COMPLETE BUSINESS STATISTICS by AMIR D. ACZE ACZEL L ACZEL & JAYAV JAY AVEL EL SOUNDER SO UNDERPANDIAN PANDIAN JAYA AVEL SOUNDERPANDIA SOU NDERPANDIAN N 7th edition. Pre!red by L"oyd J!i#in$h% Morehe!d St!te Unier#ity
Chapter 2 Probability
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2 Probability
Using Statistics Basic Definitions: Events, Sample Space, and ro!a!ilities Basic "#les for ro!a!ilit$ %onditional ro!a!ilit$ Independence of Events %om!inatorial %oncepts &'e (aw of &otal ro!a!ilit$ and Ba$es) &'eorem &'e *oint ro!a!ilit$ &a!le Using t'e %omp#ter
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2 Probability
Using Statistics Basic Definitions: Events, Sample Space, and ro!a!ilities Basic "#les for ro!a!ilit$ %onditional ro!a!ilit$ Independence of Events %om!inatorial %oncepts &'e (aw of &otal ro!a!ilit$ and Ba$es) &'eorem &'e *oint ro!a!ilit$ &a!le Using t'e %omp#ter
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2
LEARNING OBJECTIVES
After studying this chapter chapter,, you you should be able to:
Define pro!a!ilit$, sample space, and event+
Disting#is' !etween s#!ective and o!ective pro!a!ilit$+
Descri!e t'e complement of an event, t'e intersection, and t'e #nion of two events+
%omp#te pro!a!ilities of vario#s t$pes of events+
Eplain t'e concept of conditional pro!a!ilit$ and 'ow to comp#te it+
Descri!e perm#tation and com!ination and t'eir #se in certain pro!a!ilit$ comp#tations+
Eplain Ba$es) t'eorem and its applications+
'(,
'(* Prob!bi"ity i#+
. #antitative meas#re of uncertainty . meas#re of t'e strength of belief in t'e occ#rrence of an #ncertain event . meas#re of t'e degree of chance or likelihood of occurrence of an #ncertain event Meas#red !$ a n#m!er !etween 0 and 1 2or !etween 03 and 10034
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-ye# o Prob!bi"ity
Objective or Classical Probability
!ased on e#all$-li5el$ events
!ased on long-r#n relative fre#enc$ of events
not !ased on personal !eliefs
is t'e same for all o!servers 2o!ective4
eamples: toss a coin, roll a die, pic5 a card
'(3
-ye# o Prob!bi"ity 0Contin1ed2
Subjective Probability
!ased on personal !eliefs, eperiences, pre#dices, int#ition - personal #dgment
different for all o!servers 2s#!ective4
eamples: S#per Bowl, elections, new prod#ct introd#ction, snowfall
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'(' 4!#i5 Deinition#
Set - a collection of elements or objects of interest
Empt$ set 2denoted !$ ∅4
Universal set 2denoted !$ S4
a set containing no elements a set containing all possi!le elements
%omplement 2 6ot4+ &'e complement of . is
a set containing all elements of S not in .
( )
'(
Co6"e6ent o ! Set
S
. .
7enn 7ennDiagram Diagramill#strating ill#stratingt'e t'e%omplement %omplementof ofan an event event
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4!#i5 Deinition# 0Contin1ed2
∩##) ) ( ∩ ( 8 a set containing all elements in !ot' . and B
Union 29r4
Intersection 2.nd4
8
∪##) ) ( ∪ (
a set containing all elements in . or B or !ot'
'(*:
Set#+ A Inter#e5tin$ 9ith 4
S
A
∩ #
'(**
Set#+ A Union 4
S
A
∪ #
'(*'
4!#i5 Deinition# 0Contin1ed2
!utually e"clusive or disjoint sets
8 sets 'aving no elements in common, 'aving no intersection, w'ose intersection is t'e empt$ set
Partition
8 a collection of m#t#all$ ecl#sive sets w'ic' toget'er incl#de all possi!le elements, w'ose #nion is t'e #niversal set
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M1t1!""y E;5"1#ie or Di#
A
'(*,
Set#+ P!rtition
S A$
A#
A2
A% A&
'(*/
E;eri6ent
rocess t'at leads to one of several possi!le o#tcomes ;, e+g+: %oin toss
"olling a die
Heads, &ails 1, <, =, >, ?, @
ic5 a card
.H, AH, H, +++
Introd#ce a new prod#ct Eac' trial of an eperiment 'as a single o!served o#tcome+
&'e precise o#tcome of a random eperiment is #n5nown !efore a trial+
; .lsocalled calledaa!asic !asico#tcome, o#tcome,elementar$ elementar$event, event,ororsimple simpleevent event ; .lso
'(*3
Eent# + Deinition
Sample Space or Event Set
Set of all possi!le o#tcomes 2#niversal set4 for a given eperiment
E+g+: "oll a reg#lar si-sided die
Event
%ollection of o#tcomes 'aving a common c'aracteristic
E+g+: Even n#m!er
S C 1,<,=,>,?,@
. C <,>,@
Event . occ#rs if an o#tcome in t'e set . occ#rs
ro!a!ilit$ of an event
S#m of t'e pro!a!ilities of t'e o#tcomes of w'ic' it consists
2.4 C 2<4 F 2>4 F 2@4
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E=1!""y("i>e"y Prob!bi"itie# 0?yotheti5!" or Ide!" E;eri6ent#2
or eample:
"oll a die
Si possi!le o#tcomes 1,<,=,>,?,@
If eac' is e#all$-li5el$, t'e pro!a!ilit$ of eac' is 1/@ C 0+1@@ C 1@+@3
$2 e4 =
1
n2 % 4 ro!a!ilit$ of eac' e#all$-li5el$ o#tcome is 1 divided !$ t'e n#m!er of possi!le o#tcomes
Event . 2even n#m!er4
2.4 C 2<4 F 2>4 F 2@4 C 1/@ F 1/@ F 1/@ C 1/< for e in . $2 4 = ∑ $2 e4
=
n2 4 n2 % 4
=
= @
=
1 <
'(*
Pi5> ! C!rd+ S!6"e S!5e Union of Events Heart) and .ce) $ 2 Heart
ce 4 =
n 2 Heart
ce 4 =
n 2 % 4
1@
> =
?<
1=
'earts
(iamonds
Clubs
. A * 10 J K @ ? > = <
. A * 10 J K @ ? > = <
. A * 10 J K @ ? > = <
Event Heart) n 2 Heart 4 $ 2 Heart 4 =
1= =
n 2 % 4
1 =
?<
S)ades . A * 10 J K @ ? > = <
Event .ce) n 2 ce 4 $ 2 ce 4 =
> =
n 2 % 4
1 =
?<
1=
&'e intersection of t'e events Heart) and .ce) comprises t'e single point circled twice: t'e ace of 'earts
>
n 2 Heart $ 2 Heart
ce 4 =
ce 4
1 =
n 2 % 4
?<
'(*8
'() 4!#i5 R1"e# or Prob!bi"ity
*angeof of+alues +aluesfor forP,A. P,A. *ange
0 ≤ $ 2 4 ≤1
ro!a!ilit$of of not . not . Com)lements--ro!a!ilit$ Com)lements
= −
$ 2 4 1 $ 2 4 Intersection - ro!a!ilit$ of !ot' . and B Intersection - ro!a!ilit$ of !ot' . and B
$2 ∩ #4 =
n2 ∩ # 4 n2 % 4
!utuallye"clusive e"clusiveevents events 2. 2.and and%4 %4: : !utually
$2 ∩ C 4 = 0
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4!#i5 R1"e# or Prob!bi"ity 0Contin1ed2
ro!a!ilit$of of..or B or Bor or !ot' !ot'2r#le 2r#leof of#nions4 #nions4 Union--ro!a!ilit$ Union
$2 ∪ #4 =
n2 ∪ # 4 = $2 4 + $2 #4 − $2 ∩ # 4 n2 % 4
M#t#all$ecl#sive ecl#siveevents: events:IfIf..and andBBare arem#t#all$ m#t#all$ecl#sive, ecl#sive,t'en t'en M#t#all$
$ 2 ∩ #4 = 0 so $ 2 ∪ #4 = $ 2 4 + $ 2 #4
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Set#+ P0A Union 42
S
A
$ 2 ∪ # 4
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'(, Condition!" Prob!bi"ity
ro!a!ilit$of of. gi!en . gi!enBB ConditionalProbability Probability- -ro!a!ilit$ Conditional
$ 2 #4 =
$ 2 ∩ #4 , where $ 2 #4 ≠ 0 $ 2 #4
Independentevents: events: Independent
= $2 4 $2 # 4 = $2 #4 $2 #4
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Condition!" Prob!bi"ity 05ontin1ed2 *ulesof ofconditional conditional)robability. )robability. *ules $2 #4 =
$2 ∩ #4 $2 #4
so $2 ∩ #4 = $2 #4 $2 #4 = $2 # 4 $2 4
If events . and D are statistically inde)endent : $ 2 &4 = $ 2 4
so $ 2 & 4 = $ 2 &4
$ 2 ∩ &4 = $ 2 4 $ 2 &4
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Contin$en5y -!b"e ( E;!6"e '(' %o#nts A/0 /
I!
/otal
/elecommunication
>0
10
&1
Com)uters
<0
=0
&1
/otal
1
%1
#11
ro!a!ilities A/0 /
I!
/otal
/elecommunication
0+>0
0+10
13&1
Com)uters
0+<0
0+=0
13&1
/otal
131
13%1
#311
ro!a!ilit$ t'at a proect is #nderta5en !$ IBM gi!en it is a telecomm#nications proect: $ 2 I#M T 4
=
$ 2 I#M T 4
=
0+10
$ 2T 4
0+?0
= 0 +<
'('/
'(/ Indeenden5e o Eent# %onditions for t'e statistical independence of events . and B: $2 #4 = $ 2 4 $2 # 4 = $ 2 # 4 an" $2 #4 = $ 2 4 $ 2 # 4 $ 2 ce Heart 4
= =
$ 2 ce Heart 4 $ 2 Heart 4 1 ?< = 1 = $ 2 ce 4 1= 1= ?<
$ 2 ce Heart 4 =
$ 2 Heart ce 4
= =
$ 2 Heart ce 4 $ 2 ce 4 1 ?< = 1 = $ 2 Heart 4 > > ?<
> 1= 1 ; = = $ 2 ce 4 $ 2 Heart 4 ?< ?< ?<
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Indeenden5e o Eent# @ E;!6"e '(/ Events Tele!ision 2T 4 and #illboar" 2 #4 are ass#med to !e inde)endent3
== +0>;;00+0@ +0@==00+00<> +00<> ==00+0> 4 $ 22T T #44== $ $ $ 2 #44−− $ $ #44 bb4 $ # 22T T 44++ $ 2 # 22T T # +0>++00+0@ +0@−−00+00<> +00<>==00+0JH@ +0JH@ ==00+0>
4 $ 22T T #44 $ $ 4 $ 2 # 2 #44 aa4 $ # 22T T 4 $
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Prod15t R1"e# or Indeendent Eent# &'e pro!a!ilit$ of t'e intersection of several independent events is t'e prod#ct of t'eir separate individ#al pro!a!ilities: $2 ∩ ∩ ∩∩ n 4 = $2 4 $2 4 $2 4 $2 n 4 1 < = 1 < =
&'e pro!a!ilit$ of t'e union of several independent events is 1 min#s t'e prod#ct of pro!a!ilities of t'eir complements: $2 ∪ ∪ ∪∪ n 4 = 1 − $2 4 $2 4 $2 4 $2 n 4 1 < = 1 < =
Eample <-: $ 2' ∪' ∪' ∪∪' 4 =1− $ 2' 4 $ 2' 4 $ 2' 4 $ 2' 4 1 < = 10 1 < = 10 =1− 0+J010 =1− 0+=>KH = 0+@?1=
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'(3 Co6bin!tori!" Con5et# %onsider a pair of si-sided dice+ &'ere are si possi!le o#tcomes from t'rowing t'e first die 1,<,=,>,?,@ and si possi!le o#tcomes from t'rowing t'e second die 1,<,=,>,?,@+ .ltoget'er, t'ere are @;@ C =@ possi!le o#tcomes from t'rowing t'e two dice+ In general, if there are n events and the event i can ha))en in N i )ossible 4ays5 then the number of 4ays in 4hich the se6uence of n events may occur is N 1 N 2 ... N n+
ic5 ? cards from a dec5 of ?< - 4ith re)lacement
?<;?<;?<;?<;?,0=< different possi!le o#tcomes ?
ic5 ? cards from a dec5 of ?< - 4ithout re)lacement
?<;?1;?0;>J;>K C =11,K?,<00 different possi!le o#tcomes
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More on Co6bin!tori!" Con5et# 0-ree Di!$r!62
+ + ++ + + + + + +
9rder t'e letters: ., B, and %
%
B
%
B
.
%
%
.
.
B
. B %
B
.
+ ++ ++ +
.B%
.%B B.% B%. %.B %B.
'():
!5tori!" 'o4 many 4ays can you order the $ letters A5 5 and C7
&'ere are = c'oices for t'e first letter, < for t'e second, and 1 for t'e last, so t'ere are =;<;1 C @ possi!le wa$s to order t'e t'ree letters ., B, and %+ 'o4 many 4ays are there to order the letters A5 5 C5 (5 85 and 97 ,:&:%:$:2:# ; <21 9actorial: or an$ positive integer n, we define n factorial as: n(n-)*(n-2*...()*+ Le denote n factorial as n!+ &'e n#m!er n+ is t'e n#m!er of wa$s in w'ic' n o!ects can !e ordered+ B$ definition )+ ) an" 0+ ).
'()*
Per61t!tion# 0Order i# i6ort!nt2 =hat if 4e chose only $ out of the letters A5 5 C5 (5 85 and 97 &'ere are @ wa$s to c'oose t'e first letter, ? wa$s to c'oose t'e second letter, and > wa$s to c'oose t'e t'ird letter 2leaving = letters #nc'osen4+ &'at ma5es @;?;>C1<0 possi!le orderings or )ermutations+ Permutations are t'e possi!le ordered selections of r o!ects o#t of a total of n o!ects+ &'e n#m!er of perm#tations of n o!ects ta5en r at a time is denoted !$ nPr, w'ere
n> $ = n r ,n − r -> .or e-ample :
@ $ =
=
@M 2@ − =4M
=
@M =M
=
@ ; ? ; > ; = ; < ;1 = ; < ;1
= @ ; ? ; > = 1<0
'()'
Co6bin!tion# 0Order i# not I6ort!nt2 S#ppose t'at w'en we pic5 = letters o#t of t'e @ letters ., B, %, D, E, and we c'ose B%D, or BD%, or %BD, or %DB, or DB%, or D%B+ 2&'ese are t'e @ 2=4 perm#tations or orderings of t'e = letters B, %, and D+4 B#t t'ese are orderings of t'e same com!ination of = letters+ How man$ com!inations of @ different letters, ta5ing = at a time, are t'ereN Combinations are t'e possi!le selections of r items from a gro#p of n items regardless of t'e order of selection+ &'e n#m!er of com!inations is denoted and is read as n choose r + .n alternative notation is nCr. Le define t'e n#m!er of com!inations of r o#t of n elements as: n r
n = n C r = r
n! r! (n − r)!
.or e-ample :
n = @ C = = r
@M =M2@ − =4M
=
@M =M=M
=
@ ; ? ; > ; = ; < ;1 2= ; < ; 142= ; < ; 14
=
@;?; > = ; < ;1
=
1<0 @
= <0
'())
E;!6"e+ -e6"!te or C!"51"!tin$ Per61t!tion# & Co6bin!tion#
'(),
'(7 -he L!9 o -ot!" Prob!bi"ity !nd 4!ye#B -heore6 &'e law of total pro!a!ilit$: $2 4 = $2 ∩ #4 + $2 ∩ # 4
In terms of conditional pro!a!ilities: $2 4 = $2 ∩ #4 + $2 ∩ # 4 = $2 #4 $2 #4 + $2 # 4 $2 # 4
More generall$ 2w'ere Bi ma5e #p a partition4: $2 4 = ∑ $2 ∩ # 4 i = ∑ $2 # 4 $2 # 4 i i
'()/
-he L!9 o -ot!" Prob!bi"ity( E;!6"e '(8 Event U: Stoc5 mar5et will go #p in t'e net $ear Event L: Econom$ will do well in t'e net $ear $2/ 0 4 =+H? $2/ 0 4 = =0 $ 20 4 =+K0 ⇒ $ 20 4 = 1−+K =+< $ 2/ 4 = $2/ ∩ 0 4 + $2/ ∩ 0 4 = $2/ 0 4 $20 4 + $2/ 0 4 $20 4 = 2+H?42+K04 + 2+=042+<04 =+@0++0@ =+@@
'()3
4!ye#B -heore6
ayes? t'eorem ena!les $o#, 5nowing #st a little more t'an t'e pro!a!ilit$ of . given B, to find t'e pro!a!ilit$ of B given .+ Based on t'e definition of conditional pro!a!ilit$ and t'e law of total pro!a!ilit$+
$ 2 # 4
= = =
$ 2 #4 $ 2 4 $ 2 #4 $ 2 #4 + $ 2 # 4 $ 2 #4 $ 2 # 4 $ 2 #4 $ 2 # 4 + $ 2 # 4 $ 2 # 4
.ppl$ing t'e law of total pro!a!ilit$ to t'e denominator .ppl$ing t'e definition of conditional pro!a!ilit$ t'ro#g'o#t
'()7
4!ye#B -heore6 ( E;!6"e '(*:
. medical test for a rare disease 2affecting 0+13 of t'e pop#lation O P4 is imperfect:
$ 2 I 4 = 0+001
L'en administered to an ill person, t'e test will indicate so wit' pro!a!ilit$ 0+J< O P $ 2 1 I 4 =+J<⇒ $ 2 1 I 4 =+0K
&'e event 2 1 I 4 is a false negative
L'en administered to a person w'o is not ill, t'e test will erroneo#sl$ give a positive res#lt 2false positive4 wit' pro!a!ilit$ 0+0> O P
&'e event 2 1 I 4 is a false )ositive+
$ 2 1 I 4
=+0+0>⇒ $ 2 1 I 4 = 0+J@
'()
E;!6"e '(*: 05ontin1ed2 + $ 2 I 4 = 0001 + $ 2 I 4 = 0JJJ
$ 2 I 1 4
2 I 1 4 $ 2 1 4
= $ = =
$ 2 1 I 4 = 0+J< $ 2 1 I 4 = 0+0>
=
$ 2 I 1 4 $ 2 I 1 4 + $ 2 I
1 4
$ 2 1 I 4 $ 2 I 4 $ 2 1 I 4 $ 2 I 4 + $ 2 1 I 4 $ 2 I 4
2+J< 42 0+0014 2+J<42 0+0014 + 2 0+0> 42+JJJ 4
0+000J< 0+000J< + 0+0=JJ@ =+0<
=
=
0+000J< +0>0KK
'()8
E;!6"e '(*: 0-ree Di!$r!62 rior ro!a!ilities
%onditional ro!a!ilities $2 1 I 4
$2 I 4
+ = 0001
$ 2 I 4
+ = 0JJJ
= 0+J<
$ 2 1 I 4
= 0+0K
$ 2 1 I 4
= 0+0>
$2 1 I 4
= 0+J@
*oint ro!a!ilities $ 2 1 I 4
+ 420J< + 4 =+000J< = 2 0001
$ 2 1
+ 4200K + 4 =+0000K = 2 0001
I 4
$ 2 1 I 4
$ 2 1
+ 4200> + 4 =+0=JJ@ = 2 0JJJ
I 4
+ 420+J@4 =+J?J0> = 2 0JJJ
'(,:
4!ye#B -heore6 E;tended
Given a partition of events B 1,B< ,+++,Bn: $ 2 #1 4 =
=
$ 2 ∩ #1 4 $ 2 4
.ppl$ing t'e law of total pro!a!ilit$ to t'e denominator
$ 2 ∩ #1 4
∑ $2 ∩ # 4 i
=
$ 2 #1 4 $ 2 #1 4
∑ $2 # 4 $2 # 4 i
i
.ppl$ing t'e definition of conditional pro!a!ilit$ t'ro#g'o#t
'(,*
4!ye#B -heore6 E;tended ( E;!6"e '(**
.n economist !elieves t'at d#ring periods of 'ig' economic growt', t'e U+S+ dollar appreciates wit' pro!a!ilit$ 0+0Q in periods of moderate economic growt', t'e dollar appreciates wit' pro!a!ilit$ 0+>0Q and d#ring periods of low economic growt', t'e dollar appreciates wit' pro!a!ilit$ 0+<0+ D#ring an$ period of time, t'e pro!a!ilit$ of 'ig' economic growt' is 0+=0, t'e pro!a!ilit$ of moderate economic growt' is 0+?0, and t'e pro!a!ilit$ of low economic growt' is 0+?0+ S#ppose t'e dollar 'as !een appreciating d#ring t'e present period+ L'at is t'e pro!a!ilit$ we are eperiencing a period of 'ig' economic growt'N
Partition: H - Hig' growt' 2H4 C 0+=0 M - Moderate growt' 2M4 C 0+?0 ( - (ow growt' 2(4 C 0+<0
Event . − .ppreciation $2 H 4 = 0+H0 $2 M 4 = 0+>0 $2 24 = 0+<0
'(,'
E;!6"e '(** 05ontin1ed2
$ 2 H 4
= = = = = =
$ 2 H 4 $ 2 4 $ 2 H 4 $ 2 H 4
+ $2 M 4 + $2 2 4 $ 2 H 4 $ 2 H 4
$ 2 H 4 $ 2 H 4
+ $2 M 4 $2 M 4 + $2 24 $2 24
2 0+H042 0+=04 2 0+H042 0+=04 + 2 0+>042 0+?04 + 2 0+<0420+<04 0+<1 0+<1 = 0+<1 +0+<0 + 0+0> 0+>? 0+>@H
'(,)
E;!6"e '(** 0-ree Di!$r!62 rior ro!a!ilities
%onditional ro!a!ilities $ 2 H 4 = 0+H0
$ 2 H 4
=
0+=0
$ 2 H 4 = 0+=0 $ 2 M 4 = 0+>0
*oint ro!a!ilities $ 2 H 4 = 2 0+=0 42 0 +H0 4 = 0 +<1
$ 2 H 4 = 2 0+=0 42 0 +=0 4 = 0 +0J $ 2 M 4 = 2 0+?0 42 0 +>0 4 = 0 +<0
$ 2 M 4 = 0+?0
$ 2 M 4 = 0+@0 $ 2 M 4 = 2 0+?0 42 0 +@0 4 = 0 +=0 $ 2 2 4 = 0+<0
$ 2 2 4 = 0+<0
$ 2 2 4 = 0+K0
$ 2 2 4 = 2 0+<0 42 0+<0 4 = 0 +0>
$ 2 2 4 = 2 0+<0 42 0 +K0 4 = 0 +1@
'(,,
'( -he Joint Prob!bi"ity -!b"e
. oint pro!a!ilit$ ta!le is similar to a contingenc$ ta!le , ecept t'at it 'as pro!a!ilities in place of fre#encies+
&'e oint pro!a!ilit$ for Eample <-11 is s'own !elow+
&'e row totals and col#mn totals are called marginal pro!a!ilities+
'(,/
-he Joint Prob!bi"ity -!b"e
. ointpro!a!ilit$ ta!le pro!a!ilit$ ta!leisissimilar similarto toaacontingenc$ contingenc$ta!le ta!le, ,ecept eceptt'at t'atitit . oint 'aspro!a!ilities pro!a!ilitiesin inplace placeof offre#encies+ fre#encies+ 'as &'e oint pro!a!ilit$ for Eample <-11 is s'own on t'e net slide+ &'e oint pro!a!ilit$ for Eample <-11 is s'own on t'e net slide+
&'erow rowtotals totalsand andcol#mn col#mntotals totalsare arecalled calledmarginal marginalpro!a!ilities+ pro!a!ilities+ &'e
'(,3
-he Joint Prob!bi"ity -!b"e+ E;!6"e '(**
&'e oint pro!a!ilit$ ta!le for Eample <-11 is s#mmariRed !elow+ 'igh
!edium
@o4
/otal
A))reciates
132#
132
131%
13%&
(e)reciates
131B
13$
13#
13&&
/otal
13$1
13&
1321
#311
Marginal pro!a!ilities are t'e ro4 totals and t'e column totals+
'(,7
'( U#in$ Co61ter+ -e6"!te or C!"51"!tin$ the Prob!bi"ity o !t "e!#t one #155e##
'(,
'( U#in$ Co61ter+ -e6"!te or C!"51"!tin$ the Prob!bi"itie# ro6 ! Contin$en5y -!b"e(E;!6"e '(**
'(,8
'( U#in$ Co61ter+ -e6"!te or 4!ye#i!n Rei#ion o Prob!bi"itie#(E;!6"e '(**
