# Representing the Sample Space

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## Objective

SWBAT represent the sample space in a tree diagram and be able to use the fundamental counting principle to find total outcomes.

#### Big Idea

Do you know how many possible outcomes there are in rock, paper, scissors? Let's find out!

## Do Now

15 minutes

I’m using the Holt mathematics alternate opener as the do now.  I liked this worksheet because it gets the students looking at tree diagrams right away.  They have to show the sample space for tossing 3 coins.  The worksheet walks them through how to fill it out.  Then they have to represent, on their own, what it would look like to add a 4th coin.  Students may struggle with representing the sample space.  The first toss is done for them.  Direct their attention there and ask how they got that for a sample space?  (follow the branches). Then ask them for the second toss, what would be the possible sample space?  Have students work the rest out on their own.

SMP 1 and 4

## Vocabulary

10 minutes

Vocabulary

Students will be writing down 3 key vocabulary words to help them in their discovery about tree diagrams, sample spaces, and the fundamental counting principle. We will be making this vocabulary come alive throughout the lesson.

## Using Tree Diagrams and the FCP

30 minutes

Using the power point, have students visually represent the sample space for example 1 (Matt’s travel).  Allow students to show the representation either using a chart or a tree diagram.  Ultimately, the students need to show all of the possible outcomes in an organized way.  Once students have completed their representation, have them share their representations with their table mates.  In order for this to be effectively done, I will have them do a round robin share.  Once students have shared, bring them back to the power point and have students show their work on the board.  (During the round robin share, I will be looking for students that have represented the outcome correctly, but in different ways.  Those are the students I will call to do the work at the board)

Ask the students, “do these students show the same results?”  “is there a better way to show the sample space?”

Do the same for example 2 (Missy’s work outfits)

As we move in to example 3 (Rick’s pets), I want the students to visually represent the sample space and show the total number of outcomes.  My goal here is to get the students looking at the total outcomes and coming up with a way to find this without drawing the sample space.  I’m going to give the students time to create the diagram and draw their own conclusion.  This will be a good time to implement (SMP 3) by asking them to prove their conclusion by using alternate examples.

I’m going to do this with example 4 (Tina’s sandwiches) too.

Once the students have realized that you can multiply to find the total outcomes, I’m going to solidify their conclusions by telling them that we call this the Fundamental Counting Principle

Before moving to the activity, ask the following questions to check for understanding.

- What does a tree diagram show us? (shows the probability of an event occurring)

- What is the sample space? (Shows the possible outcomes from the tree diagram, it visually represents them)

- When should we use the FCP? (when we want to find total outcomes)

- When can’t we use the FCP? (when we need to find probability)

SMP 1 and 4

## Rock, Paper, Scissors

20 minutes

Divide the class into pairs and have them play the game 18 times.  A rock is a closed fist. Paper is a palm, and scissors is the number two horizontally.  The student hits their other hand twice and on the third time gives the symbol they wish.  A rock beats scissors. Paper beats rocks, and scissors beats paper.  Instruct the students to keep record of wins and losses.  Assign one person to be player A and the other player B.  Each player should keep track of their own results. Then, once the class is finished, record the results for Player A and for Player B.  Then have the students find the mean, mode and range of each data set.

Next, have them draw a tree diagram to show all possible outcomes for the game. Once the tree diagrams are complete, ask the following questions.

1. How many outcomes does the game have? (9)

2. Label each possible outcome on the tree diagram as to win for a, win for b, or  a tie

3. Count the number of wins for A (3)

4. Find the probability A will win in any round (3/9 = 1/3)

5. Count the number of wins for B (3)

6. Find the probability B will win in any round (3/9 = 1/3)

7. Is the game fair? Do both players have an equal probability of winning any round? (yes)

8. Ask students to compare the theoretical probability with the data collected from the experimental probability.  (students may want to show the comparison as a decimal or percent)

9. Have students make a list of where probability is used in the real world.  To strengthen the list, have students share out to make sure all ideas are represented.

## Closure

10 minutes

Today we have been working on visually representing possible outcomes and probability of events occurring.   Name one way to show total outcomes( FCP/sample space).  Name another way?

When do we want to use the FCP instead of a sample space? (the goal here is to have the students understand that the FCP will only show total outcomes and not probability)

Explain what a sample space is and give an example? (sample space visually represents total outcomes,  accept and reasonable visual for an example)