Theoretical and Experimental Probability LESSON 10-1 Problem of the Day If the lengths of the sides of a square increase by 10%, by what percent does the area increase? 21% 10-1

Theoretical and Experimental Probability LESSON 10-1 Check Skills You’ll Need (For help, go to Lesson 5-8.) 1. Vocabulary Review A collection of all the possible outcomes in an experiment is a(n) ? . Suppose you roll a number cube. 2. What are the possible outcomes? 3. Find P(4). 4. Find P(even number). 5. Find P(3 or 4). Check Skills You’ll Need 10-1

Theoretical and Experimental Probability LESSON 10-1 Check Skills You’ll Need Solutions 1. sample space 2. 1, 2, 3, 4, 5, 6 3. 4. 5. 1 6 1 2 1 3 10-1

Theoretical and Experimental Probability LESSON 10-1 Additional Examples A gardener plants 250 sunflower seeds and 210 germinate. Find the experimental probability that a sunflower seed will germinate. P(germinate) = number of seeds that germinate total number of seeds Write the probability ratio. = Substitute. 210 250 = 0.84 Divide. = 84% Write as a percent. The experimental probability that a sunflower seed will germinate is 84%. Quick Check 10-1

Theoretical and Experimental Probability LESSON 10-1 Additional Examples The probability of tossing one six-sided number cube and getting a 5 is , or about 17%. Is this experimental or theoretical probability? 1 6 You can calculate this probability without doing any trials because each of the six possible outcomes are equally likely. This is theoretical probability. Quick Check 10-1

Theoretical and Experimental Probability LESSON 10-1 Additional Examples Suppose you select a pen at random from 3 red ones and 4 blue ones. What are the odds in favor of selecting a red pen? Since 3 pens are red and 4 are blue, the odds of selecting a red pen at random are 3 : 4. Quick Check 10-1

Theoretical and Experimental Probability LESSON 10-1 Lesson Quiz Use this information to solve the problems: You toss two nickels 50 times. You get one head and one tail 38 times, and two heads 8 times. 1. Find the probability of getting one head and one tail. 2. Find the probability of getting two heads. 3. What kind of probability do your answers to 1 and 2 represent? 76% 16% experimental 10-1

Theoretical and Experimental Probability LESSON 10-1 Lesson Quiz 4. What are the odds in favor of getting exactly one head and one tail? 38 : 12 or 19 : 6 10-1

Making Predictions LESSON 10-2 Problem of the Day Suppose a cricket jumps a distance of 12 in. and a grasshopper jumps 16 in. If they start at the same point, how far away will be the first spot they both land on? 48 in. 10-2

1. Vocabulary Review To solve a proportion, you first Making Predictions LESSON 10-2 Check Skills You’ll Need (For help, go to Lesson 4-3.) 1. Vocabulary Review To solve a proportion, you first write the ? products. Solve each proportion. 2. 3. 4. 9 14 x 210 5 8 x 6 = 85 x 21 28 = = Check Skills You’ll Need 10-2

Solutions 1. cross 2. 135 3. 136 4. 4.5 Making Predictions LESSON 10-2 Check Skills You’ll Need Solutions 1. cross 2. 135 3. 136 4. 4.5 10-2

A restaurant promotion offers chance of Making Predictions LESSON 10-2 Additional Examples 1 48 A restaurant promotion offers chance of winning a free dessert with a meal. Out of 570 meals served, how many winners are likely? Probability of a winner A diagram can help you understand the problem. 1 48 • 570 = • 570 Find of 570. = Divide the numerator and denominator by the GCF, 6. • 1 8 95 10-2

= 11 Write the fraction as a mixed number. Making Predictions LESSON 10-2 Additional Examples (continued) 95 8 = Simplify. = 11 Write the fraction as a mixed number. 7 8 There will be about 12 winners. Quick Check 10-2

In a random survey at a school, 16 out of 25 Making Predictions LESSON 10-2 Additional Examples In a random survey at a school, 16 out of 25 students prefer reading books for leisure. If there are 400 students in the school, predict how many students prefer reading books. Method 1 Write a proportion using the survey results and the number of students. 10-2

16 • 400 = 25 • x Use the cross product property. Making Predictions LESSON 10-2 Additional Examples (continued) Set up a proportion. 16 25 = x 400 16 • 400 = 25 • x Use the cross product property. 6,400 25 = 25x Divide each side by 25. 256 = x Simplify. There are likely to be 256 students who prefer reading books. 10-2

Method 2 From the survey, find the probability that Making Predictions LESSON 10-2 Additional Examples (continued) Method 2 From the survey, find the probability that a student will prefer reading. Apply this probability to all students. The event “prefer reading books” occurred in 16 out of 25 trails. 16 25 = 64% Find 64% of 400. 64% of 400 = 0.64 x 400 Find 64% of 400. = 256 Simplify. There are likely to be 256 students who prefer reading books. Quick Check 10-2

1. The probability that a train leaving Central Station will Making Predictions LESSON 10-2 Lesson Quiz 1. The probability that a train leaving Central Station will arrive on time is . The station has 58 trains leaving each day. Predict how many trains will arrive on time. 2. A random survey of 24 buses leaving Central Station one day showed that 15 buses arrived at their destinations on time. Each day, 146 buses leave Central Station. Predict how many of these buses will arrive on time. 5 6 about 48 trains about 91 buses 10-2

3. A spinner is divided into different colored sections. The Making Predictions LESSON 10-2 Lesson Quiz 3. A spinner is divided into different colored sections. The probability of the spinner landing on the purple section is . Predict how many times the spinner lands on the purple section if it spins 98 times. 4. In a random survey of voters in a small town, 52% of the people plan to vote for Mrs. Teller for state representative. If 1,500 people vote, predict the number of votes Mrs. Teller will receive. 2 7 about 28 times about 780 10-2

Find three whole numbers whose sum is 14 and whose product is 84. Conducting a Survey LESSON 10-3 Problem of the Day Find three whole numbers whose sum is 14 and whose product is 84. 3, 4, 7 10-3

1. Vocabulary Review How do theoretical and Conducting a Survey LESSON 10-3 Check Skills You’ll Need (For help, go to Lesson 10-1.) 1. Vocabulary Review How do theoretical and experimental probability differ? There are 9 blue, 5 red, and 4 green pencils in a bag. Find the probability of randomly selecting each color. 2. P(blue) 3. P(green) 4. P(red) Check Skills You’ll Need 10-3

1. Experimental probability is based on running numerous trials or Conducting a Survey LESSON 10-3 Check Skills You’ll Need Solutions 1. Experimental probability is based on running numerous trials or experiments, whereas theoretical probability is based on the mathematical likelihood of events. 2. 3. 4. 1 2 2 9 5 18 10-3

Tell whether the survey uses a random sample. Conducting a Survey LESSON 10-3 Additional Examples Tell whether the survey uses a random sample. Describe the population of the sample. To find out how often students in your school go to movies, you select names at random from the school directory to interview. This is a random sample. The population is the students in your school. Quick Check 10-3

Determine whether the question is biased or not. Explain your answer. Conducting a Survey LESSON 10-3 Additional Examples Determine whether the question is biased or not. Explain your answer. Do you enjoy modern songs or old songs? This question is biased. “Modern” and “old” are not neutral and descriptive terms. They are terms that may make one answer choice seem better than the other. Quick Check 10-3

Malka asked three classmates from each table in Conducting a Survey LESSON 10-3 Additional Examples Malka asked three classmates from each table in the cafeteria the following question: What is your favorite shopping trip? She concluded that shopping for music was the favorite activity of people in her town. Describe a reason why her conclusions may be invalid. Malka surveyed only middle school students. She did not get a random sample of shoppers of all ages. Quick Check 10-3

For Exercises 1–3, use this situation: Conducting a Survey LESSON 10-3 Lesson Quiz For Exercises 1–3, use this situation: You want to find out how many people plan to attend the school pep rally. 1. You survey all students in your science class. Explain why this is not a good sample. 2. Explain why this is a biased question: Wouldn't you rather stay home the night of the pep rally? Sample: The sample does not include students from the other grades. Sample: The question implies that staying home might be more acceptable. 10-3

3. You survey the members of the football team. Explain Conducting a Survey LESSON 10-3 Lesson Quiz 3. You survey the members of the football team. Explain how your conclusions will be affected. 4. The owner of a fitness center wants to increase the number of members. She mails a questionnaire to current members, asking them to indicate their satisfaction with the center. Is this a good sample? Explain. Sample: The conclusions may not be valid. Students who are on the football team may be more likely to attend the pep rally than students in general. Sample: No. Recipients of the questionnaire are current members who are more likely to be satisfied with the facility than any ex-members. 10-3

Independent and Dependent Events LESSON 10-4 Problem of the Day If the population of the United States is approximately 250 million and the median age is 33 yr, roughly how many people in the United States are younger than 33 yr? 125 million 10-4

Independent and Dependent Events LESSON 10-4 Check Skills You’ll Need (For help, go to Lesson 10-2.) 1. Vocabulary Review How can you use theoretical probability to make predictions? You roll a number cube 50 times. Predict how many times the given outcome will occur. 2. 2 3. 5 4. 8 5. odd Check Skills You’ll Need 10-4

Independent and Dependent Events LESSON 10-4 Check Skills You’ll Need Solutions 1. You can set up a proportion to solve, or you can multiply the theoretical probability by the population size. 2. about 8 3. about 8 4. 0 5. about 25 10-4

Independent and Dependent Events LESSON 10-4 Additional Examples A box contains 3 red marbles and 7 blue ones. You draw a marble at random, replace it, and draw another. Find P(blue and blue). Because the first marble is replaced, these are independent events. P(blue and blue) = P(blue) • P(blue) = • Substitute. 7 10 Multiply. = 49 100 The probability of choosing a blue and then another blue is . 49 100 Quick Check 10-4

Independent and Dependent Events LESSON 10-4 Additional Examples From a class of 12 girls and 14 boys, you select two students at random. Find the probability that both students are boys. The two events are dependent. First student P(boy) = 14 26 Fourteen of the students are boys. Second student P(boy after boy) = 13 25 Thirteen boys are left of 25 students. P(boy then boy) = P(boy) • P(boy after boy) = • Substitute. 14 26 13 25 = = 182 650 Multiply and simplify. 7 25 The probability that two boys are chosen is . 7 25 Quick Check 10-4

Independent and Dependent Events LESSON 10-4 Additional Examples A piggy bank is filled with dimes, nickels, and pennies. You draw a coin at random and get a penny, which you keep. You draw at random again and get a dime. State whether the event is a dependent or independent event. After the first coin is chosen, the collection of remaining coins has changed. The event is dependent. Quick Check 10-4

Independent and Dependent Events LESSON 10-4 Lesson Quiz A bag of marbles contains 2 red, 3 green, and 1 ivory marble. Use this information to solve Questions 1 and 2. 1. You randomly choose one marble, replace it, and then choose a second marble. Find the probability that both are green. 2. You randomly choose one marble and then, without replacing the first marble, you choose a second marble. Find the probability that ivory is first and green is second. 1 4 1 10 10-4

Independent and Dependent Events LESSON 10-4 Lesson Quiz 3. What kind of event is described in Question 1? 4. What kind of event is described in Question 2? independent dependent 10-4

Permutations LESSON 10-5 Problem of the Day Jan tosses two coins. What is the probability that there will be no tails? 1 4 10-5

1. Vocabulary Review What is a sample space? Permutations LESSON 10-5 Check Skills You’ll Need (For help, go to Lesson 5-8.) 1. Vocabulary Review What is a sample space? Find the number of possible outcomes in each situation. 2. Roll a number cube and toss a coin. 3. Roll two number cubes. 4. Toss two coins. Check Skills You’ll Need 10-5

1. a collection of all possible outcomes Permutations LESSON 10-5 Check Skills You’ll Need Solutions 1. a collection of all possible outcomes 2. 12 3. 36 4. 4 10-5

In how many ways can four people form a line? Permutations LESSON 10-5 Additional Examples Quick Check In how many ways can four people form a line? Use the letters A, B, C, D to represent each person. Four people can line up in 24 different ways. This means that there are 24 permutations. 10-5

Use the counting principle. Permutations LESSON 10-5 Additional Examples Suppose you have six invitations to write. In how many different sequences can you write them? There are six invitations you can write first, five invitations you can write second, and so on. 6 • 5 • 4 • 3 • 2 • 1 = 720 Use the counting principle. There are 720 different sequences in which to write the invitations. Quick Check 10-5

The songs can be played in 362,880 different orders or permutations. LESSON 10-5 Additional Examples A CD has nine songs. In how many different orders could you play these songs? Simplify. 9! = 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 362,880 The songs can be played in 362,880 different orders or permutations. Quick Check 10-5

2. A CD has 11 songs. Find in how many orders you can play the songs. Permutations LESSON 10-5 Lesson Quiz 1. In how many different ways can you line up a half dollar, quarter, dime, nickel, and penny? 2. A CD has 11 songs. Find in how many orders you can play the songs. 3. Find 12! 4. Simplify 10P3. 120 39,916,800 479,001,600 720 10-5

How many ways can the letters of each of these names be arranged? Combinations LESSON 10-6 Problem of the Day How many ways can the letters of each of these names be arranged? a. Ruth b. Bobby c. Aaron 24 20 60 10-6

1. Vocabulary Review An arrangement of objects in a Combinations LESSON 10-6 Check Skills You’ll Need (For help, go to Lesson 10-5.) 1. Vocabulary Review An arrangement of objects in a certain order is a(n) ? . Simplify each expression. 2. 7 • 6 3. 4! 4. 10P2 5. 101P3 Check Skills You’ll Need 10-6

Solutions 1. permutation 2. 42 3. 24 4. 90 5. 999,900 Combinations LESSON 10-6 Check Skills You’ll Need Solutions 1. permutation 2. 42 3. 24 4. 90 5. 999,900 10-6

How many groups of two can be formed from a committee of six members? Combinations LESSON 10-6 Additional Examples How many groups of two can be formed from a committee of six members? Use the letters A, B, C, D, E, F to represent the six possible members. Step 1 Make an organized list of all the possible combinations of members. AB AC AD AE AF BA BC BD BE BF CA CB CD CE CF DA DB DC DE DF EA EB EC ED EF FA FB FC FD FE 10-6

Step 2 Cross out any group that is a duplicate of another. Combinations LESSON 10-6 Additional Examples (continued) Step 2 Cross out any group that is a duplicate of another. AB AC AD AE AF BA BC BD BE BF CA CB CD CE CF DA DB DC DE DF EA EB EC ED EF FA FB FC FD FE Step 3 Count the number of groups that remain. There are 15 different ways to form a group of two members. Quick Check 10-6

Write using combination notation. 8C3 = 8P3 3! Combinations LESSON 10-6 Additional Examples Find the number of ways you can choose 3 lures from a box of 8 fishing lures. Write using combination notation. 8C3 = 8P3 3! Simplify 8P3 and 3!. = 8 • 7 • 6 3 • 2 • 1 = 56 Simplify. There are 56 different ways. Quick Check 10-6

1. How many different ways can two dancers be selected for a Combinations LESSON 10-6 Lesson Quiz 1. How many different ways can two dancers be selected for a dance team out of five candidates? 2. A hairdresser schedules 10 clients for appointments. Does this situation describe a permutation or a combination? 3. Simplify 10C3. 4. How many teams of 6 players can be formed from a group of 14 players? 10 permutation 120 14 • 13 • 12 • 11 • 10 • 9 6! = 3,003 10-6