In this lesson, you will learn to identify and analyze sample spaces. The sample space is the set of all possible outcomes. For example, when you roll a number cube, you can roll numbers 1, 2, 3, 4, 5, and 6. These numbers, which represent all the possible outcomes of an experiment, make up the **sample space**. In this lesson, you will learn to make organized lists and use the fundamental counting principle to find the sample space.

### Example

A bag has a blue marble, a red marble, and a green marble. A second bag has a green marble and a blue marble. Vincent draws one marble from each bag. What are all the possible outcomes? How large is the sample space?

First, think about the various ways you could organize this information.

1. You could make an organized list to show all of the outcomes (sample space)

BAG 1 |
BAG 2 |
---|---|

Blue | Green |

Blue | Blue |

Red | Green |

Red | Blue |

Green | Green |

Green | Blue |

The possible outcomes are BG, BB, RG, RB, GG, and GB. There are six possible outcomes in the sample space

2. You could make a tree diagram to show all of the outcomes (sample space)

The possible outcomes are the same, but are organized differently. There are six possible outcomes in the sample space.

Notice that there are three marbles in one bag and two marbles in the other bag. When you multiply the number of outcomes for the first bag (3) by the number of outcomes in the second bag (2), you get six total outcomes. This is called the **Fundamental Counting Principle**, which states that you can find the total number of outcomes for two or more experiments by multiplying the number of outcomes for each separate experiment.

### Example

Marsha rolls two 1-6 number cubes. Use the Fundamental Counting Principle to find the number of outcomes.

**There are 6 outcomes in each separate trial(roll)****6 * 6 = 36 (Fundamental Counting Principle)****There are 36 possible outcomes when Marsha rolls two number cubes.**

Jeff pulls three number cubes, 1-6, from a box. After he selects a number cube, he replaces it and picks again. How many outcomes are possible?

**There are 6 outcomes in each separate trial****6 * 6 * 6 = 216 (Fundamental Counting Principle)**

Matthew’s science teacher was giving a true false quiz to her classes. There are three questions on the quiz. Use the Fundamental Counting Principle to determine how many outcomes are possible in the quiz.

**There are 2 outcomes for each question (T or F)****There are 3 questions : 2 * 2 * 2 = 8 outcomes**

Now, make an organized list of the sample space: **TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF**

Jane wanted to toss a coin four times to see how many times she could get Heads facing up. Use the Fundamental Counting Principle to determine how many outcomes are possible in the four trials.

**There are 2 outcomes for each flip of the coin (H or T)****Jane flips the coin 4 times: 2 * 2 * 2 * 2 = 16 outcomes**

Now, make an organized list of the sample space: **HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT**

Check out the video to see more examples of sample spaces and ways to display them using trees, tables, diagrams, and lists.

## Probability Sample Spaces Practice

#### Are the results found true or false?

Another way to represent the sample space is through simulations. You can use simulations to estimate probability. A simulation is a model of a real situation where a set of random numbers may be used. In other words, instead of actually doing an experiment, you can “simulate” the event in real situations. You can generate random numbers through a random number generator app, and on graphing calculators. In many situations, you will be provided with random numbers that may be used to simulate the experiment.

### Example

Blake wins a stuffed animal from the arcade machine about 30% of the time. Estimate the probability that he will win a stuffed animal 1 time out of the next 3 times he plays. Use a simulation (like a random number generator) to model this situation. Use the given digits for your simulation. The numbers 01-30 represent the win, and 31-00 represents not winning. Each group of 6 digits can represent a trial. You will need to use 10 trials.

**75 47 42**

**68 08 54 – **1 win

**26****4835 – **1 win

**468342**

**513342**

**332546 **– 1 win

**6352 16** – 1 win

**8485 17** – 1 win

**384732 **

**568967 **

**837683**

Out of 10 trials, Blake gets a win 5 times. Based on this simulation, the probability of winning at least 1 out of the next 3 games is 50%.

### Example

Hayden works at a pizza place where about 70% of the pizzas ordered have thick crust and 30% have thin crust. Use the random number table below to find the experimental probability that the next pizza ordered will have thick crust.

10 **45 **26 **64 **75 **79 **18 **58 **09 **61**

98**62**31**19**95**27**42**89**05**24**

**41728018638196911126**

**38812268938280599284**

**25595443026540974126**

To set this up, you could have thick crust pizza be represented by numbers 00-69 and the thin crust can be represented by 70-99. This represents the 70 and 30 percents. There are 50 numbers in our sample space.

How many of the 50 numbers are 00-69? Can you find 33?

which is 66%

The experimental probability that the next pizza ordered will have thick crust is 66%.

(source)