1. Sample Spaces (Combinations) Finding All Possible Outcomes

2. 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.

3. What’s for Dessert? For dessert, Sam has a choice of two kinds of pie (apple or peach) and three toppings (whipped cream, vanilla ice cream, and pecans). H How many different desserts are possible?

4. One way to find all the possible desserts is to make a tree diagram. Sam has two choices of pies: apple or peach On each type of pie, he can put one of three toppings: whipped cream, vanilla ice cream, or pecans a apple peach Whipped creamcream Vanilla Vanilla ice cream ice cream Vanilla Vanilla ice cream ice cream pecans pecans

5. Vanilla Vanilla ice cream ice cream Vanilla Vanilla ice cream ice cream How many different desserts are possible? How many different desserts are possible? ApplePeach Whipped creamcream Pecans Pecans There are 6 different possible combinations. If Sam chooses one of the combinations at random, what is probability pecans will be on his dessert? P (pecans) = 2/6 = 1/3

6. Decisions, decisions… Will is buying a new car. The salesman tells him he can choose red, blue, white, or green for the exterior, tan or gray for the interior, and a black or white roof.

7. Exterior Interior Roof redtanblack redtanwhite redgrayblack redgraywhite bluetanblack bluetanwhite bluegrayclack Bluegraywhite whitetanblack whitetanwhite whitegrayblack whitegraywhite greentanblack greentanwhite greengrayblack greengraywhite Unfortunately the car salesman spilled hot chocolate on his last copy of a table with all the possible combinations available

8. How many choices does he have and what are they? If Will chooses a red car, he can choose tan or gray for the interior. He has the same interior choices for each of the other 3 exteriors. red tangray red tangray blue tangray white tangray green tangray

9. His final choice is between a black or white roof for his new car. red tangray blackwhiteblackwhite How many choices of red cars does Will have? How many choices of blue cars does Will have? blue tangray blackwhiteblackwhite

10. What pattern do you see? red tangray blackwhiteblackwhite green tangray blackwhiteblackwhite tangray blackwhiteblackwhite blue tangray blackwhiteblackwhite How many combinations of cars will Will have? Will has 16 possible combinations.

11. Making an organized list Another way to find all possible combinations is to make an organized list. When you make an organized list, your possible outcomes are written out (often we use abbreviations to simplify the process).

12. What’s My Number? Sue has 4 cards on which one of the digits 2, 8, 5, or 4 is printed. How many different four-digit numbers can she write using each of the digits in each number? Let’s use making an organized list to answer the problem

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

14. Then list each possible second digit…. 28 25 24 82 85 84 52 58 54 42 48 45

15. Next list each possible third digit for each pair of numbers in the list…. 285 258 248 284 254 245 825 852 845 824 854 842 528 582 548 524 584 542 428 482 452 425 485 458

16. Last, add the fourth digit to complete the four- digit numbers in the list…. 2854 2584 2485 2845 2548 2458 8254 8524 8452 8245 8542 8425 5284 5824 5482 5248 5842 5428 4285 4825 4528 4258 4852 4582

17. YOUR TURN! Use a tree diagram or an organized list to show all possible three-letter combinations using the letters A, E, and T. Use a tree diagram or an organized list to show all possible three-letter combinations using the letters A, E, and T. How many three-letter combinations are there? How many three-letter combinations are there? Which combinations form actual words? Which combinations form actual words?