Probability (Part 1) Chapter55 Random Experiments Probability Rules of Probability Independent Events McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill

ProbabilityProbability(Part 1)(Part 1)

Chapter5555

Random Experiments

Probability

Rules of Probability

Independent Events

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc.

5A-2

• A A random experimentrandom experiment is an observational is an observational process whose results cannot be known in process whose results cannot be known in advance.advance.

• The set of all The set of all outcomesoutcomes ( (SS) is the ) is the sample sample spacespace for the experiment. for the experiment.

• A sample space with a countable number of A sample space with a countable number of outcomes is outcomes is discretediscrete..

Sample SpaceSample Space

Random ExperimentsRandom ExperimentsRandom ExperimentsRandom Experiments

5A-3

• For example, when CitiBank makes a For example, when CitiBank makes a consumer loan, the sample space is:consumer loan, the sample space is:

SS = {default, no default} = {default, no default}

• The sample space describing a Wal-Mart The sample space describing a Wal-Mart customer’s payment method is:customer’s payment method is:

SS = {cash, debit card, credit card, check} = {cash, debit card, credit card, check}



5A-4

• For a single roll of a die, the sample space is: S = {1, 2, 3, 4, 5, 6}

• When two dice are rolled, the sample space is the following pairs:



{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

SS = =

5A-5

Consider the sample space to describe a randomly chosen United Airlines employee by:

2 genders, 21 job classifications, 6 home bases (major hubs) and 4 education levels

It would be impractical to enumerate this It would be impractical to enumerate this sample space.sample space.

There are: 2 x 21 x 6 x 4 = 1008 possible outcomes



5A-6

• If the outcome is a If the outcome is a continuouscontinuous measurement, the measurement, the sample space can be described by a rule. sample space can be described by a rule.

• For example, the sample space for the length of a For example, the sample space for the length of a randomly chosen cell phone call would berandomly chosen cell phone call would be

S = {all X such that X > 0}

• The sample space to describe a randomly chosen The sample space to describe a randomly chosen student’s GPA would bestudent’s GPA would be

S = {X | 0.00 < X < 4.00}

or written as or written as SS = { = {XX | | X X >> 0} 0}



5A-7

• An An eventevent is any subset of outcomes in the is any subset of outcomes in the sample space.sample space.

• A A simple eventsimple event or or elementary eventelementary event, is a , is a single outcome.single outcome.

• A discrete sample space A discrete sample space SS consists of all the consists of all the simple events (simple events (EEii):):

S = {E1, E2, …, En}

EventsEvents


5A-8

• What are the chances of observing a H or T?

• These two elementary events are equally likely.

S = {H, T}

• Consider the random experiment of tossing a balanced coin.

What is the sample space?

• When you buy a lottery ticket, the sample space S = {win, lose} has only two events.

EventsEvents


Are these two events equally likely to occur?Are these two events equally likely to occur?

5A-9

• For example, in a sample space of 6 simple For example, in a sample space of 6 simple events, we could define the compound events, we could define the compound eventsevents

• A A compound eventcompound event consists of two or more consists of two or more simple events. simple events.

A = {Music, DVD, VH}B = {Newspapers, Magazines}

Events Events (Figure 5.1)


These are displayed in a Venn diagram:

5A-10

• Many different compound events could be defined.

• Compound events can be described by a rule.

S = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}

• For example, the compound event A = “rolling a seven” on a roll of two dice consists of 6 simple events:

EventsEvents


5A-11

• The The probabilityprobability of an event is a number that of an event is a number that measures the relative likelihood that the measures the relative likelihood that the event will occur.event will occur.

• The probability of event The probability of event A A [denoted [denoted PP((AA)], )], must lie within the interval from 0 to 1:must lie within the interval from 0 to 1:

0 0 << PP((AA) ) << 1 1

If P(A) = 0, then the event cannot occur.

If P(A) = 1, then the event is certain to occur.

DefinitionsDefinitions

ProbabilityProbabilityProbabilityProbability

5A-12

• In a discrete sample space, the probabilities In a discrete sample space, the probabilities of all simple events must sum to unity:of all simple events must sum to unity:

• For example, if the following number of For example, if the following number of purchases were made bypurchases were made by

P(S) = P(E1) + P(E2) + … + P(En) = 1

credit card: 32%

debit card: 20%

cash: 35%

check: 18%

Sum = 100%

DefinitionsDefinitions


P(credit card) = .32

P(debit card) = .20

P(cash) = .35

P(check) = .18

Sum = 1.0

ProbabilityProbability

5A-13

• Businesses want to be able to quantify the uncertaintyuncertainty of future events.

• For example, what are the chances that next month’s revenue will exceed last year’s average?

• How can we increase the chance of positive future events and decrease the chance of negative future events?

• The study of probabilityprobability helps us understand and quantify the uncertainty surrounding the future.


5A-14


5A-15

Three approaches to probability:Three approaches to probability:

ApproachApproach ExampleExample

EmpiricalEmpirical There is a 2 percent chance of twins in a randomly-chosen birth.

What is Probability?What is Probability?


ClassicalClassical There is a 50 % probability of heads on a coin flip.

SubjectiveSubjective There is a 75 % chance that England will adopt the Euro currency by 2010.

5A-16

• Use the Use the empiricalempirical or or relative frequencyrelative frequency approach to assign probabilities by counting approach to assign probabilities by counting the frequency (the frequency (ffii) of observed outcomes ) of observed outcomes

defined on the experimental sample space.defined on the experimental sample space.• For example, to estimate the default rate on For example, to estimate the default rate on

student loans:student loans:

P(a student defaults) = f /n

Empirical ApproachEmpirical Approach


number of defaultsnumber of loans

=

5A-17

• Necessary when there is no prior knowledge Necessary when there is no prior knowledge of events.of events.

• As the number of observations (As the number of observations (nn) increases ) increases or the number of times the experiment is or the number of times the experiment is performed, the estimate will become more performed, the estimate will become more accurate.accurate.

Empirical ApproachEmpirical Approach


5A-18

• The law of large numberslaw of large numbers is an important probability theorem that states that a large sample is preferred to a small one.

• Flip a coin 50 times. We would expect the proportion of heads to be near .50.

• A large n may be needed to get close to .50.

• However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.).

Law of Large NumbersLaw of Large Numbers


• Consider the results of 10, 20, 50, and 500 coin flips.

5A-19

ProbabilityProbability(Figure 5.2)

ProbabilityProbability(Figure 5.2)

5A-20

• Actuarial scienceActuarial science is a high-paying career is a high-paying career that involves estimating empirical that involves estimating empirical probabilities.probabilities.

• For example, actuaries For example, actuaries - calculate payout rates on life insurance,- calculate payout rates on life insurance, pension plans, and health care plans pension plans, and health care plans- create tables that guide IRA withdrawal- create tables that guide IRA withdrawal rates for individuals from age 70 to 99 rates for individuals from age 70 to 99

Practical Issues for ActuariesPractical Issues for Actuaries


5A-21

Challenges that actuaries face:Challenges that actuaries face:

- Is n “large enough” to say that f/n has become a good approximation to P(A)?

- Was the experiment repeated identically?

- Is the underlying process invariant over time?

- Do nonstatistical factors override data collection?

- What if repeated trials are impossible?

Practical Issues for ActuariesPractical Issues for Actuaries


5A-22

• In this approach, we envision the entire sample space as a collection of equally likely outcomes.

• Instead of performing the experiment, we can use deduction to determine P(A).

• a prioria priori refers to the process of assigning probabilities before the event is observed.

• a priori probabilitiesa priori probabilities are based on logic, not experience.

Classical ApproachClassical Approach


5A-23

• For example, the two dice experiment has 36 equally likely simple events. The P(7) is

• The probability is obtained a priori using the classical approach as shown in this Venn diagram for 2 dice:

number of outcomes with 7 dots 6( ) 0.1667

number of outcomes in sample space 36P A

Classical ApproachClassical Approach


5A-24

• A subjectivesubjective probability reflects someone’s personal belief about the likelihood of an event.

• Used when there is no repeatable random experiment.

• For example,- What is the probability that a new truck product program will show a return on investment of at least 10 percent?- What is the probability that the price of GM stock will rise within the next 30 days?

Subjective ApproachSubjective Approach


5A-25

• These probabilities rely on personal These probabilities rely on personal judgment or expert opinion.judgment or expert opinion.

• Judgment is based on experience with Judgment is based on experience with similar events and knowledge of the similar events and knowledge of the underlying causal processes.underlying causal processes.

Subjective ApproachSubjective Approach


5A-26

• The The complementcomplement of an event of an event AA is denoted by is denoted by AA′′ and consists of everything in the sample and consists of everything in the sample space space SS except event except event AA..

Complement of an EventComplement of an Event

Rules of ProbabilityRules of ProbabilityRules of ProbabilityRules of Probability

5A-27

• Since A and A′ together comprise the entire sample space,

P(A) + P(A′ ) = 1

• The probability of A′ is found by P(A′ ) = 1 – P(A)

• For example, The Wall Street Journal reports that about 33% of all new small businesses fail within the first 2 years. The probability that a new small business will survive is:

PP(survival) = 1 – (survival) = 1 – PP(failure) = 1 – .33 = .67 or 67% (failure) = 1 – .33 = .67 or 67%

Complement of an EventComplement of an Event


5A-28

• The The oddsodds in favor in favor of event of event AA occurring is occurring is

• The The oddsodds againstagainst event event AA occurring is occurring is

( ) ( )Odds =

( ') 1 ( )

P A P A

P A P A

Odds of an EventOdds of an Event


)(

)(1

)(

)( Odds

AP

AP

AP

AP

5A-29

• Odds are used in sports and games of Odds are used in sports and games of chance.chance.

• For a pair of fair dice, For a pair of fair dice, PP(7) = 6/36 (or 1/6). (7) = 6/36 (or 1/6). What are the odds in favor of rolling a 7?What are the odds in favor of rolling a 7?

(rolling seven) 1/ 6 1/ 6 1Odds =

1 (rolling seven) 1 1/ 6 5/ 6 5

P

P



5A-30

• On the average, for every time a 7 is rolled, On the average, for every time a 7 is rolled, there will be 5 times that it is not rolled.there will be 5 times that it is not rolled.

• In other words, the odds are 1 to 5 In other words, the odds are 1 to 5 in favorin favor of of rolling a 7.rolling a 7.

• The odds are 5 to 1 The odds are 5 to 1 againstagainst rolling a 7. rolling a 7.



• In horse racing and other sports, odds are In horse racing and other sports, odds are usually quoted usually quoted againstagainst winning. winning.

5A-31

• If the odds against event If the odds against event AA are quoted as are quoted as bb to to aa, then the implied probability of event , then the implied probability of event AA is:is:

• For example, if a race horse has a 4 to 1 For example, if a race horse has a 4 to 1 odds odds againstagainst winning, the winning, the PP(win) is (win) is

P(A) =a

a b



1 10.20

4 1 5

a

a b

P(win) = or 20%

5A-32

• The unionunion of two events consists of all outcomes in the sample space S that are contained either in event A or in event B or both (denoted A B or “A or B”).

may be read as “or” since one or the other or both events may occur.

Union of Two EventsUnion of Two Events(Figure 5.5)


• For example, randomly choose a card from a deck of For example, randomly choose a card from a deck of 52 playing cards. 52 playing cards.

• It is the possibility of drawing It is the possibility of drawing eithereither a queen (4 ways) a queen (4 ways) oror a red card (26 ways) a red card (26 ways) oror both (2 ways). both (2 ways).

• If If QQ is the event that we draw a is the event that we draw a queen and queen and RR is the event that we is the event that we draw a red card, what is draw a red card, what is Q Q RR??

Union of Two EventsUnion of Two Events


5A-33

5A-34

• The The intersectionintersection of two events of two events AA and and BB (denoted (denoted A A BB or “ or “AA and and BB”) is the event ”) is the event consisting of all outcomes in the sample consisting of all outcomes in the sample space space SS that are contained in that are contained in bothboth event event AA and event and event BB. .

may be read as “and” since both events occur. This is a joint probability.

Intersection of Two EventsIntersection of Two Events


(Figure 5.6)

• It is the possibility of getting It is the possibility of getting bothboth a queen a queen andand a red card a red card (2 ways).(2 ways).

• If If QQ is the event that we draw a is the event that we draw a queen and queen and RR is the event that we is the event that we draw a red card, what is draw a red card, what is Q Q RR??

• For example, randomly choose a card from a For example, randomly choose a card from a deck of 52 playing cards. deck of 52 playing cards.

Intersection of Two EventsIntersection of Two Events


5A-35

5A-36

• The The general law of additiongeneral law of addition states that the states that the probability of the union of two events probability of the union of two events AA and and B B is:is:

P(A B) = P(A) + P(B) – P(A B)

When you add the P(A) and P(B) together, you count the P(A and B) twice.

So, you have to subtract P(A B) to avoid over-stating the probability.

A B

A and B

General Law of AdditionGeneral Law of Addition


5A-37

• For the card example:For the card example:

P(Q) = 4/52 (4 queens in a deck)

= 4/52 + 26/52 – 2/52

P(Q R) = P(Q) + P(R) – P(Q Q)

Q4/52

R26/52

Q and R = 2/52

General Law of AdditionGeneral Law of Addition


= 28/52 = .5385 or 53.85%

P(R) = 26/52 (26 red cards in a deck)

P(Q R) = 2/52 (2 red queens in a deck)

5A-38

• Events Events AA and and BB are are mutually exclusivemutually exclusive (or (or disjointdisjoint) if ) if their intersection is the null set (their intersection is the null set () that contains no ) that contains no elements.elements. If A B = , then P(A B) = 0

• In the case of mutually In the case of mutually exclusive events, the exclusive events, the addition law reduces to:addition law reduces to:

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

Mutually Exclusive EventsMutually Exclusive Events


Special Law of AdditionSpecial Law of Addition

5A-39

• Events are Events are collectively exhaustivecollectively exhaustive if their union is if their union is the entire sample space the entire sample space SS..

• Two mutually exclusive, collectively exhaustive Two mutually exclusive, collectively exhaustive events are events are dichotomousdichotomous (or (or binarybinary) ) eventsevents..

For example, a car repair is either covered by the warranty (A) or not (B).

WarrantyWarrantyNoNo

WarrantyWarranty

Collectively Exhaustive EventsCollectively Exhaustive Events


5A-40

• More than two mutually exclusive, collectively exhaustive events are polytomous eventspolytomous events.

For example, a Wal-Mart customer can pay by credit For example, a Wal-Mart customer can pay by credit card (card (AA), debit card (), debit card (BB), cash (C) or check (), cash (C) or check (DD).).

CreditCreditCardCard

DebitDebitCardCard

CashCash

CheckCheck

Collectively Exhaustive EventsCollectively Exhaustive Events

RulesRules of Probability of ProbabilityRulesRules of Probability of Probability

5A-41

• Categorical data can be made dichotomous Categorical data can be made dichotomous (binary) by defining the second category as (binary) by defining the second category as everything everything notnot in the first category. in the first category.

Categorical Data Categorical Data Binary Binary ((Dichotomous) VariableDichotomous) Variable

Vehicle type (SUV, sedan, truck, motorcycle)

X = 1 if SUV, 0 otherwise

Categorical DataCategorical Data


A randomly-chosen NBA player’s height

X = 1 if height exceeds 7 feet, 0 otherwise

Tax return type (single, married filing jointly, married filing separately, head of household, qualifying widower)

X = 1 if single, 0 otherwise

5A-42

• The probability of event The probability of event AA givengiven that event that event BB has occurred.has occurred.

• Denoted Denoted PP((A A | | BB). ). The vertical line “ | ” is read as “given.”The vertical line “ | ” is read as “given.”

( )( | )

( )

P A BP A B

P B

for P(B) > 0 and undefined

otherwise

Conditional ProbabilityConditional Probability


• Consider the logic of this formula by looking at the Venn diagram.

( )( | )

( )

P A BP A B

P B

The sample space is restricted to B, an event that has occurred.

A B is the part of B that is also in A.

The ratio of the relative size of A B to B is P(A | B).

Conditional ProbabilityConditional Probability


5A-43

5A-44

• Of the population aged 16 – 21 and not in Of the population aged 16 – 21 and not in college:college:

Unemployed 13.5%

High school dropouts 29.05%

Unemployed high school dropouts 5.32%

• What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout?

Example: High School DropoutsExample: High School Dropouts


5A-45

• First defineFirst defineU = the event that the person is unemployedD = the event that the person is a high

school dropoutP(U) = .1350 P(D) = .2905 P(UD) = .0532

( ) .0532( | ) .1831

( ) .2905

P U DP U D

P D

• PP((U | DU | D) = .1831 > ) = .1831 > PP((UU) = .1350) = .1350

• Therefore, being a high school dropout is Therefore, being a high school dropout is related to being unemployed.related to being unemployed.

Example: High School DropoutsExample: High School Dropouts


5A-46

• Event Event A A is is independent independent of event of event BB if the if the conditional probability conditional probability PP((AA | | BB) is the same ) is the same as the marginal probability as the marginal probability PP((AA).).

• To check for independence, apply this test:To check for independence, apply this test:

If P(A | B) = P(A) then event A is independent of B.

• Another way to check for independence:Another way to check for independence:

If If PP((A A B B) = ) = PP((AA))PP((BB) then event ) then event AA is is independentindependent of event of event BB since sincePP((A | BA | B) = ) = PP((A A BB)) = = PP((AA))PP((BB)) = = PP((AA)) PP((BB) ) PP((BB))

Independent EventsIndependent EventsIndependent EventsIndependent Events

5A-47

• Out of a target audience of 2,000,000, ad Out of a target audience of 2,000,000, ad A A reaches 500,000 viewers, reaches 500,000 viewers, BB reaches 300,000 reaches 300,000 viewers and both ads reach 100,000 viewers.viewers and both ads reach 100,000 viewers.

What is P(A | B)?

500,000( ) .25

2,000,000P A

300,000( ) .15

2,000,000P B

100,000( ) .05

2,000,000P A B


Example: Television AdsExample: Television Ads

( ) .05( | ) .30

( ) .15

P A BP A B

P B

.3333 or 33%

5A-48

• So, So, PP(ad (ad AA) = .25) = .25 PP(ad (ad BB) = .15) = .15 PP((AA BB) = .05) = .05 PP((AA | | BB) = .3333 ) = .3333

• PP((AA | | BB) = .3333 ≠ ) = .3333 ≠ PP((AA) = .25) = .25

• PP((AA))PP((BB)=(.25)(.15)=.0375 ≠ )=(.25)(.15)=.0375 ≠ PP((AA BB)=.05 )=.05

• Are events Are events AA and and BB independent? independent?


Example: Television AdsExample: Television Ads

5A-49

• When When PP((AA) ≠ ) ≠ PP((AA | | BB), then events ), then events AA and and BB are are dependentdependent..

• For dependent events, knowing that event For dependent events, knowing that event BB has has occurred will affect the occurred will affect the probabilityprobability that event that event AA will will occur.occur.

• For example, knowing a person’s age would affect For example, knowing a person’s age would affect the the probabilityprobability that the individual uses text that the individual uses text messaging but causation would have to be proven messaging but causation would have to be proven in other ways.in other ways.


Dependent EventsDependent Events

• Statistical dependence does Statistical dependence does not not prove causality.prove causality.

5A-50

• An An actuaryactuary studies conditional probabilities studies conditional probabilities empirically, using empirically, using - accident statistics - accident statistics - mortality tables - mortality tables - insurance claims records- insurance claims records

• Many businesses rely on actuarial services, Many businesses rely on actuarial services, so a business student needs to understand so a business student needs to understand the concepts of conditional probability and the concepts of conditional probability and statistical independence.statistical independence.


Using Actuarial DataUsing Actuarial Data

5A-51

• The probability of n independent events occurring simultaneously is:

• To illustrate system reliability, suppose a Web site has 2 independent file servers. Each server has 99% reliability. What is the total system reliability? Let,

P(A1 A2 ... An) = P(A1) P(A2) ... P(An)if the events are independent

F1 be the event that server 1 failsF2 be the event that server 2 fails


Multiplication Law for Independent EventsMultiplication Law for Independent Events

5A-52

• Applying the rule of independence:Applying the rule of independence:

• The probability that at least one server is The probability that at least one server is “up” is: “up” is:

P(F1 F2 ) = P(F1) P(F2) = (.01)(.01) = .0001

1 - .0001 = .9999 or 99.99%

• So, the probability that both servers are So, the probability that both servers are down is .0001.down is .0001.


Multiplication Law for Independent EventsMultiplication Law for Independent Events

5A-53

• RedundancyRedundancy can increase system reliability can increase system reliability even when individual component reliability is even when individual component reliability is low.low.

• NASA space shuttle has three independent NASA space shuttle has three independent flight computers (triple redundancy). flight computers (triple redundancy).

• Each has an unacceptable .03 chance of Each has an unacceptable .03 chance of failure (3 failures in 100 missions).failure (3 failures in 100 missions).

• Let Let FFjj = event that computer = event that computer jj fails. fails.


Example: Space ShuttleExample: Space Shuttle

5A-54

• What is the probability that all three flight What is the probability that all three flight computers will fail?computers will fail?

P(all 3 fail) = P(F1 F2 F3)

= 0.000027 or 27 in 1,000,000 missions.

= P(F1) P(F2) P(F3) presuming that failures are independent

= (0.03)(0.03)(0.03)


Example: Space ShuttleExample: Space Shuttle

5A-55

• How high must reliability be?How high must reliability be?

• Public carrier-class telecommunications data Public carrier-class telecommunications data links are expected to be available 99.999% of links are expected to be available 99.999% of the time.the time.

• The The five nines rulefive nines rule implies only 5 minutes of implies only 5 minutes of downtime per year.downtime per year.

• This type of reliability is needed in many This type of reliability is needed in many business situations.business situations.


The Five Nines RuleThe Five Nines Rule

5A-56

For example,For example,


The Five Nines RuleThe Five Nines Rule

5A-57

• Suppose a certain network Web server is up Suppose a certain network Web server is up only 94% of the time. What is the probability only 94% of the time. What is the probability of it being down?of it being down?

• How many independent servers are needed How many independent servers are needed to ensure that the system is up at least to ensure that the system is up at least 99.99% of the time (or down only 99.99% of the time (or down only 1 - .9999 = .0001 or .01% of the time)?1 - .9999 = .0001 or .01% of the time)?

P(down) = 1 – P(up) = 1 – .94 = .06


How Much Redundancy is Needed?How Much Redundancy is Needed?

5A-58

• So, to achieve a 99.99% up time, 4 redundant So, to achieve a 99.99% up time, 4 redundant servers will be needed.servers will be needed.

2 servers: P(F1 F2) = (0.06)(0.06) = 0.0036

3 servers: P(F1 F2 F3) = (0.06)(0.06)(0.06) = 0.000216

4 servers: P(F1 F2 F3 F4) = (0.06)(0.06)(0.06)(0.06) =0.00001296


How Much Redundancy is Needed?How Much Redundancy is Needed?

Applied Statistics in Applied Statistics in Business & EconomicsBusiness & Economics

End of Chapter 5AEnd of Chapter 5A

5A-59

Documents

Probability (Part 1) Chapter55 Random Experiments Probability Rules of Probability Independent Events McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill