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
Methods for determining
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
Methods for determining
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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