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7676235 Chap 04 Basic Probability

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    CHAPTER 4BASIC PROBABILITY

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    Learning Objectives

    In this chapter, you learn:

    y Basic probability concepts and definitions

    y Conditional probability

    y Various counting rules

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    Important Terms

    y Probability the chance that an uncertain eventwill occur (always between 0 and 1)

    y Event Each possible outcome of a variabley Sample Space the collection of all possible

    events

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    Assessing Probability

    y There are three approaches to assessing the probabilityof an uncertain event:

    1. a prioriclassical probability

    2. empirical classical probability

    3. subjective probability

    an individual judgment or opinion about the probability of occurrence

    outcomeselementaryofnumbertotaloccurcaneventthewaysofnumber

    TXoccurrenceofyprobabilit !!

    observedoutcomesofnumbertotal

    observedoutcomesfavorableofnumberoccurrenceofyprobabilit !

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    Sample Space

    The Sample Space is the collection of all

    possible events

    e.g. All 6 faces of a die:

    e.g. All 52 cards of a bridge deck:

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    Events

    y Simple event

    An outcome from a sample space with one characteristic

    e.g., Ared card from a deck of cards

    y Complement of an event A (denoted A) All outcomes that are not part of event A

    e.g., All cards that are not diamonds

    y Joint event

    Involves two or more characteristics simultaneously

    e.g., An ace that is also red from a deck of cards

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    Visualizing Events

    y Contingency Tables

    y Tree Diagrams

    Red 2 24 26

    Black 2 24 26

    Total 4 48 52

    Ace Not Ace Total

    Full Deck

    of 52 Cards

    Sample

    Space

    Sample

    Space2

    24

    2

    24

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    Visualizing Events

    y Venn Diagrams

    Let A= aces

    Let B = red cards

    A

    B

    A B = ace and red

    A U B = ace or red

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    Mutually Exclusive Events

    y Mutually exclusive events

    Events that cannot occur together

    example:

    A= queen of diamonds; B = queen of clubs

    Events Aand B are mutually exclusive

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    Collectively Exhaustive Events

    y Collectively exhaustive events One of the events must occur The set of events covers the entire sample space

    example:A= aces; B = black cards;C = diamonds; D = hearts

    Events A, B, C and D are collectively exhaustive (but notmutually exclusive an ace may also be a heart)

    Events B, C and D are collectively exhaustive and alsomutually exclusive

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    Probability

    y Probability is the numerical measure

    of the likelihood that an event will

    occur

    y The probability of any event must bebetween 0 and 1, inclusively

    y The sum of the probabilities of all

    mutually exclusive and collectivelyexhaustive events is 1

    Certain

    Impossible

    0.5

    1

    0

    0 P(A) 1 For any event A

    1P(C)P( )P(A) !

    If A, B, and are mutually exclusive and

    collectively exhaustive

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    Computing Joint andMarginal Probabilities

    y The probability of a joint event, Aand B:

    y Computing a marginal (or simple) probability:

    Where B1, B2, , Bkare kmutually exclusive and collectivelyexhaustive events

    outcomeselementaryofnumbertotal

    BandAsatisfyingoutcomesofnumber

    )BandA(P!

    )BdanP(A)BandP(A)BandP(AP(A) k! .

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    Joint Probability Example

    P(Red and Ace)

    Black

    ColorType Red Total

    Ace 2 2 4

    Non-Ace 24 24 48

    Total 26 26 52

    52

    2

    cardsofnumbertotal

    aceandredarethatcardsofnumber!!

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    Marginal Probability Example

    P(Ace)

    Black

    ColorType Red Total

    Ace 2 2 4

    Non-Ace 24 24 48

    Total 26 26 52

    52

    4

    52

    2

    52

    2)BlackandAce(P)dReandAce(P !!!

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    P(A1 and B2) P(A1)

    TotalEvent

    Joint Probabilities Using Contingency Table

    P(A2 and B1)

    P(A1 and B1)

    Event

    Total 1

    Joint Probabilities Marginal (Simple) Probabilities

    A1

    A2

    B1 B2

    P(B1) P(B2)

    P(A2 and B2) P(A2)

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    General Addition Rule

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

    General Addition Rule:

    If A and B are mutually exclusive, then

    P(A and B) = 0, so the rule can be simplified:

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

    For mutually exclusive events A and B

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    General Addition Rule Example

    P(Red orAce) = P(Red) +P(Ace) - P(Red and Ace)

    = 26/52 + 4/52 - 2 /52 = 28/52

    Dont countthe two red

    aces twice!Black

    ColorType Red Total

    Ace 2 2 4

    Non-Ace 24 24 48

    Total 26 26 52

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    Computing Conditional Probabilities

    y Aconditional probabilityis the probability of oneevent, given that another event has occurred:

    P( )

    )P()|P( !

    P( )

    )P()|P( !

    Where P( ) = joi t prob bility of

    P( ) = m rgi l prob bility of

    P( ) = m rgi l prob bility of

    The co itio l

    prob bility of giveth t h s occurre

    The co itio l

    prob bility of give

    th t h s occurre

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    Conditional Probability Example

    y What is the probability that a car has a CD player,given that it has AC ?

    i.e., we want to find P(CD | AC)

    Of the cars on a used car lot, 70% have airconditioning (A ) and 40% have a D player( D). 20% of the cars have both.

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    Conditional Probability Example

    No CDCD TotalAC 0.2 0.5 0.7

    No AC 0.2 0.1 0.3

    Total 0.4 0.6 1.0

    Of the cars on a used car lot, 70% have air conditioning(A ) and 40% have a D player ( D).

    20% of the cars have both.

    0. 80.

    0.

    P(A )

    A )andP( DA )|P( D !!!

    (continued)

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    Conditional Probability Example

    No CDCD Total

    AC 0.2 0.5 0.7

    No AC 0.2 0.1 0.3

    Total 0.4 0.6 1.0

    Given A , we only consider the top row (70% of the cars). Of these,20% have a D player. 20% of 70% is about 2 .57%.

    0.2 570.7

    0.2

    P(A )

    A )andP( DA )|P( D !!!

    (continued)

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    Using Decision Trees

    P(A and D) = 0.2

    P(A and D) = 0.5

    P(A and D) = 0.1

    P(A and D) = 0.2

    7.

    5.

    3.

    2.

    3.

    1.

    All

    Cars

    7.

    .

    Given A or

    no A :

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    Using Decision Trees

    P( D and A ) = 0.2

    P( D and A ) = 0.2

    P( D and A ) = 0.1

    P( D and A ) = 0.5

    4.

    2.

    .

    5.

    .

    1.

    All

    Cars

    4.

    .

    Given D or

    no D:

    (continued)

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    Statistical Independence

    y Two events are independent if and onlyif:

    y Events Aand B are independent when the probabilityof one event is not affected by the other event

    P(A)B)|P(A !

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    Multiplication Rules

    y Multiplication rule for two events Aand B:

    P(B)B)|P(AB)andP(A !

    P(A)B)|P(A !Note: If A and B are independent, then

    and the multiplication rule simplifies to

    P(B)P(A)B)andP(A !

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    Marginal Probability

    y Marginal probability for event A:

    Where B1, B2, , Bkare kmutually exclusive and collectivelyexhaustive events

    )P()|P()P()|P()P()|P(P( ) kk! .

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    Counting Rules

    y Counting Rule 1:

    If there are k1 events on the first trial, k2 events on thesecond trial, and kn events on the n

    th trial, the number of

    possible outcomes is

    Example:

    You want to go to a park, eat at a restaurant, and see a movie.There are 3 parks, 4 restaurants, and 6 movie choices. Howmany different possible combinations are there?

    Answer: (3)(4)(6) = 72 different possibilities

    (k1)(k2)(kn)

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    Counting Rules

    y Counting Rule 2:

    The number of ways that n items can be arranged in order is

    Example:

    Your restaurant has five menu choices for lunch. Howmany wayscan you order them on your menu?

    Answer: 5! = (5)(4)(3)(2)(1) = 120 different possibilities

    n! = (n)(n 1)(1)

    (continued)

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    Counting Rules

    y Counting Rule 3:

    Permutations: The number of ways of arranging X objectsselected from n objects in order is

    Example:

    Your restaurant has fivemenu choices, and three are selectedfor daily specials. Howmany different ways can the specialsmenu be ordered?

    Answer: different

    possibilities

    (continued)

    X)!(

    !Pxn

    !

    600

    3)!(

    !

    X)!(n

    n!nPx !!

    !

    !

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    Counting Rules

    y Counting Rule 4: Combinations: The number of ways of selecting X objects from

    n objects, irrespective of order, is

    Example:

    Your restaurant has five menu choices, and three are selected fordaily specials. Howmany different special combinations are there,

    ignoring the order in which they are selected?

    Answer: different possibilities

    (continued)

    X)!(nX!n!

    xn

    !

    10(6)(2)

    120

    3)!(53!

    5!

    X)!(nX!

    n!xn !!

    !

    !


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