Chapter 11 PDF

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Chapter 11...

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Counting Methods and Probability Theory

11 TWO OF AMERICA’S BEST-LOVED PRESIDENTS, ABRAHAM LINCOLN AND JOHN F. KENNEDY, ARE LINKED BY A BIZARRE SERIES OF COINCIDENCES:

• Lincoln was elected president in 1860. Kennedy was elected president in 1960. • Lincoln’s assassin, John Wilkes Booth, was born in 1839. Kennedy’s assassin, Lee Harvey Oswald, was born in 1939. • Lincoln’s secretary, named Kennedy, warned him not to go to the theater on the night he was shot. Kennedy’s secretary, named Lincoln, warned him not to go to Dallas on the day he was shot. • Booth shot Lincoln in a theater and ran into a warehouse. Oswald shot Kennedy from a warehouse and ran into a theater. • Both Lincoln and Kennedy were shot from behind, with their wives present. • Andrew Johnson, who succeeded Lincoln, was born in 1808. Lyndon Johnson, who succeeded Kennedy, was born in 1908. Source: Edward Burger and Michael Starbird, Coincidences, Chaos, and All ThatMath Jazz, W.W. Norton and Company,2005.

Here’s where you’ll find these applications: Coincidences are discussed in the Blitzer Bonus on page750. Coincidences that are nearly certain are developed in Exercise77 of Exercise Set 11.7.

Amazing coincidences? A cosmic conspiracy? Not really. In this chapter, you will see how the mathematics of uncertainty and risk, called probability theory, numerically describes situations in which to expect the unexpected. By assigning numbers to things that are extraordinarily unlikely, we can logically analyze coincidences without erroneous beliefs that strange and mystical events are occurring. We’ll even see how wildly inaccurate our intuition can be about the likelihood of an event by examining an “amazing” coincidence that is nearly certain. 693

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11.1 WHAT AM I SUPPOSED TO LEARN? After studying this section, you should be able to:

1 Use the Fundamental Counting Principle to determine the number of possible outcomes in a given situation.

1

Use the Fundamental Counting Principle to determine the number of possible outcomes in a given situation.

The Fundamental Counting Principle Have you ever imagined what your life would be like if you won the lottery? What changes would you make? Before you fantasize about becoming a person of leisure with a staff of obedient elves, think about this: The probability of winning top prize in the lottery is about the same as the probability of being struck by lightning. There are millions of possible number combinations in lottery games, but there is only one way of winning the grand prize. Determining the probability of winning involves calculating the chance of getting the winning combination from all possible outcomes. In this section, we begin preparing for the surprising world of probability by looking at methods for counting possible outcomes.

The Fundamental Counting Principle with Two Groups of Items It’s early morning, you’re groggy, and you have to select something to wear for your 8 a.m. class. (What were you thinking when you signed up for a class at that hour?!) Fortunately, your “lecture wardrobe” is rather limited—just two pairs of jeans to choose from (one blue, one black) and three T-shirts to choose from (one tan, one yellow, and one blue). Your early-morning dilemma is illustrated in Figure 11.1.

FI G UR E 1 1 . 1 Selecting a wardrobe

The tree diagram, so named because of its branches, shows that you can form six different outfits from your two pairs of jeans and three T-shirts. Each pair of jeans can be combined with one of three T-shirts. Notice that the total number of outfits can be obtained by multiplying the number of choices for the jeans, 2, by the number of choices for the T-shirts, 3: 2 # 3 = 6. We can generalize this idea to any two groups of items—not just jeans and T-shirts—with the Fundamental Counting Principle. TH E FU N DAM EN TAL COU N T IN G PRIN CIPLE If you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices isM # N.

SEC TION 11.1

EXAMPLE 1

The Fundamental Counting Principle

695

Applying the Fundamental Counting Principle

The Greasy Spoon Restaurant offers 6 appetizers and 14 main courses. In how many ways can a person order a two-course meal?

SOLUTION Choosing from one of 6 appetizers and one of 14 main courses, the total number of two-course meals is 6 # 14 = 84. A person can order a two-course meal in 84 different ways.

CHECK POINT 1 A restaurant offers 10 appetizers and 15 main courses. In how many ways can you order a two-course meal?

EXAMPLE 2

Applying the Fundamental Counting Principle

This is the semester that you will take your required psychology and social science courses. Because you decide to register early, there are 15 sections of psychology from which you can choose. Furthermore, there are 9 sections of social science that are available at times that do not conflict with those for psychology. In how many ways can you create two-course schedules that satisfy the psychology–social science requirement?

SOLUTION The number of ways that you can satisfy the requirement is found by multiplying the number of choices for each course. You can choose your psychology course from 15 sections and your social science course from 9sections. For both courses you have 15 # 9, or 135 choices. Thus, you can satisfy the psychology–social science requirement in 135ways.

CHECK POINT 2 Rework Example 2 given that the number of sections of psychology and nonconflicting sections of social science each decrease by 5.

The Fundamental Counting Principle with More Than TwoGroups of Items Whoops! You forgot something in choosing your lecture wardrobe—shoes! You have two pairs of sneakers to choose from—one black and one red, for that extra fashion flair! Your possible outfits including sneakers are shown in Figure 11.2.

FI G UR E 1 1 . 2 Increasing wardrobe selections

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Counting Methods and Probability Theory

The tree diagram shows that you can form 12 outfits from your two pairs of jeans, three T-shirts, and two pairs of sneakers. Notice that the number of outfits can be obtained by multiplying the number of choices for jeans, 2, the number of choices for T-shirts, 3, and the number of choices for sneakers, 2: 2 # 3 # 2 = 12. Unlike your earlier dilemma, you are now dealing with three groups of items. The Fundamental Counting Principle can be extended to determine the number of possible outcomes in situations in which there are three or more groups of items.

FI G UR E 1 1 . 2 (repeated)

TH E FU N DAM EN TAL COU N T IN G PRIN CIPLE The number of ways in which a series of successive things can occur is found by multiplying the number of ways in which each thing can occur.

For example, if you own 30 pairs of jeans, 20 T-shirts, and 12 pairs of sneakers, you have 30 # 20 # 12 = 7200 choices for your wardrobe.

EXAMPLE 3

Options in Planning a Course Schedule

Next semester you are planning to take three courses—math, English, and humanities. Based on time blocks and highly recommended professors, there are eight sections of math, five of English, and four of humanities that you find suitable. Assuming no scheduling conflicts, how many different three-course schedules are possible?

SOLUTION The number of possible ways of playing the first four moves on each side in a game of chess is 318,979,564,000.

This situation involves making choices with three groups of items. Math

English

Humanities

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We use the Fundamental Counting Principle to find the number of threecourse schedules. Multiply the number of choices for each of the three groups. 8 # 5 # 4 = 160 Thus, there are 160 different three-course schedules.

CHECK POINT 3 A pizza can be ordered with two choices of size (medium or large), three choices of crust (thin, thick, or regular), and five choices of toppings (ground beef, sausage, pepperoni, bacon, or mushrooms). How many different one-topping pizzas can be ordered?

EXAMPLE 4

Car of the Future

Car manufacturers are now experimenting with lightweight three-wheel cars, designed for one person and considered ideal for city driving. Intrigued? Suppose you could order such a car with a choice of nine possible colors, with or without air conditioning, electric or gas powered, and with or without an onboard computer. In how many ways can this car be ordered with regard to these options?

SEC TION 11.1

The Fundamental Counting Principle

697

SOLUTION This situation involves making choices with four groups of items. Color

Air conditioning

Power

Computer

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We use the Fundamental Counting Principle to find the number of ordering options. Multiply the number of choices for each of the four groups. 9 # 2 # 2 # 2 = 72 Thus, the car can be ordered in 72 different ways.

CHECK POINT 4 The car in Example 4 is now available in ten possible colors. The options involving air conditioning, power, and an onboard computer still apply. Furthermore, the car is available with or without a global positioning system (for pinpointing your location at every moment). In how many ways can this car be ordered in terms of these options?

EXAMPLE 5

A Multiple-Choice Test

You are taking a multiple-choice test that has ten questions. Each of the questions has four answer choices, with one correct answer per question. If you select one of these four choices for each question and leave nothing blank, in how many ways can you answer the questions?

SOLUTION This situation involves making choices with ten questions. Question 1

Question 2

Question 3

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Question 9

Question 10

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We use the Fundamental Counting Principle to determine the number of ways that you can answer the questions on the test. Multiply the number of choices, 4, for each of the ten questions. 4 # 4 # 4 # 4 # 4 # 4 # 4 # 4 # 4 # 4 = 410 = 1,048,576 Use a calculator: 4 ! yx ! 10 ! = !. Thus, you can answer the questions in 1,048,576 different ways. Are you surprised that there are over one million ways of answering a ten-question multiple-choice test? Of course, there is only one way to answer the test and receive a perfect score. The probability of guessing your way into a perfect score involves calculating the chance of getting a perfect score, just one way, from all 1,048,576 possible outcomes. In short, prepare for the test and do not rely on guessing!

CHECK POINT 5 You are taking a multiple-choice test that has six questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?

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Counting Methods and Probability Theory

Blitzer Bonus

EXAMPLE 6

Running Out of TelephoneNumbers

Telephone Numbers in the United States

Telephone numbers in the United States begin with three-digit area codes followed by seven-digit local telephone numbers. Area codes and local telephone numbers cannot begin with 0 or 1. How many different telephone numbers are possible?

SOLUTION This situation involves making choices with ten groups of items. Area Code

By the year 2020, portable telephones used for business and pleasure will all be videophones. At that time, U.S. population is expected to be 323 million. Faxes, beepers, cellphones, computer phone lines, and business lines may result in certain areas running out of phone numbers. Solution: Add more digits! With or without extra digits, we expect that the 2020 videophone greeting will still be “hello,” a word created by Thomas Edison in 1877. Phone inventor Alexander Graham Bell preferred “ahoy,” but “hello” won out, appearing in the Oxford English Dictionary in 1883. Source: New York Times

Local Telephone Number

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Here are the choices for each of the ten groups of items: Area Code 8

10 10

Local Telephone Number 8

10 10

10 10 10 10

.

We use the Fundamental Counting Principle to determine the number of different telephone numbers that are possible. The total number of telephone numbers possible is 8 # 10 # 10 # 8 # 10 # 10 # 10 # 10 # 10 # 10 = 6,400,000,000 There are six billion, four hundred million different telephone numbers that are possible.

CHECK POINT 6 An electronic gate can be opened by entering five digits on a keypad containing the digits 0, 1, 2, 3, c, 8, 9. How many different keypad sequences are possible if the digit 0 cannot be used as the first digit?

Concept and Vocabulary Check Fill in each blank so that the resulting statement is true. 1. If you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices is ________. 2. The number of ways in which a series of successive things can occur is found by  _____________ the number of ways in which each thing can occur. This is called the _______________________ Principle.

In Exercises 3–4, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 3. If one item is chosen from M items, a second item is chosen from N items, and a third item is chosen from P items, the total number of three-item choices is M + N + P. _______ 4. Regardless of the United States population, we will not run out of telephone numbers as long as we continue to add new digits. _______

Exercise Set 11.1 Practice and Application Exercises 1. A restaurant offers eight appetizers and ten main courses. In how many ways can a person order a two-course meal? 2. The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or station wagon). In how many ways can you order the car?

3. A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand? 4. In how many ways can a casting director choose a female lead and a male lead from five female actors and six male actors?

SEC TION 11.1 5. A student is planning a two-part trip. The first leg of the trip is from San Francisco to New York, and the second leg is from New York to Paris. From San Francisco to New York, travel options include airplane, train, or bus. From New York to Paris, the options are limited to airplane or ship. In how many ways can the two-part trip be made? 6. For a temporary job between semesters, you are painting the parking spaces for a new shopping mall with a letter of the alphabet and a single digit from 1 to 9. The first parking space is A1 and the last parking space is Z9. How many parking spaces can you paint with distinct labels?

The Fundamental Counting Principle

699

14. A car model comes in nine colors, with or without air conditioning, with or without a sun roof, with or without automatic transmission, and with or without antilock brakes. In how many ways can the car be ordered with regard to these options? 15. You are taking a multiple-choice test that has five questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?

7. An ice cream store sells two drinks (sodas or milk shakes), in four sizes (small, medium, large, or jumbo), and five flavors (vanilla, strawberry, chocolate, coffee, or pistachio). In how many ways can a customer order a drink?

16. You are taking a multiple-choice test that has eight questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?

8. A pizza can be ordered with three choices of size (small, medium, or large), four choices of crust (thin, thick, crispy, or regular), and six choices of toppings (ground beef, sausage, pepperoni, bacon, mushrooms, or onions). How many onetopping pizzas can be ordered?

17. In the original plan for area codes in 1945, the first digit could be any number from 2 through 9, the second digit was either 0 or 1, and the third digit could be any number except 0. With this plan, how many different area codes are possible?

9. A restaurant offers the following limited lunch menu.

18. The local seven-digit telephone numbers in Inverness, California, have 669 as the first three digits. How many different telephone numbers are possible in Inverness?

Main Course

Vegetables

Beverages

Desserts

Ham

Potatoes

Coffee

Cake

Chicken

Peas

Tea

Pie

Fish

Green beans

Milk

Ice cream

Beef



Soda



19. License plates in a particular state display two letters followed by three numbers, such as AT-887 or BB-013. How many different license plates can be manufactured for this state? 20. How many different four-letter radio station call letters can be formed if the first letter must be W or K?

If one item is selected from each of the four groups, in how many ways can a meal be ordered? Describe two such orders.

21. A stock can go up, go down, or stay unchanged. How many possibilities are there if you own seven stocks?

10. An apartment complex offers apartments with four different options, designated by A through D.

22. A social security number contains nine digits, such as 074-66-7795. How many different social security numbers can be formed?

A

B

C

D

one bedroom

one bathroom

first floor

lake view

two bedrooms

two bathrooms

second floor

golf course view

three bedrooms





no special view

How many apartment options are available? Describe two such options. 11. Shoppers in a large shopping mall are categorized as male or female, over 30 or 30 and under, and cash or credit card shoppers. In how many ways can the shoppers be categorized? 12. There are three highways from city A to city B, two highways from city B to city C, and four highways from city C to city D. How many different highway routes are there from city A tocity D? 13. A person can order a new car with a choice of six possible colors, with or without air conditioning, with or without automatic transmissio...