12-3 Counting Methods and Sample Spaces Warm Up Problem of the Day Course 1 Warm Up Problem of the Day Lesson Presentation

Warm Up Tim took one marble from a bag, recorded the color, and returned it to the bag. He repeated this several times and recorded the results. 1. Find the experimental probability that a marble selected from the bag will be green. 2. Find the experimental probability that a marble selected from the bag will not be yellow. 3 5 __ 4 5 __

Problem of the Day What is the probability that the sum of four consecutive whole numbers is divisible by 4?.

Learn to make an organized list to find all possible outcomes.

Vocabulary sample space

Sample spaces are used in finding probability Sample spaces are used in finding probability. The sample space for an experiment is all the possible outcomes. You can use {} to show sample spaces. When you need to find many possible outcomes, you can make a tree diagram. A tree diagram is one way to organize information.

Additional Example 1: Problem Solving Application Matt wants to take a 3-day weekend trip to visit his grandparents. He can take either Friday or Monday off from work, and he can either fly, drive, take a train, or take a bus. How many options are available to Matt?

Understand the Problem Additional Example 1 Continued 1 Understand the Problem List the important information: Matt can take off either Friday or Monday. Matt can fly, drive, take a train, or take a bus. 2 Make a Plan You can draw a tree diagram to find all the possible outcomes.

Additional Example 1 Continued Solve 3 fly drive take a train take a bus Friday and fly Friday and drive Friday and take a train Friday and take a bus Friday fly drive take a train take a bus Monday and fly Monday and drive Monday and take a train Monday and take a bus Monday Follow each branch on the tree diagram to find all of the possible outcomes.

Additional Example 1 Continued 4 Look Back There are 8 different weekend trip combinations available to Matt. There are 8 possible outcomes listed.

Check It Out: Example 1 For her work uniform, Missy has a choice of three colors of pants—black, khaki, or navy. She has four choices for shirt colors—red, white, green, and yellow. How many different uniforms can Missy wear?

Understand the Problem Check It Out: Example 1 Continued 1 Understand the Problem List the important information: Missy has three colors of pants; black, khaki, or navy. Missy has four colors of shirts; red, white, green, and yellow. 2 Make a Plan You can draw a tree diagram to find all the possible outcomes.

Check It Out: Example 1 Continued Solve 3 red shirt white shirt green shirt yellow shirt black pants and red shirt black pants and white shirt black pants and green shirt black pants and yellow shirt black pants red shirt white shirt green shirt yellow shirt khaki pants and red shirt khaki pants and white shirt khaki pants and green shirt khaki pants and yellow shirt khaki pants

Check It Out: Example 1 Continued Solve Continued 3 red shirt white shirt green shirt yellow shirt navy pants and red shirt navy pants and white shirt navy pants and green shirt navy pants and yellow shirt navy pants Follow each branch on the tree diagram to find all of the possible outcomes.

Check It Out: Example 1 Continued 4 Look Back There are 12 different uniform combinations available to Missy. There are 12 possible outcomes listed.

Additional Example 2: Making an Organized List One girl and one boy will be chosen to go to the state science fair. The girl finalists are Alia, Brenda, Cathy, Deb, and Erika. The boy finalists are Frank, Greg, and Hal. How many different pairs of one girl and one boy can be formed?

Additional Example 2 Continued List all the pairs that begin with A. AF, AG, AH List all the pairs that begin with B. BF, BG, BH List all the pairs that begin with C. CF, CG, CH List all the pairs that begin with D. DF, DG, DH List all the pairs that begin with E. EF, EG, EH There are 5 groups of 3 pairs. 3 + 3 + 3 + 3 + 3 = 15 There are 15 pairs of one girl and one boy.

Check It Out: Example 2 One girl and one boy will be chosen to go to the movie preview. The girl finalists are Fay, Gerri, Heidi, and Ingrid. The boy finalists are Kevin, Larry, and Marc. How many different pairs of one girl and one boy can be formed?

Check It Out: Example 2 Continued List all the pairs that begin with F. FK, FL, FM List all the pairs that begin with G. GK, GL, GM List all the pairs that begin with H. HK, HL, HM List all the pairs that begin with I. IK, IL, IM There are 4 groups of 3 pairs. 3 + 3 + 3 + 3 = 12 There are 12 pairs of one girl and one boy.

Additional Example 3: Using the Fundamental Counting Principle Rick wants to buy a mammal and a reptile for pets. The pet shop has dogs, cats, rabbits, hamsters, and ferrets, which are all mammals. It also has lizards, monitors, and boa constrictors, which are all reptiles. How many combinations of one mammal and one reptile are possible? There are 5 choices for mammals and 3 choices for reptiles. Multiply the number of choices in each category. 5 • 3 = 15 There are 15 possible combinations.

Check It Out: Example 3 Tina wants to buy a sandwich and a drink. The deli offers turkey, ham, tuna, and roast beef. The deli offer water, soda, coffee, tea, and juice for drinks. How many combinations of one sandwich and one drink are possible? There are 4 choices for sandwiches and 5 choices for drinks. Multiply the number of choices in each category. 4 • 5 = 20 There are 20 possible combinations.

Lesson Quiz 1. A baseball coach has 4 pitchers, 3 catchers, and 2 shortstops on his team. How many different combinations of players can he use for the positions? 2. You take 5 shirts, 3 pairs of pants, and 2 pairs of shoes on vacation. How many different outfits can you make? 3. You are planning a small game booth at the local street fair. You have a choice of 3 games and 4 different prizes. How many combinations of games and prizes are there? 24 30 12