Jacqueline prepares the Halloween bowl for trick-or-treaters with equal numbers of Reese's, Kit Kat, Twix and M&Ms. Her first trick-or-treater randomly pulls a candy bar from the bowl, puts it back, then randomly pulls another candy bar from the bowl. Jacqueline wonders how many outcomes are possible from the sample space for the two candy bar pulls from the bowl.

In this concept, you will learn about sample spaces.

### Sample Spaces

When you conduct an experiment, there are many possible outcomes. If you are doing an experiment with a coin, there are two possible outcomes because there are two sides of the coin. You can either have heads or tails. If you experiment with a number cube, there are six possible outcomes because there are six sides of the number cube and the sides are numbered one to six. We can think of all of these possible outcomes as the **sample space**.

A **sample space** is the set of all possible outcomes for a probability experiment or activity. For example, on the spinner there are 5 different colors on which the arrow can land. The sample space, \begin{align*}S\end{align*}, for one spin of the spinner is then:

\begin{align*}S = \text{red, yellow, pink, green, blue}\end{align*}

These are the only outcomes that result from a single spin of the spinner.

Changing the spinner changes the sample space. This second spinner still has 5 equal-sized sections. But its sample space now has only 3 outcomes:

\begin{align*}S = \text{red, yellow, blue}\end{align*}

Let’s look at an example with sample spaces.

A small jar contains 1 white, 1 black, and 1 red marble. If one marble is randomly chosen, how many possible outcomes are there in the sample space?

Since only a single marble is being chosen, the total number of possible outcomes, or sample space matches the marble colors.

\begin{align*}S = \text{white, black, red}\end{align*}

Sometimes, the sample space can change if an experiment is performed more than once. If a marble is selected from a jar and then replaced and if the experiment is conducted again, then the sample space can change. The number of outcomes is altered. When this happens, we can use tree diagrams and multiplication to help us figure out the number of outcomes in the sample space.

A jar contains 1 white and 1 black marble. If one marble is randomly chosen, returned to the jar, then a second marble is chosen, how many possible outcomes are there?

This is a situation where a tree diagram is very useful. Consider the marbles one at a time. After the first marble is chosen, it is returned to the jar so now there are again two choices for the second marble.

Use a tree diagram to list the outcomes.

From the tree diagram, you can see that the sample space is:

\begin{align*}S = \text{white-white, white-black, black-white, black-black}\end{align*}

We could also multiply by multiplying the two options and the number of selections. Two marble colors and two selections:

\begin{align*}2 \times 2 = 4\end{align*}

There are four outcomes in the sample space.

### Examples

#### Example 1

Earlier, you were given a problem about Jacqueline and her curiosity about the candy from the Halloween bowl.

The trick-or-treater pulled one time randomly, returned the candy bar to the bowl, then pulled randomly once more. How many possible outcomes are there from the two pulls?

First, list the different candy choices:

Twix, Reese's, M&Ms, Kit Kat

Next, list the possible outcomes of Pull 1 and Pull 2:

Pull 1: Twix M&Ms

Pull 2: Twix Reese's M&Ms Kit Kat Twix Reese's M&Ms Kit Kat

Pull 1: Reese's Kit Kat

Pull 2: Twix Reese's M&Ms Kit Kat Twix Reese's M&Ms Kit Kat

Then, list the sample space, or all possible outcomes of the two pulls from the candy bowl:

Twix-Twix, Twix-Reese's, Twix-M&Ms, Twix-Kit Kat, M&Ms-Twix, M&Ms-Reese's, M&Ms-M&Ms, M&Ms-Kit Kat, Reese's-Twix, Reese's-Reese's, Reese's-M&Ms, Reese's-Kit Kat, Kit Kat-Twix, Kit Kat-Reese's, Kit Kat-M&Ms, Kit Kat-Kit Kat

The answer is there are 16 possible outcomes.

#### Example 2

June flipped a coin three times. How many outcomes are in the sample space?

First, make a list of the possible outcomes for each flip.

Heads

Tails

Next, count the number of the possible outcomes for each flip.

There are two outcomes for each flip of a coin: heads or tails.

Then, multiply the number of outcomes by the number of flips.

June flipped the coin three times.

\begin{align*}2 \times 3 = 6\end{align*}

The answer is there are 6 outcomes in the sample space.

#### Example 3

How many outcomes are in the sample space of three spins of a spinner with red, blue, yellow and green?

First, make a list of the possible outcomes for each spin.

Red

Blue

Yellow

Green

Next, count the number of possible outcomes for each spin.

There are four possible outcomes for each spin: red, blue, yellow, green.

Then, multiply the number of outcomes by the number of spins.

June flipped the coin three times.

\begin{align*}4 \times 3 = 12\end{align*}

The answer is there are 12 outcomes in the sample space.

#### Example 4

What is the sample space for the roll of a number cube numbered 1–6 and how many possible outcomes are there from one roll?

First, determine all possible outcomes:

1, 2, 3, 4, 5, 6

Next, count the number of possible outcomes for one roll:

There are 6 possible outcomes: 1, 2, 3, 4, 5, 6

Then, multiply the number of outcomes by the number of rolls.

Since we are only rolling once, the number of possible outcomes is 6.

The answer is the sample space is 1, 2, 3, 4, 5, 6 and the number of possible outcomes is 6.

#### Example 5

A bag with a blue and a red marble. One marble is drawn and then replaced. What is the sample space?

First, list the possible outcomes for each draw:

blue

red

Next, list the possible outcomes for Draw 1 and Draw 2:

Draw 1: red blue

Draw 2: red blue red blue

Then, list all possible outcomes from the two draws:

red-red, red-blue, blue-red, blue-blue

The answer is the sample space is red-red, red-blue, blue-red, blue-blue

### Review

Answer the following questions about sample spaces.

- What is the sample space for a single toss of a number cube?
- What is the sample space for a single flip of a coin?
- A coin is flipped two times. List all possible outcomes for the two flips.
- A coin is flipped three times in a row. List all possible outcomes for the two flips.
- A bag contains 3 ping pong balls: 1 red, 1 blue, and 1 green. What is the sample space for drawing a single ball from the bag?
- A bag contains 3 ping pong balls: 1 red, 1 blue, and 1 green. What is the sample space for drawing a single ball, returning the ball to the bag, then drawing a second ball?
- What is the sample space for a single spin of the with red, blue, yellow and green sections spinner?
- What is the sample space for 2 spins of the first spinner?
- What is the sample space for three spins of the spinner?
- A box contains 3 socks: 1 black, 1 white, and 1 brown. What is the sample space for drawing a single sock, NOT returning the sock to the box, then drawing a second sock?
- A box contains 3 socks: 1 black, 1 white, and 1 brown. What is the sample space for drawing all 3 socks from the box, one at a time, without returning any of the socks to the box?
- A box contains 3 black socks. What is the sample space for drawing all 2 socks from the box, one at a time, without returning any of the socks to the box?
- A box contains 2 black socks and 1 white sock. What is the sample space for drawing all 2 socks from the box, one at a time, without returning any of the socks to the box?
- True or false. A sample space is the total possible outcomes.
- True or false. A sample space is a percentage.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.8.