer to the FAQs below for a walkthrough on this, as well as ideas on how to respond to common student questions.

Based on student responses during the discussion, identify any areas where students may need additional clarification or examples. If necessary, reteach and provide more practice on specific concepts to ensure a solid foundation. If your class has a wide range of proficiency levels, you can pull out students for reteaching while more advanced students begin working on the practice exercises.

Have students practice finding probabilities of compound events using the practice coloring worksheet included in the resource. Walk around the classroom to answer any student questions and provide any necessary support or clarification.

Fast finishers can then move on to the maze activity provided in the resource for extra practice. You can assign it as homework for the remainder of the class.

Real-Life Application

Bring the class back together, and introduce the concept of real-life applications of finding probabilities of compound events. Explain that understanding probability is not only important in math class, but also in real-world situations.

Some examples of real-life applications include:

Weather Forecasting : Explain how meteorologists use probabilities to predict the weather. They collect data on temperature, humidity, wind speed, and other factors, and then use probability to determine the likelihood of rain, snow, or sunshine.

Sports Statistics : Discuss how probabilities are used in sports to make predictions and determine the outcomes of games. For example, sports analysts use probabilities to calculate the likelihood of a team winning or losing, and to make predictions about player performance.

Medical Testing : Talk about how probabilities play a crucial role in medical testing. For instance, doctors and scientists use probabilities to interpret the results of medical tests, such as mammograms, HIV tests, or genetic screening tests.

Encourage students to think of their own examples of real-life situations where probabilities are used. This will help them understand the practical application of the concept and its relevance in their everyday lives.

Refer to the FAQ section for more ideas on how to teach real-life applications of finding probabilities of compound events.

Additional Self-Checking Digital Practice

If you're looking for digital practice for Probability of Compound Events, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It's incredibly fun, and a powerful tool for differentiation.

Here are 2 activities to explore:

Probability of Compound Events Digital Pixel Art

Probability Digital Pixel Art Bundle

Simulations for Compound Events Pixel Art

Additional Print Practice

A fun, no-prep way to practice Probability of Compound Events is Doodle Math — they're a fresh take on color by number or color by code. It includes multiple levels of practice, perfect for a review day or sub plan.

Here are 2 activities to try:

Probability of Simple and Compound Events Doodle Math Worksheets

Probability Doodle Math Worksheets Bundle

What is the difference between independent and dependent events? Open

Independent events are events where the outcome of one event does not affect the outcome of the other event. Dependent events, on the other hand, are events where the outcome of one event does affect the outcome of the other event.

Independent events:

The occurrence of one event does not affect the occurrence of the other event.

The probability of one event does not change based on the outcome of the other event.

Dependent events:

The occurrence of one event affects the occurrence of the other event.

The probability of one event changes based on the outcome of the other event.

What is a sample space in probability? Open

A sample space in probability refers to the set of all possible outcomes of an experiment or event. It includes every possible outcome that can occur.

How do you calculate the probability of compound events? Open

To calculate the probability of compound events, you need to consider whether the events are independent or dependent.

For independent events:

Multiply the probabilities of the individual events.

For dependent events:

Determine the probability of the first event.

Use that outcome as a reduced sample space for the second event.

What is the fundamental counting principle? Open

The fundamental counting principle is a method used to determine the total number of outcomes in a sequence of events. It states that if there are "m" ways of doing one thing and "n" ways of doing another thing, then there are "m x n" ways of doing both things together.

How can I use a tree diagram to find the probability of compound events? Open

A tree diagram is a graphical representation that helps visualize the different outcomes and probabilities of compound events. To use a tree diagram:

Begin with the first event and draw branches for each possible outcome.

Repeat this process for the second event, branching off from each outcome of the first event.

Assign probabilities to each branch and multiply the probabilities along the branches to find the probability of specific outcomes.

What are mutually exclusive events? Open

Mutually exclusive events are events that cannot occur at the same time. If one event happens, the other event cannot happen simultaneously.

How can I determine if two events are mutually exclusive? Open

To determine if two events are mutually exclusive, you need to check if they share any common outcomes. If there are no common outcomes between the two events, then they are mutually exclusive.

Can you give an example of a real-life application of compound events in probability? Open

Sure! One example of a real-life application of compound events is weather forecasting. In weather forecasting, meteorologists use information from different weather models to predict the probability of both rain and wind occurring on a specific day. By considering the probabilities of independent or dependent events (such as the chance of rain and wind occurring together), meteorologists can provide more accurate forecasts to the public.

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Curriculum  /  Math  /  10th Grade  /  Unit 8: Probability  /  Lesson 3

Probability

Lesson 3 of 10

Criteria for Success

Tips for teachers, anchor problems.

• Problem Set

Target Task

Additional practice.

 Determine the probability of events without replacement using tree diagrams, addition rules for mutually exclusive events, and multiplication rules for compound events.

Common Core Standards

Core standards.

The core standards covered in this lesson

High School — Statistics and Probability

S.CP.A.2 — Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

S.CP.B.6 — Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.

S.CP.B.7 — Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

S.CP.B.8 — Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

Statistics and Probability

7.SP.C.7 — Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

Foundational Standards

The foundational standards covered in this lesson

7.SP.C.6 — Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For 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.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Describe events “without replacement” as events where the first event affects the outcome of the second event.
• Describe when an event is dependent (without replacement), and when it isn’t.
• Describe the outcomes of an event as “equally likely” or “fair” when each outcome has the same chance of occurring.
• Use a tree diagram to describe the sample space of a chance experiment when events are mutually exclusive and identify $$P(A)$$ , $$P(A \space or\space B)$$ , $$P(not A)$$ , and $$P(A,B)$$ .
• Calculate the probability of a mutually exclusive event of $$P(A \space or \space B)=P(A)+P(B)$$ .
• Calculate the compound probability of a mutually exclusive event of $$P(A,B)=P(A)∙P(B)$$ .

Suggestions for teachers to help them teach this lesson

• This is the second lesson out of two that focuses on finding the probability of mutually exclusive events. This lesson focuses on the probability of events without replacement, while Lesson 2 focused on the probability of events with replacement. 
• This lesson will prepare students to access S-CP.3 standard, which introduces the idea of conditional probability, which will be addressed in Lesson 5. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Allison has put together a simple game. She put 6 cubes in a paper bag—3 yellow and 3 blue. Allison has determined that the rules are as follows:

• Pull a cube from the bag and put it on the table.
• Pull a second cube from the bag and put it on the table.
• If the cubes are different colors, then Player A wins.
• If the cubes are the same color, then Player B wins. 

Is this game fair? How do you know? 

Guiding Questions

Dan has shuffled a deck of cards. He chooses the first card and places it on the table. He shuffles again and chooses a second card. What is the probability that Dan’s two cards are of the same suit? 

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Use the game scenario to answer the following questions.

Game Tools: Spinner 1 (three equal sectors with the number 1 in one sector, the number 2 in the second sector, and the number 3 in the third sector)          Card bag (Blue-A, Blue-B, Blue-C, Blue-D, Red-E, Red-F)

Directions: Spin Spinner 1 and randomly select two cards from the card bag (four blue cards and two red cards). 

• What is the probability of spinning an even number and choosing two blue cards? 
• If you selected and recorded the color of the first card and then placed the card back into the bag before choosing the second card, how would your probability change? 

Algebra II > Module 4 > Topic A > Lesson 1 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include problems from the Problem Set Guidance in Lesson 2 that were not used.
• Illustrative Mathematics Stay or Switch?
• Kuta Software Free Geometry Worksheets Mutually Exclusive Events — (Do not use #13 or #14)
• New Visions for Public Schools Unit 5: Probability: Big Idea 2 — Instructional Routine: Connecting Representations, "With or Without Replacement?"

Topic A: Conditional Probability and the Rules of Probability

Describe the sample space of an experiment or situation. Use probability notation to identify the “and,” “or,” and complement outcomes from a given sample space. 

7.SP.C.8 S.CP.A.1

Determine the probability of events with replacement using tree diagrams, addition rules for mutually exclusive events, and multiplication rules for compound events.

7.SP.C.7 S.CP.A.2 S.CP.A.4 S.CP.B.6 S.CP.B.7

7.SP.C.7 S.CP.A.2 S.CP.B.6 S.CP.B.7 S.CP.B.8

Determine the probability of events that are not mutually exclusive to formalize the addition rule. 

S.CP.A.1 S.CP.A.2 S.CP.B.7

Describe conditional probability and develop the rule $$P(B|A)=\frac{P(A \space \mathrm{and} \space B)}{P(A)}$$ .

Determine whether events are independent. 

S.CP.A.2 S.CP.A.3 S.CP.A.5

Calculate and analyze relative frequencies in two-way tables to make statements about the data and determine independence. 

S.CP.A.4 S.CP.A.5 S.ID.B.5

Make decisions about medical testing based on conditional probabilities. 

S.CP.A.3 S.CP.A.4 S.CP.A.5

Describe and apply the counting principle and permutations to contextual and non-contextual situations.

Describe and apply the counting principle and combinations to contextual and non-contextual situations.

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Compound Probability of Independent Events

The probabilities of simple events can be combined, or compounded, to find the probability of two or more events happening. When outcomes of these events don't depend on each other, the events are considered independent. This lecture series presents examples of calculating compound probabilities of independent events using diagrams. Watch the videos and complete the interactive exercises.