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Sample Spaces and Events

Words or diagrams that detail favorable outcomes and intersections, complements, and unions of events.
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Sample Spaces and Events

Have you ever tried to use dice to figure out a probability?

Lila rolled six dice at once. Each dice was numbered 1-6. If this is the case, how many numbers are in the sample space?

Can you figure this out?

This Concept will teach you how to do this. Pay attention and we'll revisit this question at the end of the Concept.

Guidance

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 have an 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, $S$ , for one spin of the spinner is then:

$S = \text{red, yellow, pink, green, blue}$

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:

$S = \text{red, yellow, blue}$

Let’s look at a situation 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.

$S = \text{white, black, red}$

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:

$S = \text{white-white, white-black, black-white, black-black}$

We could also multiply.

Two marble colors and two selections.

$2 \times 2 = 4$

There are four outcomes in the sample space.

Practice on your own. What is the sample space in each example?

Example A

A spinner with red, blue, yellow and green.

Solution: Sample space = red, blue, yellow, green

Example B

A number cube numbered 1–6.

Solution: Sample Space = 1, 2, 3, 4, 5, 6

Example C

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

Solution: S = red-red, blue-blue, red-blue, blue-red

Here is the original problem once again.

Lila rolled six dice at once. Each dice was numbered 1-6. If this is the case, how many numbers are in the sample space?

Can you figure this out?

To figure this out, we can think logically. Each die is numbered 1-6. If we had rolled the die one time, then we would have 6 possible outcomes. The sample space would be 1-6.

However, we rolled the die six times.

We can multiply $6 \times 6 = 36$ .

There are 36 possible outcomes. These combinations make up the sample space.

Notice that we could list them all out, but that would take a very long time. To calculate probabilities, we need to know the number of outcomes in the sample space. This is the most helpful information.

Vocabulary

Sample Space
the possible outcomes in an experiment.

Guided Practice

Here is one for you to try on your own.

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

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

June flipped the coin three times. We can multiply to figure out the number of outcomes in the sample space.

$2 \times 3 = 6$

There are 6 outcomes in the sample space.

Practice

1. What is the sample space for a single toss of a number cube?

2. What is the sample space for a single flip of a coin?

3. A coin is flipped two times. List all possible outcomes for the two flips.

4. A coin is flipped three times in a row. List all possible outcomes for the two flips.

5. 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?

6. 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?

7. What is the sample space for a single spin of the with red, blue, yellow and green sections spinner?

8. What is the sample space for 2 spins of the first spinner?

9. What is the sample space for three spins of the spinner?

10. 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?

11. 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?

12. 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?

13. 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?

14. True or false. A sample space is the total possible outcomes.

15. True or false. A sample space is a percentage.

Vocabulary Language: English

experiment

experiment

An experiment is the process of taking a measurement or making an observation.
Favorable Outcome

Favorable Outcome

A favorable outcome is the outcome that you are looking for in an experiment.
Outcome

Outcome

An outcome of a probability experiment is one possible end result.
Sample Space

Sample Space

In a probability experiment, the sample space is the set of all the possible outcomes of the experiment.
simple events

simple events

A simple event is the simplest outcome of an experiment.
tree diagrams

tree diagrams

Tree diagrams are a way to show the outcomes of simple probability events where each outcome is represented as a branch on a tree.