Compound Events & Sample Spaces

Compound Events & Sample Spaces

Coin and Number Cube

Opening

Coin and Number Cube

Suppose you conduct an experiment in which you toss a coin and roll a six-sided number cube.

  • Do you think it is likely that you will get a heads-up and roll a 3 on the first try?

Multistage Experiments and Compound Events

Opening

Multistage Experiments and Compound Events

Discuss the following with your classmates.

A multistage experiment is an experiment in which more than one action is performed. Examples include:

  • Randomly selecting two cards from a deck
  • Tossing three coins
  • Rolling two number cubes
  • Randomly choosing four people from a larger group

Compound events are the combined results of multistage experiments. The two events are said to be independent if the occurrence of one event does not affect the outcome of the other event. Examples include:

  • Selecting two cards: draw an ace, replace the ace, and draw a jack
  • Tossing heads, tails, and heads
  • Rolling a 6 and a 3 on two number cubes
  • Choosing two tenth graders from a random selection of all the students in a high school

Math Mission

Opening

Represent the outcomes of multistage experiments, and find the probability of compound events.

Sophie’s Tree Diagram

Work Time

Sophie’s Tree Diagram

Sophie conducted a multistage experiment in which she tossed a coin two times. She drew this tree diagram to represent the sample space.

  • How many outcomes does the tree diagram show?
  • What is the probability that Sophie will toss heads both times?
         P(heads, heads)
  • What is the probability that Sophie will toss heads and tails in any order?
         P(heads and tails)

Hint:

  • How many outcomes are in the sample space?
  • How many times does “heads, heads” appear in the sample space?

Jack’s Table

Work Time

Jack’s Table

Jack conducted a multistage experiment in which he rolled a number cube and tossed a coin. Jack made this table to represent the sample space.

  • How many outcomes would there be if Jack had only rolled the number cube?
  • How many outcomes would there be if Jack had only tossed the coin?
  • How many outcomes does the table show in the sample space for both events?
  • What is the probability that Jack will get a 1 and heads?
         P(1 and heads)
  • What is the probability that Jack will get a 1 and tails?
         P(1 and tails)?
  • How many outcome cells do you see in the table?
  • How many times do you see 1 and heads?
  • How can you use the quantities from the previous two questions to calculate the probability?

Lucy’s List

Work Time

Lucy’s List

Lucy conducted a multistage experiment in which she rolled two number cubes. She wanted to find the probability of rolling two numbers that add up to 7. Lucy first created a list of all the combinations of rolls. She then created a list of the totals for each combination.

  • How many total outcomes are there?
  • How many of those outcomes show two numbers that add up to 7?
  • What is the probability that Lucy will roll two numbers that add up to 7?

Tree Diagram, Lists, and Tables

Work Time

Tree Diagram, Lists, and Tables

Sophie, Jack, and Lucy used a tree diagram, a table, and an organized list to help them find probabilities of compound events in their multistage experiments.

  • Choose one of their experiments and represent the sample space using a different method. For example, Jack used a table to represent the number cube and coin outcomes. If you choose his experiment, use a tree diagram or an organized list to represent the outcomes.

Prepare a Presentation

Work Time

Prepare a Presentation

  • Prepare a presentation about the different ways of representing the probability of a compound event.

Challenge Problem

  • If you roll two ten-sided number polyhedrons (sides labeled 0–9), what is the probability that you will roll a 5 and a 7?

Make Connections

Performance Task

Ways of Thinking: Make Connections

  • Take notes about the probabilities students found and the ways they represented the outcomes.

Hint:

As your classmates present, ask questions such as:

  • Why did you approach the problem the way you did?
  • Which way of representing the outcomes do you think is easiest to work with? Explain.
  • Can you solve the problem using a different method, e.g., using a tree diagram?

Multistage Experiments and Compound Events

Formative Assessment

Summary of the Math: Multistage Experiments and Compound Events

  • Write a summary about multistage experiments and compound events.

Hint:

Check your summary:

  • Do you define multistage experiment ?
  • Do you explain what compound events are?
  • Do you define sample space ?
  • Do you describe the different ways to represent the outcomes in a multistage experiment?
  • Do you explain how to find the probability of a compound event?

Reflect on Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

Another example of a multistage experiment would be…