2 Event is equally likely to occur or not occur. When all outcomes are equally likely, the theoretical probability that an event A will occur is:


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1 10.3 TEKS a.1, a.4 Define and Use Probability Before You determined the number of ways an event could occur. Now You will find the likelihood that an event will occur. Why? So you can find reallife geometric probabilities, as in Ex. 39. Key Vocabulary probability theoretical probability odds experimental probability geometric probability When you roll a standard sixsided die, the possible results are called outcomes. The outcomes of rolling a die are 1, 2, 3, 4, 5, and 6. An event is an outcome or a collection of outcomes. For example, the event rolling an odd number consists of the outcomes 1, 3, and 5. The probability of an event is a number from 0 to 1 that indicates the likelihood the event will occur, as shown on the number line below. Probabilities can be written as fractions, decimals, or percents. Event is more likely not to occur Event is more likely to occur P 5 0 Event will not occur. KEY CONCEPT P Event is equally likely to occur or not occur. P 5 1 Event is certain to occur. For Your Notebook Theoretical Probability of an Event When all outcomes are equally likely, the theoretical probability that an event A will occur is: P(A) 5 Number of outcomes in event A }}}}}}}}}}}}}} Total number of outcomes The theoretical probability of an event is often simply called the probability of the event. all possible outcomes event A outcomes P(A) 5 3 } 8 E XAMPLE 1 Find probabilities of events You roll a standard sixsided die. Find the probability of (a) rolling a 5 and (b) rolling an even number. a. There are 6 possible outcomes. Only 1 outcome corresponds to rolling a 5. P(rolling a 5) 5 Number of ways to roll a 5 }}}}}}}}}}}}}} Number of ways to roll the die 5 1 } 6 b. A total of 3 outcomes correspond to rolling an even number: a 2, 4, or 6. P(rolling even number) 5 Number of ways to roll an even number }} Number of ways to roll the die 5 3 } } Chapter 10 Counting Methods and Probability
2 E XAMPLE 2 Use permutations or combinations ENTERTAINMENT A community center hosts a talent contest for local musicians. On a given evening, 7 musicians are scheduled to perform. The order in which the musicians perform is randomly selected during the show. a. What is the probability that the musicians perform in alphabetical order by their last names? (Assume that no two musicians have the same last name.) b. You are friends with 4 of the musicians. What is the probability that the first 2 performers are your friends? Solution a. There are 7! different permutations of the 7 musicians. Of these, only 1 is in alphabetical order by last name. So, the probability is: P(alphabetical order) 5 1 } 7! 5 1 }}} 5040 ø b. There are 7 C 2 different combinations of 2 musicians. Of these, 4 C 2 are 2 of your friends. So, the probability is: P(first 2 performers are your friends) 5 4C 2 }} 7 C }} } 7 ø GUIDED PRACTICE for Examples 1 and 2 You have an equally likely chance of choosing any integer from 1 through 20. Find the probability of the given event. 1. A perfect square is chosen. 2. A factor of 30 is chosen. 3. WHAT IF? In Example 2, how do your answers to parts (a) and (b) change if there are 9 musicians scheduled to perform? ODDS You can also use odds to measure the likelihood that an event will occur. Odds measure the chances in favor of an event occurring or the chances against an event occurring. KEY CONCEPT For Your Notebook Odds in Favor of or Odds Against an Event When all outcomes are equally likely, the odds in favor of an event A and the odds against an event A are defined as follows: Odds in favor of event A 5 Number of outcomes in A }}}}}}}}}}}}} Number of outcomes not in A Odds against event A 5 Number of outcomes not in A }}}}}}}}}}}}} Number of outcomes in A You can write the odds in favor of or against an event in the form a } b or in the form a : b Define and Use Probability 699
3 E XAMPLE 3 Find odds AVOID ERRORS Note that the odds in favor of drawing a 10, which are }} 1, do not 12 equal the probability of drawing a 10, which is }} 4 5 }} A card is drawn from a standard deck of 52 cards. Find (a) the odds in favor of drawing a 10 and (b) the odds against drawing a club. Solution a. Odds in favor of drawing a 10 5 Number of tens }}}}}}}}} Number of nontens 5 }} 4 5 }} 1, or 1: b. Odds against drawing a club 5 Number of nonclubs }}}}}}}}}} Number of clubs 5 39 }} } 1, or 3:1 EXPERIMENTAL PROBABILITY Sometimes it is not possible or convenient to find the theoretical probability of an event. In such cases, you may be able to calculate an experimental probability by performing an experiment, conducting a survey, or looking at the history of the event. KEY CONCEPT For Your Notebook Experimental Probability of an Event When an experiment is performed that consists of a certain number of trials, the experimental probability of an event A is given by: P(A) 5 Number of trials where A occurs }}}}}}}}}}}}}} Total number of trials E XAMPLE 4 Find an experimental probability SURVEY The bar graph shows how old adults in a survey would choose to be if they could choose any age. Find the experimental probability that a randomly selected adult would prefer to be at least 40 years old. Solution The total number of people surveyed is: Number of adults under Desired age Of those surveyed, would prefer to be at least 40. P(at least 40 years old) }}} 3516 ø GUIDED PRACTICE for Examples 3 and 4 A card is randomly drawn from a standard deck. Find the indicated odds. 4. In favor of drawing a heart 5. Against drawing a queen 6. WHAT IF? In Example 4, what is the experimental probability that an adult would prefer to be (a) at most 39 years old and (b) at least 30 years old? 700 Chapter 10 Counting Methods and Probability
4 GEOMETRIC PROBABILITY Some probabilities are found by calculating a ratio of two lengths, areas, or volumes. Such probabilities are geometric probabilities. E XAMPLE 5 Find a geometric probability DARTS You throw a dart at the square board shown. Your dart is equally likely to hit any point inside the board. Are you more likely to get 10 points or 0 points? 3 in. Solution P (10 points) 5 Area of smallest circle }}}}}}}}}} Area of entire board in. P (0 points) 5 5 }}} p p 18 2 }} 5 }} p ø Area outside largest circle }}}}}}}}}}}} Area of entire board (p p 9 2 ) }}}}}} p }}}}} p }}} 4 ø c Because > , you are more likely to get 0 points. at classzone.com GUIDED PRACTICE for Example 5 7. WHAT IF? In Example 5, are you more likely to get 5 points or 0 points? 10.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKEDOUT SOLUTIONS on p. WS1 for Exs. 7, 17, and 39 5 TAKS PRACTICE AND REASONING Exs. 19, 26, 27, 32, 42, 44, and 45 5 MULTIPLE REPRESENTATIONS Ex VOCABULARY Copy and complete: A probability that is the ratio of two lengths, areas, or volumes is called a(n)? probability. 2. WRITING Explain the difference between theoretical probability and experimental probability. Give an example of each. EXAMPLE 1 on p. 698 for Exs CHOOSING NUMBERS You have an equally likely chance of choosing any integer from 1 through 50. Find the probability of the given event. 3. An even number is chosen. 4. A number less than 35 is chosen. 5. A perfect square is chosen. 6. A prime number is chosen. 7. A factor of 150 is chosen. 8. A multiple of 4 is chosen. 9. A twodigit number is chosen. 10. A perfect cube is chosen Define and Use Probability 701
5 CHOOSING CARDS A card is randomly drawn from a standard deck of 52 cards. Find the probability of drawing the given card. 11. The king of diamonds 12. A king 13. A spade 14. A black card 15. A card other than a A face card (a king, queen, or jack) EXAMPLE 2 on p. 699 for Exs LOTTERIES In Exercises 17 and 18, find the probability of winning the lottery according to the given rules. Assume numbers are selected at random. 17. You must correctly select 6 out of 48 numbers. The order of the numbers is not important. 18. You must correctly select 4 numbers, each an integer from 0 to 9. The order of the numbers is important. 19. TAKS REASONING What is the probability (rounded to three decimal places) that 2 randomly selected months both have 31 days? A B C D EXAMPLE 3 on p. 700 for Exs ODDS You randomly choose a marble from a bag. The bag contains 10 black, 8 red, 4 white, and 6 blue marbles. Find the indicated odds. 20. In favor of choosing white 21. In favor of choosing blue 22. Against choosing red 23. Against choosing black ERROR ANALYSIS Describe and correct the error in calculating the odds against getting a 5 or 6 when rolling a sixsided die. 24. Odds against 5 or } } Odds against 5 or } } TAKS REASONING Flip a coin 10 times. What is the experimental probability of getting heads? 27. TAKS REASONING The probability of event A is 0.3. What are the odds in favor of event A? Explain. EXAMPLE 4 on p. 700 for Exs ROLLING A DIE The results of rolling a sixsided die 150 times are shown. Use the table to find the experimental probability of the given event. Compare your answer to the theoretical probability of the event. 28. Rolling a Rolling an even number 30. Rolling a number less than Rolling any number but a TAKS REASONING You flip a coin 80 times. You get heads 37 times and tails 43 times. What is the experimental probability of getting heads? A B 0.5 C D REASONING Find the probability that the vertex of the graph of y 5 x 2 2 6x 1 c is above the xaxis if c is a randomly chosen integer from 1 to WORKEDOUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS
6 34. CHALLENGE Suppose you throw a dart at each square target below. Assume that the dart is equally likely to hit any point inside the target. Target A Target B Target C 12 in. 12 in. 12 in. a. Calculate What is the probability that the dart lands inside the circle in target A? inside a circle in target B? inside a circle in target C? b. Generalize Consider the general case where a square target with sides 12 inches long contains n 2 identical circles arranged in n rows and n columns. Make a conjecture about the probability that a dart lands inside one of the circles. Then prove your conjecture. PROBLEM SOLVING EXAMPLE 5 on p. 701 for Exs GEOMETRIC PROBABILITY Find the probability that a dart thrown at the given target will hit the shaded region. Assume the dart is equally likely to hit any point inside the target JURY SELECTION A jury of 12 people is selected from a pool of 30 people that includes 12 men and 18 women. What is the probability that the jury will be composed of 12 women? 39. ARCHERY The standard archery target used in competition has a diameter of 80 centimeters. Find the probability that an arrow shot at the target will hit the center circle, which has a diameter of 16 centimeters. Assume the arrow is equally likely to hit any point inside the target. 40. MULTIPLE REPRESENTATIONS On a typical weekday, there are 1,181,100 oneway trips taken on the public transportation system operated by the Massachusetts Bay Transit Authority. Of these trips, 376,900 are bus rides. Suppose a oneway trip is selected at random. a. Using Fractions What is the probability, expressed as a fraction, that the trip was taken on a bus? b. Using Decimals What is the probability, expressed as a decimal, that the trip was taken on a bus? c. Using Percents What is the probability, expressed as a percent, that the trip was taken on a bus? d. Using Odds What are the odds in favor of the trip having been on a bus? 10.3 Define and Use Probability 703
7 41. GULF COAST The map shows the length of shoreline (in miles) along the Gulf of Mexico for each state that borders the body of water. What is the probability that a ship coming ashore at a random point in the Gulf of Mexico lands in the given state? a. Texas b. Florida TX 367 mi AL MS LA 44 mi 53 mi 397 mi Gulf of Mexico FL 770 mi c. Alabama 42. TAKS REASONING A magician claims to be able to read minds. To test this claim, five cards numbered 1 through 5 are used. A subject selects two cards from the five cards and concentrates on the numbers. a. What is the probability that the two numbers chosen are 3 and 4? b. What is the probability that the magician can correctly identify the two numbers by guessing? c. Suppose the magician is able to consistently identify the two numbers about half the time. Does this support the magician s claim to be a mind reader? Explain. 43. CHALLENGE In a guessing game, one player secretly places four differentcolored pegs on a board in each of four positions: A, B, C, or D. A second player guesses the configuration of the pegs by placing an identical set of pegs in slots A, B, C, and D on an identical board. The second player is then told how many of the pegs are in the correct slot. a. What is the probability that the second player has all four pegs correct on the first guess? b. What is the probability that the second player has exactly one peg correct on the first guess? c. The second player is told she has placed two pegs in the correct slot. The player then switches two of the pegs. What is the probability that the player now has all four pegs in the correct slot? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Skills Review Handbook p. 991; TAKS Workbook 44. TAKS PRACTICE What is the area of the figure shown? TAKS Obj. 6 A B C D 14 square units 18 square units 20 square units 36 square units 6 y x REVIEW Lesson 9.1; TAKS Workbook 45. TAKS PRACTICE What is the midpoint of the line segment connecting points (24, 21) and (7, 3)? TAKS Obj. 7 F 1 23 } 2, 12 G 1 3 } 2, 1 } 2 2 H 1 3 } 2, 12 J 1 3 } 2, EXTRA PRACTICE for Lesson 10.3, p ONLINE QUIZ at classzone.com
8 MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons MULTIPLE CHOICE 1. MOVIE THEATER SEATING Five people walk into a movie theater and look for empty seats in which to sit. What is the number of ways the people can be seated if there are 8 empty seats? TEKS a.1 A 20 B 56 C 336 D ARRIVAL TIMES You and a friend are meeting at a gym. You both agree to arrive between 9:00 A.M. and 9:30 A.M. and will wait for each other for up to 10 minutes. Let x be your arrival time and let y be your friend s arrival time (where x and y are in minutes after 9:00 A.M.). In the graph below, the blue region represents times such that you and your friend both arrive between 9:00 A.M. and 9:30 A.M., and the red region represents times such that the two of you arrive no more than 10 minutes apart. What is the approximate probability that you and your friend will meet if you both arrive between 9:00 A.M. and 9:30 A.M.? TEKS a.4 40 y y 2 x x 2 y SUMMER OLYMPICS The graph shows the results of a survey in 2004 that asked U.S. adults which sport they would most like to participate in at the Summer Olympics. What is the approximate probability that a randomly selected U.S. adult would most like to participate in track and field? TEKS a.1 Number of people Summer Olympics Sports 263 Swimming Track and field Gymnastics F 0.23 G Baseball Basketball H 0.3 J Other 5. OMELETS A restaurant offers 8 different ingredients for an omelet. In a deluxe omelet, you can have up to 6 ingredients. How many different combinations of ingredients can you have? TEKS a.1 A 28 B 218 C 219 D 247 classzone.com x F G H J GRADUATION REQUIREMENTS You must take 18 elective courses to meet your graduation requirements for college. There are 30 courses that you are interested in. How many different course selections are possible? TEKS a.1 A 30,045,015 B 86,493,225 C D GRIDDED ANSWER FRUIT SMOOTHIES You want to make a fruit smoothie using 3 of the fruits listed. How many different fruit smoothies can you make? TEKS a.1 Orange Banana Kiwi Canteloupe Available Fruits Strawberry Pineapple Watermelon Peach Mixed Review of Problem Solving 705
9 Investigating g Algebra ACTIVITY Use before Lesson Find Probabilities Using Venn Diagrams TEKS a.1, a.5 QUESTION How can you use a Venn diagram to find probabilities involving two events? In Lesson 10.3, you learned how to compute the probability of one event. In some situations, however, you might be interested in the probability that two events will occur simultaneously. You also might be interested in the probability that at least one of two events will occur. This activity demonstrates how a Venn diagram is useful for computing such probabilities. E XPLORE Use a Venn diagram to collect data STEP 1 Complete a Venn diagram Copy the Venn diagram shown below. Ask the members of your class if they have a sister, have a brother, have both, or have neither. Write their names in the appropriate part of the Venn diagram. No sister or brother STEP 2 Complete a table Copy and complete the frequency table. When determining the frequency for a category, be sure to include all the students who are in the category. Note that a student can belong to more than one category. Category Number of students Have a sister? Have a sister Have a brother Have a brother? Have both a sister and brother? Do not have a sister or brother? DRAW CONCLUSIONS Use your data to complete these exercises 1. A student from your class is selected at random. Find the probability of each event. Explain how you found your answers. a. The student has a sister. b. The student has a brother. c. The student has a sister and a brother. d. The student does not have a sister or a brother. 2. Find the probability that a randomly selected student from your class has either a sister or a brother. Explain how you found your answer. 3. How could you calculate the answer to Exercise 2 using your answers from Exercise 1? 706 Chapter 10 Counting Methods and Probability
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