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    Probability (Part 1)

    Random ExperimentsProbability

    Rules of Probability

    Independent Events

    Chapter

    5

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    A random experimentis an observational process

    whose results cannot be known in advance.

    The set of all outcomes (S) is the sample space for

    the experiment.

    A sample space with a countable number of

    outcomes is discrete.

    Sample Space

    Random Experiments

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    For example, when CitiBank makes a consumer

    loan, the sample space is:

    S = {default, no default}

    The sample space describing a Wal-Mart

    customers payment method is:

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

    Sample Space

    Random Experiments

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    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 isthe following pairs:

    Sample Space

    Random Experiments

    {(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)}

    S =

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    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 sample

    space.

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

    Sample Space

    Random Experiments

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    If the outcome is a continuous measurement, the

    sample space can be described by a rule.

    For example, the sample space for the length of arandomly chosen cell phone call would be

    S = {allXsuch thatX> 0}

    The sample space to describe a randomly chosen

    students GPA would be

    S = {X| 0.00 0}

    Sample Space

    Random Experiments

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    An eventis any subset of outcomes in the sample

    space.

    A simple eventorelementary event, is a singleoutcome.

    A discrete sample space S consists of all the

    simple events (Ei):

    S = {E1, E2, , En}

    Events

    Random Experiments

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    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.

    Events

    Random Experiments

    Are these two events equally likely to occur?

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    For example, in a sample space of 6 simpleevents, we could define the compound events

    A compound eventconsists of two or more simple

    events.

    These are

    displayed in a

    Venn diagram:

    A = {E1, E2}

    B = {E3, E5, E6}

    Events

    Random Experiments

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    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:

    Events

    Random Experiments

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    Theprobabilityof an event is a number that

    measures the relative likelihood that the event will

    occur. The probability of eventA [denoted P(A)], must lie

    within the interval from 0 to 1:

    0 < P(A) < 1

    IfP(A) = 0, then the

    event cannot occur.

    IfP(A) = 1, then the event

    is certain to occur.

    Defini t ions

    Probability

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    In a discrete sample space, the probabilities of all

    simple events must sum to unity:

    For example, if the following number of purchases

    were made by

    P(S) = P(E1) + P(E2) + + P(En) = 1

    credit card: 32%

    debit card: 20%

    cash: 35%

    check: 18%

    Sum = 100%

    Defini t ions

    Probability

    P(credit card) = .32

    P(debit card) = .20

    P(cash) = .35

    P(check) = .18

    Sum = 1 0

    Probability

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    Businesses want to be able to quantify theuncertaintyof future events.

    For example, what are the chances that next

    months revenue will exceed last years average?

    How can we increase the chance of positive future

    events and decrease the chance of negative future

    events? The study ofprobabilityhelps us understand and

    quantify the uncertainty surrounding the future.

    Probability

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    Three approaches to probability:

    Approach Example

    Empirical There is a 2 percent chance

    of twins in a randomly-

    chosen birth.

    What is Probabi li ty?

    Probability

    Classical There is a 50 % probability

    of heads on a coin flip.

    Subjective There is a 75 % chance that England will

    adopt the Euro currency by 2010.

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    Use the empiricalorrelative frequencyapproach

    to assign probabilities by counting the frequency

    (fi) of observed outcomes defined on theexperimental sample space.

    For example, to estimate the default rate on

    student loans:

    P(a student defaults) = f/n

    Emp irical Approach

    Probability

    number of defaults

    number of loans=

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    Necessary when there is no prior knowledge of

    events.

    As the number of observations (n) increases or the

    number of times the experiment is performed, the

    estimate will become more accurate.

    Emp irical Approach

    Probability

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    The law 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 o f Large Numbers

    Probability

    Consider the results of 10, 20, 50, and 500 coin

    flips

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    Probability

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    Actuarial science is a high-paying career that

    involves estimating empirical probabilities.

    For example, actuaries

    - calculate payout rates on life insurance,

    pension plans, and health care plans

    - create tables that guide IRA withdrawalrates for individuals from age 70 to 99

    Pract ical Issues for Actuaries

    Probability

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    Challenges that actuaries face:

    - Is nlarge enough to say thatf/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 datacollection?

    - What if repeated trials are impossible?

    Pract ical Issues for Actuaries

    Probability

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    In this approach, we envision the entire sample

    space as a collection of equally likely outcomes.

    Instead of performing the experiment, we can usededuction to determine P(A).

    a priorirefers to the process of assigning

    probabilities before the event is observed.

    a priori probabilities are based on logic, not

    experience.

    Class ical Approach

    Probability

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    For example, the two dice experiment has 36

    equally likely simple events. The P(7) is

    The probability is

    obtained a priori

    using the classicalapproach 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

    Class ical Approach

    Probability

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    A subjective probability reflects someones

    personal belief about the likelihood of an event.

    Used when there is no repeatable randomexperiment.

    For example,

    - What is the probability that a new truck

    product program will show a return oninvestment of at least 10 percent?

    - What is the probability that the price of GM

    stock will rise within the next 30 days?

    Subject ive App roach

    Probability

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    These probabilities rely on personal judgment or

    expert opinion.

    Judgment is based on experience with similar

    events and knowledge of the underlying causal

    processes.

    Subject ive App roach

    Probability

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    The complementof an eventA is denoted by

    A and consists of everything in the sample space

    S except eventA.

    Complement o f an Event

    Rules of Probability

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    SinceA andA together comprise the entire

    sample space,

    P(A) + P(A ) = 1

    The probability ofA is found by

    P(A ) = 1P(A)

    For example, The Wall Street Journalreports that

    about 33% of all new small businesses fail withinthe first 2 years. The probability that a new small

    business will survive is:

    P(survival) = 1P(failure) = 1 .33 = .67 or 67%

    Complement o f an Event

    Rules of Probability

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    The odds in favor of eventA occurring is

    Odds are used in sports and games of chance.

    For a pair of fair dice, P(7) = 6/36 (or 1/6).

    What are the odds in favor of rolling a 7?

    ( ) ( )Odds =

    ( ') 1 ( )

    P A P A

    P A P A

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

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

    P

    P

    Odds of an Event

    Rules of Probability

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    On the average, for every time a 7 is rolled, there

    will be 5 times that it is not rolled.

    In other words, the odds are 1 to 5 in favorofrolling a 7.

    The odds are 5 to 1 againstrolling a 7.

    Odds of an Event

    Rules of Probability

    In horse racing and other sports, odds are usuallyquoted againstwinning.

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    If the odds against eventA are quoted as b to a,

    then the implied probability of eventA is:

    For example, if a race horse has a 4 to 1 odds

    againstwinning, the P(win) is

    P(A) =a

    a b

    Odds of an Event

    Rules of Probability

    1 10.20

    4 1 5

    a

    a b

    P(win) = or 20%

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    The union of two events consists of all outcomes in

    the sample space S that are contained either in

    eventA or in event B or both

    (denotedA Bor A orB).

    may be readas or since

    one orthe other

    orboth events

    may occur.

    Union of Two Events

    Rules of Probability

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    For example, randomly choose a card from a deck

    of 52 playing cards.

    It is the possibility of drawing

    eithera queen (4 ways)

    ora red card (26 ways)

    orboth (2 ways).

    IfQ is the event that we draw aqueen and Ris the event that we

    draw a red card, what is Q R?

    Union of Two Events

    Rules of Probability

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    The intersection of two eventsA and B

    (denotedA Bor A and B) is the eventconsisting of all outcomes in the sample space S

    that are contained in both eventA and event B.

    may be readas and since

    both eventsoccur. This is a

    joint probability.

    Intersect ion o f Two Events

    Rules of Probability

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    It is the possibility of gettingboth a queen anda red card

    (2 ways).

    IfQ is the event that we draw aqueen and Ris the event that we

    draw a red card, what is

    Q R?

    For example, randomly choose a card from a deck

    of 52 playing cards.

    Intersect ion o f Two Events

    Rules of Probability

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    The general law of addition states that the

    probability of the union of two eventsA and B is:

    P(A B) = P(A) + P(B)P(A B)When you add

    the P(A) and

    P(B) together,

    you count theP(A and B)

    twice.

    So, you have

    to subtract

    P(A B) toavoid over-stating the

    probability.

    A B

    A and B

    General Law of Addi t ion

    Rules of Probability

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    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)

    Q

    4/52

    R

    26/52

    Q and R= 2/52

    General Law of Addi t ion

    Rules of Probability

    = 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)

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    EventsA and B are mutually exclusive (ordisjoint)

    if their intersection is the null set () that contains

    no elements. IfA B = , then P(A B) = 0

    In the case of mutually

    exclusive events, the

    addition law reducesto:

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

    Mutually Exclus ive Events

    Rules of Probability

    Special Law of Addi t ion

    l f b b l

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    Events are collectively exhaustive if their union is

    the entire sample space S.

    Two mutually exclusive, collectively exhaustive

    events are dichotomous (orbinary) events.

    For example, a car repair

    is either covered by thewarranty (A) or not (B).

    WarrantyNo

    Warranty

    Col lect ively Exhaus t ive Events

    Rules of Probability

    l f b b l

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    More than two mutually exclusive, collectively

    exhaustive events arepolytomous events.

    For example, a Wal-Mart customer can pay by credit

    card (A), debit card (B), cash (C) or check (D).

    Credit

    Card

    DebitCard

    Cash

    Check

    Col lect ively Exhaus t ive Events

    Rules of Probability

    R l f P b bili

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    Polytomous events can be made dichotomous

    (binary) by defining the second category as

    everything notin the first category.

    Polytomous Events Binary(Dichotomous) Variable

    Vehicle type (SUV, sedan, truck,

    motorcycle)

    X= 1 if SUV, 0 otherwise

    Forced Dicho tomy

    Rules of Probability

    A randomly-chosen NBA players

    height

    X= 1 if height exceeds 7 feet, 0

    otherwiseTax return type (single, married filing

    jointly, married filing separately, head

    of household, qualifying widower)

    X= 1 if single, 0 otherwise

    R l f P b bili

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    The probability of eventAgiven that event B has

    occurred.

    Denoted P(A | B).The vertical line | is read as given.

    ( )

    ( | ) ( )

    P A BP A B

    P B

    forP(B) > 0 and

    undefined otherwise

    Condit ional Probabi l i ty

    Rules of Probability

    R l f P b bilit

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    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 ofBthat is also inA.

    The ratio of the relative

    size ofA B to B isP(A | B).

    Condit ional Probabi l i ty

    Rules of Probability

    R l f P b bilit

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    Of the population aged 16 21 and not in college:

    Unemployed 13.5%

    High school dropouts 29.05%

    Unemployed high school dropouts 5.32%

    What is the conditional probability that a memberof this population is unemployed, given that the

    person is a high school dropout?

    Example: High School Dropouts

    Rules of Probability

    R l f P b bilit

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    First define

    U= the event that the person is unemployed

    D = the event that the person is a high school

    dropout

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

    ( ) .0532( | ) .1831

    ( ) .2905

    P U DP U D

    P D

    or 18.31%

    P(U | D) = .1831 > P(U) = .1350

    Therefore, being a high school dropout is related

    to being unemployed.

    Example: High School Dropouts

    Rules of Probability

    I d d t E t

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    EventA is independentof event B if the conditionalprobability P(A | B) is the same as the marginal

    probability P(A).

    To check for independence, apply this test:

    IfP(A | B) = P(A) then eventA is independentofB.

    Another way to check for independence:

    IfP(A B) = P(A)P(B) then eventA isindependentof event B since

    P(A | B) = P(A B) = P(A)P(B) = P(A)P(B) P(B)

    Independent Events

    I d d t E t

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    Out of a target audience of 2,000,000, adA

    reaches 500,000 viewers, B reaches 300,000

    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

    Independent Events

    Example: Televis ion Ads

    ( ) .05( | ) .30

    ( ) .15

    P A BP A B

    P B

    .3333 or 33%

    Independent Events

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    So, P(adA) = .25

    P(ad B) = .15

    P(AB) = .05P(A | B) = .3333

    P(A | B) = .3333 P(A) = .25

    P(A)P(B)=(.25)(.15)=.0375 P(AB)=.05

    Are eventsA and B independent?

    Independent Events

    Example: Televis ion Ads

    Independent Events

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    When P(A) P(A | B), then eventsA and B are

    dependent.

    For dependent events, knowing that event B has

    occurred will affect theprobabilitythat eventA willoccur.

    For example, knowing a persons age would affecttheprobabilitythat the individual uses text

    messaging but causation would have to be proven

    in other ways.

    Independent Events

    Dependent Events

    Statistical dependence does not prove causality.

    Independent Events

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    An actuarystudies conditional probabilities

    empirically, using

    - accident statistics

    - mortality tables

    - insurance claims records

    Many businesses rely on actuarial services, so a

    business student needs to understand the

    concepts of conditional probability and statistical

    independence.

    Independent Events

    Actuar ies Again

    Independent Events

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    The probability ofn 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(A1A2...An) = P(A1) P(A2) ... P(An)

    if the events are independent

    F1 be the event that server 1 fails

    F2be the event that server 2 fails

    Independent Events

    Mult ip l ication Law for Independent Events

    Independent Events

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    Applying the rule of independence:

    The probability that at least one server is up is:

    P(F1F2) = P(F1) P(F2)= (.01)(.01) = .0001

    1 - .0001 = .9999 or 99.99%

    So, the probability that both servers are down is

    .0001.

    Independent Events

    Mult ip l ication Law for Independent Events

    Independent Events

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    Redundancycan increase system reliability even

    when individual component reliability is low.

    NASA space shuttle has three independent flightcomputers (triple redundancy).

    Each has an unacceptable .03 chance of failure

    (3 failures in 100 missions).

    Let Fj= event that computerjfails.

    Independent Events

    Example: Space Shu tt le

    Independent Events

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    What is the probability that all three flight

    computers will fail?

    P(all 3 fail) = P(F1F2F3)

    = 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)

    Independent Events

    Example: Space Shu tt le

    Independent Events

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    How high must reliability be?

    Public carrier-class telecommunications data links

    are expected to be available 99.999% of the time.

    The five nines rule implies only 5 minutes of

    downtime per year.

    This type of reliability is needed in many businesssituations.

    Independent Events

    The Five Nines Ru le

    Independent Events

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    For example,

    Independent Events

    The Five Nines Ru le

    Independent Events

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    Suppose a certain network Web server is up only 94% of the

    time. What is the probability of it being down?

    How many independent servers are needed to ensure that

    the system is up at least 99.99% of the time (or down only

    1 - .9999 = .0001 or .01% of the time)?

    P(down) = 1P(up) = 1 .94 = .06

    Independent Events

    How Much Redundancy is Needed?

    Independent Events

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    So, to achieve a 99.99% up time, 4 redundant

    servers will be needed.

    2 servers: P(F1F2) = (0.06)(0.06) = 0.0036

    3 servers: P(F1F2F3)

    = (0.06)(0.06)(0.06) = 0.0002164 servers: P(F1F2F3F4)

    = (0.06)(0.06)(0.06)(0.06)

    =0.00001296

    Independent Events

    How Much Redundancy is Needed?

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    Applied Statistics inBusiness and Economics

    End of Part 1 of Chapter 5