chapter 3 probability and counting rules

8/15/2019 Chapter 3 Probability and Counting Rules

1/77

Chapter 3Chapter 3

Probability &Probability &

Counting RulesCounting Rules

Reference: Allan G. Bluman (2004) Elementary Statistics: A Step-by Step Approach.

New York : McGraw Hll


2/77

!"#ect$e%!"#ect$e%

&etermne 'ample 'pace% an n the pro"a"lt* oan e$ent u%n+ cla%%cal pro"a"lt* or emprcalpro"a"lt*.

,n the pro"a"lt* o compoun e$ent% u%n+ theaton rule% an the multplcaton rule%.

,n the contonal pro"a"lt* o an e$ent. ,n the total num"er o outcome% n a %e-uence o

e$ent% u%n+ the unamental countn+ rule. ,n the num"er o wa*% r r o"#ect% can "e %electe

rom nn o"#ect% u%n+ the permutaton rule. ,n the num"er o wa*% r r o"#ect% can "e %electe

rom nn o"#ect% wthout re+ar to orer u%n+ thecom"naton rule.

,n the pro"a"lt* o an e$ent u%n+ the countn+rule.


3/77

'ample 'pace% an /ro"a"lt*'ample 'pace% an /ro"a"lt*

A pro"a"lt* epermentpro"a"lt* eperment % a proce%% that lea%

to well1ene re%ult% calle outcome%.

An outcomeoutcome % the re%ult o a %n+le tral o apro"a"lt* eperment.

An e$ente$ent con%%t% o a %et o outcome% o a

pro"a"lt* eperment.

NOTE:NOTE: A tree a+ram can "e u%e a% a

%*%tematc wa* to n all po%%"le outcome% o

a pro"a"lt* eperment.


4/77

ree &a+ram or o%%n+ wo Con%ree &a+ram or o%%n+ wo Con%

First Toss

H

T

H

T H

T

Second Toss


5/77

'ample 'pace% 1'ample 'pace% 1 ample%


6/77

,ormula or Cla%%cal /ro"a"lt*,ormula or Cla%%cal /ro"a"lt*

Cla%%cal pro"a"lt* a%%ume% that all

outcome% n the %ample %pace are

e-uall* lkel* to occur. hat % e-uall* lkel*e-uall* lkel* e$ent% are e$ent%

that ha$e the %ame pro"a"lt* o

occurrn+.


7/77

,ormula or Cla%%cal /ro"a"lt*,ormula or Cla%%cal /ro"a"lt*

( ) = ( )( )

,

.

The probability of any event E is

number of outcomes in E

total number of outcomes in the sample space

This probability is denoted by

P E n E n S

This probability is called classical probability

and it uses the sample space S

.

.


8/77

Cla%%cal /ro"a"lt* 1Cla%%cal /ro"a"lt* 1 ample%

,or a car rawn rom an ornar* eck n the

pro"a"lt* o +ettn+ (a) a -ueen (") a o clu"%

(c) a 3 or a amon.

'oluton:'oluton:

(a) 'nce there are 4 -ueen% an 52 car%

P(queen) = 4/52 = 1/13P(queen) = 4/52 = 1/13.(") 'nce there % onl* one o clu"% then P(6 ofP(6 of

clubs) = 1/52 clubs) = 1/52 .


9/77


(c) here are our 3% an 63 amon% "ut

the 3 o amon% % counte twce n thel%tn+. Hence there are onl* 6

po%%"lte% o rawn+ a 3 or a amon

thu% P(3 or di!ond) = 16/52 = 4/13P(3 or di!ond) = 16/52 = 4/13.


10/77


7hen a %n+le ce % rolle n the pro"a"lt*

o +ettn+ a 8.

'oluton:'oluton: 'nce the %ample %pace % 6 2 3 4 5

an t % mpo%%"le to +et a 8.

Hence P(") = #/6 = # P(") = #/6 = # .

NOTE:NOTE: T$e su! of %$e &robbili%ies of llou%co!es in s!&le s&ce is one.


11/77

Complement o an $entComplement o an $ent

The complement of an event E is the set of outcomes in the

sample space that are not included in the outcomes

of event E The complement of E is denoted by E E bar

. ( ).

E

E


12/77

Complement o an $ent 1Complement o an $ent 1 ample

,n the complement o each e$ent.

9olln+ a ce an +ettn+ a 4. 'oluton:'oluton: Gettn+ a 6 2 3 5 or .

'electn+ a letter o the alpha"et an +ettn+

a $owel. 'oluton:'oluton: Gettn+ a con%onant.


13/77

Complement o an $ent 1Complement o an $ent 1 ample

'electn+ a a* o the week an +ettn+ a

weeka*. 'oluton:'oluton: Gettn+ 'atura* or 'una*.

'electn+ a one1chl aml* an +ettn+ a

"o*.

'oluton:'oluton: Gettn+ a +rl.


14/77

9ule or Complementar* $ent9ule or Complementar* $ent

P E P E

or

P E P E

or P E P E

( ) ( )= −1

1

1

( ) = − ( )

( ) + ( ) = .


15/77

mprcal /ro"a"lt*mprcal /ro"a"lt*

he erence "etween cla%%cal an

emprcal pro"a"lt*emprcal pro"a"lt* % that cla%%cal

pro"a"lt* a%%ume% that certan outcome%

are e-uall* lkel* whle emprcal pro"a"lt*

rele% on actual eperence to etermne the

pro"a"lt* o an outcome.


16/77

,ormula or mprcal /ro"a"lt*,ormula or mprcal /ro"a"lt*

Given a frequency distribution

the probability of an event being

in a given class is

P E frequency for the class

total frequencies in the distribution

f

n

This probability is called the empirical

probability and is based on observation

,

( ) =

= .

.


17/77

mprcal /ro"a"lt* 1mprcal /ro"a"lt* 1 ample

n a %ample o 50 people 26 ha t*pe !

"loo 22 ha t*pe A "loo 5 ha t*pe B"loo an 2 ha AB "loo. 'et up a

re-uenc* %tr"uton.


18/77


Type Frequency

A

B

AB

O

22

5

2

21

50 = n


19/77


,n the ollown+ pro"a"lte% or the

pre$ou% eample.

A per%on ha% t*pe ! "loo.

'oluton:'oluton: P (!) ; f


20/77

'u"#ect$e /ro"a"lt*

- uses a probability value used onan educated guess or estimate,employing opinions and inexactinformation. In subjectiveprobability, a person or groupmakes educated guess at te

cance tat an event !ill occur. "isguess is based on te person#sexperience and evaluation of asolution.


21/77

'u"#ect$e /ro"a"lt* 1'u"#ect$e /ro"a"lt* 1 ample

$x%

sportscaster may say tat ' Pac(uiaoas )*+ cance of !inning againstar(ue.


22/77

he Aton 9ule% or /ro"a"lt*he Aton 9ule% or /ro"a"lt*

wo e$ent% are mutuall* eclu%$emutuall* eclu%$e the*

cannot occur at the %ame tme (.e. the*ha$e no outcome% n common).


23/77

he Aton 9ule% or /ro"a"lt*he Aton 9ule% or /ro"a"lt*

A B

A and B are mutually exclusive


24/77

Aton 9ule 6 Aton 9ule 6

When two events A and B are

mutually eclusive! the probability

that A or B will occur is

P A or B P A P B ( ) ( ) ( )= +


25/77

Aton 9ule 61 Aton 9ule 61 ample

At a poltcal rall* there are 20 9epu"lcan%

(9) 63 &emocrat% (&) an nepenent%

(). a per%on % %electe n the pro"a"lt*that he or %he % ether a &emocrat or an

nepenent.

'oluton:'oluton: P (& or ) ; P (&) = P (); 63


26/77


A a* o the week % %electe at ranom.

,n the pro"a"lt* that t % a weeken. 'oluton:'oluton: P ('atura* or 'una*)

; P ('atura*) = P ('una*)

; 6 = 6 ; 2.


27/77


When two events A and B

are not mutually eclusive! the probabilityy that A or B will

occur is

P A or B P A P B P A and B

( ) ( ) ( ) ( )= + −


28/77


A B

A and B (common portion)


29/77


n a ho%ptal unt there are e+ht nur%e% an

$e ph*%can%. 'e$en nur%e% an three

ph*%can% are emale%. a %ta per%on %

%electe n the pro"a"lt* that the %u"#ect

% a nur%e or a male.

he net %le ha% the ata.


30/77

Aton 9ule 2 1 Aton 9ule 2 1 ample

STAFF FEMALES MALES TOTAL

NURSES 1 !

PHYSICIANS " 2 5

TOTAL 1# " 1"

STAFF FEMALES MALES TOTAL

NURSES 1 !

PHYSICIANS " 2 5

TOTAL 1# " 1"


31/77


'oluton:'oluton: P (nur%e or male)

; P (nur%e) = P (male) ? P (male nur%e) ;@


32/77


!n New Year% $e the pro"a"lt* that a per%on

r$n+ whle ntocate % 0.32 the pro"a"lt* o a

per%on ha$n+ a r$n+ accent % 0.08 an thepro"a"lt* o a per%on ha$n+ a r$n+ accent

whle ntocate % 0.0. 7hat % the pro"a"lt* o

a per%on r$n+ whle ntocate or ha$n+ a

r$n+ accent


33/77


'oluton:'oluton:

P (ntocate or accent); P (ntocate) = P (accent)

? /(ntocate an accent)

; 0.32 = 0.08 ? 0.0 ; 0.35.


34/77

wo e$ent% ' an are nepenentnepenent the

act that ' occur% oe% not aect thepro"a"lt* o occurrn+.

ample:ample: 9olln+ a ce an +ettn+ a

an then rolln+ another ce an +ettn+ a 3

are nepenent e$ent%.

he Multplcaton 9ule% an Contonalhe Multplcaton 9ule% an Contonal

/ro"a"lt*/ro"a"lt*


35/77

Multplcaton 9ule 6Multplcaton 9ule 6


are independent the

probability of both

occurring is

P A and B P A P B

,

( ) ( ) ( ).= ⋅


36/77

Multplcaton 9ule 6 1Multplcaton 9ule 6 1 ample

A car % rawn rom a eck an replace

then a %econ car % rawn. ,n the

pro"a"lt* o +ettn+ a -ueen an then an

ace.

'oluton:'oluton: Becau%e the%e two e$ent% are

nepenent (wh*) P (-ueen an ace) ;(4


37/77


A Harr% pole oun that 4D o Amercan% %a*

the* %uer +reat %tre%% at lea%t once a week. three people are %electe at ranom n the

pro"a"lt* that all three wll %a* that the* %uer

%tre%% at lea%t once a week.

'oluton:'oluton: Eet enote %tre%%. henP ( an an ) ; (0.4)3 ; 0.08>.


38/77


he pro"a"lt* that a %pecc mecal te%t wll

%how po%t$e % 0.32. our people are te%te

n the pro"a"lt* that all our wll %how

po%t$e.

'oluton:'oluton: Eet T enote a po%t$e te%t re%ult.

hen P (T an T an T an ) ; (0.32)4

; 0.060.


39/77

7hen the outcome or occurrence o the r%t

e$ent aect% the outcome or occurrence o the

%econ e$ent n %uch a wa* that the pro"a"lt*

% chan+e the e$ent% are %a to "e

epenent.

ample:ample: Ha$n+ h+h +rae% an +ettn+ a%cholar%hp are epenent e$ent%.


/ro"a"lt*/ro"a"lt*


40/77

he contonal pro"a"lt*contonal pro"a"lt* o an e$ent n

relaton%hp to an e$ent ' % the pro"a"lt* that an

e$ent occur% ater e$ent ' ha% alrea* occurre. he notaton or the contonal pro"a"lt* o

+$en ' % P (F ').

N!:N!: h% oe% not mean ÷ '.


/ro"a"lt*/ro"a"lt*


41/77

Multplcaton 9ule 2Multplcaton 9ule 2


are dependent the

probability of both

occurring is

P A and B P A P B A

,

( ) ( ) ( | ).= ⋅


42/77

n a %hpment o 25 mcrowa$e o$en% two are

eect$e. two o$en% are ranoml* %electe

an te%te n the pro"a"lt* that "oth are

eect$e the r%t one % not replace ater t

ha% "een te%te.

'oluton:'oluton: 'ee net %le.


/ro"a"lt* 1/ro"a"lt* 1 ample


43/77

'oluton:'oluton: 'nce the e$ent% are

epenentP (*6 an *2) ; P (*6)⋅P (*2F *6)

; (2


44/77

he 77 n%urance Compan* oun that 53D o

the re%ent% o a ct* ha homeowner% n%urance

wth t% compan*. ! the%e clent% 2>D al%o haautomo"le n%urance wth the compan*. a

re%ent % %electe at ranom n the pro"a"lt*

that the re%ent ha% "oth homeowner% an

automo"le n%urance.




45/77

'oluton:'oluton: 'nce the e$ent% are

epenentP (+ an ') ; P (+ )⋅P ( 'F+ ) ; (0.53)(0.2>)

; 0.6436.




46/77

Bo 6 contan% two re "all% an one "lue "all.

Bo 2 contan% three "lue "all% an one re"all. A con % to%%e. t all% hea% up "o 6

% %electe an a "all % rawn. t all% tal%

up "o 2 % %electe an a "all % rawn. ,n

the pro"a"lt* o %electn+ a re "all.




47/77

ree &a+ram orree &a+ram or ample

P ( B1) 1$2

%ed

%ed

Blue

Blue

Box 1

P ( B2) 1$2 Box 2

P ( R& B1) 2$"

P ( B& B1) 1$"

P ( R& B2) 1$'

P ( B& B2) "$'

(1$2)(2$")

(1$2)(1$")

(1$2)(1$')

(1$2)("$')


48/77

'oluton:'oluton: P (re) ; (6


49/77

Contonal /ro"a"lt* 1Contonal /ro"a"lt* 1 ,ormula,ormula

.

( | ) =( )

( )

The probability that the event B occurs

given that the first event A has occurred can be

found by dividing the probability that both events

occurred by the probability that the first event has

occurred The formula is

P B A P A and B

P A

second

.


50/77

he pro"a"lt* that 'am park% n a no1parkn+

one nd +et% a parkn+ tcket % 0.0 an the

pro"a"lt* that 'am cannot n a le+al parkn+%pace an ha% to park n the no1parkn+ one %

0.2. !n ue%a* 'am arr$e% at %chool an ha% to

park n a no1parkn+ one. ,n the pro"a"lt* that

he wll +et a tcket.

Contonal /ro"a"lt* 1Contonal /ro"a"lt* 1 ample


51/77

'oluton:'oluton: Eet N ; parkn+ n a no1parkn+

one an ; +ettn+ a tcket. hen P (T FN ) ; P (N an T ) I


52/77

A recent %ur$e* a%ke 600 people the*

thou+ht women n the arme orce%%houl "e permtte to partcpate n

com"at. he re%ult% are %hown n the

ta"le on the net %le.



53/77


(ender )es *o Total

+ale "2 1! 5#

Female ! '2 5#

Total '# ,# 1##

(ender )es *o Total

+ale "2 1! 5#

Female ! '2 5#

Total '# ,# 1##


54/77

,n the pro"a"lt* that the re%ponent an%were

J*e%K +$en that the re%ponent wa% a emale.

'oluton:'oluton: Eet , ; re%ponent wa% a male

- ; re%ponent wa% a emale

; re%ponent an%were J*e%KN ; re%ponent an%were JnoK.



55/77

P ( F- ) ; P ( - an ) I


56/77

ree &a+ram%ree &a+ram%

A tree a+ramtree a+ram % a e$ce u%e to l%t all

po%%"lte% o a %e-uence o e$ent% n a%*%tematc wa*.


57/77

ree &a+ram% 1ree &a+ram% 1 ample

'uppo%e a %ale% per%on can tra$el rom

New York to /tt%"ur+h "* plane tran or"u% an rom /tt%"ur+h to Cncnnat "*

"u% "oat or automo"le. &%pla* the

normaton u%n+ a tree a+ram.


58/77

ree &a+ram% 1ree &a+ram% 1 ample

-incinnati

Bus

*e.

or/

0ittsur3

Plane

Train

Bus

Boat

Auto

BusBoat

Boat

Bus

Auto

Auto

0lane4 Bus

0lane4 oat

0lane4 auto

Train4 us

Train4 oat

Train4 auto

Bus4 us

Bus4 oat

Bus4 auto


59/77

he Multplcaton 9ule or Countn+he Multplcaton 9ule or Countn+

Multplcaton 9ule :Multplcaton 9ule : n a %e-uence o nn

e$ent% n whch the r%t one ha% 66

po%%"lte% an the %econ e$ent ha% 22

an the thr ha% 33 an %o orth the total

po%%"lte% o the %e-uence wll "e

66×× 22×× 33× ⋅⋅⋅ ×× ⋅⋅⋅ × nn.


60/77

he Multplcaton 9ule or Countn+ 1he Multplcaton 9ule or Countn+ 1

ample

A nur%e ha% three patent% to $%t. How

man* erent wa*% can %he make herroun% %he $%t% each patent onl*

once


61/77


ampleample

'he can choo%e rom three patent% or the r%t

$%t an choo%e rom two patent% or the

%econ $%t %nce there are two let. !n the

thr $%t %he wll %ee the one patent who %

let. Hence the total num"er o erent

po%%"le outcome% % 3×

2×

6

; .


62/77


ample

mplo*ee% o a lar+e corporaton are to "e

%%ue %pecal coe entcaton car%.he car con%%t% o 4 letter% o the

alpha"et. ach letter can "e u%e up to 4

tme% n the coe. How man* erent &

car% can "e %%ue


63/77

he Multplcaton 9ule% or Countn+ 1he Multplcaton 9ule% or Countn+ 1

ample

'nce 4 letter% are to "e u%e there are 4

%pace% to ll ( L L L L ). 'nce there are 2

erent letter% to %elect rom an each

letter can "e u%e up to 4 tme% then the

total num"er o entcaton car% that can

"e mae % 2 × 2 × 2 × 2 ; 458>.


64/77


ample

he +t% 0 6 2 3 an 4 are to "e u%e n a

41+t & car. How man* erent car% are

po%%"le repetton% are permtte

olu%ion:olu%ion: 'nce there are our %pace% to ll

an $e choce% or each %pace the %oluton

% 5 × 5 × 5 × 5 ; 54 ; 25.


65/77


ample

7hat the repetton% were not permtte n

the pre$ou% eample

olu%ion:olu%ion: he r%t +t can "e cho%en n $e

wa*%. But the %econ +t can "e cho%en n

onl* our wa*% %nce there are onl* our +t%

let etc. hu% the %oluton % 5 × 4 × 3 × 2 ;620.


66/77

/ermutaton%/ermutaton%

Con%er the po%%"le arran+ement% o the letter%

bb an c c .

he po%%"le arran+ement% are: bc0 cb0 bc0 bc0bc0 cb0 bc0 bc0cb0 cbcb0 cb..

the order of %$e rrne!en% is i!&or%n% order of %$e rrne!en% is i!&or%n% then we

%a* that each arran+ement % a permutaton o the

three letter%. hu% there are % permutaton% othe three letter%.


67/77


An arran+ement o n %tnct o"#ect% n a%pecc orer % calle a permutatonpermutaton o the

o"#ect%.

No%e:No%e: o etermne the num"er o po%%"lte%

mathematcall* one can u%e the multplcaton

rule to +et:3 × 2 × 6 ; permutaton%.


68/77


/ermutaton 9ule :/ermutaton 9ule : he arran+ement o nn

o"#ect% n a %pecc orer u%n+ r o"#ect% at

a tme % calle a permutaton o nn o"#ect%

taken r r o"#ect% at a tme. t % wrtten a% nnP P r r

an the ormula % +$en "*

nnP P r r = n / (n r) = n / (n r) .


69/77

/ermutaton% 1/ermutaton% 1 ample

How man* erent wa*% can a charper%on

an an a%%%tant charper%on "e %electe or a

re%earch pro#ect there are %e$en %cent%t%a$ala"le

olu%ion:olu%ion: Num"er o wa*%

; P P 2 2 = / ( 2) = / ( 2) = /5 = 42 = /5 = 42 .


70/77

/ermutaton% 1/ermutaton% 1 ample

How man* erent wa*% can our "ook%

"e arran+e on a %hel the* can "e%electe rom nne "ook%

olu%ion:olu%ion: Num"er o wa*%

;""P P 44

= " / (" 4) = " / (" 4)

= "/5 = 3#24= "/5 = 3#24.


71/77

Com"naton%Com"naton%

Con%er the po%%"le arran+ement% o the letter%

bb an c c .

he po%%"le arran+ement% are: bc0 cb0 bc0 bc0bc0 cb0 bc0 bc0cb0 cbcb0 cb..

the order of %$e rrne!en% is no% i!&or%n% order of %$e rrne!en% is no% i!&or%n%

then we %a* that each arran+ement % the %ame.

7e %a* there % one com"naton o the threeletter%.


72/77

Com"naton%Com"naton%

Com"naton 9ule :Com"naton 9ule : he num"er o

com"naton% o o r r o"#ect% rom nn o"#ect% % enote "* nn r r an the ormula

% +$en " nn r r = n / (n r)r7 = n / (n r)r7 .


73/77

Com"naton% 1Com"naton% 1 ample

How man* com"naton% o our o"#ect%

are there taken two at a tme olu%ion:olu%ion: Num"er o com"naton%: 44 2 2

= 4 / (4 2) = 4 / (4 2) 27 = 4/227 = 6 27 = 4/227 = 6 ..


74/77


n orer to %ur$e* the opnon% o cu%tomer% at

local mall% a re%earcher ece% to %elect 5 mall%

rom a total o 62 mall% n a %pecc +eo+raphcarea. How man* erent wa*% can the %electon

"e mae

olu%ion:olu%ion: Num"er o com"naton%: 12 12 5 5 = 12 /= 12 /

(12 5) (12 5) 57 = 12/57 = "2 57 = 12/57 = "2 ..


75/77


n a clu" there are > women an 5 men. A

commttee o 3 women an 2 men % to "e cho%en.

How man* erent po%%"lte% are there olu%ion:olu%ion: Num"er o po%%"lte%: (num"er o

wa*% o %electn+ 3 women rom >) × (num"er o

wa*% o %electn+ 2 men rom 5) ; 33 ×× 5 5 2 2 = (35)= (35)

(1#) = 35# (1#) = 35# ..


76/77


A commttee o 5 people mu%t "e

%electe rom 5 men an @ women. Howman* wa*% can the %electon "e mae

there are at lea%t 3 women on the

commttee


77/77


olu%ion:olu%ion: he commttee can con%%t o 3

women an 2 men or 4 women an 6 man or

5 women. o n the erent po%%"lte% neach %eparatel* an then a them:

8 8 33 ×× 5 5 2 2 99 8 8 44 ×× 5 5 11 99 8 8 5 5 ×× 5 5 # #

= (56)(1#) 9 (#)(5) 9 (56)(1)= (56)(1#) 9 (#)(5) 9 (56)(1)= "66 = "66 ..

chapter 3 probability and counting rules

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