Probability (Tugh)

8/17/2019 Probability (Tugh)

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PROBABILITY

Introduction



Life is full of uncertainties.

Our need to cope with this uncertainty

of life has led to the study of

Probability theory.



What is probability?

In simple words, probability is the lielihood

or chance that a particular e!ent will occur

or not"

#$amples%



Importance o& Probability

Probability constitutes the &oundation o&statistical theory and application"

It has become increasin'ly essential in

(uantitati!e analysis o& business and

economic problems"



A thorou'h understandin' o& the &undamentals o&

probability theory will help a businessman to deal

with uncertainty in business situation in such a way

that he can asses systematically the riss in!ol!ed

in each alternati!e and conse)uently act to

minimise the riss"



Basic Terminology/Concepts

*" Random #$periment

+" #!ent

" #)ually liely #!ents

-" .utually #$clusi!e #!ents

/" #$hausti!e #!ents



#$periment

An acti!ity 0or process1 which produces

results 0outcomes1"

2or e$ample, I& we toss a &air coin, we may

obtain either a head or a tail"



Another #$ample%

3imilarly, when we roll a die, si$ possible

Outcomes can arise, that is , turnin' o& any

o& the si$ numbers *,+,,-,/,4 on the upper

&ace o& the dice"



*"Random #$periment

Random e$periment is an e$periment whoseoutcomes can not be predicted in ad!ance"

2or e$ample, i& we toss a coin either head

appears or tail appears" But we can not be

certain about which one is 'oin' to happen

in ad!ance"



3ample 3pace

The set o& all possible outcome o& a random

e$periment is called its sample space"

Let us consider the e$periment o& tossin' a

coin , then the possible outcomes are either

5ead 051 or Tail 0T1" Thus the sample spaceis% 36 5,T



.ore #$amples

In case o& an e$periment where two coins

are tossed simultaneously, sample space is



+"#!ent

#!ent is the outcome o& the e$periment"

2or e$ample, i& the e$periment is to toss a

&air coin, then o& the e!ents are%

#*6 #!ent o& 'ettin' a 5ead"#+ 6 #!ent o& 'ettin' a Tail"



3imilarly i& we roll a die , the e!ents

are %#*6 obtainin' * on the upper &ace o& the die"



"#)ually liely outcomes

The outcomes o& a random e$periment are

3aid to be e)ually liely i& each o& them has

e)ual chance o& occurrence"

In other words, two or more outcomes are

said to be e)ually liely i& any o& them can

not e$pected to occur in pre&erence to other"



2or #$ample,

In an unbiased coin tossin' e$periment,

both the outcomes, that is, head and tail,

ha!e an e)ual chance o& occurrence"



-" .utually #$clusi!e #!ents

Two or more e!ents are said to be Mutually

Exclusive i& the occurrence o& one

implies non7 occurrence o& other"

2or e$ample, i& an unbiased coin is tossed,either head or tail will occur, but these two

e!ents can not occur simultaneously"



/" 8ollecti!e #$hausti!e #!ents

The combination o& all the possible e!ents o&

a random e$periment is called #$hausti!e

#!ents"



9e&initions o& Probability

*" The classical 9e&inition

The probability o& an e!ent :A; is de&ined as

P0A16 m <m6no" o& outcomes &a!ourable

n to the e!ent A" n6total no"

o& outcomes o& the

e$periment"



3ome #$amples%

Ex 1.If a card is randomly drawn from a well-shuffled pack of 52

playing cards, what is the probability of

a) Drawing a queen b) Drawing a red card

c) Drawing a card no ! of diamond.



Ex 2

"here are # balls numbered to # in a bag. If a person selects

one at random what is the probability that the number printed on

the ball will be an odd number greater than !$



3et Theory

Set: A set is well-defined collection of distinct objects/ elements.

"he members of the set are distinct in the sense that repetition of

elements are not permitted in a gi%en set.

"he totality&i.e. collection of elements) is denoted by '. (ni%ersal set

'ubset



3et Operations

&i) Intersection of sets

Illustration

U= x/ x is positive integer

A = !"!#!$!%!&!'!(!)!*

A+, =



9is=oint 3ets

A B

>

U= !"!#!$!%!&!'!(!)!*

A = !"!#!

,= %!&

A+, =φ



>nion o& 3ets

U= x/ x is positive integer

A = !"!#

,= #!$!%!&

AU,=



8omplement o& A

>60*,+,,-1

A60*,+,1

A; 6



9i&&erence o& Two 3ets

Let A60 *,,/,,@,**1

B60/,@,*,*1

Then, A7B6



Rules of Probability

Addition theorem

ii1 I& A and B are any two Mutually Exclusive e!ents then the probability o& the occurrence o&

either A or B or both is 'i!en by,

P0A>B1 6 P0A1 P 0B1

A B



#$ample *

A ba' contains */ ticets mared with

numbers * to */" One ticet is drawn at

Random" 2ind the probability that the

number on it is multiple o& or "



Example

student went to college stationary store to buy a pen . *or

writing he can either buy a ball pen or gel pen or inc pen with

equal probability. 'uppose he chose only one item, what is the probability that the student bought either a ball pen or gel pen.



Rules o& Probability

A" i1 Addition Rule &or e!ents which are not

.utually #$clusi!e

f A and B are any t!o events then the probability

of the occurrence of either A or B or both is given

by"

P0A>B1 6 P0A1 P 0B1 P0A B1



illustration

+ity residents were sur%eyed recently to determine readership of

newspapers a%ailable. ! of the residents read the morning

aper, ! read the e%ening paper, and 2! read bothnewspapers. *ind the probability that a resident selected reads

either the morning or e%ening or both the papers.



#$ample

A ba' contains */ ticets mared with numbers * to */" One ticet is drawn

at Random" 2ind the probability that thenumber on it is multiple o& + or /"



#$ample

/hat is the probability that a leap year selected at random will

contain either 50 "hursdays or 50 *ridays$



P0A>B>81 6 P0A1 P0B1P0817 P0ACB17P0AC817 P0B C 81

7P0A CB C81



#$ample

age no 010



Independent #!ents

"wo or more e%ents are said to be independent in the sense that

occurrence of one e%ent is not influenced by the occurrence of

others.3ample 4 toss a coin twice.

3ample 24 e%ents with replacement are also treated as

independent. Drawing a card from a pack of cards withreplacement.



9ependent #!ents

Dependent e%ents are such that occurrence of which is not

influenced by the occurrence of others.

3ample4 Drawing two cards from a pack without replacing back

the first card.



B# Multiplication Rule

This rule states that if t!o events A and B are

independent " then the probability that they !ill

occur is given by"

P$ A and B%& P$A B%&P$A%#P$B%



Illustration

A man wants to marry a 'irl ha!in' &ollowin'

)ualities% !hite complexion' the probability o&

'ettin' such a 'irl is one in twenty< handsome

do!ry 7 the probability o& 'ettin' this is one in

&i&ty < !esterni(ed manners and eti)uettes

the probability here is one in hundred" 2ind out

the probability o& his 'ettin' married to such a'irl when the possession o& these three attributes

Is independent"



#$

0 fair coins are tossed. /hat is the probability of getting frst

head, second head and third tail.



#$ample

A candidate is selected &or inter!iew o& mana'ement

trainees &or companies" 2or the &irst company there are

*+ candidates, &or the second there are */ candidates and&or the third there are *D candidates"

i1 What is probability o& 'ettin' =obs in all these

companies"

ii1 What is probability that the candidate does not 'et any

=ob in any one o& the three companies"



8onditional Probability

The multiplication rule as mentioned abo!e

will not remain same in case o& dependence

e!ents" Two e!ents A and B are said to be

dependent when A can only when B has

already occurred and !ice7!ersa" The

probability attached to such an e!ent is called

Conditional Probability.

It is denoted by P 0AEB1 or P0BEA1"



2ormula



P0BEA16 P0A B1

P0A1



Multiplication Rule for dependent events

P0A B16P0B1"P0AEB1

6P0A1" P0BEA1



Example on Conditional Probability

3uppose that we select a person

randomly in the world" #!eryone has an

e)ual chance o& bein' selected" Let A bethe e!ent that the person is an IIT student,

and let B be the e!ent that the person

li!es in 9elhi" What are the probabilities o&

these e!ents?



Set of people livingat Delhi

Set of Delhi IIT

students



The !ast ma=ority o& people in the world neither

li!e in 9elhi nor are IIT students, so e!ents A and

B both ha!e low probability"

But what is the probability that a person is an

9elhi7IIT student, 'i!en that the person li!es in

9elhi?



What we are asin' &or is called a that is,conditional probability< the probability that

One e!ent happens, 'i!en that some other

e!ent de&initely happens"



#$ *

"wo cards are drawn one after another without replacement from a

well-shuffled pack of cards.

&i) +alculate the probability that the second card is a club gi%en

that the first card is a club.

&ii) +alculate the probability that the first card is club and the

second card is also a club.



Illustration +

A ba' contains / white and blac balls" Two

balls are drawn at random one a&ter the other

without replacement" 2ind the probability that

i1 both balls drawn are blac"

ii1 both balls drawn are white"

iii1One is blac and other is white"



3olution

Probability o& drawin' a blac ball in &irst

Attempt is P 0A1 6 EF"

Probability o& drawin' a second blac ball

'i!en that the &irst ball drawn is blac

P 0BEA1 6+E"Probability that both are blac P0AB16 P 0A1 " P 0BEA1



iii1 5int% The e!ent Gone is blac and other is

white; is the same as e!ent : either the &irst

is blac and second is white or the &irst iswhite and second is blac"H



#$

A ba' contains / white and blac balls"Two balls are drawn at random one a&ter

the other with replacement" 2ind the

probability that

i1 both balls drawn are blac"

ii1 both balls drawn are white"

iii1One is blac and other is white"



#$

contractor is trying to get both plumbing and electric contract.

"he probability that the contractor will get a plumbing contract is

260 and the probability that he will not get an electric contract is

561. If the probability of getting at least one contract is 65, what

is the probability that he will get both$

ns !.0



#$ /"

A problem in business statistics is 'i!en to

&i!e students A, B,8,9 and #" Their chances

o& sol!in' it are ,*E,*E-,*E/ and *E4" What

is the chance that no one can sol!e the

problem ? What is the chance that problem

is sol!ed?



#$ 4"wo persons and 7 were interrogated. Duringthe interrogation process the probability that

speaks the truth is 065 and he speaks lie is 265.

8n the other hand, the probability that 7 speaksthe truth is 56# and he speaks lie is 06#. If they

are interrogated independently on common issue

in what percentage are they likely to contradict$

9.5



#$

Delhi "raffic police found that 55 of all road accidents

in Delhi are caused by trucks, 9! occur at night and

# are caused by trucks at night.a) :i%en that an accident has occurred at night what is the

probability that it was caused by a truck$

b) :i%en that a truck has caused the accident what is the

probability that it has occurred at night $



3olution

T6accident caused by truc

J 6 accident occurred at ni'ht



3

3plain whether or not each of the following claims could be

correct4

0i) businessman claims the probability that he will get

contract is !.5 and that he will get contract 7 is

!.2!. *urthermore, he claims that the probability ofgetting or 7 is !.5!.

&ii) market analyst claims that the probability of

selling ten thousand shares of company or fi%e

thousand shares of company 7 is !.!. ;e alsoclaims that the probability of selling ten thousand

shares of company and fi%e thousand shares of

company 7 is !.5.



#$+

In a certain town, male and female each form 5! of the population.

It is known that 2! of the males and 5 of the females are

unemployed. research student studying the employment situation

selects an unemployed person at random. /hat is the probability that

the person selected is &a) male &b) female.



3olution

*nemployed Employed Total

.ales D"*D D"-D D"/D

2emales D"D+/ D"-/ D"/D

Total D"*+/ D"F/ *"DD

>nemployment 9ata

P0.E>1 6 D"F

P02E>1 6 D"+



#$

In a locality, out of 5!!! people residing, 2!! are abo%e 0!

years and 0!!! are women. 8ut of this 2!! & who are more than

0!) , 2!! are women. 8ne person is chosen from out of 5!!

people who is women. <nowing this calculate probability that

she is abo%e 0! years of age.

.!9



Baye;s Theorem

Let, B* , B+, K"B n are n mutually e$clusi!e

e!ents,

P0 B i EA1 6 & = 7 i )

& = 7 )> & = 7 2)> ?.. & = 7 n)

"he

abo%e equation calculates the probability of

7 & cause) by which the e%ent has resulted.



Illustration *

In a company three people are assigned to process incoming

or recei%ing mail. "he st person named 7 processes ! , the

2nd person named 72 processes 05 and the 0rd person named

70 processes 25. . "he st person has an error rate !.! , the

2nd has an error rate !.! and the third has an error rate !.!0.

mail selected at random from a day@s output is found to ha%e an

error. "he owner wants to calculate the probability that it was processed by st , 2nd and 0rd person respecti%ely.



3olution

&76 ) A !.0

&726 ) A !.9

&706 ) A !.9"hese probabilities are called posterior probability because they were calculated after it

was known that the mail was containing an error.



#$ +

manufacturing firm produces steel pipes in three plants with

daily production %olume of 5!!, ,!!! and 2!!! units

respecti%ely. ccording to past e3perience it is known that the

probability of defective outputs produced by the three plants are

respecti%ely !.!!5, !.!!# and !.!!. If a pipe is selected from a

day@s output and found to be defecti%e, find out &i) /hat is the

probability that it came from the first plant$II) *rom which the selected defecti%e pipe line is likely to come$



P0A * E#1 6 /E4*

P0A + E#1 6 *4E4*

P0A E#1 6 -DE4*

Probability (Tugh)

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