Chapter 4 Probability and Counting Rules 1 Probability

Chapter 4 Probability and Counting Rules 1

Probability and Counting Rules Outline 4 -1 Sample Spaces and Probability 4 -2 The Addition Rules for Probability 4 -3 The Multiplication Rules and Conditional Probability 4 -4 Counting Rules 4 -5 Probability and Counting Rules 4

Probability and Counting Rules Objectives 1 2 3 4 4 Determine sample spaces and find the probability of an event, using classical probability or empirical probability. Find the probability of compound events, using the addition rules. Find the probability of compound events, using the multiplication rules. Find the conditional probability of an event.

Probability and Counting Rules Objectives 5 6 7 8 4 Find the total number of outcomes in a sequence of events, using the fundamental counting rule. Find the number of ways that r objects can be selected from n objects, using the permutation rule. Find the number of ways that r objects can be selected from n objects without regard to order, using the combination rule. Find the probability of an event, using the counting rules.

Probability • Probability can be defined as the chance of an event occurring. It can be used to quantify what the “odds” are that a specific event will occur. Some examples of how probability is used everyday would be weather forecasting, “ 75% chance of snow” or for setting insurance rates. 5

4 -1 Sample Spaces and Probability • A probability experiment is a chance process that leads to well-defined results called outcomes. • An outcome is the result of a single trial of a probability experiment. • A sample space is the set of all possible outcomes of a probability experiment. • An event consists of outcomes. 6

Sample 7"Spaces Experiment Toss a coin Roll a die Answer a true/false question Toss two coins Sample Space Head, Tail 1, 2, 3, 4, 5, 6 True, False HH, HT, TH, TT 7

Chapter 4 Probability and Counting Rules Section 4 -1 8

Example 4 -1: Rolling Dice Find the sample space for rolling two dice. 9

Section 4 -1 Exercise #1 If two dice are rolled one time, find the probability of getting these results. a. b. c. d. e. A sum of 6 Doubles A sum of 7 or 11 A sum greater than 9 A sum less than or equal to 4

a. A sum of 6 Total of 36 outcomes b. Doubles There are six ways to get doubles. They are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6).

c. A sum of 7 or 11 Total of 36 outcomes There are six ways to get a sum of 7. They are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). There are two ways to get a sum of 11. They are (5, 6) and (6, 5). d. A sum of greater than 9 To get a sum greater than nine, one must roll a 10, 11, or 12. There are six ways to get a 10, 11, or 12. They are (4, 6), (5, 5), (6, 4), (6, 5), (5, 6), and (6, 6).

e. The patient has had 1 or 2 tests done. Number of Tests Performed 0 1 2 3 4 or more Numbe r of Patient s 12 8 2 3 5

Chapter 4 Probability and Counting Rules Section 4 -1 14

Example 4 -3: Gender of Children Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl. BBB BBG BGB BGG GBB GBG GGB GGG 15

Chapter 4 Probability and Counting Rules Section 4 -1 16

Example 4 -4: Gender of Children Use a tree diagram to find the sample space for the gender of three children in a family. B B G G B BBB G BBG B BGB G BGG B GBB G GBG B GGB G GGG

Sample Spaces and Probability There are three basic interpretations of probability: • Classical probability • Empirical probability • Subjective probability

Sample Spaces and Probability Classical probability uses sample spaces to determine the numerical probability that an event will happen and assumes that all outcomes in the sample space are equally likely to occur. 19

Sample Spaces and Probability Rounding Rule for Probabilities should be expressed as reduced fractions or rounded to two or three decimal places. When the probability of an event is an extremely small decimal, it is permissible to round the decimal to the first nonzero digit after the decimal point. 20

Chapter 4 Probability and Counting Rules Section 4 -1 Example 4 -6 21

Example 4 -6: Gender of Children If a family has three children, find the probability that two of the three children are girls. Sample Space: BBB BBG BGB BGG GBB GBG GGB GGG Three outcomes (BGG, GBG, GGB) have two girls. The probability of having two of three children being girls is 3/8. 22

Probability Rule 1 The probability of any event E is a number (either a fraction or decimal) between and including 0 and 1. This is denoted by 0 P(E) 1.

Probability Rule 2 If an event E cannot occur (i. e. , the event contains no members in the sample space), its probability is 0.

Probability Rule 3 If an event E is certain, then the probability of E is 1.

Probability Rule 4 The sum of the probabilities of all the outcomes in the sample space is 1.

Chapter 4 Probability and Counting Rules Section 4 -1 Exercise 4 -9 27

Exercise 4 -9: Rolling a Die When a single die is rolled, what is the probability of getting a number less than 7? Since all outcomes— 1, 2, 3, 4, 5, and 6—are less than 7, the probability is The event of getting a number less than 7 is certain. 28

Sample Spaces and Probability 29

Chapter 4 Probability and Counting Rules Section 4 -1 Example 4 -10 30

Example 4 -10: Finding Complements Find the complement of each event. Event Complement of the Event Rolling a die and getting a 4 Getting a 1, 2, 3, 5, or 6 Selecting a letter of the alphabet and getting a vowel Getting a consonant (assume y is a consonant) Selecting a month and getting a month that begins with a J Getting February, March, April, May, August, September, October, November, or December Selecting a day of the week and getting a weekday Getting Saturday or Sunday 31

Chapter 4 Probability and Counting Rules Section 4 -1 Example 4 -11 32

Example 4 -11: Residence of People If the probability that a person lives in an industrialized country of the world is , find the probability that a person does not live in an industrialized country. 33

Sample Spaces and Probability There are three basic interpretations of probability: • Classical probability • Empirical probability • Subjective probability 34

Sample Spaces and Probability Empirical probability relies on actual experience to determine the likelihood of outcomes. 35

Chapter 4 Probability and Counting Rules Section 4 -1 Example 4 -13 36

Example 4 -13: Blood Types In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. a. A person has type O blood. Type A B AB O Frequency 22 5 2 21 Total 50 37

Example 4 -13: Blood Types In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. b. A person has type A or type B blood. Type A B AB O Frequency 22 5 2 21 Total 50 38

Example 4 -13: Blood Types In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. c. A person has neither type A nor type O blood. Type A B AB O Frequency 22 5 2 21 Total 50 39

Example 4 -13: Blood Types In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. d. A person does not have type AB blood. Type A B AB O Frequency 22 5 2 21 Total 50 40

Sample Spaces and Probability There are three basic interpretations of probability: • Classical probability • Empirical probability • Subjective probability 41

Sample Spaces and Probability Subjective probability uses a probability value based on an educated guess or estimate, employing opinions and inexact information. Examples: weather forecasting, predicting outcomes of sporting events 42

4. 2 Addition Rules for Probability • Two events are mutually exclusive events if they cannot occur at the same time (i. e. , they have no outcomes in common) 43

Chapter 4 Probability and Counting Rules Section 4 -2 Example 4 -15 44

Example 4 -15: Rolling a Die Determine which events are mutually exclusive and which are not, when a single die is rolled. a. Getting an odd number and getting an even number Getting an odd number: 1, 3, or 5 Getting an even number: 2, 4, or 6 Mutually Exclusive 45

Example 4 -15: Rolling a Die Determine which events are mutually exclusive and which are not, when a single die is rolled. b. Getting a 3 and getting an odd number Getting a 3: 3 Getting an odd number: 1, 3, or 5 Not Mutually Exclusive 46

Example 4 -15: Rolling a Die Determine which events are mutually exclusive and which are not, when a single die is rolled. c. Getting an odd number and getting a number less than 4 Getting an odd number: 1, 3, or 5 Getting a number less than 4: 1, 2, or 3 Not Mutually Exclusive 47

Example 4 -15: Rolling a Die Determine which events are mutually exclusive and which are not, when a single die is rolled. d. Getting a number greater than 4 and getting a number less than 4 Getting a number greater than 4: 5 or 6 Getting a number less than 4: 1, 2, or 3 Mutually Exclusive 48

Chapter 4 Probability and Counting Rules Section 4 -2 Example 4 -18 49

Example 4 -18: R&D Employees The corporate research and development centers for three local companies have the following number of employees: U. S. Steel 110 Alcoa 750 Bayer Material Science 250 If a research employee is selected at random, find the probability that the employee is employed by U. S. Steel or Alcoa. 50

Example 4 -18: R&D Employees 51

Chapter 4 Probability and Counting Rules Section 4 -2 Example 4 -21 52

Example 4 -21: Medical Staff In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male. Staff Females 7 Nurses 3 Physicians Total 10 Males 1 2 Total 3 13 8 5 53

4. 3 Multiplication Rules n. Two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring. 54

Chapter 4 Probability and Counting Rules Section 4 -3 Example 4 -23 55

Example 4 -23: Tossing a Coin A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die. This problem could be solved using sample space. H 1, H 2, H 3, H 4, H 5, H 6, T 1, T 2, T 3, T 4, T 5, T 6 56

Chapter 4 Probability and Counting Rules Section 4 -3 Example 4 -26 57

Example 4 -26: Survey on Stress A Harris poll found that 46% of Americans say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three will say that they suffer great stress at least once a week. 58

Chapter 4 Probability and Counting Rules Section 4 -3 Example 4 -28 59

Example 4 -28: University Crime At a university in western Pennsylvania, there were 5 burglaries reported in 2003, 16 in 2004, and 32 in 2005. If a researcher wishes to select at random two burglaries to further investigate, find the probability that both will have occurred in 2004. 60

4. 3 Conditional Probability n. Conditional probability is the probability that the second event B occurs given that the first event A has occurred. 61

Chapter 4 Probability and Counting Rules Section 4 -3 Example 4 -33 62

Example 4 -33: Parking Tickets The probability that Sam parks in a no-parking zone and gets a parking ticket is 0. 06, and the probability that Sam cannot find a legal parking space and has to park in the no-parking zone is 0. 20. On Tuesday, Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a parking ticket. N = parking in a no-parking zone T = getting a ticket 63

Chapter 4 Probability and Counting Rules Section 4 -3 Example 4 -34 64

Example 4 -34: Women in the Military A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown. 65

Example 4 -34: Women in the Military a. Find the probability that the respondent answered yes (Y), given that the respondent was a female (F). 66

Example 4 -34: Women in the Military b. Find the probability that the respondent was a male (M), given that the respondent answered no (N). 67

Chapter 4 Probability and Counting Rules Section 4 -3 Example 4 -37 68

Example 4 -37: Bow Ties The Neckware Association of America reported that 3% of ties sold in the United States are bow ties (B). If 4 customers who purchased a tie are randomly selected, find the probability that at least 1 purchased a bow tie. 69

EXAMPLE If one card is drawn from an ordinary deck of cards, find the probability of getting the following: a. b. c. d. e. A king or a queen or a jack. A club or a heart or a spade. A king or a queen or a diamond. An ace or a diamond or a heart. A 9 or a 10 or a spade or a club.

If one card is drawn from an ordinary deck of cards, find the probability of getting the following: a. A king or a queen or a jack. There are 4 kings, 4 queens, and 4 jacks, hence: P (king or queen or jack)

If one card is drawn from an ordinary deck of cards, find the probability of getting the following: b. A club or a heart or a spade. There are 13 clubs, 13 hearts, and 13 spades, hence: P(club or heart or spade)

If one card is drawn from an ordinary deck of cards, find the probability of getting the following: c. A king or a queen or a diamond. There are 4 kings, 4 queens, and 13 diamonds but the king and queen of diamonds were counted twice, hence: P(king or queen or diamond) P(king) + P(queen) + P(diamond) – P(king or queen of diamonds)

If one card is drawn from an ordinary deck of cards, find the probability of getting the following: d. An ace or a diamond or a heart. There and 13 are 4 aces, 13 diamonds hearts. There is one ace of diamonds and one ace of hearts, hence: P(ace or diamond or heart)

If one card is drawn from an ordinary deck of cards, find the probability of getting the following: e. A 9 or a 10 or a spade or a club. There are 4 nines, 4 tens, 13 spades, and 13 clubs. There is one nine of spades, one ten of spades, one nine of clubs, and one ten P ( 9 oforclubs, 10 or hence: spade or club)

If one card is drawn from an ordinary deck of cards, find the probability of getting the following: e. A 9 or a 10 or a spade or a club. P ( 9 or 10 or spade or club)

EXAMPLE At a local university 54. 3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities. a. None have computers b. At least one has a computer c. All have computers

At a local university 54. 3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities. a. None have computers

At a local university 54. 3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities. b. At least one has a computer

At a local university 54. 3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities. c. All have computers

EXAMPLE 0. 8 0. 2 0. 1 D (0. 8)(0. 1) = 0. 08 0. 9 ND 0. 18 D (0. 2)(0. 18) = 0. 036 0. 82 ND

Finally, use the addition rule, since the item is chosen at random from model I or model II. 0. 8 0. 2 0. 1 D (0. 8)(0. 1) = 0. 08 0. 9 ND 0. 18 D (0. 2)(0. 18) = 0. 036 0. 82 ND

EXAMPLE In Rolling Acres Housing Plan, 42% of the houses have a deck and a garage; 60% have a deck. Find the probability that a home has a garage, given that it has a deck.

EXAMPLE Corporati Governme Individu on nt al U. S. 70, 894 921 6129 Forei 63, 182 104 6267 Considergnthis table concerning utility patents granted for a specific year. Select one patent at random. a. What is the probability that it is a foreign patent, given that it was issued to a corporation? b. What is the probability that it was issued to an individual, given that it was a U. S. patent?

Corporati Governme Individu on nt al U. S. 70, 894 921 6129 63, 182 104 that it 6267 a. Forei What is the probability is a gn foreign patent, given that it was issuedpatent to a corporation? P(foreign | corporation)

Corporati Governme Individu on nt al U. S. 70, 894 921 6129 Forei 63, 182 104 6267 P(foreign patent | corporation) gn

Corporati Governme Individu on nt al U. S. 70, 894 921 6129 Forei 63, 182 104 6267 P(foreign patent | corporation) gn

Corporati Governme Individu on nt al U. S. 70, 894 921 6129 63, 182 104 that it 6267 b. Forei What is the probability was gn issued to an individual, given that it was a U. S. patent? P (individual | U. S. )

Corporati Governme Individu on nt al U. S. 70, 894 921 6129 Forei 63, 182 104 6267 P(individual | U. S. ) gn

Corporati Governme Individu on nt al U. S. 70, 894 921 6129 Forei 63, 182 104 6267 P(individual | U. S. ) gn

Chapter 4 Probability and Counting Rules Section 4 -4 Counting Rules

4. 4 Counting Rules n. The fundamental counting rule is also called the multiplication of choices n. In a sequence of n events in which the first one has k 1 possibilities and the second event has k 2 and the third has k 3, and so forth, the total number of possibilities of the sequence will be k 1 · k 2 · k 3 · · · kn 92

Chapter 4 Probability and Counting Rules Section 4 -4 Example 4 -39 93

Example 4 -39: Paint Colors A paint manufacturer wishes to manufacture several different paints. The categories include Color: red, blue, white, black, green, brown, yellow Type: latex, oil Texture: flat, semigloss, high gloss Use: outdoor, indoor How many different kinds of paint can be made if you can select one color, one type, one texture, and one use? 94

Counting Rules n Factorial is the product of all the positive numbers from 1 to a number. n Permutation is an arrangement of objects in a specific order. Order matters. 95

Counting Rules n. Combination is a grouping of objects. Order does not matter. 96

Chapter 4 Probability and Counting Rules Section 4 -4 Example 4 -42/4 -43 97

Example 4 -42: Business Location Suppose a business owner has a choice of 5 locations in which to establish her business. She decides to rank each location according to certain criteria, such as price of the store and parking facilities. How many different ways can she rank the 5 locations? Using factorials, 5! = 120. Using permutations, 5 P 5 = 120. 98

Example 4 -43: Business Location Suppose the business owner in Example 4– 42 wishes to rank only the top 3 of the 5 locations. How many different ways can she rank them? Using permutations, 5 P 3 = 60. 99

Chapter 4 Probability and Counting Rules Section 4 -4 Example 4 -44 100

Example 4 -44: Television Ads The advertising director for a television show has 7 ads to use on the program. If she selects 1 of them for the opening of the show, 1 for the middle of the show, and 1 for the ending of the show, how many possible ways can this be accomplished? Since order is important, the solution is Hence, there would be 210 ways to show 3 ads. 101

Chapter 4 Probability and Counting Rules Section 4 -4 Example 4 -45 102

Example 4 -45: School Musical Plays A school musical director can select 2 musical plays to present next year. One will be presented in the fall, and one will be presented in the spring. If she has 9 to pick from, how many different possibilities are there? Order matters, so we will use permutations. 103

Chapter 4 Probability and Counting Rules Section 4 -4 Example 4 -48 104

Example 4 -48: Book Reviews A newspaper editor has received 8 books to review. He decides that he can use 3 reviews in his newspaper. How many different ways can these 3 reviews be selected? The placement in the newspaper is not mentioned, so order does not matter. We will use combinations. 105

Chapter 4 Probability and Counting Rules Section 4 -4 Example 4 -49 106

Example 4 -49: Committee Selection In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there? There are not separate roles listed for each committee member, so order does not matter. We will use combinations. There are 35· 10 = 350 different possibilities. 107

4. 5 Probability and Counting Rules The counting rules can be combined with the probability rules in this chapter to solve many types of probability problems. By using the fundamental counting rule, the permutation rules, and the combination rule, you can compute the probability of outcomes of many experiments, such as getting a full house when 5 cards are dealt or selecting a committee of 3 women and 2 men from a club consisting of 10 women and 10 men. 108

Chapter 4 Probability and Counting Rules Section 4 -5 Example 4 -52 109

Example 4 -52: Magazines A store has 6 TV Graphic magazines and 8 Newstime magazines on the counter. If two customers purchased a magazine, find the probability that one of each magazine was purchased. TV Graphic: One magazine of the 6 magazines Newstime: One magazine of the 8 magazines Total: Two magazines of the 14 magazines 110

Chapter 4 Probability and Counting Rules Section 4 -5 Example 4 -53 111

Example 4 -53: Combination Lock A combination lock consists of the 26 letters of the alphabet. If a 3 -letter combination is needed, find the probability that the combination will consist of the letters ABC in that order. The same letter can be used more than once. (Note: A combination lock is really a permutation lock. ) There are 26· 26 = 17, 576 possible combinations. The letters ABC in order create one combination. 112

EXAMPLE How many different 3 - digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted? If digits can be used more than once: Since there are three spaces to fill and 10 choices for each space, the solution is:

How many different 3 - digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted: There is only one way to assign the first digit, 9 ways to assign the second, and 8 ways to assign the third:

EXAMPLE How many different ID cards can be made if there are 6 digits on a card and no digit can be used more than once? Since order is important, the solution is:

EXAMPLE How many ways can a committee of 4 people be selected from a group of 10 people? Since order is not important, the solution is:

EXAMPLE How many ways can a foursome of 2 men and 2 women be selected from 10 men and 12 women in a golf club?

EXAMPLE In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. a. All 3 selected will be women. b. All 3 selected will be men. c. 2 men and 1 woman will be selected. d. 1 man and 2 women will be selected.

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. a. All 3 selected will be women.

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. b. All 3 selected will be men.

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. c. 2 men and 1 woman will be selected.

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. d. 1 man and 2 women will be selected.

EXAMPLE A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of: a. All dentists. b. 2 dentists and 2 doctors. c. All doctors. d. 3 doctors and 1 dentist. e. 1 doctor and 3 dentists.

A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of: a. All dentists.

A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of: b. 2 dentists and 2 doctors.

A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of: c. All doctors.

A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of: d. 3 doctors and 1 dentist.

A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of: e. 1 doctor and 3 dentists.

EXAMPLE A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark. a. What is the probability that you get a pair of red socks? b. What is the probability that you get a pair of black socks? c. What is the probability that you get 2 unmatched socks? d. Where did the other red sock go?

A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark. a. What is the probability that you get a pair of red socks?

A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark. b. What is the probability that you get a pair of black socks?

A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark. c. What is the probability that you get 2 unmatched socks?

A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark. d. Where did the other red sock go? It probably got lost in the wash!

EXAMPLE Find the probability that if 5 different- sized washers are arranged in a row, they will be arranged size. to There arein 5!order = 120 ofways arrange 5 washers in a row and 2 ways to have them in correct order, small to large or large to small; hence, the probability is: