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Chapter 6. Probability: Understanding Random Situations. Introduction. The study of Uncertainty Changes “I’m not sure …” to “I’m positive we’ll succeed … with probability 0.8 ” Can’t predict “for sure” what will happen next But can quantify the likelihood of what might happen

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Chapter 6

Chapter 6

Probability: Understanding Random Situations

© Andrew F. Siegel, 2003


Introduction
Introduction

  • The study of Uncertainty

    • Changes “I’m not sure …”

      • to “I’m positive we’ll succeed … with probability 0.8”

    • Can’t predict “for sure” what will happen next

      • But can quantify the likelihood of what might happen

      • And can predict percentages well over the long run

      • e.g., a 60% chance of rain

      • e.g., success/failure of a new business venture

  • New terminology (words and concepts)

    • Keeps as much as possible Certain (not random)

    • Put the randomness in only at the last minute

© Andrew F. Siegel, 2003


Terminology
Terminology

  • Random Experiment

    • A procedure that produces an outcome

      • Not perfectly predictable in advance

    • There are many random experiments (situations)

      • We will study them one at a time

    • Example: Record the income of a random family

      • Random telephone dialing in a target marketing area, repeat until success (income obtained), round to nearest $thousand

  • Sample Space

    • A list of all possible outcomes

      • Each random experiment has one (i.e., one list)

    • Example: {0, 1,000, 2,000, 3,000, 4,000, …}

© Andrew F. Siegel, 2003


Terminology continued
Terminology (continued)

  • Event

    • Happens or not, each time random experiment is run

      • Formally: a collection of outcomes from sample space

      • A “yes or no” situation: if the outcome is in the list, the event “happens”

      • Each random experiment has many different events of interest

    • Example: the event “Low Income” ($15,000 or less)

      • The list of outcomes is {0, 1, 2, …, 14,999, 15,000}

    • Example: the event “Six Figures”

      • The list of outcomes is {100,000, 100,001, 100,002, …, 999,999}

    • Example: the event “Ten to Forty Thousand”

      • The list of outcomes is {10,000, 10,001, …, 39,900, 40,000}

© Andrew F. Siegel, 2003


Terminology continued1
Terminology (continued)

  • Probability of an Event

    • A number between 0 and 1

      • The likelihood of occurrence of an event

    • Each random experiment has many probability numbers

      • One probability number for each event

    • Example: Probability of event “Low Income” is 0.17

      • Occurs about 17% over long run, but unpredictable each time

    • Example: Probability of event “Six Figures” is 0.08

      • Not very likely, but reasonably possible

    • Example: Probability of “10 to 40 thousand” is 0.55

      • A little more likely to occur than not

always happens

never happens

© Andrew F. Siegel, 2003


Sources of probabilities
Sources of Probabilities

  • Relative Frequency

    • From data

    • What percent of the time the event happened in the past

  • Theoretical Probability

    • From mathematical theory

    • Make assumptions, draw conclusions

  • Subjective Probability

    • Anyone’s opinion, perhaps even without data or theory

    • Bayesian analysis uses subjective probability with data

© Andrew F. Siegel, 2003


Relative frequency
Relative Frequency

  • From data. Run random experiment n times

    • See how often an event happened

      • (Relative Frequency of A) = (# of times A happened)/n

    • e.g., of 12 flights, 9 were on time.

      • Relative frequency of the event “on time” is 9/12 = 0.75

  • Law of Large Numbers

    • If n is large, then the relative frequency will be close to the probability of an event

      • Probability is FIXED. Relative frequency is RANDOM

    • e.g., toss coin 20 times. Probability of “heads” is 0.5

      • Relative frequency is 12/20 = 0.6 , or 9/20 = 0.45 , depending

© Andrew F. Siegel, 2003


Relative frequency continued

0.5

Relative frequency

0.0

1

2

3

4

5

Number n of times random experiment was run

Relative Frequency (continued)

  • Suppose event has probability 0.25

  • In n = 5 runs of random experiment

    • Event happens: no, yes, no, no, yes

    • Relative frequency is 2/5 = 0.4

  • Graph of relative frequencies for n = 1 to 5

© Andrew F. Siegel, 2003


Relative frequency continued1

0.5

Relative frequency

Probability = 0.25

0

0

50

100

150

200

Number n of times random experiment was run

Relative Frequency (continued)

  • As n gets larger

    • Relative frequency gets closer to probability

  • Graph of relative frequencies for n = 1 to 200

    • Relative frequency approaches the probability

© Andrew F. Siegel, 2003


Relative frequency continued2

Probability Probability Probability

0.500.25 or 0.750.10 or 0.90

n = 10 0.16 0.14 0.09

25 0.10 0.09 0.06

50 0.07 0.06 0.04

100 0.05 0.04 0.03

1,000 0.02 0.01 0.01

Relative Frequency (continued)

Table 6.3.1

  • About how far from the probability will the relative frequency be?

    • The random relative frequency will be about one of its standard deviations away from the (fixed) probability

    • Depends upon the probability and n

      • Farther apart when more uncertainty (probability near 0.5)

Relative frequency will be within about 0.09 of the probability, if n = 25 and probability is 0.75

© Andrew F. Siegel, 2003


Theoretical probability
Theoretical Probability

  • From mathematical theory

  • One example: The Equally Likely Rule

    • If all N possible outcomes in the sample space are equally likely, then the probability of any event A is

      • Prob(A) = (# of outcomes in A) / N

    • Note: this probability is not a random number. The probability is based on the entire sample space

      • e.g., Suppose there are 35 defects in a production lot of 400. Choose item at random. Prob(defective) = 35/400 = 0.0875

      • e.g., Toss coin. Prob(heads) = 1/2

      • But: Tomorrow it may snow or not. Prob(snow)  1/2

        • because “snow” and “not snow” are not equally likely

© Andrew F. Siegel, 2003


Subjective probability
Subjective Probability

  • Anyone’s opinion

    • What do you think the chances are that the U.S. economy will have steady expansion in the near future?

    • An economist’s answer

  • Bayesian analysis

    • Combines subjective probability with data to get results

    • Non-Bayesian “Frequentist” analysis computes using only the data

      • But subjective opinions (prior beliefs) can still play a background role, even when they are not introduced as numbers into a calculation, when they influence the choice of data and the methodology (model) used

© Andrew F. Siegel, 2003


Bayesian and nonbayesian analysis

Data

Bayesian

Analysis

Results

Prior Probabilities

Model

Frequentist

Analysis

Results

Bayesian and Nonbayesian Analysis

Fig 6.3.3

  • Bayesian Analysis

  • Frequentist (non-Bayesian) Analysis

Data

Prior

Beliefs

Model

© Andrew F. Siegel, 2003


Combining events
Combining Events

  • Complement of the event A

    • Happens whenever A does not happen

  • Union of events A and B

    • Happens whenever either Aor B or both events happen

  • Intersection of A and B

    • Happens whenever both A and B happen

  • Conditional Probability of A Given B

    • The updated probability of A, possibly changed to reflect the fact that B happens

© Andrew F. Siegel, 2003


Complement of an event

not A

A

Complement of an Event

  • The event “not A” happens whenever A does not

  • Venn diagram: A (in circle), “notA” (shaded)

  • Prob(not A) = 1 – Prob(A)

    • If Prob(Succeed) = 0.7, then Prob(Fail) = 1–0.7 = 0.3

© Andrew F. Siegel, 2003


Union of two events

A

B

Union of Two Events

  • Union happens whenever either (or both) happen

  • Venn diagram: Union “A orB” shaded)

    • e.g., A= “get Intel job offer”, B = “get GM job offer”

      • Did the union happen? Congratulations! You have a job

    • e.g., Did I have eggs or cereal for breakfast? “Yes”

© Andrew F. Siegel, 2003


Intersection of two events

A

B

Intersection of Two Events

  • Intersection happens whenever both events happen

  • Venn diagram: Intersection “A andB” shaded)

    • e.g., A= “sign contract”, B = “get financing”

      • Did the intersection happen? Great! Project has been launched!

    • e.g., Did I have eggs and cereal for breakfast? “No”

© Andrew F. Siegel, 2003


Relationship between and and or
Relationship Between and and or

  • Prob(AorB) = Prob(A)+Prob(B)–Prob(A and B)

    = + –

  • Prob(AandB) = Prob(A)+Prob(B)–Prob(A or B)

  • Example: Customer purchases at appliance store

    • Prob(Washer) = 0.20

    • Prob(Dryer) = 0.25

    • Prob(WasherandDryer) = 0.15

  • Then we must have

    • Prob(WasherorDryer) = 0.20+0.25–0.15 = 0.30

© Andrew F. Siegel, 2003


Conditional probability
Conditional Probability

  • Examples

    • Prob (WingivenAhead at halftime)

      • Higher than Prob (Win) evaluated before the game began

    • Prob (SucceedgivenGood results in test market)

      • Higher than Prob (Succeed) evaluated before marketing study

    • Prob (Get jobgivenPoor interview)

      • Lower than Prob (Get job given Good interview)

    • Prob (Have AIDSgivenTest positive)

      • Higher than Prob (Have AIDS) for the population-at-large

© Andrew F. Siegel, 2003


Conditional probability continued

Prob (AandB)

Prob (AgivenB) =

Prob (B)

A

B

B

A and B

Conditional Probability (continued)

  • Given the extra information that B happens for sure, how must you change the probability for A to correctly reflect this new knowledge?

    • This is a (conditional) probability about A

    • The event B gives information

  • Unconditional

    • The probability of A

  • Conditional

    • A new universe, since B must happen

© Andrew F. Siegel, 2003


Conditional probability continued1
Conditional Probability (continued)

  • Key words that may suggest conditional probability

    • By restricting attention to a particular situation where some condition holds (the given information)

      • Given …

      • Of those …

      • If …

      • When …

      • Within (this group) …

© Andrew F. Siegel, 2003


Conditional probability continued2
Conditional Probability (continued)

  • Example: appliance store purchases

    • Prob(Washer) = 0.20

    • Prob(Dryer) = 0.25

    • Prob(WasherandDryer) = 0.15

  • Conditional probability of buying a Dryergiven that they bought a Washer

    • Prob(DryergivenWasher)

    • = Prob(WasherandDryer)/Prob(Washer) = 0.15/0.20 = 0.75

    • 75% of those buying a washer also bought a dryer

  • Conditional probability of WashergivenDryer

    • = Prob(WasherandDryer)/Prob(Dryer) = 0.15/0.25 = 0.60

    • 60% of those buying a dryer also bought a washer

Watch the denominator!

© Andrew F. Siegel, 2003


Independent events
Independent Events

  • Two events are Independent if information about one does not change the likelihood of the other

    • Three equivalent ways to check independence

      • Prob (AgivenB) = Prob (A)

      • Prob (BgivenA) = Prob (B)

      • Prob (Aand B) = Prob (A)  Prob (B)

  • Two events are Dependent if not independent

    • e.g., Prob(WasherandDryer) = 0.15

      • Prob (Washer)  Prob (Dryer) = 0.20 0.25 = 0.05

    • Washer and Dryer are not independent

      • They are dependent

If independent,

all three are true.

Use any one to check.

Not equal

© Andrew F. Siegel, 2003


Mutually exclusive events

A

B

Mutually Exclusive Events

  • Two events are Mutually Exclusive if they cannot both happen, that is, if

    Prob(A and B) = 0

    • No overlap

      in Venn diagram

  • Examples

    • Profit and Loss (for a selected business division)

    • Green and Purple (for a manufactured product)

    • Country Squire and Urban Poor (marketing segments)

  • Mutually exclusive events are dependent events

  • © Andrew F. Siegel, 2003


    Probability trees
    Probability Trees

    • A method for solving probability problems

      • Given probabilities for some events (perhaps union, intersection, or conditional)

        • Find probabilities for other events

      • Record the basic information on the tree

        • Usually three probability numbers are given

          • Perhaps two probability numbers if events are independent

        • The tree helps guide your calculations

          • Each column of circled probabilities adds up to 1

          • Circled prob times conditional prob gives next probability

          • For each group of branches

            • Conditional probabilities add up to 1

            • Circled probabilities at end add up to probability at start

    © Andrew F. Siegel, 2003


    Probability tree continued

    Event B

    P(B given A)

    P(A and B)

    Event A

    Yes

    P(A)

    No

    P(“not B” given A)

    Yes

    P(A and “not B”)

    No

    P(“not A” and B)

    P(B given “not A”)

    Yes

    P(not A)

    No

    P(“not A” and “not B”)

    P(“not B” given “not A”)

    Probability Tree (continued)

    • Shows probabilities and conditional probabilities

    © Andrew F. Siegel, 2003


    Example appliance purchases

    Washer?

    Dryer?

    These add

    up to P(Dryer)

    = 0.25

    0.15

    Yes

    0.20

    No

    Yes

    No

    Yes

    No

    Example: Appliance Purchases

    • First, record the basic information

      • Prob(Washer) = 0.20, Prob(Dryer) = 0.25

      • Prob(WasherandDryer) = 0.15

    © Andrew F. Siegel, 2003


    Example continued

    Washer?

    Dryer?

    These add

    up to P(Dryer)

    = 0.25

    0.15

    Yes

    0.20

    No

    Yes

    No

    0.10

    Yes

    0.80

    No

    Example (continued)

    • Next, subtract: 1–0.20 = 0.80, 0.25–0.15 = 0.10

    © Andrew F. Siegel, 2003


    Example continued1

    Washer?

    Dryer?

    0.15

    Yes

    0.20

    No

    Yes

    0.05

    No

    0.10

    Yes

    0.80

    No

    0.70

    Example (continued)

    • Now subtract: 0.20–0.15 = 0.05, 0.80–0.10 = 0.70

    © Andrew F. Siegel, 2003


    Example completed tree

    Washer?

    Dryer?

    0.75

    0.15

    Yes

    0.20

    No

    0.25

    Yes

    0.05

    No

    0.125

    0.10

    Yes

    0.80

    No

    0.875

    0.70

    Example (completed tree)

    • Now divide to find conditional probabilities

      0.15/0.20 = 0.75, 0.05/0.20 = 0.25

      0.10/0.80 = 0.125, 0.70/0.80 = 0.875

    © Andrew F. Siegel, 2003


    Example finding probabilities

    Washer?

    Dryer?

    0.75

    0.15

    Yes

    0.20

    No

    0.25

    Yes

    0.05

    No

    0.125

    0.10

    Yes

    0.80

    No

    0.875

    0.70

    Example (finding probabilities)

    • Finding probabilities from the completed tree

      P(Washer) = 0.20

      P(Dryer) = 0.15+0.10 = 0.25

      P(WasherandDryer) = 0.15

      P(WasherorDryer) =

      0.15+0.05+0.10 = 0.30

      P(Washerand not Dryer) = 0.05

      P(DryergivenWasher) = 0.75

      P(Dryergiven notWasher) = 0.125

      P(WashergivenDryer) = 0.15/0.25 = 0.60

      (using the conditional probability formula)

    © Andrew F. Siegel, 2003


    Example venn diagram

    Washer

    Dryer

    0.15

    0.05

    0.10

    0.70

    Example: Venn Diagram

    • Venn diagram probabilities correspond to right-hand endpoints of probability tree

    P(Washer andDryer)

    P(“notWasher” andDryer)

    P(Washer and “notDryer”)

    P(“notWasher”and “notDryer”)

    © Andrew F. Siegel, 2003


    Example joint probability table

    Washer

    Yes

    No

    Yes

    0.15

    0.10

    0.25

    Dryer

    0.05

    0.70

    0.75

    No

    0.20

    0.80

    1

    Example: Joint Probability Table

    • Shows probabilities for each event, their complements, and combinations using and

      • Note: rows add up, and columns add up

    P(“notWasher” andDryer)

    P(Washer andDryer)

    P(Dryer)

    P(notDryer)

    P(Washer and “notDryer”)

    P(“notWasher”and “notDryer”)

    P(Washer)

    © Andrew F. Siegel, 2003