Probability. favorable outcome resultados favorables desired results in a probability experiment. English Español

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1 Probability CHAPTER 12 Probability connected.mcgraw-hill.com Investigate Animations Vocabulary The BIG Idea How can you determine if a situation involves dependent or independent events? Multilingual eglossary Learn Personal Tutor Virtual Manipulatives Graphing Calculator Make this Foldable to help organize information about probability. Probability Audio Foldables Practice Self-Check Practice Worksheets Assessment Review Vocabulary favorable outcome resultados favorables desired results in a probability experiment If you want to spin blue, there are 3 favorable outcomes. Key Vocabulary English Español dependent events eventos dependientes Fundamental Counting Principio Fundamental Principle de Contar independent events eventos independientes probability probabilidad sample space espacio muestral 708 Probability

2 When Will I Use This? How many photos did you take? I took 3 of the drama club. I took 2 awesome photos of the basketball game. I even got a picture of Jack s slam dunk! I have 4 great shots of the band concert. How much room is on the cover? We only have room for 1 photo. How in the world can we decide who gets the cover? Let s just pick at random. What is the probability that a drama club photo is chosen? Hmm... Your Turn! You will solve this problem in the chapter. Probability 709

3 Are You Ready for the Chapter? You have two options for checking prerequisite skills for this chapter. Text Option Take the Quick Check below. Refer to the Quick Review for help. Choose from certain, impossible, likely, or unlikely to describe each probability. 1. If you toss a coin, it will land on either heads or tails. 2. If you choose a letter from the word TEACHER, it will be a T. 3. If you spin the spinner at the right, B the letter will be a A C G. F D 4. If you spin the E spinner at the right, the letter will be a consonant. EXAMPLE 1 Taylor chooses a marble from the bag without looking. Choose certain, impossible, likely, or unlikely to describe the probability that the marble she chooses will be yellow. There are 4 yellow marbles and only 1 marble that is not yellow. So, the probability that Taylor chooses a yellow marble is likely. Write each fraction in simplest form. 5. _ _ 7. 5_ _ _ _ SPELLING Sydney spelled _ 14 of her 20 spelling words correctly. Write _ as a fraction in simplest form. 12. FOOD John ate 1 _ 6 pieces of cake. 12 Write 1 _ 6 in simplest form. 12 EXAMPLE 2 14_ Write in simplest form. 21 Factors of 14: 1, 2, 7, 14 The GCF of Factors of 21: 1, 3, 7, and 21 is 7. _ = _ = _ 2 Divide by the GCF, 7. 3 EXAMPLE 3 5_ Write 2 in simplest form. 15 Factors of 5: 1, 5 The GCF of Factors of 15: 1, 3, 5, 15 5 and 15 is _ 15 = 2 _ = 2 _ 1 Divide by the GCF, 5. 3 Online Option Take the Online Readiness Quiz. 710 Probability

4 Multi-Part Lesson 1 Probability PART A B C D E Main Idea Find and interpret the probability of a simple event. Vocabulary outcomes simple event probability random complementary events odds in favor Get ConnectED GLE Understand the meaning of probability and how it is expressed. SPI Determine the theoretical probability of simple and compound events in familiar contexts. Probability of Simple Events FLOWERS A flower shop sells carnations in several different colors. Morgan is selecting a carnation for her mom from the colors shown. She decides to close her eyes and pick a carnation. 1. Write a ratio that compares the number of yellow carnations to the total number of carnations. 2. What percent of the carnations are yellow? 3. Does Morgan have a good chance of selecting a yellow carnation? 4. What would happen to her chances of picking a yellow carnation if a green, lilac, orange, dark purple, and teal carnation were added to the flowers shown? 5. What happens to her chances if there is only one yellow carnation and one pink carnation? It is equally likely to select any one of the five carnations. The five carnations represent the possible outcomes. A simple event is one outcome or a collection of outcomes. possible outcomes event or favorable outcome Probability is the chance that some event will occur. Probability Words Symbols The probability of an event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. P(event) = number of favorable outcomes number of possible outcomes Lesson 1A Probability 711

5 The probability that an event will occur is a number from 0 to 1, including 0 and 1. The closer a probability is to 1, the more likely it is that an event will happen. Impossible Unlikely As likely to happen as not Likely Certain % 25% 50% 75% 100% Outcomes occur at random if each outcome is equally likely to occur. Probability Probability The notation P(6) is read the probability of rolling a 6. There are six equally likely outcomes if a number cube with sides labeled 1 through 6 is rolled. Find the probability of rolling a 6. There is only one 6 on the number cube. number of favorable outcomes P(6) = number of possible outcomes = _ 1 6 The probability of rolling a 6 is _ 1 6. Find the probability of rolling a 2, 3, or 4. The word or indicates that the number of favorable outcomes needs to include the numbers 2, 3, and 4. P(2, 3, or 4) = number of favorable outcomes number of possible outcomes = _ 3 6 or _ 1 2 Simplify. The probability of rolling a 2, 3, or 4 is 1 _ 2. Angles on Spinners You know that the outcomes of spinning each letter at the right are equally likely since the angles formed are the same size. The spinner at the right is spun once. Find the probability of each event. Write each answer as a fraction. a. P(F) b. P(D or G) c. P(vowel) I H G J F A E B C D 712 Probability

6 Everyday Use Complement the quantity required to make something complete Math Use Complementary Events two events that are the only ones that can happen Complementary events are two events in which either one or the other must happen, but they cannot happen at the same time. For example, a coin can either land on heads or not land on heads. The sum of the probability of an event and its complement is 1 or 100%. Find Probability of the Complement Find the probability of not rolling a 6 in Example 1. The probability of not rolling a 6 and the probability of rolling a 6 are complementary. So, the sum of the probabilities is 1. P(6) + P(not 6) = 1 1_ 6 + P(not 6) = 1 1_ Replace P(6) with 6. 1_ 6 + _ 5 6 = 1 1_ THINK plus what number equals 1? 6 So, the probability of not rolling a 6 is 5 _ 6. A bag contains 5 blue, 8 red, and 7 green marbles. A marble is selected at random. Find the probability of each event. d. P(not red) e. P(not blue or green) EYE COLOR Mr. Harada surveyed his class and discovered that 30% of his students have blue eyes. Identify the complement of this event. Then find its probability. The complement of having blue eyes is not having blue eyes. The sum of the probabilities is 100%. P(blue eyes) + P(not blue eyes) = 100% 30% + P(not blue eyes) = 100% Replace P(blue eyes) with 30%. 30% + 70% = 100% THINK 30% plus what number equals 100%? So, the probability that a student does not have blue eyes is 70%. MOVIES Wahlid surveyed his classmates about their favorite type of movies. Identify the complement of each event. Then find the probability of the complement. f. adventure g. romance or thriller Type of Movie Percent of Students comedy 46 romance 22 adventure 18 thriller 14 Lesson 1A Probability 713

7 Examples 1 3 A letter tile is chosen randomly. Find the probability of each event. Write each answer as a fraction. 1. P(D) 2. P(I) 3. P(B or E) 4. P(S, V, or L) 5. P(not a vowel) 6. P(not a consonant) Example 4 7. GAMES The probability of choosing a Go Back 1 Space card in a board game is 25%. Describe the complement of this event and find its probability. Write the answer as a fraction, decimal, and percent. = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2. Examples 1 3 The spinner shown is spun once. Find the probability of each event. Write each answer as a fraction. 8. P(blue) 9. P(orange) 10. P(red or yellow) 11. P(red, yellow, or green) 12. P(not brown) 13. P(not green) RED GREEN GREEN YELLO YELLOW BLUE BLUE RED GREEN Ten cards numbered 1 through 10 are mixed together and then one card is drawn. Find the probability of each event. Write each answer as a fraction. 14. P(8) 15 P(7 or 9) 16. P(less than 5) 17. P(greater than 3) 18. P(odd) 19. P(even) 20. P(not a multiple of 4) 21. P(not 5, 6, 7, or 8) 22. AIR TRAVEL Use the table on air Air Travel travel at selected airports. Write each answer as a fraction, decimal, and percent. a. Suppose a flight that arrived at El Centro is selected at random. What is the probability that the flight did not arrive on time? b. Suppose a flight that arrived at Islip is selected at random. What is the probability that the flight did arrive on time? Airport Arrivals (Percent on-time) El Centro (CA) 80 Baltimore (MD) 82 Charleston (SC) 77 Islip (NY) 83 Milwaukee (WI) 76 B One jelly bean is picked, without looking, from the dish shown. Write a sentence stating how likely it is for each event to happen. Justify your answer. 23. black 24. purple 25. purple, red, or yellow 26. green 714 Probability

8 27 SCHOOL Of the students at Grant Middle School, 63% are girls. The school newspaper is randomly selecting a student to be interviewed. Describe the complement of selecting a girl and find the probability of the complement. Write the answer as a fraction, decimal, and percent. 28. GEOMETRY The probability of landing in a certain section on a spinner can be found by considering the size of the angle formed by that section. In spinner A, the angle formed by the blue section is one-fourth of the angle formed by the entire circle. So, P(blue) = _ 1, 0.25, or 25%. 4 Spinner A Spinner B Spinner C GREEN BLUE ORANGE ORANGE YELLOW BLUE GREEN BLUE PINK GREEN YELLOW PURPLE ORANGE a. Determine P(green) for each spinner. Write the probabilities as fractions, decimals, and percents. Justify your response. b. Determine P(not orange) for each spinner. Write the probabilities as fractions, decimals, and percents. Justify your response. 29. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a b. I think my photo of the basketball game will look great on the front page. a. Noah has 2 basketball photos, Amanda has 3 drama club photos, and Luis has 4 concert band photos. They select one photo at random. What is the total number of outcomes? b. Find the probability that a drama club photo is chosen. C 30. CHALLENGE A game s spinner has more than three sections, all of equal size. The probability of it stopping on blue is 0.5. Design two possible spinners for the game. Explain why each makes sense. 31. E WRITE MATH Explain the relationship between the probability of an event and its complement. Give an example. Lesson 1A Probability 715

9 Test Practice 32. Joel has a bowl containing the mints shown in the table. Color Number Red 5 Orange 3 Yellow 1 Green 6 If he randomly chooses one mint from the bowl, what is the probability that the mint will be orange? A. B. 1_ 5 2_ 3 C. 11_ 15 D. 4 _ What is the probability of the spinner landing on an A, C, or D? 34. SHORT RESPONSE Max has 50 songs on his MP3 player. If he plays a song at random, what is the probability that it is a rock song? Number of Songs Types of Music 10 Country 25 Rock Genre 12 R & B 3 Classical F. 1_ 4 G. _ 3 8 D H A B F G C E H. _ 1 2 I. 3_ A miniature golf course has a bucket with 7 yellow golf balls, 6 green golf balls, 3 blue golf balls, and 8 red golf balls. If Tamika draws a golf ball at random from the bucket, what is the probability that she will NOT draw a green golf ball? A. B. 1_ 4 1_ 3 C. 2_ 3 D. 3 _ 4 More About Probability Another way to describe the chance of an event occurring is with odds. The odds in favor of an event is the ratio that compares the number of ways the event can occur to the ways that the event cannot occur. ways to occur ways to not occur odds of rolling a 3 or a 4 on a number cube 2 : 4 or 1 : 2 Find the odds in favor of each outcome if a number cube is rolled. 36. a 2, 3, 5, or a number less than an even number 39. a number greater than Probability

10 Multi-Part Lesson 1 Probability PART A B C D E Main Idea Compare experimental probability to theoretical probability. Vocabulary theoretical probability experimental probability Experimental and Theoretical Probability Theoretical probability is based on what should happen in an experiment. Experimental probability is based on what actually happens in an experiment. Let s investigate the relationship between these two types of probabilities. Get ConnectED GLE Understand the meaning of probability and how it is expressed. Place 3 blue cubes and 5 red cubes in a paper bag. Without looking, draw a cube out of the bag. If the cube is blue, record a B in a table like the one shown. If the cube is red, record an R. Trials Outcome R B R 30 Replace the cube and repeat step 2 for a total of 30 trials. the Results 1. Find the ratio of the number of times a blue cube was selected to the number of trials. This is the experimental probability of selecting a blue cube. 2. What is the theoretical probability of selecting a blue cube? How does this probability compare to the experimental probability you found in Exercise 1? Explain any differences. 3. Compare your results to the results of other groups in your class. Do the experimental probabilities vary? Explain why or why not. 4. Find the experimental probability for the entire class s trials. How do the experimental and theoretical probability compare? 5. MAKE A CONJECTURE Explain why the experimental probability obtained in Exercise 4 may be closer in value to the theoretical probability than the experimental probability in Exercise 1. Lesson 1B Probability 717

11 1_ The theoretical probability of spinning blue on the spinner shown is. So, if 3 you spin the spinner 30 times, you would expect to land on blue 10 times. If you spin 60 times, you would expect 20 blue outcomes. If you spin 90 times, you would expect 30 blue outcomes. Is this what really happens? Spin the spinner 30 times and record the result of each spin in a table like the one shown. Number of Trials Spinner Results Blue Red Yellow Continue spinning 30 more times, and then another 30 times. Record the results in the table. the Results 6. Did the spinner land on one of the colors for a much greater amount of time than the other colors? 7. What is the experimental probability of spinning blue after 30, 60, and 90 spins? 8. Is the experimental probability closer to the theoretical probability after 30, 60, or 90 spins? and Apply 9. How many times would you expect to spin yellow on each spinner if you completed 24 spins? Test your predictions by conducting experiments. 10. COLLECT THE DATA Work with a partner. Have your partner place some red and blue cubes into a bag. Use experimental probability to guess the correct number of red and blue cubes in the bag. Explain your reasoning. 11. E WRITE MATH In six rolls of a number cube, Josh rolled the number 4 three times. He concluded that the probability of rolling a 4 is higher than rolling any other number. Explain what is wrong with Josh s conclusion. 718 Probability

12 Multi-Part Lesson 1 Probability PART A B C D E Main Idea Construct sample spaces using tree diagrams or lists. Vocabulary sample space tree diagram Get ConnectED Sample Spaces MOVIES A movie theater s concession stand sign is shown. 1. List all the possible ways to choose a soft drink, a popcorn, and a candy. 2. How do you know you have accounted for all possible combinations? SOFT DRINK Jumbo, Large, Medium POPCORN Giant, Large, Small CANDY Chocolate GLE Understand the meaning of probability and how it is expressed. The set of all possible outcomes is called the sample space. The sample space for choosing a marble and the sample space for picking a card are shown below. {red, blue, orange, purple, green, yellow} {1, 2, 3, 4} Use a List to Find Sample Space ASSEMBLIES The three students chosen to represent Mr. Balderick s class in a school assembly are shown. All three of them need to sit in a row on the stage. In how many different ways can they sit in a row? Use a list. Use A for Adrienne, C for Carlos, and G for Greg. Use each letter exactly once. ACG AGC CAG CGA GAC GCA There are 6 ways the students can sit on the stage. Students Adrienne Carlos Greg a. FOOD How many chicken and sauce combinations are possible if you can choose from crispy or grilled chicken with ranch or barbecue sauce? Use a list to show the sample space. Lesson 1C Probability 719

13 A tree diagram can also be used to show a sample space. A tree diagram is a diagram that shows all possible outcomes of an event. Find a Sample Space Use a tree diagram to find how many ice cream cones are possible from a choice of a waffle cone or sugar cone and a choice of chocolate, mint, or peanut butter ice cream. Outcomes The outcome WC means a waffle cone and chocolate ice cream. There are six possible ice cream cones. b. Use a tree diagram to find words that can be made using the words fast, slow, old, and young and the suffixes -er and -est. Use a Tree Diagram to Find Probability Suppose you toss a quarter, a dime, and a nickel. Draw a tree diagram to show the sample space. What is the probability of getting three tails? favorable outcome P(3 tails) = 1 _ 8 number of favorable outcomes number of possible outcomes So, the probability of getting three tails is 1 _ 8. c. Use the tree diagram above to find the probability of tossing one heads and two tails. 720 Probability

14 Example 1 1. LIBRARY How many ways can Ramiro, Garth, and Lakita line up to check out library books? Use a list to show the sample space. Example 2 2. Use a tree diagram to find how many different backpacks can be made if the backpack comes in nylon or leather and red, green, or black. Example 3 3 Lucy is choosing a screen saver and a background photo for her Computer Options computer. Draw a tree diagram Screensaver Background Photo to show the sample space. If she stars family chooses at random, what is the geometry puppies probability that she chooses the flowers kittens waves scenic waves screen saver and a background photo of her family? = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2. Example 1 Use a list to show the sample space for each situation. 4. AMUSEMENT PARK The names of three roller Roller Coasters coasters at an amusement park are shown. In Superman-Ultimate Flight how many different ways can Felipe and his Deja Vu friend ride each of the three roller coasters, one The Georgia Cyclone time on each roller coaster? 5. MUSIC In how many ways can Kame listen to 4 CDs assuming he listens to each CD once? 6. MATH IN THE MEDIA Use the Internet or another source to find the top five best-selling animated movies. Then create a list of the possibilities for choosing a movie and choosing a wide-screen or full-screen version. Example 2 7. BOOKS Refer to the table at the right. Best-Selling Children s Hardcover Ms. Collins plans on buying one of the Books of All Time books listed for her nephew. She can also 1. The Poky Little Puppy (1942) choose from yellow or green gift bags. 2. The Tale of Peter Rabbit (1902) How many book and gift bag selections 3. Tootle (1945) are possible? 4. Green Eggs and Ham (1960) Draw a tree diagram to show the sample space for each situation. Then tell how many outcomes are possible. 8. jeans or khakis and a yellow, white, or blue shirt 9. sesame, raisin, or onion bagel with butter, jelly, cream cheese, or peanut butter 10. spin a spinner with 4 equal sections and roll a number cube 11. select a letter from the word FUN, toss a coin, and spin a spinner with 2 sections Lesson 1C Probability 721

15 Example To win a carnival prize, you need to choose one of three doors labeled 1 through 3. There is one red, one black, one yellow, and one blue box behind each door. The prize is located behind the second door in the yellow box. Draw a tree diagram to illustrate the sample space. What is the probability of winning the prize? 13. A coin is tossed and one card is chosen from a set of four, shown at the right. Draw a tree diagram to show the sample space. What is the probability of getting heads on the coin and the card with a heart? B 14. DELI Use a tree diagram to find how many different sandwiches can be made from a choice of white or multigrain bread, a choice of ham, turkey, or roast beef, and a choice of American or provolone cheese. 15 SCHOOL A science quiz has one multiple-choice question with answer choices A, B, and C, and two true/false questions. Draw a tree diagram that shows all of the ways a student can answer the questions. Then find the probability of answering all three questions correctly by guessing. Probability The notation P(shorts, T-shirt, white socks) is read the probability of selecting shorts, a T-shirt, and white socks. 16. Use the required clothing list at the right for Camp Wood Springs. a. How many different outfits are possible? b. What is P(shorts, T-shirt, white socks)? c. Find the probability of selecting an outfit consisting of shorts, a T-shirt, and socks with no green articles of clothing. Camp Wood Springs Required Clothing List 1 pair of green shorts 1 pair of white shorts 1 pair of jeans 1 green Wood Springs T-shirt 1 white Wood Springs T-shirt 1 white Wood Springs sweatshirt 1 pair of green socks 1 pair of white socks C 17. CHALLENGE The names of 5 students are placed in a hat. In how many ways can three students be chosen so that the first student is captain, the second student is co-captain, and the third student is the group secretary? Explain. Kaylaa Jeremy Chi-Wei Martin Sunil 18. OPEN ENDED Write a probability question that requires a tree diagram to solve and has an answer of _ 1. Use the spinner shown and a coin E WRITE MATH Describe a situation in which there are 12 possible outcomes. 722 Probability

16 Test Practice 20. Claire is deciding between a red shirt and a blue shirt. The shirt also comes in small, medium, and large sizes. Which diagram shows all of the possible combinations of shirt color and size? A. Red Blue C. Red Blue Small Medium Large Small Large Small Medium Large Small Medium Large B. Red Blue D. Small Medium Large Small Large Medium Large Red Blue Red Blue Red 21. Joey s Pizza Parlor offers 3 kinds of toppings and 3 sizes of pizza. Which table shows all the possible 1-topping pizzas? F. Size Toppings Small Pepperoni Medium Pepperoni Large Pepperoni Small Cheese Medium Cheese Large Cheese H. Size Toppings Small Pepperoni Medium Cheese Large Veggie G. Size Toppings Small Pepperoni Small Pepperoni Small Pepperoni Medium Cheese Medium Cheese Medium Cheese Large Veggie Large Veggie Large Veggie I. Size Toppings Small Pepperoni Small Cheese Small Veggie Medium Pepperoni Medium Cheese Medium Veggie Large Pepperoni Large Cheese Large Veggie 22. SHORT RESPONSE Parker plans on buying one toy listed below for his little sister. He can wrap the gift in pink or yellow paper. How many toy and paper combinations are possible? Toy Doll Sidewalk chalk Stuffed animal 23. EXTENDED RESPONSE Describe what the tree diagram below is showing. Lesson 1C Probability 723

17 Multi-Part Lesson 1 Probability PART A B C D E Main Idea Use probability to decide whether a game is fair or unfair. Vocabulary fair game unfair game Get ConnectED GLE Understand the meaning of probability and how it is expressed. Fair and Unfair How can you tell if a game is fair or unfair? If a game is fair, each player has an equal chance of winning. A game is unfair if one player has a better chance of winning. In this activity, you will analyze two games and determine whether each game is fair or unfair. In a coin-toss game, players toss two coins. The winner of the game is determined by the results of both coins. Play this game with a partner. Player 1 tosses both coins at the same time. If both coins land on heads, Player 1 wins. If one coin lands on heads and the other on tails, Player 2 wins. If both coins land on tails, no one wins. Record the winner. Game 1 2 Player 1 Player 2 Player 2 then tosses both coins. Record the winner. Continue alternating the tosses until each player has tossed the coins 10 times. the Results 1. Make a tree diagram to find the possible outcomes in Activity 1. Write the total number of outcomes resulting from tossing two coins and explain your method. 2. Find the experimental probability that Player 1 wins. 3. Find the experimental probability that Player 2 wins. 4. MAKE A CONJECTURE Based on the probabilities of each player winning, is this a fair or unfair game? Explain your reasoning. 5. MAKE A PREDICTION Predict the number of times Player 1 would win if the game were played 100 times. 724 Probability

18 While playing a game, the spinner shown is spun and a coin is tossed. The winner of the game is determined by the results of both the spinner and the coin. Play this game with a partner. Player 1 spins the spinner and tosses a coin. If the spinner lands on 2, 4, or 6 and the coin lands on tails, Player 1 gets a point. If the numbers 1, 3, or 5 are spun and the coin lands on heads, Player 2 gets a point. No one receives a point for the other outcomes. Record the results Game 1 2 Player 1 Player 2 Player 2 spins the spinner and tosses a coin. Record the results. Continue taking turns until each player has spun the spinner and tossed the coin 10 times. the Results 6. Make a tree diagram to find the possible outcomes in Activity 2. Write the total number of outcomes. 7. What is the probability of Player 1 receiving a point? Player 2? 8. MAKE A CONJECTURE Based on the probabilities of each player winning, is this a fair or unfair game? Explain your reasoning. 9. MAKE A PREDICTION Predict the number of times no one would receive a point if the game were played 100 times. 10. GAMES In the Rock-Paper-Scissors game, two players Scissors show a hand-sign at the same time. If the players make cut paper the same hand-sign, it is a tie. The rest of the rules are shown in the diagram. a. What is the probability of winning? losing? tying? b. Is the game fair? Explain. Game Rules Paper wraps rock 11. E WRITE MATH Use a number cube to create a game. Write two sets of rules to your game. Make one set of rules fair and the other unfair. Rock beats scissors Lesson 1D Probability 725

19 PURPLE Multi-Part Lesson 1 Probability PART A B C D E Main Idea Use the Fundamental Counting Principle to count outcomes and find probabilities. Vocabulary Fundamental Counting Principle Get ConnectED GLE Understand the meaning of probability and how it is expressed. SPI Determine the theoretical probability of simple and compound events in familiar contexts. Also addresses GLE The Fundamental Counting Principle SALES MP3 players come in different colors and styles. 1. According to the table, how many colors of MP3 Color Style players are available? styles? Black Sleek 2. Make a list to find the number of different color Silver Slim and style combinations. Blue Regular Pink 3. Find the product of the two numbers you found Red in Exercise 1. How does the number of outcomes in the list compare to the product? Another way to find the total number of outcomes is to use the Fundamental Counting Principle. Fundamental Counting Principle If event M has m possible outcomes and event N has n possible outcomes, then event M followed by event N has m n possible outcomes. Find the Number of Outcomes Find the total number of outcomes when a coin is tossed and the spinner is spun. RED GREEN BLUE ORANGE coin outcomes = spinner outcomes = total outcomes Fundamental Counting Principle There are 10 different outcomes. a. A spinner with four equal sections marked A, B, C, and D is spun, and a card is picked from cards numbered Find the total number of possible outcomes. 726 Probability

20 PIZZA Casey s Pizza House offers the choices shown. Use the Fundamental Counting Principle to find the total number of 1-topping pizzas you can order. Casey s Pizza House Crust Toppings Deep-dish Hand-tossed Thin Green pepper Ham Mushroom Onion Pepperoni Sausage Real-World Link Pepperoni is America s favorite topping. Americans eat approximately 251,770,000 pounds of pepperoni each year. number of outcomes for crust choice number of outcomes for = topping choice = There are 18 possible 1-topping pizzas. total number of outcomes b. Casey decided to add a whole-wheat crust. How many 1-topping pizzas are possible now? Find Probabilities Teresa rolls a number cube and spins the spinner shown. Find the total number of outcomes. Then determine the probability that she will roll a 3 and the spinner will land on yellow. Step 1 Find the number of possible outcomes. number of choices for the number cube number of choices for = the spinner = total number of outcomes Step 2 Find the probability of rolling a 3 and spinning yellow. Of the 24 outcomes, only one is favorable. So, the probability of rolling a 3 and spinning yellow is 1 _ 24. c. Two number cubes are rolled. What is the probability of rolling two 6s? Lesson 1E Probability 727

21 Examples 1 and 2 Use the Fundamental Counting Principle to find the total number of outcomes in each situation. 1. rolling a number cube with sides labeled 1 6 and a letter from the bag shown 2. choosing a shoe color from brown, black, or gray and picking a style from high-, mid-, or low-tops 3. BICYCLES A bicycle shop sells a bicycle that comes as a mountain bike or a racing bike. The same bicycle also comes in black, silver, white, tan, blue, or red. How many possible bicycles are there? Example 3 4. How many possible outcomes are there if a coin is tossed and the spinner shown is spun? 5. A spinner with 8 equal sections labeled 1 8 is spun twice. What is the probability that it will land on 2 after the first spin and on 5 after the second spin? TOYS A box of cars contains 5 different colored cars. The colors of the cars are blue, orange, yellow, red, and black. A separate box contains a male and a female action figure. What is the probability of randomly choosing an orange car and a female action figure? = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2. Examples 1 and 2 Use the Fundamental Counting Principle to find the total number of possible outcomes in each situation. 7. rolling a number cube and spinning a spinner with eight equal sections 8. tossing a coin and selecting one letter from the word RACE 9. selecting one sweatshirt from a choice of five sweatshirts and one pair of pants from a choice of four pairs of pants 10. selecting one entrée from a choice of nine entrées and one dessert from a choice of three desserts 11 selecting either Mark, Padma, or Terrence to be captain and either Flora, Miguel, or Derek to be co-captain 12. selecting one month of the year and one day of the week 13. choosing a color from 5 colors and a number from 1 to Probability

22 Example SHIRTS Geoff has one gray, one black, one red, and one purple dress shirt. Bryan has one purple, one yellow, one red, one black, and one blue dress shirt. Each boy picks a shirt at random to wear to the banquet. What is the probability the boys will both wear a black shirt? 15. FOOD A cafeteria offers oranges, apples, or bananas as its fruit option. It offers peas, green beans, or carrots as the vegetable option. If the fruit and the vegetable are chosen at random, what is the probability of getting an orange and carrots? Real-World Link Board games are still very popular in American households. However, the board game industry sales represent less than 6% of the video game industry sales. 16. BOARD GAMES A game requires you to toss a 6-sided number cube and spin the spinner at the right to determine how to move on the game board. Find the probability of getting the same number on the number cube and the spinner. B 17 MULTIPLE REPRESENTATIONS The table shows sandwiches sold at a restaurant. a. MODEL Make a tree diagram to find all possible outcomes of one bread and one meat. b. WORDS Explain how you can use the Fundamental Counting Principal to find the outcomes. c. NUMBERS If the bread and meat are picked at random, explain how to find the probability that the sandwich will use white bread and ham. Write your answer as a fraction, decimal, and percent SURVEY A city survey found that 45% of teenagers have a part-time job. The same survey found that 80% plan to attend college. If a teenager is chosen at random, what is the probability that the teenager has a part-time job and plans to attend college? C 19. OPEN ENDED Write a real-world problem that can be solved using the Fundamental Counting Principle. Then solve the problem. 20. NUMBER SENSE Samantha has a choice of a purple, white, pink, or red shirt to wear with a choice of tan, brown, black, or denim pants. Without calculating the number of possible outcomes, how many more outfits can she make if she buys a yellow shirt? Explain. 21. E WRITE MATH Explain a possible benefit of using the Fundamental Counting Principle instead of a using a tree diagram to find the probability of a situation. Lesson 1E Probability 729

23 Test Practice 22. A candy store sells vanilla, chocolate, strawberry, cherry, and mint candies separately in 4 different-sized packages. How many possible boxes of candy can the store make? A. 9 C. 20 B. 16 D The spinner is spun two times. What is the probability that it will land on 2 after the first spin and on 1 after the second spin? F. 1_ 512 G. 1_ H. I. 3_ 64 1_ A movie theater offers a combo special with 3 different drink sizes and 3 different popcorn sizes. From how many different combos can Erica choose? A. 6 B. 9 C. 15 D SHORT RESPONSE Percy chooses one shirt and one pair of pants from the table. How many different combinations are possible? Shirts Yellow Dark blue Orange Green Pants Khaki Denim Dress For each situation, find the sample space using a list or tree diagram. (Lesson 1C) 26. Two coins are tossed. 27. A coin is tossed and the spinner at the right is spun. 28. The spinner shown is spun, and a letter from the word MATH is randomly selected. A drawer of tableware contains 6 forks, 5 knives, and 3 spoons. Tableware is selected at random. Find the probability of each event. (Lesson 1A) 29. P(fork) 30. P(spoon) 31. P(fork or spoon) 32. P(knife or spoon) 33. Mr. Hartman has a box of the geometric figures shown. If a figure is chosen at random, what is the probability of getting a rectangular prism or a cube? (Lesson 1A) A C B D 730 Probability

24 Mid-Chapter Check The spinner is spun once. Find the probability of each event. Write each answer as a fraction. (Lesson 1A) P(4) 2. P(2 or 6) 4 3. P(even) 4. P(1, 3, 4, or 5) 5. VOLLEYBALL The volleyball coach randomly selects a player to lead the team onto the court before each game. Of the players on the team, 58% of them are eighth graders. What is the probability that the coach will not pick an eighth grader? Write your answer as a fraction, decimal, and percent. (Lesson 1A) The table shows the different snacks in a box. One snack is chosen at random. Find the probability of each event. Write each answer as a fraction. (Lesson 1A) Peanuts Pretzels Popcorn 3 Snacks Chips Dried fruit 6. P(popcorn) 7. P(chips or pretzels) 8. MULTIPLE CHOICE A bag contains 3 chocolate chip cookies, 4 sugar cookies, and 2 peanut butter cookies. Kevin picks one at random. What is the probability that he will select a chocolate chip cookie? (Lesson 1A) A. _ 1 24 B. 1_ 12 C. _ 1 3 D. _ GO-KARTS In how many ways can Dirk, Natasha, and Corey finish the go-kart race in first, second, and third place? Use a list to show the sample space. (Lesson 1C) Draw a tree diagram to show the sample space for each situation. Then tell how many outcomes are possible. (Lesson 1C) 10. choosing a red or blue marble and rolling a number cube 11. white, wheat, or rye bread with ham, turkey, roast beef, bologna, or salami 12. MULTIPLE CHOICE How many possible outcomes are there if a number cube with sides labeled 1 6 is rolled and a coin is tossed? (Lesson 1C) F. 8 H. 12 G. 10 I The spinner is spun twice. What is the probability that it will land on 2 after the first spin and on 5 after the second spin? (Lesson 1E) Use the Fundamental Counting Principle to find the total number of possible outcomes for each situation. (Lesson 1E) 14. spinning a spinner with 4 equal sections and selecting a letter from the word SAMPLE 15. selecting a T-shirt from a choice of 7 colors and a pair of shoes from a choice of 6 pairs Mid-Chapter Check 731

25 Multi-Part Lesson 2 Independent and Dependent Events PART A B C D E Probability of Independent Events Main Idea Explore the probability of independent events. Are you an independent person? An independent person does not rely or depend on other people or things. In probability, two events are independent if they do not rely or depend on each other. Let s investigate two independent events. Get ConnectED GLE Understand the meaning of probability and how it is expressed. SPI Determine the theoretical probability of simple and compound events in familiar contexts. Make a tree diagram that shows the sample space for tossing a coin and choosing a cube from the bag. the Results 1. How many outcomes are in the sample space? 2. Find the probability of tossing a coin and landing on heads. 3. What is P(red), the probability of choosing a red cube? 4. Use the tree diagram to find P(heads and red), the probability of landing on heads and choosing a red cube. 5. MAKE A CONJECTURE Describe the relationship between P(heads), P(red), and P(heads and red). and Apply Draw a tree diagram to find the sample space of each situation. Then, find each probability A B 8. A X I C D Z U V P(heads and 5) P(A and blue) P(3 and a vowel) 9. MAKE A CONJECTURE Explain how you can find the probability of rolling an even number on a number cube and tossing heads on a coin without using a tree diagram. 732 Probability

26 Multi-Part Lesson 2 Independent and Dependent Events PART A B C D Main Idea Find the probability of independent events. Vocabulary compound event independent events Get ConnectED Probability of Independent Events STATES Four Corners USA is the only point in the United States where the boundaries of four states intersect. Katie and Joel each randomly select one of the four states to write about in their reports. 1. Draw a tree diagram to show P(Joel and Katie choosing a state). UTAH ARIZONA COLORADO NEW MEXICO GLE Understand the meaning of probability and how it is expressed. SPI Determine the theoretical probability of simple and compound events in familiar contexts. 2. What is the probability of Joel and Katie each choosing to write their reports about Arizona? The combined action of two simple events is called a compound event. Compound events in which the outcome of one event does not affect the outcome of the other event are independent events. Probability of Independent Events Words Symbols The probability of two independent events can be found by multiplying the probability of the first event by the probability of the second event. P(A and B) = P(A) P(B) Probability of Independent Events A coin is tossed and the spinner is spun. Find the probability of tossing heads and spinning a number less than 4. First, find the probability of each event. P(heads) = _ 1 number of ways to toss heads 2 number of possible outcomes P(less than 4) = 3 _ 4 number less than four number of possible outcomes Then find the probability of both events occurring. P(heads and less than 4) = _ 1 2 _ 3 4 or _ 3 8 The probability of tossing heads and spinning less than 4 is _ 3 8. Lesson 2B Independent and Dependent Events 733

27 Blake placed 6 pencils and 4 pens inside his desk. He selected one writing tool without looking, replaced it, and then selected a second writing tool. What is the probability that each writing tool selected was a pencil? Multiply Fractions To multiply fractions, multiply the numerators and multiply the denominators. Find the probability that the first writing tool selected is a pencil. P(pencil) = _ 6 10 or _ 3 5 Since Blake replaces the writing tool back into his desk after the first trial, the probability that the second writing tool selected is a pencil is also _ 6 10 or _ 3 5. P(pencil and pencil) = _ 3 5 _ 3 P(pencil) P(pencil) 5 = _ 9 Multiply. 25 So, the probability of selecting two pencils is _ a. In Example 2, what is the probability of selecting 2 pens? Refer to the coin and spinner from Example 1. Find each probability. b. P(tails and even) c. P(heads and 4) Example 1 One letter is selected from the bag without looking and the spinner is spun. Find the probability of each event. 1. P(red and S) 2. P(green and Z) red red red red green blue green orange 3 P(orange and A) 4. P(green and not A) Example 2 5. COINS Demarcus has 3 quarters, 2 dimes, 3 nickels, and 2 pennies in his pocket. He reaches in and pulls out one coin. Then he replaces it and pulls out another coin. What is the probability that each coin is a quarter? Write your answer as a fraction, decimal, and percent. 734 Probability

28 = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2. Example 1 A card is chosen and a number cube is rolled. Find the probability of each event. 6. P(triangle and 5) 7. P(green shape and even) 8. P(not a rectangle and less than 3) 9. P(blue triangle and odd) Example 2 One color tile is chosen from the bag without looking. Then it is replaced and another color tile chosen. Find each probability. 10. P(yellow and red) 11 P(green and blue) 12. P(blue and yellow) 13. P(red and red) B 14. COMMUNITY SERVICE Lana and Anthony volunteer to read to children at the library. Each student has the books shown in the table. Find each probability if they select the books at random. Write the probability as a fraction, decimal, and percent. Name Comic Animal Books Sports Books Lana Anthony a. P(Lana chooses an animal book and Anthony chooses a sports book) b. P(Lana chooses a sports book and Anthony chooses an animal book) c. P(Lana chooses a comic or animal book and Anthony chooses a sports book) DVDs 15 MOVIES The graph shows the 12 kinds of DVDs on a shelf. 10 Suppose one DVD is chosen without looking. Then it is 8 replaced, and another DVD is 6 chosen. Find each probability. 4 a. P(comedy and not comedy) 2 b. P(drama and action) c. P(comedy or drama and action) Number of DVDs 0 Comedy Action Drama Types of DVDs Lesson 2B Independent and Dependent Events 735

29 C 16. FIND THE ERROR Pilar is finding the probability of rolling a 1 or 2 on a number cube and tossing heads on a coin. Find her mistake and correct it. 2_ 6 + _ 1 2 = _ CHALLENGE Two spinners are spun. The probability of both spinners landing on blue is _ 1. The probability of the second spinner landing on 5 blue is _ 1. What is the probability of the first spinner landing on blue? E WRITE MATH Explain how to find the probability of two independent events. Test Practice 19. SHORT RESPONSE A coin is tossed and the spinner below is spun. Find P(tails and Z). Y Z A B X W V C E D 20. Two number cubes are rolled. Find the probability of rolling a 6 on the first number cube and a number less than 4 on the second number cube. 1_ A. B. 18 1_ 12 C. 1_ 9 D. 1 _ 2 Use the Fundamental Counting Principle to find the total number of possible outcomes for each situation. (Lesson 1E) 21. rolling a number cube and spinning a spinner with twelve equal sections 22. selecting one drink from a choice of five kinds and one dessert from a choice of seven options 23. The table shows the folder colors and writing tools in Brady s Folder Color Writing Tools desk. Brady selects a folder and a writing tool at random. What is the probability of selecting a blue folder and a pen? (Lesson 1E) Yellow Red Blue Green Pencil Pen Marker 736 Probability

30 Multi-Part Lesson 2 Independent and Dependent Events PART A B C Main Idea Explore the probability of dependent events. Get ConnectED D Probability of Dependent Events Five girls and five boys volunteered to work at the community center after school. Ms. Mason will pick two students at random. What is the probability that she will pick two boys? GLE Understand the meaning of probability and how it is expressed. SPI Determine the theoretical probability of simple and compound events in familiar contexts. Also addresses GLE Write 5 girl names and 5 boy names on separate small pieces of paper. Place them inside a bag. Without looking, remove a name from the bag. Make a tally mark to show the result in a table like the one below. Do not replace the name into the bag. Pick 1 2 Boy Girl Without looking, remove a second name from the bag and make another tally mark in the table. Repeat this experiment 20 times. the Results 1. What is the theoretical probability of choosing a girl s name as the first name out of the bag? 2. Suppose a boy s name is chosen first. Is the probability that a girl s name will be chosen next more likely, equally likely, or less likely? 3. Suppose a boy s name is chosen first. Is the probability that a boy s name will be chosen next more likely, equally likely, or less likely? 4. Why are these events called dependent events? 5. MAKE A CONJECTURE The probability of choosing a boy s name first is _ 1. The probability of choosing another boy s name second 2 is _ 4. What is the probability of choosing a boy s name as the first 9 two names? Lesson 2C Independent and Dependent Events 737

31 Multi-Part Lesson 2 Independent and Dependent Events PART A B C D E Main Idea Find the probability of dependent events. Vocabulary dependent events Get ConnectED GLE Understand the meaning of probability and how it is expressed. SPI Determine the theoretical probability of simple and compound events in familiar contexts. Probability of Dependent Events MONEY Tara has five coins in a coin purse. She needs two quarters to buy a snack. There are 2 quarters, 2 pennies, and 1 nickel inside the coin purse. 1. What is the probability of drawing a quarter on the first draw? 2. What is the probability of drawing a quarter on the second draw after drawing a quarter that was not replaced on the first draw? 3. Multiply the probability from Exercise 1 and the probability from Exercise 2. This is the probability of choosing two quarters in a row without replacing the first quarter. If the outcome of one event affects the outcome of a second event, the events are called dependent events. The probability of the second event depends on the fact that the first event has already occurred. Probability of Dependent Events Words If two events A and B are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. Symbols P(A and B) = P(A) P(B following A) Probability of Dependent Events A bag contains 4 blue chips, 4 red chips, and 2 green chips. A chip is selected and not replaced. Another chip is then chosen. Find P(two greens). P(first chip is green) = 2_ 10 P(second chip is green) = _ 1 9 P(two greens) = _ 2 10 _ 1 9 = _ number of green chips total number of chips number of green chips left total number of chips left P(A and B) = P(A) P(B following A) So, the probability of choosing two green chips is 1 _ Probability

32 Mrs. Harris is randomly choosing two students to help with the class recycling project. There are 15 boys and 10 girls in the class. Find the probability that Mrs. Harris selects two boys in a row. number of boys in class 15 P(boy) = _ 25 Reasonable Answer Since at least one possibility is eliminated in a second event, the probability of two dependent events is less than the probability of two similar independent events. total number of students 14 P(boy) = _ number of students after a boy is selected total number of students after a boy is selected _ P(two boys) = _ =_ or _ 60 P(A and B) = P(A) P(B following A) 20 Multiply. 7 So, the P(two boys) is _. 20 a. MARBLES Quinton draws a marble from the jar. Without replacing the first marble, he draws another. What is the probability that he draws a red marble and then a yellow marble? Independent and Dependent Events Sherry spins the two spinners. Tell whether spinning a 2 on each spinner is an example of independent or dependent events Since spinning the first spinner does not affect the results of the second spinner, these events are independent events. There are 3 raisin, 5 chocolate chip, and 2 peanut butter cookies in a container. Pierre randomly selects two cookies without replacing the first cookie. Tell whether choosing two chocolate chip cookies is an example of independent or dependent events. Explain. Since the first cookie is not replaced, the first choice affects the second choice. These are dependent events. Tell whether the events are independent or dependent. Explain. b. A coin is tossed and a number cube is rolled. c. A plate of different kinds of apples is on the table. One apple is selected and not replaced. Another apple is then selected. Lesson 2D Independent and Dependent Events 739

33 Example 1 Example 2 Examples 3 and 4 A ball is drawn from the number balls shown and not replaced. Then, a second ball is drawn. Find each probability. 1. P(two even numbers) 2. P(two odd numbers) 3. P(two numbers less than 14) 4. P(a number less than 15 followed by a number greater than 15) 5. MONEY Seth has 5 one-dollar bills, 2 five-dollar bills, and 3 ten-dollar bills in his wallet. He pulls out one bill and does not replace it. Then he pulls out another bill. What is the probability that both bills are five-dollar bills? Tell whether the events are independent or dependent. Explain. 6. drawing a card labeled with a letter of the alphabet and not replacing it before a second card is drawn 7 spinning a spinner and rolling a number cube = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2. Example 1 Example 2 A drawer contains 4 blue socks, 5 white socks, and 8 black socks. After a sock is chosen, it is not replaced. Find each probability. 8. P(two blue socks) 9. P(two black socks) 10. P(two white socks) 11. P(blue sock, black sock) 12. PROJECTS Five boys and two girls are volunteering for a class project. Their names are written on slips of paper and placed in a bowl. The teacher will choose two names without replacing the first name chosen. What is the probability that two girls names will be chosen? 13. BOARD GAMES The letter tiles shown below are placed in a bag. Find the probability that two consonants will be chosen if the first tile is not replaced. Examples 3 and 4 Tell whether the events are independent or dependent. Explain. 14. drawing a marble out of a bag and not replacing it before drawing another marble 15. rolling a number cube twice 16. choosing any book from a list for a book report and then having another student choose any book from the same list 17 choosing a partner to work with on a science project and then having another student choose a partner from the remaining students 740 Probability

34 B 18. MATH IN THE MEDIA Find an example of a dependent event in a newspaper or magazine, on television, or on the Internet. Explain why it is a dependent event. 19 ANNOUNCEMENTS Three girls and two boys want to be the two students who announce the morning news at school. To be fair, the teacher decides to place their names in a hat and select two students. Find the probability of selecting a girl and then a boy. Write your answer as a fraction, decimal, and percent. 20. TABLES Rocia randomly chooses Paper Clips paper clips one at a time from the Small drawer without replacing them. Green 5 What is the probability that the Yellow 5 first three paper clips Rocia Black 10 chooses are small and black? Large GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a b. I think I have a better chance of getting two of my photos placed in the newspaper! a. Noah, Amanda, and Luis are randomly selecting two photos for the back page of the paper. Noah has 2 basketball photos, Amanda has 3 drama club photos, and Luis has 4 concert band photos. What is the probability that two concert band photos will be selected? b. What is the probability that two drama club photos will be selected? C 22. OPEN ENDED Write a real-world problem involving two dependent events. Then solve the problem. 23. CHALLENGE Javier spun two spinners. The probability of both spinners landing on red is _ 3. The probability of just the first spinner landing on 32 red is _ 1. What is the probability of the second spinner landing on red? E WRITE MATH Compare and contrast independent events and dependent events. Give an example of each. Lesson 2D Independent and Dependent Events 741

35 TTest Practice 25. A jar contains 8 yellow cubes, 4 blue cubes, and 2 red cubes. Fred randomly selects two cubes without replacing the first cube. What is the probability that two yellow cubes are selected? 2 A. _ 7 4 B. _ 13 6 C. _ 91 1 D. _ EXTENDED RESPONSE Matthew tosses a coin 4 times. Part A What is the probability that the first 2 tosses are tails? Part B Tell whether the situation describes independent events or dependent events. Explain. 27. GRIDDED RESPONSE A jar contains 17 marbles. Courtney picks a marble at random. Without replacing it, she randomly chooses a second marble. Find the probability of choosing a red marble and then a white marble. 28. The digits of a locker combination consist of 3 one-digit numbers, 1 9. If the digits are not repeated, what is the probability of the combination being 1-2-3? 1 F. _ 1 H. _ 720 _ G GAMES A game involves randomly choosing a card from the set of 5 cards shown and spinning the spinner. (Lesson 2B) a. What is the probability of choosing a four? spinning green? I. _ 168 b. What is the product of P(4) and P(green)? c. Draw a tree diagram to determine P(4 and green). Use the Fundamental Counting Principle to find the total number of outcomes in each situation. (Lesson 1E) 30. choosing a number from 1 to 10 and rolling a number cube 31. choosing between 5 entrées and 6 side dishes 32. picking a basketball from 4 different types of basketballs; indoor, indoor/outdoor, or outdoor; and rubber or leather 33. choosing a pair of shoes that comes in 5 different styles, 4 different colors, and 6 different sizes 742 Probability

36 Multi-Part Lesson 3 Collect Data PART A B C Main Idea Predict the actions of a larger group using a sample. Vocabulary survey population sample Get ConnectED GLE Understand the meaning of probability and how it is expressed. GLE Interpret representations of data from surveys and polls, and describe sample bias and how data representations can be misleading. Also addresses SPI Make Predictions In this activity, you will predict the number of students in your school that walk to school. Have one student in each group copy the table shown. Count the number of students in your group for each transportation type. Record the results. Transportation Type Students walk bus car Predict the number of students in your school that walk to school. Combine your results with the other groups in your class. Make a prediction based on the class data. 1. When working in a group, how did your group predict the number of students that walk to school? 2. Compare your group s prediction with the class prediction. Which do you think is more accurate? Explain. A survey is a method of collecting information. The group being studied is the population. Sometimes, the population is very large. To save time and money, part of the group, called a sample, is surveyed. A classroom is a subset of the entire school. The entire group of people in a school is an example of a population. The group of people in one classroom in a school is a sample of the population. A good sample is: selected at random, or without preference representative of the population large enough to provide accurate data. Lesson 3A Collect Data 743

37 The ratios of the responses of a good sample are often the same as the ratios of the responses of the population. So, you can use the results of a survey or past actions to predict the actions of a larger group. Make Predictions Using Ratios Real-World Link The three basic types of camera lenses are standard, wide angle, and telephoto. The power of a camera lens is measured in millimeters and is known as the focal length of a lens. The 50mm lens is usually the standard lens for a 35mm camera. PHOTOS The students in Mr. Blackwell s class brought photos from their summer break. The table shows how many students brought each type of photo. What is the probability that a student brought a photo taken at a theme park? number of students with theme park photos P(theme park) = number of students with a photo = _ So, the probability that a student brought a photo taken at a theme park is _ There are 560 students at the school where Mr. Blackwell teaches. Predict how many students would bring in a photo taken at a theme park. Let s represent the number of students who would bring in a photo taken at a theme park. 11_ 28 = s_ Write an equivalent ratio _ 28 = s_ _ 28 = _ Since = 560, multiply 11 by 20 to find s. s = 220 Summer Break Photos Location Students beach 6 campground 4 home 7 theme park 11 Of the 560 students, you can expect about 220 to bring a photo from a theme park. INTERNET A survey at a school found that 6 out of every 10 students have an Internet blog. a. What is the probability that a student at the school has a blog? b. If there are about 250 students at the school, about how many have a blog? 744 Probability

38 Example 1 Example 2 CAREERS Every tenth student entering Hamilton Middle School is asked what career field he or she may pursue upon finishing school. 1. Find the probability that a student will choose public service as a career field. 2. Predict how many students out of 400 will enter the education field. Career Field Students Entertainment 17 Education 14 Medicine 11 Public service 6 Sports 2 = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2. Example 1 Example 2 MAGAZINES Three out of every 10 students ages 6 14 have a magazine subscription. 3. Find the probability that Annabelle has a magazine subscription. 4. Suppose there are 30 students in Annabelle s class. About how many will have a magazine subscription? VIDEO GAMES Luther won 12 of the last 20 video games he played. 5. Find the probability of Luther winning the next game he plays. 6. Suppose Luther plays a total of 60 games with his friends over the next month. Predict how many of these games Luther will win. SPORTS Use the table to predict the number of Sport Students students out of 500 that would participate in Baseball/softball 6 each sport. Basketball 5 7. football 8. tennis Football 9 Gymnastics 2 9 gymnastics 10. basketball Tennis 3 B 11. VOLUNTEERING Use the graph. a. About 300,000 kids ages live in Virginia. Predict the number of kids who volunteer a few times a year. b. Tennessee has about 250,000 kids ages Predict the number of kids in this age group who volunteer once a week. c. About 240,000 kids ages live in Missouri. Predict the number of kids in this age group who volunteer once a year. Lesson 3A Collect Data 745

39 12. BOOKS The school librarian recorded the types of books students checked out on a typical day. If there are 605 students enrolled at the school, predict the number of students that prefer humor books. Compare this to the number of students at the school who prefer nonfiction. Number of Students Types of Books Checked Out Humor Nonfiction Mystery Romance Adventure Type of Book 13 BASKETBALL The probability of Jaden making a free throw is 15%. Predict the number of free throws that he can expect to make if he attempts 40 free throws. 13 C 14. FIND THE ERROR A survey of a sixth-grade class showed that 4 out of every 10 students are taking a trip during spring break. There are 150 students in the sixth grade. Elisa is trying to determine how many of the sixth-grade students can be expected to take a trip during spring break. Find her mistake and correct it = 150 x 4_ 10 = _ x = 375 students 15. CHALLENGE One letter tile is drawn from the bag and replaced 300 times. Predict how many times a consonant will not be picked. 16. OPEN ENDED Give an example of a situation in which you would make a prediction. 17. E WRITE MATH Three out of four of Mitch s sixth-grade friends say that they will not attend the school dance. Based on this information, Mitch predicts that only 25 of 100 sixth graders at his school will attend the dance. Is this a valid prediction? Explain your reasoning. 746 Probability

40 Test Practice 18. At the school carnival, Jesse won the balloon dart game 1 out of every 5 times he played. If he plays the game 15 more times, about how many times can he expect to win? A. 3 B. 4 C. 5 D SHORT RESPONSE If 7 out of 30 students are going on the ski trip, predict the number of students out of 150 that are going on the ski trip. 20. The table shows the results of a survey of sixth-grade students in the lunch line. Favorite Drink Drink Students Chocolate Milk 15 Soda 12 Milk 6 Water 2 If there are 245 sixth graders in the school, how many can be expected to prefer chocolate milk? F. 45 G. 84 H. 90 I A magazine rack contains 5 sports magazines, 7 news magazines, and 10 fashion magazines. After a magazine is chosen, it is not replaced. Find the probability of randomly choosing two fashion magazines. (Lesson 2D) 22. Each week, Ryan s mother has him randomly choose a chore that he must complete from the list shown. The first week he chose washing the dishes. What is the probability that Ryan will choose washing the dishes two weeks in a row? (Lesson 2B) Weekly Chores Collecting the trash Folding the laundry Cleaning the house Washing the dishes Cutting the grass 23. VIDEOS How many ways can a person watch 3 different videos? Use a list to show the sample space. (Lesson 1C) Juanita randomly turns to a page in a 15-page booklet. Find the probability of each event. (Lesson 1A) 24. P(odd page) 25. P(even page) 26. P(page that is a composite number) 27. P(3, 5, or 7) Lesson 3A Collect Data 747

41 Multi-Part Lesson 3 Collect Data PART A B C Unbiased and Biased Samples Main Idea Determine if samples are biased. Vocabulary unbiased sample biased sample Get ConnectED GLE Interpret representations of data from surveys and polls, and describe sample bias and how data representations can be misleading. SPI Determine whether or not a sample is biased. You have learned that samples should be selected at random, represent the entire population, and be large enough to provide accurate data. Samples that meet these qualifications are called unbiased samples. Samples that do not meet these qualifications are called biased samples. The principal wants Which Type of Entertainment students to decide what do You Prefer? entertainment to have for the school s Family Fun Band music Night. Students can choose from the options listed in the survey. Sports activities To save time, the principal is only going to survey a sample of the students. He is deciding Magic which of the samples shown below to use. A. Survey members of the school band. B. Survey the first two students he sees. C. Survey students as they leave the school. With a partner, create a list of pros and cons for each possible sample. the Results 1. Would surveying the members of the band be biased or unbiased? Explain. 2. Would surveying the first two students he sees be biased or unbiased? Explain. 3. Would surveying the members of the basketball team be biased or unbiased? Explain. 4. Would surveying students as they leave the school be biased or unbiased? Explain. 5. MAKE A CONJECTURE Name another sample the principal could use that would be unbiased. Explain. 748 Probability

42 It is also important that the questions asked on surveys are unbiased. If surveys have biased survey questions, the data collected will be invalid. and Apply 6. SURVEYS The school newspaper published the two questions shown. Do you prefer the cafeteria s delicious spaghetti, or would you rather eat the hot dogs? Spaghetti Hot Dogs Which animal should be our school mascot? Bulldog Tiger a. Is the first survey question biased or unbiased? If the question is biased, rewrite it so that it is unbiased. b. Is the second survey question biased or unbiased? If the question is biased, rewrite it so that it is unbiased. c. Give an example of a biased survey question about choosing a school mascot. 7. The parks department wants to survey citizens to see if they would like a dog park in their city. Decide which sample would be unbiased. Explain. Survey 1: people at a dog show Survey 2: every 5th person listed in a phone book Survey 3: the students at a nearby middle school Determine whether each survey question is biased or unbiased. If the question is biased, explain how to make it unbiased. 8. Would you rather listen to a baseball game on the radio or watch all of the action on a huge TV? 9. How much time do you spend reading every night? 10. Do you spend more time on the computer or watching television? 11. More students in the school are using Emma s beauty products than any other brand. Do you use Emma s beauty products? 12. Do you want to eat a hamburger or the popular veggie burger? 13. E WRITE MATH Create your own survey questions about a topic of interest to you. Then write a sentence explaining what unbiased sample could take your survey. Lesson 3B Collect Data 749

43 Multi-Part Lesson 3 PART Collect Data A B Problem-Solving Investigation C Main Idea Solve problems by acting them out. Act It Out LISETA: I am in a contest to sell magazines for the Comets Drama Club. For every subscription I sell, I will receive a card with a letter on it. The first student to spell the word COMETS wins. Everyone has an equally likely chance of getting a different card each time. I wonder how many subscriptions I should sell to win the contest? YOUR MISSION: Predict how many subscriptions Liseta needs to sell. Understand You know that she will receive a letter for each subscription she sells. Plan Act out the problem. Create a spinner that will help you do a simulation of the contest. A simulation is a way of using models to act out events which would be difficult to act out in real life. S T C E O M Solve Check Spin the spinner and make a table of the results. Check to see if the results agree by doing several more trials. 1. Explain how the act it out strategy is related to the experimental probability. 2. E WRITE MATH Write a problem that could be solved by acting it out. Then use the strategy to solve the problem. Explain your reasoning. GLE Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution. Also addresses GLE Probability

44 = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2. Act it out. Make a table. Make a list. Choose an operation. 8. PATTERNS Find the next figure in the pattern. Use the act it out strategy to solve Exercises SWIMMING Five students are entered in a 200-meter backstroke race. Assuming there are no ties, in how many different ways can first and second places be awarded? 4. RESTAURANT At a certain restaurant, prizes are given with each children s meal that is ordered. During a promotion, three different prizes are given at random. Estimate how many children s meals should be purchased to get all three prizes. 5. MONEY Vaughn bought a granola bar and fruit punch for $3.65. If he paid the cashier $4, in how many different ways can he receive his change if the cashier only gives him quarters, dimes, and nickels? Use any strategy to solve Exercises EYE COLOR The table shows the eye color of each student in a class. Make a table of the data. How many more students have brown eyes than green eyes? Eye Color BR G BR H BL BR BR BL BR H G BR BR BL G H BR BL BR G H BR = brown BL = blue G = green H = hazel 7. NUMBERS Isaac is thinking of three numbers from 1 to 9 that have a sum of 20. Find all of the possible numbers. 9 MAIL On each day Monday through Saturday, about 2,300 pieces of mail are delivered by each mail carrier in a certain town. About how many pieces of mail are delivered by each carrier in five years? 10. SHOES Naomi has three different pairs of basketball shoes. She has a white pair with red stripes, a white pair with blue stripes, and a red pair with white stripes. She picks the pairs at random to wear at each game. Estimate the probability that she wears the white pair with red stripes more than once during a five-day basketball tournament. Shoes Tally Frequency White with red stripes White with blue stripes Red with white stripes 11. GRADES The teacher notices that about 6 out of every 10 of her students usually get an A on the weekly math quiz. If 50 students take the quiz, how many As can the teacher expect in all? 12. NUMBER SENSE The probability of getting a a red piece of candy from a jar is _ 1 5. The probability of getting a green piece of candy is _ 1. The number of blue pieces of 2 candy in the jar is double the number of yellow candies. What is one possibility for the number of each color of candy in the jar? Lesson 3C Collect Data 751

45 in Medicine On Call for Kids Do you have compassion, a sense of humor, and the ability to analyze data? You might want to consider a career in medicine. Pediatricians care for the health of infants, children, and teenagers. They diagnose illnesses, interpret diagnostic tests, and prescribe and administer treatment. Are you interested in a career as a pediatrician? Take some of the following courses in high school. Algebra Biology Calculus Get ConnectED 752 Probability Chemistry Psychology

46 200 Patients Tested for Strep Throat Patients Have Strep Throat Patients do Not Have Strep Throat Test is Positive True Positive (TP) 90 False Positive (FP) 17 Test is Negative False Negative (FN) 8 True Negative (TN) 85 GLE Use the information in the table to solve each problem. Round to the nearest tenth, if necessary. Write each answer as a percent rounded to the nearest whole number. 1. What is the probability that one of the patients tested has strep throat? 2. If a patient has strep throat, what is the probability that they have a positive test? 3. What is the probability that a patient with the disease has a negative test? 4. If a patient does not have the disease, what is the probability that they have a positive test? 5. What is the probability that a patient that does not have strep throat tested negative for the disease? 6. The positive predictive value, or PPV, is the probability that a patient with a positive test result will have the disease. What is the PPV? 7. The negative predictive value, or NPV, is the probability that a patient with a negative test result will not have the disease. What is the NPV? Problem Solving in Medicine 753

47 Probability Chapter Study Guide and Review Key Concepts Be sure the following Key Concepts are noted in your Foldable. Probability Probability (Lesson 1) The probability of an event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. Fundamental Counting Principle (Lesson 1) If event M has m possible outcomes and is followed by event N that has n possible outcomes, then the event M followed by N has m n possible outcomes. Independent and Dependent Events (Lesson 2) The probability of two independent events can be found by multiplying the probability of the first event by the probability of the second event. If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. Making Predictions (Lesson 3) You can use the results of a survey to predict the probability of the actions of a larger group. Key Vocabulary biased sample complementary event compound event experimental probability fair game Fundamental Counting Principle independent event outcome population probability Vocabulary Check random sample sample space simple event survey theoretical probability tree diagram unbiased sample unfair game State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. Compound events consist of two or more simple events. 2. A random outcome is an outcome that occurs by chance. 3. The Fundamental Counting Principle counts the number of possible outcomes using the operation of addition. 4. Events in which the outcome of the first event does not affect the outcome of the other event(s) are simple events. 5. The sample space of an event is the set of outcomes not included in the event. 6. An unfair game is a game where there is not a chance of each player being equally likely to win. 7. A sample is any one of the possible results of an action. 754 Probability

48 Multi-Part Lesson Review Lesson 1 Probability Probability (Lesson 1A) One marble is pulled from the bag without looking. Find the probability of each event. Write each answer as a fraction. 8. P(purple) 9. P(not red) 10. P(yellow or green) 11. P(purple or red) 12. SOCKS A drawer contains 14 socks. Of these, 4 are black, 2 are brown, 2 are blue, and the rest are white. Find the probability of randomly selecting a white sock. EXAMPLE 1 The spinner 10 1 shown is spun once. Find 9 2 the probability of landing 8 3 on an even number There are 10 equally likely outcomes on the spinner. Five of the numbers are even numbers. Those numbers are 2, 4, 6, 8, and 10. P(2, 4, 6, 8, or 10) = _ 5 10 or _ 1 2 So, P(2, 4, 6, 8, or 10) is _ 1, 0.5, or 50%. 2 Sample Spaces (Lesson 1C) For Exercises 13 and 14, use a list to show the sample space for the situation. Then tell how many outcomes are possible. 13. a choice of black or blue jeans in classic fit, stretch, or bootcut style 14. a choice of a comedy, action, horror, or science fiction DVD in wide-screen or full-screen format Draw a tree diagram to show the sample space for each situation. Then tell how many outcomes are possible. 15. apple, peach, or cherry pie with milk, juice, or tea 16. toss a dime, quarter, and penny A coin is tossed and a number cube is rolled. 17. How many outcomes are possible? 18. Find P(tails, not odd). 19. Find P(heads, less than 5). EXAMPLE 2 Suppose you have a choice of a plain (P) or frosted (F) doughnut with cream (C), jelly (J), or custard (S) filling. How many different doughnuts are possible? Use a tree diagram. There are 6 possible doughnuts. Chapter Study Guide and Review 755

49 Chapter Study Guide and Review Lesson 1 Probability (continued) The Fundamental Counting Principle (Lesson 1E) 20. HOUSES When building a house, Bob can choose from four different house styles and five different floor plans. How many different possibilities are there for building a house? 21. BICYCLE A bicycle shop offers bicycles in two different styles: rugged-terrain and road racing. Those bicycles come in three colors: red, blue, or yellow. What is the probability that the bicycle will be a rugged-terrain blue bicycle? EXAMPLE 3 Find the probability that in a family of three children, all three children are girls. outcomes outcomes outcomes of 1 st child of 2 nd child of 3 rd child There are 8 outcomes, but there is only one possible outcome resulting in three girls. So, the probability that all three children are girls is _ 1 8. Lesson 2 Independent and Dependent Events Probability of Independent Events (Lesson 2B) 22. SPORTS Anna is choosing from a crate of 7 baseballs, 3 softballs, and 6 tennis balls. Another crate has 4 footballs, 3 golf balls, and 2 basketballs. If Anna chooses a ball from each crate at random, what is the probability that she will select a baseball and a golf ball? EXAMPLE 4 A number cube is rolled, and the spinner shown is spun. Find the probability of rolling an odd number and then landing on blue. P(odd and blue) = _ 1 2 _ 1 5 = _ 1 10 The probability of rolling an odd number and landing on blue is _ RED GREEN BLUE PURPLE ORANGE Probability of Dependent Events (Lesson 2D) 23. FRUIT A bag of fruit contains 4 strawberries, 7 blueberries, and 3 cherries. Maria randomly selects two pieces of fruit without replacing the first piece. What is the probability that she selects two blueberries? EXAMPLE 5 Two cards are drawn from a deck of 6 cards numbered 1 to 6. The first card is not replaced. Find the probability of drawing a 1 and then a 4. P(1 and 4) = _ 1 6 _ 1 5 = _ 1 30 So, P(1 and 4) = _ Probability

50 Multi-Part Lesson Review Lesson 3 Collect Data Make Predictions (Lesson 3A) PIZZA Out of 32 students, 8 prefer cheese only on their pizza. 24. What is the probability that a student in this group likes cheese only on a pizza? Write the answer as a fraction, decimal, and percent. 25. If there were 240 students, how many would you expect to like cheese only on their pizza? EXAMPLE 6 If 9 out of 25 people prefer rock music, how many people out of 1,000 would you expect to prefer rock music? Let p represent the number of people who prefer rock music. 9_ 25 = p_ Use an equivalent ratio. 1, _ 25 = _ 360 Since = 1,000, multiply 9 by 40. 1, So, of the 1,000 people, you would expect 360 to prefer rock music. PSI: Act It Out (Lesson 3C) Solve each problem. Use the act it out strategy. 26. DESKS In how many ways can Mrs. Rolloson arrange five desks in one row? She wants to place Iris s desk and Emily s desk next to each other. She wants to put Thomas s desk in the front. 27. MONEY In how many ways can $0.34 be represented using only nickels and dimes? Use pennies only for the $ GOLF Will s golf ball is 25 feet from the hole. He putts the ball 20 feet in a straight line toward the hole on his first shot. On his next shot, he putts the ball 8 feet in a straight line, but the ball rolls past the hole. How far is Will s ball from the hole? EXAMPLE 7 How many ways can two girls and two boys sit in a row of four seats at a movie theater if the girls must sit in the first two seats? You can do a simulation of the situation by creating cards to represent each child. Determine how many ways the cards can be arranged. Girl 1 Girl 1 Girl 2 Girl 2 Girl 2 Girl 2 Girl 1 Girl 1 Boy 1 Boy 2 Boy 1 Boy 2 Boy 2 Boy 1 Boy 2 Boy 1 Each row of cards represents 1 seating arrangement. So, there are 4 possible arrangements. Chapter Study Guide and Review 757

51 Practice Chapter Test A set of 20 cards is numbered One card is chosen without looking. Find each probability. Write each probability as a fraction, decimal, and percent. 1. P(8) 2. P(prime) 3. P(3 or 10) 4. P(not odd) 5. FOOD A food cart offers a choice of iced tea or soda and nachos, popcorn, or pretzels. a. Draw a tree diagram that shows all of the choices for a beverage and a snack. b. Find the probability that the next customer who orders a beverage and a snack will choose iced tea and popcorn. 6. VACATION Julio asked every second sixth-grade student what they enjoy doing most on an extended break from school. Activity Students playing outside 31 shopping 24 traveling 16 playing video games 15 sports 14 a. Find the probability that a student enjoys playing outside most. b. If there are 280 students in the sixth grade, how many can be expected to enjoy playing outside most? Use the Fundamental Counting Principle to find the total number of possible outcomes in each situation. 7. A number cube is rolled and a spinner with four equal sections is spun. 8. A password uses a 3-digit number. The digits can repeat. 9. MULTIPLE CHOICE Jose is practicing free throws. He has made 10 out of 12 shots. If he shoots 24 more free throws, about how many of these free throws can he expect to make? A. 30 C. 24 B. 28 D A marble is selected from the bag and not replaced. Find the probability of selecting two red marbles. 11. One letter tile is selected and a number cube is rolled. What is the probability of selecting a vowel and an odd number? 12. EXTENDED RESPONSE Serena s Boutique is having a sale. If you pick one item from each category, you get all three for a total of $25. Polo Shirt Hat Socks Teal Mauve White Blue Green Red Striped Polka dots Checkered a. What are the possible outcomes you can buy to get the sale price? Show these outcomes in a tree diagram. b. If the mauve shirt is removed from the choices, how many fewer outcomes will there be? c. Suppose you choose an outfit at random. What is the probability that it will contain a white shirt, a blue or green hat, and striped socks? 758 Probability

52 Preparing for Standardized Tests Multiple Choice: Use Key Words When solving multiple-choice questions, pay attention to words like most, least, and NOT. These words may be boldfaced or uppercase. Music Unlimited is having a musical instrument sale. The sales from last week are shown in the table. What is the probability that the next purchase will NOT be a trumpet? A. 1 _ 5 B. 2 _ 5 C. 3 _ 5 D. 4 _ 5 P(a trumpet) = _ 8 40 or _ 1 5 P(not a trumpet) = 1 - _ 1 5 or _ 4 5 The correct answer is D. Music Unlimited Weekly Sales Instrument Number Sold Saxophone 11 Trumpet 8 Flute 7 Clarinet 14 Devon made the spinner below as an interesting way to determine in which order to do his homework each night after school. What is the probability that on the first spin, the arrow will NOT land on science? SCIENCE HISTORY HISTORY LANGUAGE ARTS MATH SCIENCE Check every answer choice in a multiplechoice question. Each time you find an incorrect answer, cross it off so that you remember that you ve eliminated it. F. 1_ 6 G. 1_ 3 H. 2_ 3 I. 5_ 6 Preparing for Standardized Tests 759

53 Test Practice Read each question. Then fill in the correct answer on the answer sheet provided by your teacher or on a sheet of paper. 1. What is the total number of outcomes if you can choose from 8 different styles of cell phones that come in 4 different colors? A. 12 B. 24 C. 32 D The circle graph shows the percentages of the colors of cars purchased at a car dealership. What fraction of the cars purchased are red? 2. The table shows the number of wins the Bears baseball team had in the last 4 seasons. What is the mean number of wins? 1_ F. 2 G. _ H. _ I. 8_ 25 Year Wins F. 6 H. 10 G. 9 I Titus ate _ 3 of a lasagna. If the lasagna was 8 divided into 16 equal sections, how many sections did Titus eat? A. 6 B. 8 C. 10 D SHORT RESPONSE What is the probability of spinning red and then spinning green on the spinner below? red red red blue blue blue green green 8. Malik was estimating the area of his room to replace the carpet. Which measurement would be a reasonable estimate of the area of his room? 4. SHORT RESPONSE Ethan and two friends are going to split the cost of a video game equally. How much will each person pay if the cost of the video game is $48.36? 7.5 ft Malik s room 5. GRIDDED RESPONSE A bakery offers 5 kinds of muffins and 4 kinds of juice. How many possible orders of one muffin and one juice are available? 8 ft A. 15 square feet C. 64 square feet B. 20 square feet D. 72 square feet 760 Probability