eco module 1 part iii

8/11/2019 eco Module 1 Part III

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Probability :Introductory Ideas

Event : one or more of the possibleoutcomes of doing something If we toss a coin getting tail would be one event and

getting head would be another event.

Experiment Process that produces outcomes

In probability theory, the activity that produces suchan event is referred to as an experiment.

Sample space The set of all possible outcomes of an experiment is

called the sample space of the experiment

S=( head, tail) 2


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The sample space, S, of a probability experimentis the collection of all possible outcomes.

An event is any collection of outcomes from aprobability experiment.

An event may consist ofone outcome or more than one outcome.

denote events with one outcome,

sometimes called simple events,ei.

In general, events are denoted using capital letterssuch as E.


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Probability and Statistics: Basics

Collectively exhaustive: Events are said to becollectively

exhaustive if they exhaust all possible outcomes of anexperiment.

Example:In a two coin tossing experiment {HH,HT,TH,TT}are the possible outcomes, they are (collectively) exhaustive

events.

Random Variable (r.v.): A variable that stands for theoutcome of a random experiment is called a random

variable. A random variable satisfies four properties:

it takes a single, specific value;

do not know in advance what value it happens to take;

however do know all of the possible values it may take; and

do know the probability that it will take any one of those

possible values.


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Probability and Statistics: Basics

Random Experiment:A random experiment isa process leading to at least two possibleoutcomes with uncertainty as to which will occur. Examples:Tossing a coin, throwing a pair of dice,

drawing a card from a pack of cards are allexperiments.


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Consider the probability experiment of having twochildren.

(a) Identify the outcomes of the probability

experiment.(b) Determine the sample space.(c) Define the event E= have one girl.

EXAMPLE Identifying Events and the Sample Space of aProbability Experiment

(a) e1= girl, girl, e2= boy, boy, e3= boy, girl, e4= girl, boy,

(b){(girl, girl), (boy, boy), (boy, girl), (girl, boy)}

(c) {(boy, girl), (girl, boy)}


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A probability model lists the possible outcomes of aprobability experiment and each outcomes probability. Aprobability model must satisfy rules 1 and 2 of the rules ofprobabilities.


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EXAMPLE A Probability Model

In a bag of peanut M&M milk chocolate

candies, the colors of the candies can bebrown, yellow, red, blue, orange, or green.Suppose that a candy is randomlyselected from a bag. The table showseach color and the probability of drawing

that color. Verify this is a probabilitymodel.

Color Probability

Brown 0.12

Yellow 0.15

Red 0.12

Blue 0.23

Orange 0.23

Green 0.15

All probabilities are between 0 and 1, inclusive.

Because 0.12 + 0.15 + 0.12 + 0.23 + 0.23 + 0.15 = 1, rule 2 (the sum of all

probabilities must equal 1) is satisfied.


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If an event is a certainty, the probability of the event is 1.

If an event is impossible, the probability of the event is 0.

An unusual event is an event that has alow probability of occurring.


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Three methods for determining the

probability of an event:(1) the empirical method

Methods for determining

probability


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Three methods for determining theprobability of an event:

(1) the empirical method

(2) the classical method


probability


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Three methods for determining the

probability of an event:

(1) the empirical method

(2) the classical method

(3) the subjective method


probability


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EXAMPLE : Empirical approach

The following data represent the number ofhomes with various types of fuels used forcooking based on a survey of 1,000 homes.

Type of fuel Frequency

LPG 631

Electricity 57

Kerosene 278

Coal or coke 2

Wood 27

Solar energy 1

Other fuels 4


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Type of fuel Frequency

LPG 631

Electricity 57Kerosene 278

Coal or coke 2

Wood 27

Solar energy 1

Other fuels 4

(a) Approximate the probability that a randomly selectedhome uses LPG as its cooking fuel.

(b) Would it be unusual to select a home that uses coalor coke as its cooking fuel?


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Equally likely events

Equally likely events: Two events are said to be

equally likelyif we are confident that one event is as

likely to occur as the other event.

Example:In a single toss of a coin a head isas likely to appear as a tail.


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The classical method of computing probabilitiesrequires equally likely outcomes.

An experiment is said to have equally likelyoutcomes when each simple event has the sameprobability of occurring.


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The subjective probabilityof an outcome

is a probability obtained on the basis ofpersonal judgment.

For example, an economist predicts that there is a 20%chance for Rupee downfall to $65 in the next year wouldbe a subjective probability.


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An article in a Sports Magazine investigates the probabilities that a

particular horse will win a race. The magazine reports that theseprobabilities are based on the amount of money bet on each horse. Whena probability is given that a particular horse will win a race, is thisempirical, classical, or subjective probability?

Question? Empirical, Classical, or Subjective Probability

Subjective because it is based upon peoples feelings about whichhorse will win the race. The probability is not based on a probabilityexperiment or counting equally likely outcomes.


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Suppose that a pair of dice are thrown. Let E= the first die

is a two and let F= the sum of the dice is less than or equalto 5. Find P(Eor F) using the General Addition Rule.

EXAMPLE Illustrating the General Addition Rule


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( )( )

( )

636

1

6

N EP E

N S

( )( )

( )

1036

5

18

N FP F

N S

( and )( and )

( )

336

1

12

N E FP E F

N S

( or ) ( ) ( ) ( and )

6 10 3

36 36 3613

36

P E F P E P F P E F


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Mutually exclusive events

Mutually exclusive events: Events are said to

be mutually exclusive if the occurrence of

one event prevents the occurrence of another

event at the same time.

X

Y

X Y

1 7 9

2 3 4 5 6

, ,

, , , ,

FC

AppleGrapeF

C

,

HP,DELL,IBM

YX

P X Y( ) 0


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Two events are disjoint if they have nooutcomes in common. Another name fordisjoint events is mutually exclusiveevents.


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We often draw pictures of eventsusing Venn diagrams. Thesepictures represent events ascircles enclosed in a rectangle.The rectangle represents thesample space, and each circlerepresents an event. For example,suppose we randomly select achip from a bag where each chip inthe bag is labeled 0, 1, 2, 3, 4, 5,

6, 7, 8, 9. Let E represent theevent choosea number less thanor equal to 2,and let F representthe event choose a numbergreater than or equal to 8. Theseevents are disjoint as shown in the

figure.


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Number of Roomsin Housing Unit

Probability

One 0.010

Two 0.032

Three 0.093

Four 0.176

Five 0.219

Six 0.189

Seven 0.122

Eight 0.079

Nine or more 0.080

The probability model shows thedistribution of the number of

rooms in housing units.

(A) What is the probability a

randomly selected housing unit

has two? or three rooms?

P(two or three)

= P(two) + P(three)

= 0.032 + 0.093

= 0.125

EXAMPLE The Addition Rule for Disjoint Events


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(B) What is the probability a

randomly selected housing unit

has one or two or three rooms?

Number of Rooms

in Housing Unit

Probability

One 0.010

Two 0.032

Three 0.093

Four 0.176

Five 0.219

Six 0.189

Seven 0.122

Eight 0.079

Nine or more 0.080

P(one or two or three)

= P(one) + P(two) + P(three)

= 0.010 + 0.032 + 0.093

= 0.135


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Complement of an Event

Let Sdenote the sample space of a probability experimentand let Edenote an event. The complement of E, denoted E,

is all outcomes in the sample space Sthat are not outcomes

in the event E.


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Complementary Events

All elementary events not in the event A are

in its complementary event.

Sample

Space

A

1)( SpaceSampleP

P A P A( ) ( ) 1

A


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Complement Rule

If E represents any event and E represents the

complement of E, then

P(E) = 1P(E)


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Independent Events

Occurrence of one event does not affect the

occurrence or non-occurrence of the other

event

The conditional probability of X given Y is

equal to the marginal probability of X.

The conditional probability of Y given X is

equal to the marginal probability of Y.

P X Y P X and P Y X P Y( | ) ( ) ( | ) ( )


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Two events E and F are independent if theoccurrence of event E in a probability experimentdoes not affect the probability of event F. Twoevents are dependent if the occurrence of eventEin a probability experiment affects the probability

of event F.

Example: Suppose you draw a card from a standard 52-

card deck of cards and then roll a die. The eventsdraw a heart and roll an even number areindependent because the results of choosing a card donot impact the results of the die toss.


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A manufacturer knows that 10% of their products are defective. They also knowthat only 30% of their customers will actually use the product in the first year afterit is purchased. If there is a one-year warranty on the equipment, what proportionof the customers will actually make a valid warranty claim?

We assume that the defectiveness of the equipment is independent of the

use of the equipment. So,

defective and used defective used

(0.10)(0.30)

0.03

P P P


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Conditional Probability

Conditional Probability:The probability that event Bwill takeplace provided that eventAhas taken place (is taking place orwill with certainty take place) is called the condi t ionalprobabi l i tyBrelative toA. Symbolically, it is written as P(B|A)

to be read the probability of B, givenA.

If A and B are mutually exclusive events, then P(B|A) = 0

and P(A|B) = 0.


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Problem?

Example:Suppose we flip two identical coinssimultaneously. What is the probability of obtaininga head on the first coin (call eventA) and a headon the second coin (call event B)?

Notice that probability of obtaining a head on thefirst coin is independent of the probability of

obtaining a head on the second coin.


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Rules of Probability

1. A probability cannot be negative or larger than 1.

That is, for any event

2. If the probability of the occurrence of event is ,

the probability that will not occur is .

3. Special rule of addition: If and are mutually exclusive events,

the probability that one of them will occur is or = .

Similarly, if are mutually exclusive events,

the probability that one of them will occur is or or ... or =

0 1

1

1 2

1 2

1 2 1 2

P A A

A P(A)

A -P(A)

A B

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

A ,A ,...,A

P(A A A )

P( A A A ) P( A ) +P( A )

n

n

n

.

... ...P( A )n .


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Rules of Probability

4. If are mutually exclusive and collectively

exhaustive set of events, the sum of the probabilities of

their individual occurences is . That is,

.

A ,A ,...,A

P( A A A ) P( A ) P( A ) P( A )

n

n n

1 2

1 2 1 2

1

1 ... ...

Example: The probability of any of the six numbers on a die is

1/6 since there are six equally likely outcomes and each one of

them has an equal chance of turning up. Since the numbers

{1,2,3,4,5,6} form an exhaustive set of events

P(1+2+3+4+5+6) =P(1)+P(2)+P(3)+P(4)+P(5)+P(6) = 1.


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A deck of playing cards


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The event the king of hearts isselected

1/52


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The event a king is selected

4/52 =1/13


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The event a heart is selected

13/52= 1/4

Th t f d i


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The event a face card isselected

12/52=3/13


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Venn diagram for event E


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Relationships Among Events

(not E): The event that Edoes not occur.

(A& B): The event that both Aand Boccur.

(Aor B): The event that either Aor Bor both occur.

A t d it l t


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An event and its complement

For any event E,

P(E) = 1P(~ E).

In words, the probability that an event occurs equals 1 minus the probability

that it does not occur.


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Combinations of Events

The Addition RuleOr The special addition rule (mutually exclusive events)

The general addition rule (non-mutually exclusive

events)

The Multiplication RuleAnd

The special multiplication rule (for independent

events)

The general multiplication rule (for non-independentevents)


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eco module 1 part iii

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