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CHAPTER 8 Probability and Predictions Chapter 8: Section 1: Part A Probability and Simple Events Book pages 429 - 434 8-1-A: Probability and Simple Events A cheesecake has four equal slices of each piece as shown. What fraction of cheesecake is chocolate? Write in simplest form. Suppose your friend gives you the first piece of cheesecake without asking you which type you prefer. Are your chances of getting original the same as getting raspberry? Explain. 8-1-A: Probability and Simple Events An outcome is any one of the possible results of an action. A simple event has one outcome or a collection of outcomes. For example, getting a piece of chocolate cheesecake is a simple event. The chance of that event happening is called its Favorable outcomes are probability. the outcomes for which you are finding the probability. PROBABILITY If all outcomes are equally likely, the probability of a simple event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. SYMBOLS: 8-1-A: Probability and Simple Events Find Probability What is the probability of drawing an ace from a deck of cards? What is the probability of drawing a red card? So the probability is ½, 0.5, or 50%. Now you try! P(drawing a heart) P(drawing a face card) 1/4 3/13 8-1-A: Probability and Simple Events Outcomes occur at random if each outcomes occurs by chance. For example, the number that results when rolling a number cube is a random outcome. The probability that an event will happen can be any number from 0 to 1, including 0 and 1. Notice that probabilities can be written as fractions, decimals, and percents. 8-1-A: Probability and Simple Events Simone and her three friends were deciding how to pick the song they will sing for their school’s talent show. They decide to roll a number cube. The person with the lowest number chooses the song. If her friends rolled a 6, 5, and 2, what is the probability that Simone will get to choose the song? The table shows the number of brass instrument players in the New York Philharmonic. Suppose one brass player is randomly selected to be a featured performer. Find the probability of each event. Write your New York Philharmonic answer as a fraction in simplest form. Brass Instrument Players P(trumpet) P(brass) P(flute) P(horn or tuba) Horn 6 Trombone 4 Trumpet 3 Tuba 1 8-1-A: Probability and Simple Events In one of the previous problems, either Simone will go first or she will not go first. These two events are complementary events. Two events are complementary events if they are the only two possible outcomes. The sum of the probability of an event and its complement is 1 or 100%. SYMBOLS: P(A) + P(not A) = 1 Example: Ricardo’s teacher uses a spinner similar to the one shown at the right to determine the order in which each group will make its presentation. Use the spinner to find each probability. Write as a fraction in simplest form. P(not group 4) P(not group 1 or group 3) 8-1-A: Probability and Simple Events MORE ABOUT Probability Geometric probability deals with the areas of figures instead of the number of outcomes. For example, the probability of landing in a specific region of a target is the ratio of the area of the specific region to the area of the target. Try these! Self Assessment: On your own, complete # 1 – 10 on page 431. Chapter 8: Section 1: Part B Sample Spaces Book pages 435 - 439 8-1-B: Sample Spaces The set of all possible outcomes in a probability experiment is called the sample space. For instance, the sample space for rolling a number cube is 1, 2, 3, 4, 5, and 6 because those are the possible outcomes. A tree diagram is a display that represents a sample space. A vendor sells vanilla and chocolate ice cream. Customers can choose from a waffle or sugar cone. Find the sample space for all possible orders of one scoop of ice cream in a cone. NOTE: When creating a tree diagram, you must start the diagram with one of the possibilities. YOU TRY! The animal shelter has both male and female Labradors in yellow, brown, or black. Find the sample space for all possible Labradors at the shelter. 8-1-B: Sample Spaces Find the Probability Delmar tosses three coins. If all three coins show up heads, Delmar wins. Otherwise, Kara wins. Find the sample space. Then find the probability that Delmar wins. COIN 1 COIN 2 COIN 3 Sample Space: Heads Heads Probability: Look at the last column of the tree diagram. Here is where you can find the total number of Heads outcomes. In this case, there are 8. Out of those 8, there is only ONE option with three heads. The probability of Delmar winning is 1/8. Now you try! Tails Tails Heads Heads Tails Heads Tails Elba tosses a quarter, a dime, and a nickel. Tails Heads If tails comes up at least twice, Steve wins. Tails Otherwise, Elba wins. Find the sample space for the Tails game. Then find the probability that Elba will win. Ming rolls a number cube, tosses a coin, and chooses a card from two cards marked A and B. If an even number and a heads appears, Ming wins no matter which card is chosen. Otherwise, Lashonda wins. Find the sample space. Then find the probability that Ming will win. 8-1-B: Sample Spaces MORE ABOUT Sample Spaces Another way to describe the chance of an event occurring is with odds. The odds in favor of an event is a ratio that compares the number of ways the even can occur to the number of ways that the event cannot occur. Find the odds in favor of rolling a 5 or 6 on a number cube. How many ways can the event occur? How may ways can the event NOT occur? 4 ways: 1, 2, 3, or 4 What are the odds in favor? 2 ways: a 5 or a 6 2:4 1:2 You try! A letter from the alphabet is chosen at random. Find the odds in favor of picking an A, E, U, or O. What is the difference between finding a probability and finding odds in favor? Self Assessment: On your own, complete # 1 – 4 on page 436. Chapter 8: Section 1: Part C Count Outcomes Book pages 440 - 443 8-1-C: Count Outcomes Off Broadway Shoes sells sandals in different colors and styles. 1. 2. 3. 4. According to the table, how may colors of sandals are available? How many styles are available? Draw a tree diagram to find the number of different color and style combinations. Find the product of the two numbers you found in 1 and 2. How does this number compare to the number of combinations you found in 3. 8-1-C: Count Outcomes On the previous slide, you discovered that multiplication, instead of a tree diagram, can be used to find the number of possible outcomes in a sample space. This is called the Fundamental Counting Principle. THE FUNDAMENTAL COUNTING PRINCIPLE If event M has m possible outcomes and event N has n c possible outcomes, then event M followed by event N has m × n possible outcomes. The counting principle gives the number of possible outcomes whereas the tree diagram gives the entire sample space. 8-1-C: Count Outcomes Find the Number of Outcomes Find the total number of outcomes when a coin is tossed and a number cube is rolled. You try! Find the total number of outcomes when choosing from bike helmets that come in three colors and two styles. 6 Find the total number of outcomes when a number from 0 to 9 is picked randomly and when a letter from A to D is picked randomly. 40. 8-1-C: Count Outcomes The Gap sells young men’s jeans in different sizes, styles, and lengths. Find the number of jeans available. Then find the probability of randomly selecting a size 32 × 34 slim fit. Is it likely or unlikely that the size would be chosen? There are 45 different types of jeans to choose from. Out of the 45 possible outcomes, only one is favorable. So the probability of selecting 32 × 34 slim fit is 1/45 or about 2%. It is very unlikely that the size would be chosen at random. 8-1-C: Count Outcomes If the Gap adds relaxed fit jeans to its selection, find the number of available jeans. Then find the probability of randomly selecting a 36 × 30 relaxed fit pair of jeans. Is it likely or unlikely that the size would be chosen? A shoe store sells shoes in different sizes, widths, and styles, as shown in the table. Find the number of pairs of shoes available. Then find the probability of selecting a size 8 in medium width sneakers. Is it likely or unlikely that the size would be chosen at random? Self Assessment: On your own, complete # 1 – 3 on page 441. Chapter 8: Section 1: Part D Permutations Book pages 444 - 447 8-1-D: Permutations In how many different ways can you arrange the letters in the word DOG? Let’s list them on the board! DOG, DGO, ODG, OGD, GOD, GDO When we first started making the list, how many choices did we have for our first letter? Once our first letter was selected, how many choices did we have for the second letter? For the third? 3 2, 1 A permutation is an arrangement, or listing, of objects in which order is important. The arrangement of DGO is a permutation of DOG because the order of the letters is different. You can use the Fundamental Counting Principle to find the number of permutations. 8-1-D: Permutations Find a Permutation How many different ways are there to arrange your first three classes if they are math, science, and language arts? Using the Counting Principle, find the number of possible arrangements. You try! In how many ways can the starting six players of a volleyball team stand in a row for a picture? A team of bowlers has five members, who bowl one at a time. In how many orders can they bowl? 720 120 How many 3-digit numbers can be formed using the digits 9, 3, 4, 7, and 6? 60 8-1-D: Permutations Find Probability The finals of the Northwest Swimming League features 8 swimmers. If each swimmer has an equally likely chance of finishing in the top two, what is the probability that Yumii will be in first place and Paquita in second place? There are 56 possible arrangements of the first and second place finishers. Of those 56, there is only one arrangement in which Yumii is first and Paquita is second. So, the probability is 1/56. You try! Two letters are randomly selected from the letters in the word math. What is the probability that the first letter selected is m and the second letter is h? 1/12 A school fair holds a raffle with 1st, 2nd, and 3rd prizes. Seven people enter the raffle, including Mark, Lilly, and Heather. What is the probability that Mark will win the 1st prize, Lilly will win the 2nd prize, and Heather will win the 3rd prize? 1/210 8-1-D: Permutations MORE ABOUT Outcomes An arrangement, or listing, of objects in which order is not important is called a combination. You can find the number of combinations of objects by dividing the number of permutations of the entire set by the number of ways the smaller set can be arranged. This just means that DOG would be the same as DGO, and OGD, etc. because the same three letters are used. A one-on-one basketball tournament has 6 players. How many different pairings are there in the first round. Find the number of ways 2 players can be chosen from a group of 6. So there are 15 different pairings in the first round. You try! Of 12 websites, in how many ways can you choose to visit 6? 924 There are 8 members on the debate team. How many different 5-player teams are possible? 56 Self Assessment: On your own, complete # 1 – 4 on page 445. Chapter 8: Section 2: Part A EXPLORE Independent and Dependent Events Book page 449 8-2-A: Explore Independent and Dependent Events Place two red counters and two white counters in a paper bag. Then complete the following: 1. 2. 3. 1. 2. 3. 1st Experiment Without looking, remove a counter from the bag and record its color. Place the counter back in the bag. Without looking, remove a second counter from the bag and record its color. The two colors are one trial. Place the counter back in the bag. Repeat until you have 50 trials. Count and record the number of times you choose a red counter followed by a white counter. 2nd Experiment Without looking, remove a counter from the bag and record its color. This time, do not replace the counter. Without looking, remove a second counter and record its color. The two colors are one trial. Place both counters back in the bag. Repeat until you have 50 trials. Count and record the number of times you choose a red counter followed by a white counter. Answer questions #1 – 3 on page 449 with your partner. We will discuss your findings as a class. Chapter 8: Section 2: Part B Independent and Dependent Events Book pages 450 - 455 8-2-B: Independent and Dependent Events A sale advertises that if you buy an item from the column of the left, you get a tote bag for free. Suppose you choose items at random. 1. 2. 3. Type of Item T-shirt jacket hat beach towel visor polo shirt Tote Bag Colors green red white What is the probability of buying a beach towel? receiving a red tote bag? What is the product of the probabilities in question 1? Draw a tree diagram to determine the probability that someone buys a beach towel and receives a red tote bag. The combined action of buying an item and receiving the free tote bag is a compound event. 8-2-B: Independent and Dependent Events A compound event consists of two or more simple events. These events are independent events because the outcome of one event does not affect the other event. Probability of Independent Events The probability of two independent events can be found by multiplying the cprobability of the first event by the probability of the second event. P(A and B) = P(A) ∙ P(B) 8-2-B: Independent and Dependent Events One letter tile is selected and the spinner is spun. What is the probability that both will be a vowel? P(selecting a vowel) = 2/7 P(spinning a vowel) = 2/6 or 1/3 P(both letters are vowels) = A spinner and a number cube are used in a game. The spinner has an equal chance of landing on one of five colors: red, yellow, blue, green, or purple. The faces of the cube are labeled 1 through 6. What is the probability of a player spinning blue and then rolling a 3 or 4? What is P(spinning blue)? What is P(rolling a 3 or 4)? Then what is P(blue then 3 of 4)? 8-2-B: Independent and Dependent Events You try! A game requires players to roll two number cubes to move the game pieces. The faces of the cubes are labeled 1 through 6. What is the probability of rolling a 2 or 4 on the first number cube and then rolling a f5 on the second? 1/18 The two spinners are spun. What is the probability that both spinners will show a number greater than 6? 9/100 8-2-B: Independent and Dependent Events If the outcome of one event affects the outcome of another event, the events are called dependent events. Probability of Dependent Events If two events A and B are dependent, then the probability of both events occurring is c the product of the probability of A and the probability of B after A occurs. P(A and B) = P(A) ∙ P(B following A) Independent: outcome does not rely on another event Dependent: relying on another quantity or action 8-2-B: Independent and Dependent Events There are 4 oranges, 7 bananas, and 5 apples in a fruit basket. Ignacio selects a piece of fruit at random and then Terrance selects a piece of fruit at random. Find the probability that two apples are chosen. Since the first piece of fruit is not replaced, the first event affects the second event. These are dependent events. There are 4 red, 8 P(first piece is an apple) = 5/16 P(second piece is an apple) = 4/15 yellow, and 6 blue socks mixed up in a drawer. P(two apples) = Once a sock is You try! selected, it is not P(two bananas) 7/40 replaced. Find the P(orange then apple) 1/12 probability of reaching into the drawer without P(apple then banana) 7/48 looking at choosing two 1/20 P(two oranges) blue socks. 5/51 8-2-B: Independent and Dependent Events MORE ABOUT Compound Events A number cube is rolled. What is the probability of rolling a multiple of 3 or a 1? Sometimes, two events cannot happen at the same time. For example, when a coin is tossed, either heads or tails will turn up. Tossing heads and tossing tails are examples of disjoint events, or events that cannot happen at the same time. Disjoint events are also called mutually exclusive events. There are three favorable outcomes, 3, 6, or 1. So the probability of rolling a multiple of 3 or a 1 is ½. You try! A number cube is rolled. Find each probability. Write as a fraction in simplest form. 1/3 P(4 or 5) 2/3 P(3 or even number) 2/3 P(1 or multiple of 2) Self Assessment: On your own, complete # 1 – 4 on page 445. Chapter 8: Section 3: Part A Theoretical and Experimental Probability Book pages 458 - 462 8-3-A: Theoretical and Experimental Probability EXPLORE Follow the steps to determine how many times doubles are expected to turn up when two number cubes are rolled. 1. 2. Use the table to help you find the expected number of times doubles should turn up when rolling two number cubes 36 times. The top row represents one number cube, and the left column represents the other number cube. Roll two number cubes 36 times. Record the number of times doubles turn up. With your partner, answer the following questions about the experiment. 1. 2. Compare the number of times you expected to roll doubles with the number of times you actually rolled doubles. Write the probability of rolling doubles out of 36 rolls using the number of times you expected to roll doubles from Step 1. Then write the probability of rolling doubles out of 36 rolls using the number of times you actually rolled doubles from Step 2. 8-3-A: Theoretical and Experimental Probabilityc Theoretical probability is based on what should happen when conducting an experiment. Experimental probability is based on what actually occurred during such an experiment. This is the probability you wrote down based on what you expected to roll. This is the probability you wrote down based on what you actually rolled. The theoretical probability and the experimental probability of an event may or may not be the same. As the number of times an experiment is conducted increases, the theoretical probability and the experimental probability should become closer together. 8-3-A: Theoretical and Experimental Probability Experimental Probability When two number cubes are rolled together 75 times, a sum of 9 is rolled 10 times. What is the experimental probability of rolling a sum of 9? The experimental probability of rolling a sum of 9 is 2/15. You try! In the above experiment, what is the experimental probability of rolling a sum that is not 9? 13/15 A spinner is spun 100 times, and it lands on green 32 times. What is the experimental probability of spinning green? 8/25 8-3-A: Theoretical and Experimental Probability Experimental and Theoretical Probability The graph shows the results of an experiment in which a spinner with 3 equal sections is spun sixty times. Find the experimental probability of spinning red for this experiment. The graph indicated that the spinner landed on red 24 times, blue 15 times, and green 21 times. The experimental probability is 2/5. Compare the experimental probability you found above to its theoretical probability. The spinner has three equal sections: red, blue, and green. So, the theoretical probability of spinning red is 1/3. Since 2/5 is close to 1/3, then the experimental probability is close to the theoretical probability. 8-3-A: Theoretical and Experimental Probability You try! Refer to the table again. If the spinner was spun three more times and landed on green each time, find the experimental probability of spinning green for this experiment. Compare the experimental probability you found above to its theoretical probability. 8/21 8/21 is close to 1/3 The graph shows the results of an experiment in which a number cube is rolled 50 times. Find the experimental probability of rolling a 4. 9/50 Compare the experimental probability you found above to its theoretical probability. 1/6 is close to 9/50. 8-3-A: Theoretical and Experimental Probability Theoretical and experimental probability can be used to make predictions about future events. Last year’s DVD sales at the local video store are shown. What is the probability that a customer bought a comedy DVD last year? Suppose a medial buyer expects to sell 5,000 DVDs this year. How many comedy DVDs should she buy? She should buy 1,450 comedy DVDs. 8-3-A: Theoretical and Experimental Probability You try! Mrs. Ramirez surveys her seventh-grade classes about what they are going to choose for lunch this week. Thirty students choose pizza, 17 students choose macaroni and cheese, 12 students choose hamburgers, and 5 students choose chicken fingers. What is the experimental probability of someone choosing hamburgers? 3/16 Suppose Mrs. Ramirez surveys all 1,200 students in the school. How many students can she expect to choose hamburgers for lunch? 225 students Self Assessment: On your own, complete # 1 – 2 on page 460. Chapter 8: Section 3: Part E Use Data to Predict Book pages 468 - 471 8-3-E: Use Data to Predict The circle graph shows the results of a survey in which children ages 8 to 12 were asked whether they have a television in their bedroom. Can you tell how many were surveyed? Explain. Describe how you could use the graph to predict how many students in your school have a television in their bedroom? A survey is designed to collect data about a specific group of people, call the population. A smaller group of people called a sample must be chosen. A sample is used to represent a population. If a survey is conducted at random, or without preference, you can assume that the survey represents the population. In this lesson, you will use the results of randomly conducted surveys to make predictions about the population. 8-3-E: Use Data to Predict Refer to the circle graph again. Predict how many out of 1,725 students would not have a television in their bedroom. You can use the percent equation and the survey results to predict what part p of the 1,725 students have no TV in their bedroom. About 932 students do not have a television in their bedroom. Refer to the same graph. Predict how many out of 1,370 students have a television in the bedroom? About 630 students 8-3-E: Use Data to Predict Use this information to predict how many of the 2,450 students at Washington Middle School use emoticons on their instant messengers. You need to predict how many of the 2,450 students use emoticons. n = 0.85 ∙ 2,450 n = 2,082.5 About 2,083 students use emoticons. 8-3-E: Use Data to Predict You try! This same survey found that 59% of people use sound on their instant messengers. Predict how many of the 2,450 students use sound on their instant messengers. The table shows the results of a survey in which people were asked whether their house pets watch television. There are 540 students at McCloskey Middle School who owns pets. Predict how many of them would say their pets watch TV. About 1,446 students About 205 According to one survey, 25% of high school students reported that they would not get summer jobs. Predict how many of the 948 students at Mohawk High School will not get summer jobs. About 237 students Self Assessment: On your own, complete # 1 – 2 on page 469. Chapter 8: Section 3: Part F Unbiased and Biased Samples Book pages 472 - 476 8-3-F: Unbiased and Biased Samples The manager of a News Channel 5 wants to conduct a survey to determine which sport people consider their favorite to watch. 1. 2. 3. Suppose she surveys a group of 100 people at a basketball game. Do the results represent all of the people in the viewing area? Explain. Suppose she surveys 100 students at your middle school. Do the results represent all of the people in the viewing area? Explain. Suppose she calls every 100th household in the telephone book. Do the results represent all of the people in the viewing area? Explain. 8-3-F: Unbiased and Biased Samples The manager of News Channel 5 cannot survey everyone in the viewing area. To get valid results, a sample must be chosen very carefully. An unbiased sample is selected so that it accurately represents the entire population. Two ways to pick an unbiased sample are below. Unbiased Samples Type Description Example Simple Random Sample Each item of person in the population is as likely to be chosen as any other. Each student’s name is written on a piece of paper. The names are placed in a bowl, and names are picked without looking. Systematic Random Sample The items or people are selected according to a specific time or item interval. Every 20th person is chosen from an alphabetical list of all students attending a school. 8-3-F: Unbiased and Biased Samples In a biased sample, one of more parts of the population are favored over others. Two ways to pick a biased sample are listed below. Biased Samples Type Description Example Convenience Sample A convenience sample consists of members of a population that are easily accessed. To represent all the students attending a school, the principal surveys students in one math class. Voluntary Response Sample A voluntary response sample Students at a school who wish involves only those who want to express their opinions to participate in the complete an online survey. sampling. 8-3-F: Unbiased and Biased Samples Determining the Validity of Conclusions A radio station asks its listeners to indicate their preference for one or two candidates in an upcoming election. Seventy-two percent of the listeners who responded preferred candidate A, so the radio station announced that candidate A would win the election. Is the conclusion valid? Justify your answer. Every tenth students who walked into the cafeteria was surveys to determine his of her favorite lunch. Out of 40 students, 19 students stated that they liked the burgers best. The cafeteria staff concluded that about 50% of the students like burgers best. Is the conclusion valid? Justify your answer. Janet surveyed the student athletes on the girls’ field hockey team to determine which sports teenagers liked best. Of these, 65% said that they liked field hockey the best. Janet concluded that over half of teenagers like field hockey best. Is the conclusion valid? Justify your answer. 8-3-F: Unbiased and Biased Samples A valid sampling method uses unbiased samples. If a sampling method is valid, you can make generalizations about the population. A store sells 4 styles of pants: jeans, capris, cargos, and khakis. The store workers survey 50 customers at random. The survey responses are indicated at the right. If 450 pairs of pants are to be ordered, how many should be jeans? First, determine whether the sampling method was valid. The sample is a simple random sample since customers were randomly selected. Thus the sample method is valid. 25/50 or 50% of the customers prefer jeans. So, find 50% of 450. 0.5 × 450 = 225, so about 225 pairs of jeans should be ordered. You try! An instructor at a swimming pool asker her students if they would be interested in an advanced swimming course, and 60% stated that they would. Is the sample method valid? If so, suppose there are 870 pool members. How many people can the instructor expect to take the course?