Probability Understanding Random Situations

8/3/2019 Probability Understanding Random Situations

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Chapter 6

Probability: Understanding

Random Situations


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Introduction

The study ofUncertainty

Changes Im not sure

to Im positive well succeed with probability 0.8

Cant predict for sure what will happen next

But can quantify the likelihood of what mighthappen

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


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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, }


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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}


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Terminology (continued)

Probability of an Event

A number between 0 and 1

The likelihood of occurrence of an event

Each random experiment has manyprobability 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


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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 Anyones opinion, perhaps even without data or theory

Bayesian analysis uses subjective probability with data


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Relative Frequency

From data. Run random experiment n times

See how often an event happened

(Relative Frequency ofA) = (# of timesA happened)/n

e.g., of12 fligh

ts, 9 were on time. Relative frequency of the event on time is 9/12 = 0.75

Law of Large Numbers

Ifn is large, then the relative frequency will be close to

th

eprobability 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 , or9/20 = 0.45 , depending


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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 forn = 1 to 5

0.5

1 2 3 4 5

0.0

Relativefre

quency

Numbern of times random experiment was run


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As n gets larger

Relative frequency gets closer to probability

Graph of relative frequencies forn = 1 to 200

Relative frequency approach

es th

e probability

0

0.5

0 50 100 150 200

Numbern of times random experiment was run

Relativefrequency

Probability = 0.25


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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 near0.5)

Table 6.3.1

Probability Probability Probability

0.50 0.25 or 0.75 0.10 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


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Theoretical Probability

From mathematical theory

One example: The Equally Likely Rule

If allNpossible outcomes in thesample space are

equally likely, then the probability of any eventA is

Prob(A) = (# of outcomes inA) / N

Note: this probability is nota random number. The

probability is based on the entire sample space

e.g., Suppose there are 35 defects in a production lot of400.

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 notequally likely


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Subjective Probability

Anyones opinion

What doyou think the chances are that the U.S.

economy will have steady expansion in the near future?

An economists 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


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Bayesian and Nonbayesian Analysis

Bayesian Analysis

Frequentist (non-Bayesian) Analysis

Bayesian

Analysis

Data

Prior Probabilities

Model

Results

DataPrior

Beliefs

Model

Frequentist

AnalysisResults

Fig 6.3.3


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Combining Events

Complement of the eventA

Happens wheneverA does not happen

Union of eventsA andB

Happens wh

enever eith

erA orB or both

eventsh

appen Intersection ofA andB

Happens whenever bothA andB happen

Conditional Probability ofA GivenB

The updated probability ofA, possibly changed toreflect the fact thatB happens


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Complement of an Event

The event not A happens wheneverA does not

Venn diagram:A (in circle), not A (shaded)

Prob(not A) = 1 Prob(A)

If Prob(Succeed) = 0.7, then Prob(Fail) = 10.7 = 0.3

Anot

A


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Union ofTwo 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

A B


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Intersection ofTwo 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

A B


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Relationship Betweenandand or

Prob(A orB) = Prob(A)+Prob(B)Prob(A andB)

= +

Prob(A andB) = Prob(A)+Prob(B)Prob(A orB)

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+0.250.15 = 0.30


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Conditional Probability

Examples

Prob (Wingiven Ahead at halftime)

Higher than Prob (Win) evaluated before the game began

Prob (Succeedgiven Good results in test market) Higher than Prob (Succeed) evaluated before marketing study

Prob (Get jobgiven Poor interview)

Lower than Prob (Get job given Good interview)

Prob (Have AIDSgiven Test positive) Higher than Prob (Have AIDS) for the population-at-large


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Conditional Probability (continued)

Given the extra information thatB happens forsure, how must you change the probability forA to

correctly reflect this new knowledge?

This is a (conditional) probability aboutA

The eventB gives information

Unconditional

The probability ofA

Conditional

A new universe, sinceB must happen

Prob (AgivenB) =

Prob (A andB)

Prob (B)

A B

A andB B


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Key words that may suggest conditionalprobability

By restricting attention to a particular situation where

some condition holds (thegiven information) Given

Of those

If

When

Within (this group)


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Example: appliance store purchases Prob(Washer) = 0.20

Prob(Dryer) = 0.25


Conditional probability of buying a Dryergiven thatthey bought a Washer

Prob(Dryergiven Washer)

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

75% of those buying a washer also bought a dryer

Conditional probability ofWashergiven Dryer

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

60% of those buying a dryer also bought a washer


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Independent Events

Two events are Independent if information aboutone does not change the likelihood of the other

Three equivalent ways to check independence

Prob (AgivenB) = Prob (A)

Prob (BgivenA) = Prob (B)

Prob (A andB) = Prob (A) v Prob (B)

Two events are Dependent if not independent

e.g., Prob(WasherandDryer) = 0.15 Prob (Washer) v Prob (Dryer) = 0.20 v 0.25 = 0.05

Washerand Dryerare not independent

They are dependent


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Mutually Exclusive Events

Two events are Mutually Exclusive if they cannotbothhappen, that is, if

Prob(A andB) = 0

No overlapin 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 dependentevents

A B


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Probability Trees

A method for solving probability problems

Given probabilities for some events (perhaps union,

intersection, or conditional)

Find probabilities for other events

Record t he basic information on the tree Usually three probability numbers are given

Per haps 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


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Probability Tree (continued)

Shows probabilities and conditional probabilities

P(A andB)

P(A andnotB)

P(notA andB)

P(notA andnotB)

P(A)

P(notA)

EventB

EventA


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Example: Appliance Purchases

First, record the basic information Prob(Washer) = 0.20, Prob(Dryer) = 0.25


0.15

0.20

Dryer?Washer?


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Example (continued)

Next, subtract: 10.20 = 0.80, 0.250.15 = 0.10

0.15

0.10

0.20

0.80

Dryer?Washer?


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Example (continued)

Now subtract: 0.200.15 = 0.05, 0.800.10 = 0.70

0.15

0.05

0.10

0.70

0.20

0.80

Dryer?Washer?


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Example (completed tree)

Now divide to find conditional probabilities0.15/0.20 = 0.75, 0.05/0.20 = 0.25

0.10/0.80 = 0.125, 0.70/0.80 = 0.875

0.15

0.05

0.10

0.70

0.20

0.80

Dryer?Washer?


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Example (finding probabilities)

Finding probabilities from the completed treeP(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 notDryer) = 0.05

P(Dryergiven Washer) = 0.75

P(Dryergiven notWasher) = 0.125

P(Washergiven Dryer) = 0.15/0.25 = 0.60

(using the conditional probability formula)

0.15

0.05

0.10

0.70

0.20

0.80

Dryer?Washer?


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Example: Venn Diagram

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

Washer Dryer

0.05

0.15

0.10

0.70

P(WasherandDryer)

P(WasherandnotDryer )

P(notWasher andDryer)

P(notWasher andnotDryer )


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Example: Joint Probability Table

Shows probabilities for each event, theircomplements, and combinations using and

Note: rows add up, and columns add up

Washer

Yes No

D

ryer Yes

No

0.15 0.10

0.700.05

0.20 0.80 1

0.75

0.25

P(WasherandDryer)

P(WasherandnotDryer )

P(notWasher andDryer)

P(notWasher andnotDryer )

P(Dryer)

P(notDryer)

P(Washer)

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Probability Understanding Random Situations