Basic Basic concepts concepts of of probabilityprobability

Numerical Numerical measure of likelihoodmeasure of likelihood that that

the event (Simple Event / Joint Event / the event (Simple Event / Joint Event /

Compound Event) will occur.Compound Event) will occur.

What is Probability?What is Probability?

Three concepts of probabilityThree concepts of probability

1.1. Classical probabilityClassical probability

2.2. Relative frequency probabilityRelative frequency probability

3.3. Subjective probability Subjective probability

(There are 2 ways to get one 6 and the other 4)

e.g. P( ) = 2/36

Computing ProbabilitiesComputing ProbabilitiesThe probability of an event E:The probability of an event E:

Each of the outcomes in the sample space is Each of the outcomes in the sample space is equally likely to occur equally likely to occur

number of event outcomes( )

total number of possible outcomes in the sample space

P E

X

T

Properties of ProbabilityProperties of Probability

1.1. P(A) lies between 0 & 1. i.e.P(A) lies between 0 & 1. i.e.1

.5 0

Certain

Impossible

If A any event in the sample space S then If A any event in the sample space S then

1)(0 AP

2.2. Let A be any event and Let A be any event and iodenote the basic outcomes. Then denote the basic outcomes. Then

)()( iOPAP

4.4. P(S)=1P(S)=1

3.3. Probability of sum of events is 1Probability of sum of events is 1

Compound ProbabilityCompound ProbabilityAddition RuleAddition Rule

1.1. P(A or B)P(A or B)= P(A = P(A B) B)

= = P(A) + P(B) P(A) + P(B) P(A P(A B) B)

2.2. For For Mutually ExclusiveMutually Exclusive Events: Events:

P(A or B) = P(A P(A or B) = P(A B) = P(A) + P(B) B) = P(A) + P(B)

3.3. Probability of ComplementProbability of Complement

P(A) + P(Ā) = 1. So, P(Ā) = 1 P(A) + P(Ā) = 1. So, P(Ā) = 1 P(A) P(A)

Conditional ProbabilityConditional Probability

Event Probability Event Probability GivenGiven that Another Event that Another Event OccurredOccurred

P(A | B) = P(A | B) = P(A and B)P(A and B) , P(B)>0 , P(B)>0 P(B) P(B)

)(

)()(

BP

BAPBAP

SimilarlySimilarly

)(

)()(

AP

BAPABP

, P (A)>0

Statistical IndependenceStatistical Independence

1.1. Event Occurrence Event Occurrence Does Does NotNot Affect Probability Affect Probability of Another Event.of Another Event.

e.g. Toss 1 Coin Twice, Throw 3 Dice e.g. Toss 1 Coin Twice, Throw 3 Dice

2.2. Tests For Independence Tests For Independence

P(A | B) = P(A), or P(B | A) = P(B), P(A | B) = P(A), or P(B | A) = P(B),

or P(A and B) = P(A)P(B)or P(A and B) = P(A)P(B)

Multiplication RuleMultiplication Rule1.1. Joint Probabilities for Joint Probabilities for IntersectionIntersection of Events (Joint of Events (Joint Events)Events)

2.2. P(A and B) = P(A P(A and B) = P(A B) B) P(A P(A B) = P(A)P(B|A) B) = P(A)P(B|A) = P(B)P(A|B) = P(B)P(A|B)

3.3. For For IndependentIndependent Events: Events:P(A and B) = P(AP(A and B) = P(AB) = P(A)P(B)B) = P(A)P(B)

A sample of executives were surveyed about A sample of executives were surveyed about

their loyalty to the company. One of the their loyalty to the company. One of the

questions was, “ If you were given an offer by questions was, “ If you were given an offer by

another company equal to or slightly better than another company equal to or slightly better than

your present position, would you remain with your present position, would you remain with

the company or take the other position?” the the company or take the other position?” the

responses of the 200 executives in the survey responses of the 200 executives in the survey

were cross-classified with their length of service were cross-classified with their length of service

with the company and their loyalty.with the company and their loyalty.

EXERCISE

LoyaltyLoyalty

Length of serviceLength of service

Less Less than 1 than 1 yearyear

BB11

1 – 5 1 – 5 yearyearss

BB22

6 – 6 – 10 10 yearsyears

BB33

More More than 10 than 10 yearsyears

BB44

Would remain, AWould remain, A1 1

Would not remain, Would not remain, AA22

1010

25253030

151555

10107575

3030

a.a.What is the probability of randomly selecting an What is the probability of randomly selecting an executive who is loyal to the company (would executive who is loyal to the company (would remain) and who has more than 10 years of remain) and who has more than 10 years of service?service?

b.b.What is the probability of randomly selecting an What is the probability of randomly selecting an executive who would remain with the company or executive who would remain with the company or has less than 1 year of experience?has less than 1 year of experience?

c.c.What is the probability of randomly selecting What is the probability of randomly selecting

an executive with more than 10 years of an executive with more than 10 years of

service?service?

d.d.What is the probability of randomly selecting What is the probability of randomly selecting

an executive who would not remain with the an executive who would not remain with the

company, given that he or she has more than company, given that he or she has more than

10 years of service?10 years of service?

e.e.What is the probability of randomly selecting What is the probability of randomly selecting

an executive with more than 10 years of an executive with more than 10 years of

service or one who would not remain with the service or one who would not remain with the

company?company?

Joint and marginal probabilitiesJoint and marginal probabilities

In the context of bivariate probabilities, In the context of bivariate probabilities,

the intersection probabilities the intersection probabilities

are called joint probabilities and the are called joint probabilities and the

probabilities of individual events, P(Aprobabilities of individual events, P(Aii) )

or P(Bor P(Bii) are called marginal ) are called marginal

probabilities.probabilities.

)( ii BAP

P(A1 and B2) P(A1)

TotalEvent(A)

Joint Probability Using Contingency TableJoint Probability Using Contingency Table

P(A2 and B1)

P(A1 and B1)

Event (B)

Total 1

Joint ProbabilityJoint ProbabilityJoint ProbabilityJoint Probability Marginal (Simple) ProbabilityMarginal (Simple) Probability

A1

A2

B1 B2

P(B1) P(B2)

P(A2 and B2) P(A2)

A survey carried out for a super market shows the A survey carried out for a super market shows the

number of customers according to 2 characteristicsnumber of customers according to 2 characteristics

Frequency Frequency of visitof visit

Purchase of generic productsPurchase of generic products

Often Often Sometimes Sometimes Never Never

Frequent Frequent 1212 4848 1919

Infrequent Infrequent 77 66 88

• What is the probability that a customer is both a What is the probability that a customer is both a

frequent shopper and often purchases generic frequent shopper and often purchases generic

products?products?• What is the probability that a customer who never What is the probability that a customer who never

buys generic products visits the store frequently?buys generic products visits the store frequently?• Are the events “ never buys generic products” and Are the events “ never buys generic products” and

“visits the store frequently” independent?“visits the store frequently” independent?

• What is the probability that a customer who What is the probability that a customer who

infrequently visits the store often buys generic infrequently visits the store often buys generic

products?products?

• What is the probability that a customer frequently What is the probability that a customer frequently

visits the store?visits the store?

• What is the probability that the customer often What is the probability that the customer often

buys generic products?buys generic products?

• What is the probability that a customer either What is the probability that a customer either

frequently visits the store or never buys generic frequently visits the store or never buys generic

products or do both?products or do both?

Consumers are surveyed on the relative Consumers are surveyed on the relative

number of visits to a circuit city store and if number of visits to a circuit city store and if

the store was conveniently located. the store was conveniently located.

Visits Visits

Convenient Convenient

Yes Yes NoNo

Often Often

OccasionalOccasional

NeverNever

6060

2525

55

2020

3535

5050

EXERCISE

a.a.What is this table called?What is this table called?

b.b.Are visits and convenience Are visits and convenience

independent? Interpret your independent? Interpret your

conclusion.conclusion.

c.c.Draw a tree diagram and Draw a tree diagram and

determine the joint determine the joint

probabilities.probabilities.

Tree diagramA tree diagram is a diagram that branches out and ends in leaves that correspond to the final variety.                   

Often

Occasional

Never P(Y

|O)

P(N|O)

P(Y|O)

P(N|O)

P(Y|n)

P(N|n)

.410

.308

.282

.308

.103

.128

.179

.026

.256

.751

.416

.092

.042

.581

.908

Bayes’ TheoremBayes’ Theorem

)|()()|()(

)|()()|(

2211 ABPAPABPAP

ABPAPBAP ii

i

Prior probabilityPrior probability

The initial The initial probability based on probability based on the present level of the present level of information.information.

Posterior ProbabilityPosterior Probability

A revised probability A revised probability based on additional based on additional information.information.

Example Bayes’ theoremExample Bayes’ theorem

Suppose 5% of a population have a disease. There is Suppose 5% of a population have a disease. There is

a diagnostic technique to detect the disease , but it a diagnostic technique to detect the disease , but it

is not very accurate. Historical evidence shows that is not very accurate. Historical evidence shows that

if a person has actually if a person has actually has the diseasehas the disease, the , the

probability that the test will indicate the probability that the test will indicate the presence of presence of

the diseasethe disease is 0.9. Further the probability for a is 0.9. Further the probability for a

person who actually person who actually does not have the diseasedoes not have the disease but but

the test will indicate the the test will indicate the disease is presentdisease is present is 0.15. If is 0.15. If

a person is randomly selected and the test is a person is randomly selected and the test is

performed. The results indicate the performed. The results indicate the disease is disease is

presentpresent. What is the probability the person actually . What is the probability the person actually

has the diseasehas the disease ? ?

Solution Solution

Let,Let,

AA11 refer to the event ‘ refer to the event ‘ has the diseasehas the disease’. ’.

A2A2 refer to the event ‘ refer to the event ‘does not have the diseasedoes not have the disease’. ’.

BB denote the event “ denote the event “ test shows the disease is presenttest shows the disease is present”.”.

So P ( ASo P ( A11 ) = 0.05, ) = 0.05,

P (A2 ) = 0.95, P (A2 ) = 0.95,

P ( B I AP ( B I A11)= 0.90)= 0.90

and P( B I Aand P( B I A22) = 0.15) = 0.15

So P ( ASo P ( A11 I B ) = 0.24 I B ) = 0.24

Example Bayes’ theoremExample Bayes’ theorem

EventPrior prob.

P (Ai)

Conditional prob.

P ( B I Ai)

Joint prob.

P (Ai and B)

Posterior prob.

P (Ai I B)

Disease A1 .05 .90 .0450 .24

No Disease A2 .95 .15 .1425 .76

The credit department of Lion’s Department The credit department of Lion’s Department Store in Anaheim, California, reported that Store in Anaheim, California, reported that 30 percent of their sales are cash, 30 percent 30 percent of their sales are cash, 30 percent are paid for by check at the time of the are paid for by check at the time of the purchase, and 40 percent are charged. 20% purchase, and 40 percent are charged. 20% of the cash purchasers, 90% of the checks, of the cash purchasers, 90% of the checks, and 60% of the charges are for more than and 60% of the charges are for more than $50. Ms. Tina Stevens just purchased a new $50. Ms. Tina Stevens just purchased a new dress that cost $120. what is the probability dress that cost $120. what is the probability that she paid cash? that she paid cash?

Exercise

Assignment #1

Lind, Marchal and Wathen. Statistical Techniques in Business and Economics.

Page # 174

Problem # 67, 70, 71, 74, and 79

Random variableRandom variable

A variable that can take numerical A variable that can take numerical

values determined by the outcomes of values determined by the outcomes of

a random experiment.a random experiment.

Discrete r. v

A random variable that can A random variable that can

assume only certain clearly assume only certain clearly

separated values i.e. countable separated values i.e. countable

number of values.number of values.

e.g. number of runs of a cricketer in a match.e.g. number of runs of a cricketer in a match.

e.g. number of heads after tossing a coin 4 times.e.g. number of heads after tossing a coin 4 times.

Continuous r. vContinuous r. vA random variable that can A random variable that can

assume one of an infinitely assume one of an infinitely

large number of values, large number of values,

within certain limitations.within certain limitations.

e.g. height, weight, temperature etc.e.g. height, weight, temperature etc.

What is probability distribution?What is probability distribution?

A listing of all the outcomes of A listing of all the outcomes of

an experiment and the an experiment and the

probability associated with probability associated with

each outcome. each outcome.

What is probability distribution What is probability distribution function?function?

Let X is a discrete random variable Let X is a discrete random variable

and P(x) is called the probability and P(x) is called the probability

distribution function if and only ifdistribution function if and only if

1. 1.

2.2.

0)( xP

1)( xP

Mathematical Expectation Mathematical Expectation

Sum of the product of the values of Sum of the product of the values of

a discrete r.v. with the a discrete r.v. with the

corresponding probabilities.corresponding probabilities.

The supervisor of traffic signals must decide whether to The supervisor of traffic signals must decide whether to

install a traffic light at the reportedly dangerous install a traffic light at the reportedly dangerous

intersection of 2 roads. The policy is to install a traffic intersection of 2 roads. The policy is to install a traffic

light at an intersection at which the monthly expected light at an intersection at which the monthly expected

number of accidents is higher than 7. Should he number of accidents is higher than 7. Should he

recommend that traffic light be installed at this recommend that traffic light be installed at this

intersection?intersection?

Year Year

Number of accidentsNumber of accidents

J J F F M M AA

M M J J J J A A S S O O N N D D

20072007 1010 88 1010 66 99 1212 22 1010 1010 00 77 1010

20082008 1212 99 77 88 44 33 77 1414 88 88 88 44

Exercise

TheThe supervisorsupervisor

has has collectedcollecteddata on data on

Accidents Accidents at the intersection:at the intersection:

What is the average number of accidents? What is the average number of accidents? Standard deviation?Standard deviation?

Two types of losses are incurred in Two types of losses are incurred in business : business :

i)i) Obsolescence losses, Obsolescence losses, caused by caused by stocking too much stocking too much

ii)ii) Opportunity losses , Opportunity losses , caused by caused by

being out of stockbeing out of stock

Neither of the losses occurs when Neither of the losses occurs when the demand and supply are same.the demand and supply are same.

Combining probabilities and Monetary values Combining probabilities and Monetary values

Example Example

Levin and Rubin. Levin and Rubin. Statistics for Statistics for ManagementManagement. . Page # 233-235Page # 233-235

A fruit and vegetable wholesaler sells strawberries. A fruit and vegetable wholesaler sells strawberries.

This product has a very limited useful life. If This product has a very limited useful life. If not soldnot sold on on

the day of delivery it is the day of delivery it is worthlessworthless. One case of . One case of

strawberries strawberries cost $20cost $20, and the wholesaler , and the wholesaler receives $50receives $50

for it. The wholesaler can not specify the number of for it. The wholesaler can not specify the number of

cases customers will call for on any one day, but his cases customers will call for on any one day, but his

analysis of past records produced the following analysis of past records produced the following

information information

Sales Sales during during

100 100 daysdays

Daily Daily salessales

Number of Number of days solddays sold

1010

1111

1212

1313

1515

2020

4040

2525

i.i. Calculate probability of each Calculate probability of each number being sold.number being sold.

ii.ii. Find the conditional loss table.Find the conditional loss table.

iii.iii.Calculate expected lossesCalculate expected losses

iv.iv.Comment on the wholesaler actionComment on the wholesaler action

Mario, owner of Mario’s Pizza emporium, has a Mario, owner of Mario’s Pizza emporium, has a difficult decision on his hands. He has found that he difficult decision on his hands. He has found that he always sells between one and four of his famous always sells between one and four of his famous ‘everything but’ pizzas per night. These pizzas take ‘everything but’ pizzas per night. These pizzas take so long to prepare, however, that Mario prepares all so long to prepare, however, that Mario prepares all of them in advance and stores them in the of them in advance and stores them in the refrigerator. Because the ingredients go bad within refrigerator. Because the ingredients go bad within one day, Mario always throws out any unsold pizzas one day, Mario always throws out any unsold pizzas at the end of each evening. The cost of preparing at the end of each evening. The cost of preparing each pizza is $7, and Mario sells each one for $12. each pizza is $7, and Mario sells each one for $12. He has the following informationHe has the following information

Number of pizzas demanded Number of pizzas demanded 11 22 3344ProbabilityProbability .4.4 .3.3 .2.2 .1.1

1.1. Find the conditional loss table.Find the conditional loss table.2.2. How many “everything but” pizzas should Mario How many “everything but” pizzas should Mario

stock each night in order to minimize expected stock each night in order to minimize expected loss?loss?

Probability Probability DistributionDistribution

Binomial Binomial DistributionDistribution

We call a distribution a We call a distribution a binomial distribution binomial distribution if all of if all of

the following are true the following are true

1.1. There are a There are a fixed number of trialsfixed number of trials, n, which are all , n, which are all

independent.independent.

2.2. There must be There must be exactly two mutually exclusive exactly two mutually exclusive

outcomes in a trialoutcomes in a trial, such as True or False, yes or , such as True or False, yes or

no, success or failure.no, success or failure.

3.3. The The probability of successprobability of success is the is the samesame for each for each

trial. trial.

What is binomial distribution?What is binomial distribution?

The binomial distribution is used to obtain the The binomial distribution is used to obtain the

probability of observing probability of observing xx successes in successes in nn trials, with trials, with

the probability of success on a single trial denoted the probability of success on a single trial denoted

by by pp. .

The formula for the binomial probability mass The formula for the binomial probability mass

function is function is

                                                                                                             

where where

                                     

Probability mass functionProbability mass function

PP((SS) = ) = pp. . PP((FF) = ) = qq = 1- = 1-pp. . nn indicates the fixed number of trials. indicates the fixed number of trials. xx indicates the number of successes (any indicates the number of successes (any

whole number [0,whole number [0,nn]). ]). pp indicates the probability of success for any indicates the probability of success for any

one trial. one trial. qq indicates the probability of failure (not indicates the probability of failure (not

success) for any one trial. success) for any one trial. PP((xx) indicate the probability of getting exactly ) indicate the probability of getting exactly

xx successes in successes in nn trials. trials.

Some features of binomial distributionSome features of binomial distribution

Some statistics

Average = Mean = np

Standard Deviation = )1( pnp

The Bernoulli distribution is a special The Bernoulli distribution is a special

case of the binomial distribution, case of the binomial distribution,

where n=1. where n=1.

Bernoulli distributionBernoulli distribution

Suppose that each time you take a free throw shot, you have a 25% chance of making it.  If you take 15 shots, a. what is the probability of making exactly 5 of them.

SolutionSolution

We haveWe have                n  =  15,        r  =  5        p  = .25        q  =  .75n  =  15,        r  =  5        p  = .25        q  =  .75

P(5) =  0.165 P(5) =  0.165 

There is a 16.5 percent chance of making exactly 5 There is a 16.5 percent chance of making exactly 5 shots.shots.

Example

 

b.  What is the probability of making fewer than 3

shots?

 

Solution

The possible outcomes that will make this happen

are 2 shots, 1 shot, and 0 shots.  Since these are

mutually exclusive, we can add these probabilities.

        P(2)+P(1)+P(0)

        =  .156 + .067 + .013  =  0.236

There is a 24 percent chance of sinking fewer than

3 shots.

Shape of binomial distribution for different Shape of binomial distribution for different values of values of nn and and r.r.

•When p is small (<0.5) , the binomial distribution is When p is small (<0.5) , the binomial distribution is skewed to the right.skewed to the right.

•As p increases and approaches to 0.5, the As p increases and approaches to 0.5, the skewness is less noticeable.skewness is less noticeable.

•When p = 0.5, the binomial distribution is When p = 0.5, the binomial distribution is symmetrical.symmetrical.

•When p is larger than 0.5, the distribution is When p is larger than 0.5, the distribution is skewed to the left.skewed to the left.

•When n increases binomial distribution When n increases binomial distribution approaches to symmetrical.approaches to symmetrical.

Problems with binomial distributionProblems with binomial distribution

The requirement that the probability of the outcome The requirement that the probability of the outcome must be fixed overtime is very difficult to meet in must be fixed overtime is very difficult to meet in practice. practice. In many industrial processes, however, it is In many industrial processes, however, it is extremely difficult to guarantee that this is indeed the extremely difficult to guarantee that this is indeed the case.case.

Again, the outcome of one trial may affect in any way Again, the outcome of one trial may affect in any way the outcome of any other trial. the outcome of any other trial. Consider an Consider an interviewing process in which high-potential interviewing process in which high-potential candidates are being screened for top positions. If the candidates are being screened for top positions. If the interviewer has talked to five unacceptable candidates interviewer has talked to five unacceptable candidates in a row, he may not view the sixth with complete in a row, he may not view the sixth with complete partiality. Therefore, independence is violated.partiality. Therefore, independence is violated.

Hypergeometric DistributionHypergeometric Distribution

1.1. An outcome on each trial of an experiment is classified An outcome on each trial of an experiment is classified

into into one of two mutually exclusiveone of two mutually exclusive categories-a categories-a

success or a failure success or a failure

2.2. The random variable is the number of successes in a The random variable is the number of successes in a

fixed number of trialsfixed number of trials

3.3. The trials are The trials are not independentnot independent. .

4.4. We assume that we sample from a finite population We assume that we sample from a finite population

without replacement. So, the without replacement. So, the probability of a success probability of a success

change for each trialchange for each trial. .

P.M.F of the Hypergeometric Distribution

If If XX is the number of is the number of SS’s in a completely random ’s in a completely random sample of size sample of size nn drawn from a population consisting drawn from a population consisting of of MM SS’s and (’s and (NN––MM)) FF’s, then the probability ’s, then the probability distribution is hypergeometric distribution and is distribution is hypergeometric distribution and is given bygiven by : :

n

N

xn

MN

x

M

NMnxhxXP ),,;()(

Mean and VarianceThe mean and variance of a hypergeometric rv The mean and variance of a hypergeometric rv XX

having pmf having pmf hh((x;n,M,Nx;n,M,N) are:) are:

N

M

N

Mn

N

nNXV

N

MnXE 1

1)(,)(

ExampleIn a lot of 20 units out of the production line, 2 units In a lot of 20 units out of the production line, 2 units

are known to be defective. If the inspector picks a are known to be defective. If the inspector picks a

sample of 3 units at random, what is the distribution sample of 3 units at random, what is the distribution

of the number of defectives in the sample?of the number of defectives in the sample?

What is the probability of having 0, 1, 2 and 3 What is the probability of having 0, 1, 2 and 3

defectives in the sample?defectives in the sample?What is the probability that none of the chosen What is the probability that none of the chosen

sample are defective?sample are defective?