Sample SpacesSample Spaces(Combinations)(Combinations)

Finding All Possible OutcomesFinding All Possible Outcomes

What is a sample space?What is a sample space?

The set of all possible outcomes is called the sample space.

A tree diagram is a diagram used to show the total number of possible outcomes. It can be used to show the sample space.

When you make a tree diagram, you have an organized diagram that shows all possible outcomes.

When you know the number of outcomes, you can easily find the probability that an event will occur.

What’s for Dessert?What’s for Dessert?

For dessert, Sam has a choice of two For dessert, Sam has a choice of two kinds of pie (apple or peach) and three kinds of pie (apple or peach) and three toppings (whipped cream, vanilla ice toppings (whipped cream, vanilla ice cream, and pecans).cream, and pecans).

How many different desserts are How many different desserts are possible?possible?

One way to find all the possible One way to find all the possible desserts is to make a desserts is to make a tree tree

diagramdiagram.. Sam has two choices of pies: apple or Sam has two choices of pies: apple or

peachpeach On each type of pie, he can put one of On each type of pie, he can put one of

three toppings: whipped cream, vanilla three toppings: whipped cream, vanilla ice cream, or pecansice cream, or pecans

apple peachapple peach

Whipped Whipped cream cream VanillaVanilla Vanilla Vanilla

ice creamice cream ice cream ice cream

pecans pecans pecanspecans

VanillaVanilla Vanilla Vanilla ice cream ice cream ice cream ice cream

How many different desserts are How many different desserts are possible?possible? Apple Peac

h

Whipped Whipped

cream cream

Pecans Pecans

There are 6 different possible There are 6 different possible combinations.combinations.If Sam chooses one of the combinations at If Sam chooses one of the combinations at random, what is probability pecans will be random, what is probability pecans will be on his dessert?on his dessert?

P (pecans) = 2/6 = P (pecans) = 2/6 = 1/31/3

Decisions, decisions…Decisions, decisions…

Will is buying a new car. Will is buying a new car. The salesman tells him he can choose red, The salesman tells him he can choose red,

blue, white, or green for the exterior, tan or blue, white, or green for the exterior, tan or gray for the interior, and a black or white gray for the interior, and a black or white roof. roof.

Exterior Interior RoofExterior Interior Roofredred tantan blackblack

redred tantan whitewhite

redred graygray blackblack

redred graygray whitewhite

blueblue tantan blackblack

blueblue tantan whitewhite

blueblue graygray clackclack

BlueBlue graygray whitewhite

whitewhite tantan blackblack

whitewhite tantan whitewhite

whitewhite graygray blackblack

whitewhite graygray whitewhite

greengreen tantan blackblack

greengreen tantan whitewhite

greengreen graygray blackblack

greengreen graygray whitewhite

Unfortunately the car salesman spilled hot chocolate on his last copy of a table with all the possible combinations available

How many choices does he have How many choices does he have and what are they?and what are they?

If Will chooses a If Will chooses a red red car, he can car, he can choose choose tantan or or graygray for the interior. for the interior.

He has the same interior choices for He has the same interior choices for each of the other 3 exteriors.each of the other 3 exteriors.

red

tan gray

red

tan gray

blue

tan gray

white

tan gray

green

tan gray

His final choice is between a His final choice is between a blackblack or or whitewhite roof for his new car. roof for his new car.

red

tan gray

black white black white

How many choices of red cars does Will have?

How many choices of blue cars does Will have?

blue

tan gray

black white black white

What pattern do you see?What pattern do you see?

red

tan gray

black white black white

green

tan gray

black white black white

white

tan gray

black white black white

blue

tan gray

black white black white

How many combinations of cars will Will have?How many combinations of cars will Will have?

Will has 16 possible combinations. Will has 16 possible combinations.

Making an organized listMaking an organized list

Another way to find all possible Another way to find all possible combinations is to combinations is to make an make an organized listorganized list..

When you make an organized list, When you make an organized list, your possible outcomes are written your possible outcomes are written out (often we use abbreviations to out (often we use abbreviations to simplify the process).simplify the process).

What’s My Number?What’s My Number?

Sue has 4 cards on which one of the Sue has 4 cards on which one of the digits 2, 8, 5, or 4 is printed.digits 2, 8, 5, or 4 is printed.

How many different four-digit How many different four-digit numbers can she write using each of numbers can she write using each of the digits in each number?the digits in each number?

Let’s use making an Let’s use making an organized listorganized list to to answer the problemanswer the problem

Sue has 4 cards on which one of the digits 2, Sue has 4 cards on which one of the digits 2, 8, 5, or 4 is printed.8, 5, or 4 is printed.How many different four-digit numbers can How many different four-digit numbers can she write using each of the digits in each she write using each of the digits in each number?number? First list all the ways the numbers could First list all the ways the numbers could

begin…begin…

22

88

55

44

Then list each possible Then list each possible secondsecond digit….digit….

2288 2 255 2 244

8822 8 855 8 844

5522 5 588 5 54 4

4422 4 488 4 455

Next list each possible Next list each possible third third digit for each digit for each pair of numbers in the list….pair of numbers in the list….

228855 2 25588 2 24488 228844 2 25544 2 24455

882255 8 85522 8 84455 882244 8 85544 8 8442 2

552288 5 58822 5 54488 552244 5 58844 5 54422

442288 4 48822 4 4552 2 442255 4 48855 4 4558 8

Last, add the Last, add the fourthfourth digit to complete the digit to complete the four-digit numbers in the list….four-digit numbers in the list….

22885544 2 2558844 2 2448855 22884455 2 2554488 2 2445588

88225544 8 8552244 8 844552 2 88224455 8 855442 2 8 8442255

55228844 5 5882244 5 544882 2 55224488 5 588442 2 5 5442288

44228855 4 4882255 4 4552288 44225588 4 488552 2 4 455882 2

YOUR TURN!YOUR TURN!

Use a Use a tree diagramtree diagram or an or an organized organized listlist to show all possible three-letter to show all possible three-letter combinations using the letters combinations using the letters A, E, A, E, and Tand T..

How many three-letter combinations How many three-letter combinations are there?are there?

Which combinations form actual Which combinations form actual words?words?