## 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?

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?

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

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

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)

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

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 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?

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 about the different ways of representing the probability of a compound event.

- 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?

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

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?

- Write a summary about multistage experiments and compound events.

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?

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…**