3 Probability

Mathematics and Statistics

Permutations and Combinations

Mathematics and Statistics

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.

Mathematics and Statistics

In order to determine the number of permutations, we introduce n! (read: n factorial) which is defined as follows: n! = 1・2・ . . . ・(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

Mathematics and Statistics

Mathematics and Statistics

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 nondistinguishable (copies of the same textbook) elements.

Mathematics and Statistics

In the arrangements of a,b,c,d,e taken all at a time, how many begin with a?

Mathematics and Statistics

How many ways can you arrange the letters in REFERENCE?

Mathematics and Statistics

Mathematics and Statistics

Mathematics and Statistics

Mathematics and Statistics

How many ways 4 fruits can be selected out of 10 fruits to exclude the smallest fruit?

Mathematics and Statistics

Probability Basics

Mathematics and Statistics

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)

Mathematics and Statistics

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

P( E ) 

n

e

N

Where : N  total number of outcomes

n

e

 number of outcomes in E

Mathematics and Statistics

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

P( E ) 

n

e

N

Where : N  total number of trials

n

e

 number of outcomes

producing E

Mathematics and Statistics

Subjective Probability • Comes from a person’s 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

Mathematics and Statistics

Set Theory

Mathematics and Statistics

Union of Sets • The union of two sets contains an instance of each element of the two sets. X  1,4,7,9 Y  2,3,4,5,6

X

Y

X  Y  1,2,3,4,5,6,7,9 C   IBM , DEC , Apple

F   Apple, Grape, Lime

C  F   IBM , DEC , Apple, Grape, Lime

XY

Mathematics and Statistics

Intersection of Sets • The intersection of two sets contains only those element common to the two sets. X  1,4,7,9 Y 

2,3,4,5,6

X

Y

X  Y   4

C  IBM , DEC , Apple 

F  Apple , Grape , Lime 

C  F  Apple 

XY

Mathematics and Statistics

Mutually Exclusive Events • Events with no common outcomes • Occurrence of one event precludes the occurrence of the other event C  IBM , DEC , Apple F  Grape, Lime CF  

P( X  Y )  0 X

Y

X  1,7,9 Y  2,3,4,5,6 X Y  



Mathematics and Statistics

Types of Probability

Mathematics and Statistics

Four Types of Probability • • • •

Marginal Probability Union Probability Joint Probability Conditional Probability

Mathematics and Statistics

Four Types of Probability Marginal

Union

Joint

Conditional

P( X )

P( X  Y )

P( X  Y )

P( X| Y )

The probability of X occurring

X

The probability of X or Y occurring

X Y

The probability of X and Y occurring

The probability of X occurring given that Y has occurred

X Y

Y

Mathematics and Statistics

General Law of Addition P( X  Y )  P( X )  P(Y )  P( X  Y ) X

Y

Mathematics and Statistics

Office Design Problem Probability Matrix

Noise Reduction

Yes No Total

Increase Storage Space Yes No .14 .56 .19 .11 .33 .67

Total .70 .30 1.00

What is the probability that a randomly selected employee wanted more space or noise reduction?

Mathematics and Statistics

General Law of Addition -- Example P( N  S )  P( N )  P( S )  P( N  S ) S

N .70

.56

.67

P( N ) P( S ) P( N  S ) P( N  S )

.70 .67 .56 .70.67 .56  0.81

Mathematics and Statistics

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?

Mathematics and Statistics

Suppose that 25% of the population in a given area is exposed to a television commercial for Ford automobiles, and 34% is exposed to Ford’s 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?

Mathematics and Statistics

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?

Mathematics and Statistics

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?

Mathematics and Statistics

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?

Mathematics and Statistics

Law of Multiplication • General Law

P( X  Y )  P( X )  P(Y| X )  P(Y )  P( X| Y ) • Special Law If events X and Y are independent, P( X )  P( X | Y ), and P(Y )  P(Y | X ). Consequently , P( X  Y )  P( X )  P( Y )

Mathematics and Statistics

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  Y ) P(Y | X )  P( X ) P( X| Y )   P( Y ) P(Y )

Mathematics and Statistics

Office Design Problem Probability Matrix

Noise Reduction

Yes No Total

Increase Storage Space Yes No .14 .56 .19 .11 .33 .67

Total .70 .30 1.00

What is the probability of an employee wants space given that he wants noise reduction?

Mathematics and Statistics

Law of Conditional Probability

N

S .56

.70

P ( N ) .70 P ( N  S ) .56 P( N  S ) P( S | N )  P( N ) .56  .70 .80

Probability of an employee wants space given that he wants noise reduction

Mathematics and Statistics

Law of Multiplication what is the probability that employee is supervisor if he is married?

Probability Matrix of Employees Supervisor Yes No Total

Married Yes No Total .1143 .1000 .2143 .4571 .3286 .7857 .5714 .4286 1.00

Mathematics and Statistics

Law of Multiplication what is the probability that employee is supervisor if he is married?

Probability Matrix of Employees Supervisor Yes No Total

Married Yes No Total .1143 .1000 .2143 .4571 .3286 .7857 .5714 .4286 1.00

30 P( S )   0.2143 140 80 P( M )   0.5714 140 P( S | M )  0.20 P( M  S )  P( M )  P( S| M )  ( 0. 5714)( 0. 20)  0.1143

Mathematics and Statistics

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?

Mathematics and Statistics

If a large competitor will buy a small firm, the firm’s 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 firm’s stock will rise?

Mathematics and Statistics

A bank loan officer knows that 12% of the bank’s mortgage holders lose their jobs and default on the loan in the course of 5 years. She also knows that 20% of the bank’s 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?

Mathematics and Statistics

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 analyst’s 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?

Mathematics and Statistics

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.

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?

Mathematics and Statistics

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.

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?

Mathematics and Statistics

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.

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?

Mathematics and Statistics

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-topurchase 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-topurchase responses of individual households?

Mathematics and Statistics

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.

Mathematics and Statistics

Find the probability that a randomly selected household that purchased a big-screen television also purchased a plasma screen television and a DVR.

Mathematics and Statistics

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:

Mathematics and Statistics

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 or actually purchased is:-

Mathematics and Statistics

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 or actually purchased:-

Mathematics and Statistics

Table below presents the results of the sample of 1,000 households in terms of purchase behavior for big-screen televisions. Find the probability the probability that a household actually purchased the big-screen television given that he or she planned to purchase:-

Mathematics and Statistics

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?

Mathematics and Statistics

* 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? * What is the probability that a household that purchases a big-screen television will also purchase a DVR? * What is the probability that a household that purchases a big-screen television will be satisfied with the purchase?

Mathematics and Statistics

Thank You