3 Probability

Mathematics and Statistics

Permutations and

Combinations


In this section, we summarize some basics on combinations. In

particular, we investigate two questions:

(1) How many possibilities exist of sequencing the elements

of a set?

(2) How many possibilities exist of selecting a certain number

of elements from a set?

Let us start with the determination of the number of possible

sequences formed with a set of elements. To this end, we first

introduce the notion of a permutation.

Definition 1.19 Let M = {a1, a2, . . . , an}. Any sequence (ap1 , ap2 ,

. . . , apn ) of all elements of set M is called a permutation.


In order to determine the number of permutations, we

introduce n! (read: n factorial) which is defined as follows: n!

= 12 . . . (n 1) n for n 1. For n = 0, we define 0! = 1.

Given 6 cities, how many possibilities exist of organizing a tour

starting from one city visiting each of the remaining cities exactly

once and to return to the initial city? Assume that 1 city is the

starting point, all remaining 5 cities can be visited in arbitrary

order.

P(5) = 5! = 1 2 . . . 5 = 120


In a school, a teacher wishes to put 13 textbooks of three

types (mathematics, physics, and chemistry textbooks)

on a shelf. How many possibilities exist of arranging the

13 books on a shelf when there are 4 copies of a

mathematics textbook, 6 copies of a physics textbook and

3 copies of a chemistry textbook? The problem is to find

the number of possible permutations with non-

distinguishable (copies of the same textbook) elements.


In the arrangements of a,b,c,d,e taken all at

a time, how many begin with a?


How many ways can you arrange the letters

in REFERENCE?


How many ways 4 fruits can be selected

out of 10 fruits to exclude the smallest

fruit?


Probability Basics


Methods of Assigning Probabilities

Classical method of assigning probability (rules and laws)

Relative frequency of occurrence

(cumulated historical data) Subjective Probability (personal intuition or

reasoning)


Classical Probability

Number of outcomes leading to the event divided by the total number of outcomes possible

Each outcome is equally likely

Determined a priori -- before performing the experiment

Applicable to games of chance

Objective -- everyone correctly using the method assigns an identical probability

Ein outcomes ofnumber

outcomes ofnumber total

:

)(

e

n

n

N

Where

NEP e


Relative Frequency Probability

Based on historical data

Computed after performing the experiment

Number of times an event occurred divided by the number of trials

Objective -- everyone correctly using the method assigns an identical probability

E producing

outcomes ofnumber

trialsofnumber total

:

)(

e

n

n

N

Where

NEP e


Subjective Probability

Comes from a persons intuition or reasoning

Subjective -- different individuals may (correctly) assign different numeric probabilities to the same event

Degree of belief Useful for unique (single-trial) experiments

New product introduction Initial public offering of common stock Site selection decisions Sporting events


Set Theory


Union of Sets

The union of two sets contains an instance of each element of the two sets.

X

Y

X Y

1 4 7 9

2 3 4 5 6

1 2 3 4 5 6 7 9

, , ,

, , , ,

, , , , , , ,

C IBM DEC Apple

F Apple Grape Lime

C F IBM DEC Apple Grape Lime

, ,

, ,

, , , ,

Y X

X Y


Intersection of Sets

The intersection of two sets contains only those element common to the two sets.

X

Y

X Y

1 4 7 9

2 3 4 5 6

4

, , ,

, , , ,

Y X

C IBM DEC Apple

F Apple Grape Lime

C F Apple

, ,

, ,

X Y


Mutually Exclusive Events

Events with no common outcomes

Occurrence of one event precludes the occurrence of the other event

X

Y

X Y

1 7 9

2 3 4 5 6

, ,

, , , ,

FC

LimeGrapeF

AppleDECIBMC

,

,,

Y X

P X Y( ) 0


Types of Probability


Four Types of Probability

Marginal Probability

Union Probability

Joint Probability

Conditional Probability


Four Types of Probability

Marginal

The probability

of X occurring

Union

The probability

of X or Y

occurring

Joint

The probability

of X and Y

occurring

Conditional

The probability

of X occurring

given that Y

has occurred

Y X Y X

Y

X

P X( ) P X Y( ) P X Y( ) P X Y( | )


General Law of Addition

P X Y P X P Y P X Y( ) ( ) ( ) ( )

Y X


Office Design Problem

Probability Matrix

.11 .19 .30

.56 .14 .70

.67 .33 1.00

Increase

Storage Space

Yes No Total

Yes

No

Total

Noise

Reduction

What is the probability that a randomly selected

employee wanted more space or noise reduction?


General Law of Addition -- Example

P N S P N P S P N S( ) ( ) ( ) ( )

S N

.56 .67 .70

P N

P S

P N S

P N S

( ) .

( ) .

( ) .

( ) . . .

.

70

67

56

70 67 56

081


According to an article in Fortune, institutional

investors recently changed the proportions of

their portfolios toward public sector funds. The

article implies that 8% of investors studied invest

in public sector funds and 6% in corporate funds.

Assume that 2% invest in both kinds. If an

investor is chosen at random, what is the

probability that this investor has either public or

corporate funds?


Suppose that 25% of the population in a given area is

exposed to a television commercial for Ford automobiles,

and 34% is exposed to Fords radio advertisements. Also,

it is known that 10% of the population is exposed to both

means of advertising.

If a person is randomly chosen out of the entire population

in this area, what is the probability that he or she was

exposed to at least one of the two modes of advertising?


A firm has 550 employees; 380 of them have had at

least some college education, and 412 of the employees

underwent a vocational training program. Furthermore,

357 employees both are college-educated and have had

the vocational training.

If an employee is chosen at random, what is the

probability that he or she is college-educated or has had

the training or both?


A machine produces components for use in cellular phones. At any

given time, the machine may be in one, and only one, of three states:

operational, out of control, or down. From experience with this

machine, a quality control engineer knows that the probability that the

machine is out of control at any moment is 0.02, and the probability

that it is down is 0.015.

a. What is the relationship between the two events machine is out of

control and machine is down?

b. When the machine is either out of control or down, a repair person

must be called. What is the probability that a repair person must be

called right now?

c. Unless the machine is down, it can be used to produce a single

item. What is the probability that the machine can be used to produce

a single component right now?


According to The New York Times, 5 million

BlackBerry users found their devices nonfunctional on

April 18, 2007. If there were 18 million users of

handheld data devices of this kind on that day, what is

the probability that a randomly chosen user could not

use a device?

Assume that 3 million out of 18 million users could not

use their devices as cellphones, and that 1 million

could not use their devices as a cellphone and for data

device. What is the probability that a randomly chosen

device could not be used either for data or for voice

communication?


Law of Multiplication

General Law

Special Law

P X Y P X P Y X P Y P X Y( ) ( ) ( | ) ( ) ( | )

If events X and Y are independent,

and P X P X Y P Y P Y X

Consequently

P X Y P X P Y

( ) ( | ), ( ) ( | ).

,

( ) ( ) ( )


Law of Conditional Probability

The conditional probability of X given Y is the joint probability of X and Y divided by the marginal probability of Y.

P X YP X Y

P Y

P Y X P X

P Y( | )

( )

( )

( | ) ( )

( )


Office Design Problem

Probability Matrix

.11 .19 .30

.56 .14 .70

.67 .33 1.00

Increase

Storage Space

Yes No Total

Yes

No

Total

Noise

Reduction

What is the probability of an employee wants space

given that he wants noise reduction?


Law of Conditional Probability

N S

.56 .70

P N

P N S

P S NP N S

P N

( ) .

( ) .

( | )( )

( )

.

.

.

70

56

56

70

80

Probability of an employee wants space given that he

wants noise reduction



what is the probability that employee is supervisor if he is married?

Total

.7857

Yes No

.4571 .3286

.1143 .1000 .2143

.5714 .4286 1.00

Married

Yes

No

Total

Supervisor

Probability Matrix

of Employees



what is the probability that employee is supervisor if he is married?

Total

.7857

Yes No

.4571 .3286

.1143 .1000 .2143

.5714 .4286 1.00

Married

Yes

No

Total

Supervisor

Probability Matrix

of Employees

20.0)|(

5714.0140

80)(

2143.0140

30)(

MSP

MP

SP

P M S P M P S M( ) ( ) ( | )

( . )( . ) .

0 5714 0 20 0 1143


Twenty-one percent of the executives in a large advertising

firm are at the top salary level. It is further known that 40%

of all the executives at the firm are women. Also, 6.4% of

all executives are women and are at the top salary level.

Recently, a question arose among executives at the firm as

to whether there is any evidence of salary inequity.

Assuming that some statistical considerations (explained in

later chapters) are met, do the percentages reported above

provide any evidence of salary inequity?


If a large competitor will buy a small firm, the firms

stock will rise with probability 0.85. The purchase of

the company has a 0.40 probability.

What is the probability that the purchase will take

place and the firms stock will rise?


A bank loan officer knows that 12% of the banks

mortgage holders lose their jobs and default on the

loan in the course of 5 years. She also knows that

20% of the banks mortgage holders lose their jobs

during this period.

Given that one of her mortgage holders just lost his

job, what is the probability that he will now default

on the loan?


An investment analyst collects data on stocks and notes

whether or not dividends were paid and whether or not

the stocks increased in price over a given period. Data are

presented in the following table.

a. If a stock is selected at random out of the analysts list

of 246 stocks, what is the probability that it increased in

price?

b. If a stock is selected at random, what is the probability

that it paid dividends?






c. If a stock is randomly selected, what is the probability

that it both increased in price and paid dividends?

d. What is the probability that a randomly selected stock

neither paid dividends nor increased in price?






e. Given that a stock increased in price, what is the

probability that it also paid dividends?

f. If a stock is known not to have paid dividends, what is

the probability that it increased in price?



whether or not dividends were paid and whether or not the

stocks increased in price over a given period. Data are


g. What is the probability that a randomly selected stock

was worth holding during the period in question; that is,

what is the probability that it increased in price or paid

dividends or did both?


As the marketing manager for the Consumer Electronics Company, you

are analyzing the survey results of an intent-to-purchase study. This study

asked the heads of 1,000 households about their intentions to purchase a

big-screen television (defined as 36 inches or larger) sometime during the

next 12 months. Investigations of this type are known as intent-to-

purchase studies. As a follow-up, you plan to survey the same people 12

months later to see whether such a television was purchased. In addition,

for households purchasing big-screen televisions, you would like to know

whether the television they purchased was a plasma screen, whether they

also purchased a digital video recorder (DVR) in the past 12 months, and

whether they were satisfied with their purchase of the big-screen

television.

You are expected to use the results of this survey to plan a new marketing

strategy that will enhance sales and better target those households likely to

purchase multiple or more expensive products. What questions can you

ask in this survey?

How can you express the relationships among the various intent-to-

purchase responses of individual households?


In the survey, additional questions were asked of the 300 households

that actually purchased big-screen televisions. Table below indicates

the consumers responses to whether the television purchased was a

plasma screen and whether they also purchased a DVR in the past 12

months.

Find the probability that if a household that purchased a big-screen

television is randomly selected, the television purchased is a plasma

screen.


Find the probability that a randomly selected household

that purchased a big-screen television also purchased a

plasma screen television and a DVR.


Table below presents the results of the sample of 1,000

households in terms of purchase behavior for big-screen

televisions. Find the probability of planned to purchase a

big-screen television:




televisions. Find the probability of planned to purchase or

actually purchased is:-




televisions. Find the probability of planned to purchase or

actually purchased:-




televisions. Find the probability the probability that a

household actually purchased the big-screen television

given that he or she planned to purchase:-


Table below is a contingency table for whether the

household purchased a plasma screen television and

whether the household purchased a DVR. Of the

households that purchased plasma-screen televisions, what

is the probability that they also purchased DVRs?


* What is the probability that a household is planning to purchase a

big-screen television in the next year?

* What is the probability that a household will actually purchase a

big-screen television?

* What is the probability that a household is planning to purchase a

big-screen television and actually purchases the television?

* Given that the household is planning to purchase a big-screen

television, what is the probability that the purchase is made?

* Does knowledge of whether a household plans to purchase the

television change the likelihood of predicting whether the household

will purchase the television?

* What is the probability that a household that purchases a big-screen

television will purchase a plasma-screen television?


television will also purchase a DVR?


television will be satisfied with the purchase?


Thank You

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3 Probability