# Sample Spaces

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.

1. There are 6 outcomes in each separate trial(roll)
2. 6 * 6 = 36 (Fundamental Counting Principle)
3. 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?

1. There are 6 outcomes in each separate trial
2. 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.

1. There are 2 outcomes for each question (T or F)
2. 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.

1. There are 2 outcomes for each flip of the coin (H or T)
2. 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?

1. Cheryl flips a coin and rolls a 1–6 number cube at the same time. She listed the possible outcomes: (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)
2. Jamal has a choice of wheat bread or rye bread and a choice of turkey, ham, or chicken salad for lunch. Jamal says there are 5 possible sandwiches he can make from these choices.
3. Allison has to decide whether to study German, French, or Spanish, and whether to take golf, tennis, or archery in gym class. Listed below are all possible choices that Allison has for these classes: German and golf, French and golf, Spanish and golf, German and tennis, French and tennis, French and archery, Spanish and archery.
4. Chad and Victoria are playing a game with a quarter and a spinner divided into sixths, numbered 1–6. Each player spins the spinner and tosses the coin. There are 12 outcomes possible in the game.
5. Cheyenne has a spinner divided into eighths and a 1–6 number cube. She spins the spinner and rolls the number cube. There are 64 possible outcomes in the sample space.
6. For a snack, Julia can choose milk, soda, orange juice, or lemonade. To go with her drink, she can choose a brownie, cookie, or crackers. There are 12 outcomes in the sample space.
7. Larry has a choice of vanilla, chocolate, or strawberry ice cream. The choices of toppings are nuts, sprinkles, or chocolate sauce. The sample space has 9 outcomes.

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 31-99 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

264835 – 1 win

468342

513342

332546 – 1 win

635216 – 1 win

848517 – 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

98 62 31 19 95

41 72 80 18 63

38 81 22 68 93

25 95 54 43 02

To set this up, you could have thick crust pizza be represented by numbers 01-70 and the thin crust can be represented by 71-100. This represents the 70 and 30 percents. There are 50 numbers in our sample space.

How many of the 25 numbers are 01-70? Can you find 17?

17 out of 25 which is 68%  Equivalent fractions: 17 times 4 is 68 and 25 times 4 is 100. 68 hundredths is 0.68, which is 68%

The experimental probability that the next pizza ordered will have thick crust is 68%.

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