Holt CA Course 1

8-4 Sample Spaces

Warm UpWarm Up

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Holt CA Course 1

8-4 Sample Spaces

Holt CA Course 1

8-4 Sample Spaces

Holt CA Course 1

8-4 Sample Spaces

Holt CA Course 1

8-4 Sample Spaces

Warm Up

1. A dog catches 8 out of 14 flying disks thrown. What is the experimental probability that it will catch the next one?

2. If Ted popped 8 balloons out of 12 tries, what is the experimental probability that he will pop the next balloon?

47

23

Holt CA Course 1

8-4 Sample Spaces

SDAP3.1 Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.Also covered: SDAP3.3

California Standards

Holt CA Course 1

8-4 Sample Spaces

Objective:Objective: You will learn You will learn how to (YWLHT) use how to (YWLHT) use counting methods to counting methods to determine possible determine possible outcomes.outcomes.

Holt CA Course 1

8-4 Sample Spaces

Vocabulary

sample space

compound eventFundamental Counting Principle

Holt CA Course 1

8-4 Sample Spaces

Together, all the possible outcomes of an experiment make up the sample space. For example, when you toss a coin, the sample space is landing on heads or tails.

A compound event includes two or more simple events. Tossing one coin is a simple event; tossing two coins is a compound event. You can make a table to show all possible outcomes of an experiment involving a compound event.

Holt CA Course 1

8-4 Sample Spaces

One bag has a red tile, a blue tile, and a green tile. A second bag has a red tile and a blue tile. Vincent draws one tile from each bag. Use a table to find all the possible outcomes. What is the theoretical probability of each outcome?

Example 1: Using a Table to Find a Sample Space

Holt CA Course 1

8-4 Sample Spaces

Let R = red tile, B = blue tile, and G = green tile.

Record each possible outcome.

Example 1 Continued

Bag 1 Bag 2

R R

R B

B R

B B

G R

G B

RR: 2 red tiles

RB: 1 red, 1 blue tile

BR: 1 blue, 1 red tile

BB: 2 blue tiles

GR: 1 green, 1 red tile

GB: 1 green, 1 blue tile

Holt CA Course 1

8-4 Sample Spaces

Find the probability of each outcome.

Example 1 Continued

P(2 red tiles) = 16

P(1 red, 1 blue tile) = 13

P(2 blue tiles) = 16

P(1 green, 1 red tile) = 16

P(1 green, 1 blue tile) = 16

Bag 1 Bag 2

R R

R B

B R

B B

G R

G B

Holt CA Course 1

8-4 Sample Spaces

Check It Out! Example 2

Darren has two bags of marbles. One has a green marble and a red marble. The second bag has a blue and a red marble. Darren draws one marble from each bag. Use a table to find all the possible outcomes. What is the theoretical probability of each outcome?

Holt CA Course 1

8-4 Sample Spaces

Check It Out! Example 2 Continued

Let R = red marble, B = blue marble, and G = green marble.

Record each possible outcome.Bag 1 Bag 2

G B

G R

R B

R R

GB: 1 green, 1 blue marble

GR: 1 green, 1 red marble

RB: 1 red, 1 blue marble

RR: 2 red marbles

Holt CA Course 1

8-4 Sample Spaces

Find the probability of each outcome.

Check It Out! Example 2 Continued

P(1 green, 1 blue marble) = 14

P(1 red, 1 blue marble) = 14

P(2 red marbles) = 14

P(1 green, 1 red marble) = 14

Bag 1 Bag 2

G B

G R

R B

R R

Holt CA Course 1

8-4 Sample Spaces

When the number of possible outcomes of an experiment increases, it may be easier to track all the possible outcomes on a tree diagram.

Holt CA Course 1

8-4 Sample Spaces

There are 4 cards and 2 tiles in a board game. The cards are labeled N, S, E, and W. The tiles are numbered 1 and 2. A player randomly selects one card and one tile. Use a tree diagram to find all the possible outcomes. What is the probability that the player will select the E card and the 2 card?

Example 3: Using a Tree Diagram to Find a Sample Space

Holt CA Course 1

8-4 Sample SpacesExample 3 Continued

List each letter on the cards. Then list each numberon the tiles.

N

1 2N1 N2

S

1 2S1 S2

E

1 2E1 E2

W

1 2W1 W2

There are eight possible outcomes in the sample space.

18

=

The probability that the player will select the E and 2 card is .18

P(E and 2 card) = number of ways the event can occurtotal number of equally likely outcomes

Holt CA Course 1

8-4 Sample Spaces

Check It Out! Example 4

There are 3 cubes and 2 marbles in a board game. The cubes are numbered 1, 2, and 3. The marbles are pink and green. A player randomly selects one cube and one marble. Use a tree diagram to find all the possible outcomes. What is the probability that the player will select the cube numbered 1 and the green marble?

Make a tree diagram to show the sample space.

Holt CA Course 1

8-4 Sample SpacesCheck It Out! Example 4 Continued

List each number on the cubes. Then list each colorof the marbles.

1

Pink Green1P 1G

2

Pink Green2P 2G

3

Pink Green3P 3G

There are six possible outcomes in the sample space.

16

=

The probability that the player will select the cube numbered 1and the green marble is .1

6

P(1 and green) = number of ways the event can occurtotal number of equally likely outcomes

Holt CA Course 1

8-4 Sample Spaces

The Fundamental Counting Principle states that you can find the total number of outcomes for a compound event by multiplying the number of outcomes for each simple event.

Holt CA Course 1

8-4 Sample Spaces

Carrie rolls two 1–6 number cubes. How many outcomes are possible?

Example 5: Recreation Application

The first number cube has 6 outcomes.The second number cube has 6 outcomes

List the number of outcomes for each simple event.

6 · 6 = 36

There are 36 possible outcomes when Carrie rollstwo number cubes.

Use the Fundamental Counting Principle.

Holt CA Course 1

8-4 Sample SpacesCheck It Out! Example 6

A sandwich shop offers wheat, white, and sourdough bread. The choices of sandwich meat are ham, turkey, and roast beef. How many different one-meat sandwiches could you order?

There are 3 choices for bread.There are 3 choices for meat.

List the number of outcomes for each simple event.

3 · 3 = 9

There are 9 possible outcomes for sandwiches.

Use the Fundamental Counting Principle.

Holt CA Course 1

8-4 Sample Spaces

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Holt CA Course 1

8-4 Sample SpacesLesson Quiz

1. Ian tosses 3 pennies. Use a tree diagram to find all the possible outcomes. What is the probability that all 3 pennies will land heads up?

What are all the possible outcomes? How many outcomes are in the sample space?

2. a three question true-false test

3. choosing a pair of co-captains from the following athletes: Anna, Ben, Carol, Dan, Ed, Fran

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT;

15 possible outcomes: AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF

1 8

8 possible outcomes: TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF

Holt CA Course 1

8-4 Sample Spaces

Holt CA Course 1

8-4 Sample Spaces

Holt CA Course 1

8-4 Sample Spaces

Holt CA Course 1

8-4 Sample Spaces