Representing Sample Spaces You calculated experimental probability Learning

Representing Sample Spaces You calculated experimental probability. Learning Intention • Use lists, tables, and tree diagrams to represent sample spaces. • Use the Fundamental Counting Principle to count outcomes.

Experiments, Outcomes, and Events There are many situations that involve more than one outcome. Finding probabilities is easier if all possible outcomes are shown. The sample space of an experiment is the set of all possible outcomes. Sample space can be shown as a list, a table, a Venn diagram or Tree diagrams

Tree diagrams • Tree diagram is an organized table of line segments (branches) which shows possible experiment outcomes

Tables/Arrays • A visual way of recording probabilities of two events. The tables include the possible outcomes of one event listed across and one event listed down.

Does this look familiar? ?

One red token and one black token are placed in a bag. A token is drawn and the color is recorded. It is then returned to the bag and a second draw is made. Represent the sample space for this experiment by making an organized list, a table, and a tree diagram. Organized List Pair each possible outcome from the first drawing with the possible outcomes from the second drawing. R, R R, B B, R Table List the outcomes of the first drawing in the left column and those of the second drawing in the top row. Tree Diagram

One yellow token and one blue token are placed in a bag. A token is drawn and the color is recorded. It is then returned to the bag and a second draw is made. Choose the correct display of this sample space. A. B. C. D. Y, Y; B, B; Y, B

Worked example using table (array)

Worked example using tree diagram

Experiment Stages Two-stage experiment – an experiment with two stages or events (like the 1 st problem). Multi-stage experiment – experiments with more than two stages.

Multi-Stage Tree Diagrams CHEF’S SALAD A chef’s salad at a local restaurant comes with a choice of French, ranch, or blue cheese dressings and optional toppings of cheese, turkey, and eggs. Draw a tree diagram to represent the sample space for salad orders. Answer: The sample space is the result of 4 stages. ● Dressing (F, R, or BC) ● Cheese (C or NC) ● Turkey (T or NT) ● Eggs (E or NE) Draw a tree diagram with 4 stages.

BASEBALL GAME In the bleachers at a major league game you can purchase a hotdog, bratwurst, or tofu dog. This comes with the optional choices of ketchup, mustard, onions, and/or relish. How many stages are in the sample space? A. 3 B. 4 C. 5 D. 6

The Fundamental Counting Principle • If you have 2 events: 1 event can occur m ways and another event can occur n ways, then the number of ways that both can occur is m*n • Event 1 = 4 types of meats • Event 2 = 3 types of bread • How many different types of sandwiches can you make? • 4*3 = 12

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3 or more events: • 3 events can occur m, n, & p ways, then the number of ways all three can occur is m*n*p • 4 meats • 3 cheeses • 3 breads • How many different sandwiches can you make? • 4*3*3 = 36 sandwiches

• At a restaurant at Cedar Point, you have the choice of 8 different entrees, 2 different salads, 12 different drinks, & 6 different deserts. • How many different dinners (one choice of each) can you choose? • 8*2*12*6= • 1152 different dinners

Use the Fundamental Counting Principle CARS New cars are available with a wide selection of options for the consumer. One option is chosen from each category shown. How many different cars could a consumer create in the chosen make and model? Use the Fundamental Counting Principle. exterior interior doors possible color× ×color × outcomes seat engine computer wheels × × × = Answer: 83, 160 different 11 So, a 7 consumer 5 can 3 create 6 4 3 possible cars. 83, 160

BICYCLES New bicycles are available with a wide selection of options for the rider. One option is chosen from each category shown. How many different bicycles could a consumer create in the chosen model? A. 3, 888 B. 3, 912 C. 4, 098 D. 4, 124

13 -1 Assignment p. 918, 6 -8, 15 -18, 20