Slide 5-2

Chapter 5

Probability and Random Variables

Slide 5-3

Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of insurance, investments, and weather forecasting, and in various areas. There are three types: Classical, Empirical, and Subjective.

Classical is more theoretical.

Empirical is obtained through observations.

Subjective is based on inexact methods such as guesswork or opinion

Chapter 5 Section 1 – Probability Basics

Slide 5-4

Rounding Rule for Probabilities

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.

Slide 5-5

Experiment – is an action whose outcome cannot be predicted with certainty

An outcome is the result of a single trial of a probability experiment

Event – is some specified result that may or may not occur when an experiment is performed.

Slide 5-6

Empirical Probability

Empirical probability relies on actual experience (or observations) to determine the likelihood of outcomes.

Empirical is obtained through observations.

Given a frequency distribution, the probability of an event being in a given class is:

Empirical Exp: The chance depends on what you observed on several tosses of a coin. Flip coin 4 times…heads 1 out of 4 therefore ¼.

Slide 5-7

When two balanced dice are rolled, 36 equally likely outcomes are possible:

The sum of the dice can be 11 in two ways. The probability the sum is 11 is f/N = 2/36 or 0.056.

Doubles can be rolled in six ways. The probability of doubles is f/N = 6/36 or 0.167.

Contains a three. The probability contains a three is f/N = 11/36 or 0.306.

Slide 5-8

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

Example: Rolling a 7 on a die.

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

Example: rolling a number less than 7 on a die.

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

06

0)7( P

16

6)7( numberP

Slide 5-9

5.8 Oklahoma State Officials:Governor G

Lieutenant Governor L

Secretary of State S

Attorney General A

Treasurer T

a) List the possible samples without replacement of size 3 that can be obtained from the population of five officials

G,L,S G,L,A G,L,T G,S,A G,S,T G,A,T L,S,A L,S,T L,A,T S,A,T

If a simple random sample without replacement of three officials is taken from the five officials, determine the probability that the Governor, Attorney General, and Treasurer are obtained.

One sample includes the governor, attorney general, and treasurer. Therefore the probability is f/N = 1/10 or 0.1

Slide 5-10

5.8 Oklahoma State Officials:Governor G

Lieutenant Governor L

Secretary of State S

Attorney General A

Treasurer T

a) List the possible samples without replacement of size 3 that can be obtained from the population of five officials

G,L,S G,L,A G,L,T G,S,A G,S,T G,A,T L,S,A L,S,T L,A,T S,A,T

If a simple random sample without replacement of three officials is taken from the five officials, determine the probability that the Governor, and Treasurer are included in the sample.

Three samples included the governor and treasurer. Therefore, the probability is f/N = 3/10 or 0.3

Slide 5-11

5.8 Oklahoma State Officials:Governor G

Lieutenant Governor L

Secretary of State S

Attorney General A

Treasurer T

a) List the possible samples without replacement of size 3 that can be obtained from the population of five officials

G,L,S G,L,A G,L,T G,S,A G,S,T G,A,T L,S,A L,S,T L,A,T S,A,T

If a simple random sample without replacement of three officials is taken from the five officials, determine the probability that the Governor, is included in the sample.

Six samples include the governor. Therefore, the probability is f/N = 6/10 or 0.6

Slide 5-12

Property 1 - states that probabilities cannot be negative or greater than one.

16

6)7( numberP

Property 3 - If an event E is certain, then the probability of E is 1. Example: rolling a number less than 7 on a die.

Property 2 - If an event E cannot occur (i.e., the event contains no members in the sample space), the probability is zero. Example: Rolling a 7 on a die.

06

0)7( P

Slide 5-13

Chapter 5 Section 2 - Events

Slide 5-14Figure 5.9

Venn diagrams are used to represent probabilities pictorially

Slide 5-15

Two events are mutually exclusive if they cannot occur at the same time (i.e., they have no outcomes in common).

The probability of two or more events can be determined by the addition rules.

Slide 5-16

Figure 5.14

Figure 5.15

Two mutually exclusive events

Two non-mutually exclusive events

Three mutually exclusive events

Three non-mutually exclusive events

Three non-mutually exclusive events

Slide 5-17

Addition Rules

Addition Rule 1—When two events A and B are mutually exclusive, the probability that A or B will occur is:

Addition Rule 2—If A and B are not mutually exclusive, then:

Slide 5-18

EXAMPLES

Determine whether these event are mutually exclusive (cannot occur at the same time).

a) Roll a die: Get an even number, and get a number less than 3. NO – can occur at the same time. Can get a 2.

b) Roll a die: Get a prime number (2,3,5) and get an odd number. NO – can occur at the same time. Can get a 3 or 5.

c) Roll a die: Get a number greater than 3, and get a number less than 3. YES – cannot occur at the same time.

d) Select a student in class: The student has blond hair, and the student has blue eyes. NO – can occur at the same time. A student can have blond hair and blue eyes.

Slide 5-19

EXAMPLES

Determine whether these event are mutually exclusive (cannot occur at the same time).

e) Select a student at Rio: The student is a sophomore, and the student is a business major. NO – can occur at the same time. A student can be a Sophomore with a business major.

f) Select any course: It is a calculus course, and it is an English course. YES – cannot occur at same time. A course is one or the other not both.

g) Selected a registered voter: The voter is a Republican, and the voter is a Democrat. YES – cannot occur at same time. A voter is one or the other not both.

Slide 5-20

Chapter 5 Section 3 – Some Rules of Probability

Probability Notation

If E is an event, then P(E) represents the probability that event E occurs. It is read “the probability of E.”

Slide 5-21

Addition Rule 2If A and B are not mutually exclusive

Slide 5-22

Roll a six-sided die. What is the probability of rolling

1 = 1/6 2 = 1/6 3 = 1/6

4 = 1/6 5 = 1/6 6 = 1/6P(roll a number > 4) = P(roll 5 or roll 6)

P(roll a number ≤ 4) = P(roll 1 or roll 2 or roll 3 or roll 4)

P(roll a number ≤ 4) + P(roll a number > 4) = 1

How are the statements above related?

= 1/6 + 1/6 = 2/6 or 1/3

= P(roll 1) + P(roll 2) + P(roll 3) + P(roll 4) = 4/6 or 2/3

= P(roll 5) + P(roll 6)

Examples

Slide 5-23

EXAMPLES

An automobile dealer had 10 Fords, 7 Buicks, and 5 Plymouths on her used car lot. If a person purchased a used car, find the probability that it is a Ford or Buick.

Total = 10 + 7 + 5 = 22

P(Ford) = 10/22

P(Buick) = 7/22

P(Ford or Buick) = P(Ford) + P(Buick) =

10/22 + 7/22 = 17/22

The probability that a student owns a car is 0.65, and the probability that a student owns a computer is 0.82. If the probability that a student owns both is 0.55, what is the probability that a given student owns neither a car nor a computer?

P(car or computer) = 0.65 + 0.82 – 0.55 = 0.92

P(neither) = 1 – 0.92 = 0.08

Slide 5-24

EXAMPLES A single card is drawn from a deck. Find the probability of

selecting the following:

a) A 4 or a diamond.P(4 or diamond) = P(4) + P(diamonds) – P(4 of diamonds)

= 4/52 +13/52 – 1/52 = 16/52 or 4/13

There are four 4’s and 13 diamonds, but the 4 of diamonds is counted twice.

b) A club or a diamond.

P(club or diamond) = 13/52 + 13/52 = 26/52 or ½

c) A jack or a black card.

P(jack or black) = 4/52 +26/52 – 2/52 = 28/52 or 7/13

Slide 5-25

EXAMPLES

In a certain geographic region, newspapers are classified as being published daily morning, daily evening, and weekly. Some have a comic section and some do not. The distribution is shown here.

If a newspaper is selected at random, find these probabilities.

a) The newspaper is a weekly publication.

P(weekly) = 3/15 or 1/5

Have Comic Section Morning Evening Weekly TOTAL

Yes 2 3 1 6

No 3 4 2 9

Total 5 7 3 15

Slide 5-26

EXAMPLES In a certain geographic region, newspapers are classified as being published daily morning, daily evening, and weekly. Some have a comic section and some do not. The distribution is shown here.

If a newspaper is selected at random, find these probabilities.

b) The newspaper is a daily morning publication or has comics.

P(morning or has comics) =

P(morning) + P(has comics) – P(morning with comics) =

5/15 + 6/15 - 2/15 = 9/15 or 3/5

Have Comic Section Morning Evening Weekly TOTAL

Yes 2 3 1 6

No 3 4 2 9

Total 5 7 3 15

Slide 5-27

EXAMPLES In a certain geographic region, newspapers are classified as being published daily morning, daily evening, and weekly. Some have a comic section and some do not. The distribution is shown here.

If a newspaper is selected at random, find these probabilities.

c) The newspaper is published weekly or it does not have comics.

P(weekly or no comics) =

P(weekly) + P(no comics) – P(weekly with no comics) =

3/15 + 9/15 – 2/15 = 10/15 or 2/3

Have Comic Section Morning Evening Weekly TOTAL

Yes 2 3 1 6

No 3 4 2 9

Total 5 7 3 15

Slide 5-28

Independent Events

Two events A and B are independent if the fact that A occurs does not affect the probability of B occurring.

Module P3 The Multiplication Rule: Independence

Slide 5-29

Multiplication Rule 1

There are two multiplication rules they can be used to find the probability of two or more events that occur in sequence. Such as roll a die and toss a coin at the same time.

Multiplication Rule 1—When two events are independent, the probability of both occurring is:

Slide 5-30

Example -- Independent

A washer and dryer have been purchased; both are under warranty.

outcome 1 -- the washer needs service

outcome 2 -- the dryer needs service

Does knowing that the washer needs service affect the likelihood that the dryer needs service? What is the probability that both the washer and the dryer need service while under warranty?

P(washer needs service and dryer needs service) = P(washer needs service) * P(dryer needs service)

NO

Slide 5-31

Examples - - independent event

1. Tossed a coin twice, the probability of getting two heads is ½ * ½ = ¼

NOTE: The sample space would be. HH, HT, TH, TT; then P(HH) = ¼

2. Toss a coin and roll a die. What is the probability of getting heads on the coin and a 4 on the die. P(H and 4) = P(H) * P(4) =

NOTE: The sample space for coin is H, T and for Die is 1, 2, 3, 4, 5, 6. The sample space for both is H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6; then P(head and 4) =

3. A urn contains 3 red balls, 2 blue balls, and 5 white balls. Find probability of selecting 2 blue balls. P(B and B) = P(B) * P(B) = 2/10 * 2/10 = 4/100 = 1/25 selecting 1 blue and 1 white. P(B and W) = P(B) * P(W) = 2/10 * 5/10 = 10/100 = 1/10selecting 1 red and 1 blue. P(R and B) = P(R) * P(B) = 3/10 * 2/10 = 6/100 = 3/50

12

1

6

1

2

1

12

1

Slide 5-32

Dependent Events

When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent events.

Slide 5-33

Multiplication Rule 2

The conditional probability of an event B in relationship to an event A is the probability that event B occurs after event A has already occurred. The notation for conditional probability is P(B|A). Does NOT mean divide it means the probability that event B occurs given that event A has already occurred.

Multiplication Rule 2—When two events are dependent, the probability of both occurring is:

Slide 5-34

Formula for Conditional Probability

The probability that the second event B occurs given that the first event A has occurred can be found dividing the probability that both events occurred by the probability that the first event has occurred. The formula is:

Slide 5-35

Example - Dependent

Select a card from a deck of 52 cards. What is the likelihood of selecting an ace?

Do not replace the first card. Select another card. What is the likelihood of selecting an ace?

The second outcome depends on the first.

What is the likelihood of selecting an ace knowing that the first was an ace?

What is the likelihood of selecting two aces if the first and second cards are not replaced?

P(ace) = 4/52

P(ace) = 4/51

P(ace|ace) = 3/51

P(two aces) = 4/50

Slide 5-36

Tree Diagram

A tree diagram is a device used to list all possibilities of a sequence of events in a systematic way.

Can be used for independent or dependent and can also be used for sequences of three of more events.

Slide 5-37

Slide 5-38

EXAMPLE

If 18% of all Americans are underweight, find the probability that if three Americans are selected at random, all will be underweight.

P(all three underweight)

= 0.18 * 0.18 * 0.18 = (0.18)3 = 0.005832 or 0.6%

Slide 5-39

EXAMPLE

If 25% of U.S. federal prison inmates are not U.S. citizens, find the probability that two randomly selected federal prison inmates will not be U.S. citizens.

P(two inmates are not citizens)

= 0.25 * 0.25 = (0.25)2 = 0.0625 or 6.3%

Slide 5-40

EXAMPLES

A Flashlight has six batteries, two of which are defective. If two are selected at random without replacement, find the probability that both are defective.

P(both are defective) = 15

1

30

2

5

1

6

2or

Slide 5-41

EXAMPLES

In a scientific study there are 8 guinea pigs, 5 of which are pregnant. If three are selected at random without replacement, find the probability that all are pregnant.

P(all are pregnant) =

Find the probability that none are pregnant.

P(none are pregnant) =

28

5

6

3

7

4

8

5

56

1

336

6

6

1

7

2

8

3or

Slide 5-42

EXAMPLESAn automobile manufacturer has three factories A, B, and C.

They produce 50%, 30%, and 20%, respectively, of a specific model of car.

30% of the cars produced in factory A are white,

40% of those produced in factory B are white and

25% produced in factory C are white.

If an automobile produced by the company is selected at random, find the probability that it is white.

Slide 5-43

0.5

0.3

0.2

A

B

C

FactoryW

NWW

NWW

NW

0.3

0.7

0.4

0.60.25

0.75

(0.5)(0.3) = 0.15

(0.3)(0.4) = 0.12

(0.2)(0.25) = 0.05

P(white) = 0.15 + 0.12 + 0.05 = 0.32

Slide 5-44

EXAMPLES

In a pizza restaurant, 95% of the customers order pizza. If 65% of the customers order pizza and a salad, find the probability that a customer who orders pizza will also order a salad.

P(pizza|salad) = 68.4%or 684.095.0

65.0

Slide 5-45

EXAMPLESAt a teachers’ conference, there were 4 English teachers, 3 mathematics teachers, and 5 science teachers. If 4 teachers are selected for a committee, find the probability that at least one is a science teacher.

4+3+5 = 12

P(at least one science) = 1 – P(no science) =

99

92

99

71

11880

8401

9

4

10

5

11

6

12

71

Slide 5-46

EXAMPLESEighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results are shown in the table.

If a student is selected at random, find these probabilities:

a) Given that the student is a freshman, he or she opposes the ban.

P(fr and oppose) =

Class Favor Oppose No Opinion Total

FR 15 27 8 50

SOPH 23 5 2 30

TOTAL 38 32 10 80

50

27

4000

2160

50

80

80

27

80

50

80

27

Slide 5-47

EXAMPLESEighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results are shown in the table.

If a student is selected at random, find these probabilities:

b) Given that the student favors the ban, the student is a sophomore.

P(favor and soph) =

Class Favor Oppose No Opinion Total

FR 15 27 8 50

SOPH 23 5 2 30

TOTAL 38 32 10 80

38

23

3040

1840

38

80

80

23

80

38

80

23

Slide 5-48

Fundamental Counting Rule Many times we wish to know the number of outcomes for a sequence

of events. We can use the fundamental counting rule to determine this number.

The fundamental counting rule can be used to determine the total number of outcomes in a sequence of events.

In a sequence of n events in which the first one has k1 possibilities and the second event has k2 and the third has k3, and so forth, the total number of possibilities of the sequence will be:

Note: “And” in this case means to multiply.

Module P3 - The Counting Rule

Slide 5-49

Consider the digits 0,1,2,3, and 4. If they are used on a four-digit ID card, How many different cards are possible?

(5 * 5 * 5 * 5) = 54 = 625

How does the fundamental counting rule apply here?

Since there are 4 spaces to fill and 5 digits for each space: = 5 * 5 * 5 * 5 In this example, repetition is allowed. If repetition is allowed then number stays the same from left to right. If not allowed then the number decreases by 1 from left to right.

If not allowed then we would have the first digit can be chosen 5 ways, but the second digit can be chosen 4 ways, since there are only four digits left, etc.

5 * 4 * 3 * 2 = 120

nkkkk 321

Slide 5-50

EXAMPLE

The call letters of a radio station must have 4 letters. The first letter must be a K or a W. How many different station call letters can be made if repetitions are not allowed? Note: 26 letters in the alphabet.

2 * 25 * 24 * 23 = 27,600

If repetitions are allowed?

2 * 26 * 26 * 26 = 35,152

Slide 5-51

Permutations

In order to understand the permutation and combination rules, we need a special definition of 0! Factorial Formula for any counting n is:

A permutation is an arrangement of n objects in a specific order.

The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taking r objects at a time. It is written as nPr, and the formula is:

1!0

1)......2)(1(!

nnnn

Slide 5-52

Combinations

A selection of distinct objects without regard to order is called a combination.

The number of combinations of r objects selected from n objects is denoted nCr and is given by the formula:

Slide 5-53

Suppose an organization has 4 members: A, B, C, D.How many ways can a president and a VP be selected? In other words, select 2 from 4 when order is important.

How many ways can a committee of two be selected? In other words, select 2 from 4 when order is not important.

Permutation:

Combination:

Slide 5-54

EXAMPLES

In a board of directors composed of 8 people.

How many ways can 1 chief executive officer, 1 director, and 1 treasurer be selected?

Which rule will we need to use?

Permutation

336120

40320

12345

12345678

!5

12345678

)!38(

!838

P

Slide 5-55

EXAMPLES

If a person can select three presents form 10 presents under a Christmas tree.

How many different combinations are there?

Which rule will we need to use?

Combination

1206

720

123!7

!78910

!3!7

!10310

C

Slide 5-56

EXAMPLESIn a train yard there are 4 tank cars, 12 boxcar, and 7 flatcars. How many ways can a train be made up consisting of 2 tank cars, 5 boxcars, and 3 flatcars? (In this case order is not important.)

320,166

357926

6

210

120

040,95

2

12

123!4

!4567

12345!7

!789101112

12!2

!234

!3)!37(

!7

!5)!512(

!12

!2)!24(

!4

3751224

CCCWe need to use thecombination rule.

Slide 5-57

EXAMPLESThere are 7 women and 5 men in a department store.

How many way can a committee of 4 people be selected?

How many ways can this committee be selected if there must be 2 men and 2 women on the committee?

How many ways can this committee be selected if there must be at least 2 women on the committee?

49524

11880

1234!8

!89101112

!4)!412(

!12412

C

2104

840

2

20

2

42

12!3

!345

12!5

!567

!2!3

!5

!2!5

!7

!2)!25(

!5

!2)!27(

!72527

CC

4203517521035535102124

840

1

5

6

210

2

20

2

421234!3

!34567

1!4

!45

123!4

!4567

12!3

!345

12!5

!567

!4)!47(

!7

!1)!15(

!5

!3)!37(

!7

!2)!25(

!5

!2)!27(

!74715372527

CCCCC

Slide 5-58

Summary

The three types of probability are classical, empirical, and subjective.

Classical probability uses sample spaces.

Empirical probability uses frequency distributions and is based on observations.

In subjective probability, the researcher makes an educated guess about the chance of an event occurring.

Slide 5-59

Summary (cont’d.) An event consists of one or more outcomes of a

probability experiment.

Two events are said to be mutually exclusive if they cannot occur at the same time.

Events can also be classified as independent or dependent.

If events are independent, whether or not the first event occurs does not affect the probability of the next event occurring.

Slide 5-60

Summary (cont’d.) If the probability of the second event occurring is

changed by the occurrence of the first event, then the events are dependent.

The complement of an event is the set of outcomes in the sample space that are not included in the outcomes of the event itself.

Complementary events are mutually exclusive.

Slide 5-61

Summary (cont’d.)

Rule Definition

Multiplication rule

Permutation rule

Combination rule

The number of ways a sequence of n events can occur; if the first event can occur in k1 ways, the second event can occur in k2 ways, etc.

The arrangement of n objects in a specific order using r objects at a time

The number of combinations of r objects selected from n objects (order is not important)

Slide 5-62

Conclusions

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.

Slide 5-63

Conclusions (cont’d)

A tree diagram can be used when a list of all possible outcomes is necessary. When only the total number of outcomes is needed, the multiplication rule, the permutation rule, and the combination rule can be used.