Introduction to Probability and Statistics Chapter 7 Sampling Distributions.
Introduction to Statistics Chapter 1 Web
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Introductionto
Statistics
Dr. P MurphyDr. P Murphy
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We like to think that we haveWe like to think that we havecontrol over our lives.control over our lives.
But in reality there are manyBut in reality there are many
things that are outside ourthings that are outside ourcontrol.control.
Everyday we are confrontedEveryday we are confronted
by our own ignorance.by our own ignorance.
According to Albert Einstein:According to Albert Einstein:
““od does not !lay dice."od does not !lay dice."
But we all should knowBut we all should knowbetter than #rof. Einstein.better than #rof. Einstein.
$he world is governed by $he world is governed by
%uantum &echanics where%uantum &echanics where
#robability reigns su!reme.#robability reigns su!reme.
Why study Statistics'
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(ou wake u! in the morning (ou wake u! in the morning
and the sunlight hits yourand the sunlight hits your
eyes. $hen suddenly withouteyes. $hen suddenly withoutwarning the world becomeswarning the world becomes
an uncertain !lace.an uncertain !lace.
)ow long will you have to)ow long will you have towait for the *umber +, Buswait for the *umber +, Bus
this morning'this morning'
When it arrives will it beWhen it arrives will it be
full'full'
Will it be out of service'Will it be out of service'
Will it be raining while youWill it be raining while you
wait'wait'
onsider a day in thelife of an average
/0 student.
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It is used by #hysicists toIt is used by #hysicists to
!redict the behaviour of!redict the behaviour of
elementary !articles.elementary !articles.It is used by engineers toIt is used by engineers to
build com!uters.build com!uters.
It is used by economists toIt is used by economists to!redict the behaviour of the!redict the behaviour of the
economy.economy.
It is used by stockbrokersIt is used by stockbrokersto make money on theto make money on the
stockmarket.stockmarket.
It is used by !sychologistsIt is used by !sychologists
to determine if you shouldto determine if you should
#robabilityis the
Science of /ncertainty.
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Statistics is the Science ofStatistics is the Science of
0ata.0ata.
$he Statistics you have $he Statistics you have
seen before has beenseen before has been
!robably been 0escri!tive!robably been 0escri!tive
Statistics.Statistics.And 0escri!tive StatisticsAnd 0escri!tive Statistics
made you feel like this 2.made you feel like this 2.
What about Statistics'
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It is a disci!line that allowsIt is a disci!line that allows
us to estimate unknownus to estimate unknown
3uantities by making some3uantities by making someelementary measurements.elementary measurements.
/sing these estimates we/sing these estimates we
can thencan thenmake #redictions andmake #redictions and
4orecast the 4uture4orecast the 4uture
What isInferential Statistics'
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ha!ter +
#robability
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an you make moneyan you make money
!laying the 5ottery'!laying the 5ottery'
5et us calculate chances of5et us calculate chances of
winning.winning.
$o do this we need to learn $o do this we need to learn
some basic rules aboutsome basic rules about!robability.!robability.
$hese rules are mainly 1ust $hese rules are mainly 1ust
ways of formalising basicways of formalising basiccommon sense .common sense .
E6am!le: What are theE6am!le: What are the
chances that you get a )EA0chances that you get a )EA0
when you toss a coin'when you toss a coin'
onsidera 8eal #roblem
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AnAn EE6!eriment6!eriment leads to aleads to a
single outcome whichsingle outcome which
cannot be !redicted withcannot be !redicted with
certainty.certainty. E6am!lesE6am!les
$oss a coin $oss a coin:: head or tailhead or tail
8oll a die8oll a die:: +; ; ?;+; ; ?;@@
$ake medicine $ake medicine:: worse;worse;
same; bettersame; better
Set of allSet of all outcomesoutcomes
SSampleample SSpacepace..
+.+ E6!eriments
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$he $he ##robabilityrobability of aof ann
outcome is aoutcome is a numbernumber
between , and +between , and + thatthat
measures themeasures the likelihoodlikelihoodthat the outcome willthat the outcome will
occuroccur when thewhen the
e6!eriment is !erformed.e6!eriment is !erformed.D,im!ossible; +certain.D,im!ossible; +certain.
#robabilities of all sam!le#robabilities of all sam!le
!oints must sum to +.!oints must sum to +.
5ong run relative5ong run relative
fre3uency inter!retation.fre3uency inter!retation.
+.< #robability
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AnAn eventevent is a s!eciGcis a s!eciGc
collection of sam!lecollection of sam!le
!oints.!oints.
$he !robability of an $he !robability of an
event A is calculated byevent A is calculated bysumming the !robabilitiessumming the !robabilities
of theof the outcomesoutcomes in thein the
sam!le s!ace for A.sam!le s!ace for A.
+.= Events
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0eGne the e6!eriment.0eGne the e6!eriment.
5ist the sam!le !oints.5ist the sam!le !oints.Assign !robabilities to theAssign !robabilities to the
sam!le !oints.sam!le !oints.
0etermine the collection of0etermine the collection ofsam!le !oints contained insam!le !oints contained in
the event of interest.the event of interest.
Sum the sam!le !ointSum the sam!le !oint!robabilities to get the event!robabilities to get the event
!robability.!robability.
+.> Ste!s forcalculating
#robailities
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E6am!le: $)E A&E Hf
8A#SIn Craps one rolls two fair dice.In Craps one rolls two fair dice.
What is the probability of theWhat is the probability of thesum of the two dice showing 7?sum of the two dice showing 7?
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(1,6)
(2,5)
(3,4)
(4,3)(5,2)
(6,1)
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
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+.? E3ually likelyoutcomes
So the Probability of 7 whenSo the Probability of 7 when
rolling two dice is 1/rolling two dice is 1/
!his e"ample illustrates the!his e"ample illustrates the
following rule#following rule#
In a Sample Space S of e$uallyIn a Sample Space S of e$ually
li%ely outcomes. !heli%ely outcomes. !he
probability of the e&ent ' is probability of the e&ent ' is
gi&en bygi&en by
P(') * +' / +SP(') * +' / +S
!hat is the number of outcomes!hat is the number of outcomes
in ' di&ided by the total numberin ' di&ided by the total number
of e&ents in S.of e&ents in S.
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AA compound event compound event is ais a
com!osition of two or morecom!osition of two or more
other events.other events.
AA:: $he $he Complement Complement of A isof A is
the event thatthe event that A does notA does not
occuroccur
AA∪∪BB :: $he $he UUnionnion of twoof two
events A and B is the eventevents A and B is the event
that occurs ifthat occurs if either A or Beither A or B
or both occuror both occur; it consists of; it consists ofall sam!le !oints thatall sam!le !oints that
belong to A or B or both.belong to A or B or both.
AA∩∩BB:: $he $he IIntersectionntersection ofof
+.@ Sets
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+.7 Basic#robability 8ules
P('P('cc)*1,P('))*1,P(')
P(P(''∪∪--)*P(')P(-),P()*P(')P(-),P(''∩∩--))
utually 0"clusi&e 0&ents areutually 0"clusi&e 0&ents are
e&ents which cannot occur ate&ents which cannot occur at
the same time.the same time.
PP(''∩∩--)*)* for utuallyfor utually
0"clusi&e 0&ents.0"clusi&e 0&ents.
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+. onditional#robability
P(' 2 -) 3 Probability of 'P(' 2 -) 3 Probability of '
occuring gi&en that - hasoccuring gi&en that - has
occurred.occurred.
P(' 2 -) *P(' 2 -) * PP(''∩∩--) / P(-)) / P(-)
ultiplicati&e 4ule#ultiplicati&e 4ule#
PP(''∩∩--))
* P('2-)P(-)* P('2-)P(-)
* P(-2')P(')* P(-2')P(')
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+.- Inde!endentEvents
' and - are independent e&ents' and - are independent e&ents
if the occurrence of one e&entif the occurrence of one e&ent
does not affect the probabilitydoes not affect the probability
of the othe e&ent.of the othe e&ent.
If ' and - are independent thenIf ' and - are independent then
P('2-)*P(')P('2-)*P(')
P(-2')*P(-)P(-2')*P(-)
PP(''∩∩--)*P(')P(-))*P(')P(-)
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ha!ter +#robability
EFAES
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Probability as
a matter oflife and death
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Positi&e !est for 5isease
1 in e&ery 1 people in Ireland1 in e&ery 1 people in Ireland
suffer from 'I5Ssuffer from 'I5S
!here is a test for 6I/'I5S!here is a test for 6I/'I5S
which is 89: accurate.which is 89: accurate.
;ou are not feeling well and you;ou are not feeling well and you
go to hospital where yourgo to hospital where your
Physician tests you.Physician tests you.6e says you are positi&e for 'I5S6e says you are positi&e for 'I5S
and tells you that you ha&e 1
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Positi&e !est for 5isease
=et 5 be the e&ent that you=et 5 be the e&ent that you
ha&e 'I5Sha&e 'I5S
=et ! be the e&ent that you test=et ! be the e&ent that you test
positi&e for 'I5S positi&e for 'I5S
P(5)*.1P(5)*.1
P(!25)*.89P(!25)*.89
P(52!)*?P(52!)*?
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Positi&e !est for 5isease
)8888.)(9.()1.)(89.(
)1.)(89.(
)()2()()2()()2(
)()(
)()2(>)>(
)()2(
)()()2(
+
=
+
=
+
=
=
=
C C
C
C
D P DT P D P DT P D P DT P
DT P DT P
D P DT P
DT DT P
D P DT P
T P T D P T D P
1
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ha!ter +E6am!les
0"ample 1.10"ample 1.1
S*'@-@C>S*'@-@C>
P(') * AP(') * A
P(-) * 1/BP(-) * 1/B
P(C) * 1/P(C) * 1/
What is P('@->)?What is P('@->)?
What is P('@-@C>)?What is P('@-@C>)?=ist all e&ents such that=ist all e&ents such that
P() * A.P() * A.
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ha!ter +E6am!les
0"ample 1.D0"ample 1.D
Suppose that a lecturer arri&esSuppose that a lecturer arri&es
late to class 1: of the time@late to class 1: of the time@
lea&es early D: of the timelea&es early D: of the time
and both arri&es late 'E5and both arri&es late 'E5
lea&es early 9: of the time.lea&es early 9: of the time.
Fn a gi&en day what is theFn a gi&en day what is the
probability that on a gi&en day probability that on a gi&en day
that lecturer will either arriðat lecturer will either arri&elate or lea&e early?late or lea&e early?
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ha!ter +E6am!les
0"ample 1.B0"ample 1.B
Suppose you are dealt 9 cardsSuppose you are dealt 9 cards
from a dec% of 9D playing cards.from a dec% of 9D playing cards.
Gind the probability of theGind the probability of the
following e&entsfollowing e&ents
1. 'll four aces and the %ing of
spades
D. 'll 9 cards are spades
B. 'll 9 cards are different
H. ' Gull 6ouse (B same@ D
same)
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ha!ter +E6am!les
0"ample 1.H0"ample 1.H
!he -irthday Problem!he -irthday Problem
Suppose there are E people in aSuppose there are E people in a
room.room.
6ow large should E be so that6ow large should E be so that
there is a more than 9: chancethere is a more than 9: chance
that at least two people in thethat at least two people in the
room ha&e the same birthday?room ha&e the same birthday?
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Number in Room Prob at least 2 have same birthday
1 0.00
2 0.00
3 0.01
4 0.02
5 0.03
6 0.04
7 0.06
8 0.07
9 0.09
10 0.1211 0.14
12 0.17
13 0.19
14 0.22
15 0.25
16 0.28
17 0.32
18 0.35
19 0.38
20 0.41
21 0.4422 0.48
23 0.51
24 0.54
25 0.57
26 0.60
27 0.63
28 0.65
29 0.68
30 0.71
31 0.73
32 0.7533 0.77
34 0.80
35 0.81
36 0.83
37 0.85
38 0.86
39 0.88
40 0.89
41 0.90
42 0.91
43 0.9244 0.93
45 0.94
46 0.95
47 0.95
48 0.96
49 0.97
50 0.97
51 0.97
52 0.98
53 0.98
54 0.9855 0.99
56 0.99
57 0.99
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ha!ter +E6am!les
0"ample 1.H0"ample 1.H
Children are born e$ually li%elyChildren are born e$ually li%ely
as -oys or irlsas -oys or irls
y brother has two childreny brother has two children
(not twins)(not twins)
Fne of his children is a boyFne of his children is a boy
named =u%enamed =u%e
What is the probability that hisWhat is the probability that his
other child is a girl?other child is a girl?
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0"ample 1.9
!he onty 6all Problem ame Showame Show
B doorsB doors
1 Car J D oats1 Car J D oats
;ou pic% a door , e.g. +1;ou pic% a door , e.g. +1
6ost %nows whatKs behind all6ost %nows whatKs behind all
the doors and he opens anotherthe doors and he opens another
door@ say +B@ and shows you adoor@ say +B@ and shows you a
goatgoat
6e then as%s if you want to6e then as%s if you want tostic% with your original choicestic% with your original choice
+1@ or change to door +D?+1@ or change to door +D?
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's% arilyn.Parade agaLine Sept 8 188
arilyn &os Sa&antarilyn &os Sa&ant
uinness -oo% of 4ecordsuinness -oo% of 4ecords,6ighest I,6ighest I
MM;es you should switch. !he;es you should switch. !hefirst door has a 1/B chance offirst door has a 1/B chance ofwinning while the second has awinning while the second has aD/B chance of winning.ND/B chance of winning.N
Ph.5.s , Eow two doors@ 1 goatPh.5.s , Eow two doors@ 1 goatJ 1 car so chances of winningJ 1 car so chances of winningare 1/D for door +1 and 1/D forare 1/D for door +1 and 1/D for
door +D.door +D. MM;ou are the goatN;ou are the goatN , Western, WesternState Oni&ersity.State Oni&ersity.
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WhoKs right? 't the start@ the sample space is#'t the start@ the sample space is#
CCGG,GG, GGCG,CG, GGGCGC>> Pic% a door e.g. +1Pic% a door e.g. +1
1 in B chance of winning1 in B chance of winning
6ost shows you a goat so now6ost shows you a goat so now
CCGGGG,, GGCCGG,, GGGGCC>> So arilyn was right@ you shouldSo arilyn was right@ you shouldswitch.switch.
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Eot con&inced? Imagine a game with 1 doors.Imagine a game with 1 doors.
1 GHB Gerrari@ 88 oats.1 GHB Gerrari@ 88 oats.
;ou pic% a door.;ou pic% a door.
6ost opens 8< of the 88 other6ost opens 8< of the 88 other
doors.doors.
5o you stic% with your original5o you stic% with your original
choice?choice? Prob * 1/1Prob * 1/1
Fr mo&e to the unopened door.Fr mo&e to the unopened door.
Prob * 88/1Prob * 88/1
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-oys@ irls
and onty 6all Sample Space ( listing oldest childSample Space ( listing oldest childfirst)first)
@ -@ -@ -->@ -@ -@ -->
0$ually li%ely e&ents0$ually li%ely e&ents
Fne child is a boy#Fne child is a boy# is impossible is impossible
-@ -@ --> *-@ -@ --> *
P(FC * ) * D/BP(FC * ) * D/B
=u%e is months old.=u%e is months old.
-@ --> *-@ --> * P(FC * ) * 1/DP(FC * ) * 1/D
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Hdd Socks
It is winter and the ESBIt is winter and the ESBare on strike. $hisare on strike. $his
morning when you wokemorning when you woke
u! it was dark. In youru! it was dark. In your
sock drawer there wassock drawer there wasone !air of two blackone !air of two black
socks and one odd brownsocks and one odd brown
one.one.
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EFA&SCampusCampus FemaleFemale
Pass RatePass Rate
MaleMale
Pass RatePass Rate
BelfieldBelfield 40%40% 33%33%
E!E!
Ca"#sf$"tCa"#sf$"t
et&et&
'5%'5% '1%'1%
Seeing this evidenceSeeing this evidence
amale student takesamale student takes
/0 to court saying/0 to court sayingthere is disciminationthere is discimination
against male students.against male students.
/0 gathers all itJs/0 gathers all itJse6am informatione6am information
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EFA&
#ass 8atesHverall 4emale !ass rateHverall 4emale !ass rate
is ?@Kis ?@KHverall &ale !ass rate isHverall &ale !ass rate is
@,K@,K
)HW A* $)IS BE')HW A* $)IS BE'
learly /0learly /0 areare 5(I* L5(I* L
CampusCampus FemaleFemale
Pass RatePass Rate
MaleMale
Pass RatePass Rate
BelfieldBelfield 40%40% 33%33%
E!E!
Ca"#sf$"tCa"#sf$"t
et&et&
'5%'5% '1%'1%
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Sim!sonJs
#arado6Hverall 4emale !ass rateHverall 4emale !ass rate
is ?@Kis ?@KHverall &ale !ass rate isHverall &ale !ass rate is
@,K@,KCampusCampus FemaleFemale
Pass RatePass Rate
MaleMale
Pass RatePass Rate
BelfieldBelfield 40%40%
20!50 20!50
33%33%
10!30 10!30
E!E!
Ca"#sf$"tCa"#sf$"t
et&et&
30!4030!40
'5%'5%
50!'050!'0
'1% '1%
50!050!0
56% 56%
60!10060!100
60%60%
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)it and 8/*Hnce u!on a time inHnce u!on a time in
)icksville; /SA there was)icksville; /SA there was
a nighttime hit and runa nighttime hit and run
accident involving a ta6i.accident involving a ta6i. $here are two ta6i $here are two ta6i
com!anies in )icksville;com!anies in )icksville;
reen and Blue. ?K ofreen and Blue. ?K of
ta6is are reen and +?Kta6is are reen and +?K
are Blue. A witnessare Blue. A witness
identiGed the ta6i asidentiGed the ta6i as
being Blue. In thebeing Blue. In thesubse3uent court case thesubse3uent court case the
1udge ordered that the 1udge ordered that the
witnessJs observationwitnessJs observation
under the conditions thatunder the conditions that
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)it and 8/*What is the !robabilityWhat is the !robability
that it was indeed a bluethat it was indeed a blue
ta6i that was involved inta6i that was involved in
the accident'the accident'
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0*A (ou are holiday in Belfast (ou are holiday in Belfast
and an e6!losion destroysand an e6!losion destroys
the Hdessey arena.the Hdessey arena.
(ou are seen running from (ou are seen running from
the e6!losion and arethe e6!losion and are
arrested.arrested.
(ou are subse3uently (ou are subse3uently
charged with being acharged with being a
member of a !rescribedmember of a !rescribed
!aramilitary organisation!aramilitary organisationand with causing theand with causing the
e6!losion.e6!losion.
In court you !rotest yourIn court you !rotest your
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0*A $heir forensic scientist $heir forensic scientist
delivers the following vitaldelivers the following vital
evidence.evidence.
$he forensic scientist $he forensic scientist
indicates that 0*A foundindicates that 0*A found
on the bomb matcheson the bomb matches
your 0*A.your 0*A.
(our lawyer at Grst (our lawyer at Grst
dis!utes this evidence anddis!utes this evidence and
hires an inde!endenthires an inde!endentscientist.scientist.
)owever the second)owever the second
forensic scientist also saysforensic scientist also says
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0*AWhat do you do'What do you do'
It a!!ears as if you areIt a!!ears as if you are
going to s!end the rest ofgoing to s!end the rest of
your days in 1ail.your days in 1ail.
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$he *ational
5ottery
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*+ lied, eated a-d st$le t$
.e$me a milli$-ai"e& /$a-#.$d# at all a- i- te
l$tte"# a-d .e$me a
milli$-ai"e
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GME 1 6!42
What are the chance of winningWhat are the chance of winning
with one selection of numbers?with one selection of numbers?
atchesatches Chances of WinningChances of Winning
1 in 9@DH9@7
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GME 1 6!42
0"pected Winnings0"pected Winnings
Fnly consider Qac%potFnly consider Qac%pot
1 0uro get 1 play1 0uro get 1 play
0(win)* Qac%potR(1/9@DH9@7
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6!42
!he a&erage time to win each of the priLes is!he a&erage time to win each of the priLes is
gi&en by#gi&en by#
atch B with -onusatch B with -onus 2
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$ossing a fair coin
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$ossing a coinL
;ou are To%ingU;ou are To%ingU
!hat is boring V no $uestion about itU!hat is boring V no $uestion about itU
1897 Second edition of William GellerKs1897 Second edition of William GellerKs
!e"tboo% includes a chapter on coin,!e"tboo% includes a chapter on coin,tossing.tossing.
Introduction#Introduction# M!he results concerningM!he results concerningVcoin,tossing show that widely heldVcoin,tossing show that widely held
beliefs V are fallacious. !hese results beliefs V are fallacious. !hese results
are so amaLing and so at &ariance withare so amaLing and so at &ariance with
common intuition that e&encommon intuition that e&ensophisticated colleagues doubted thatsophisticated colleagues doubted that
coins actually misbeha&e as theorycoins actually misbeha&e as theory
predicts.N predicts.N
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$ossing a coinL
!oss a coin DE times.!oss a coin DE times.
=aw of '&erages#=aw of '&erages#
's E increases the chances that's E increases the chances thatthere are e$ual numbers of headsthere are e$ual numbers of heads
and tails among the DE tossesand tails among the DE tosses
increases.increases.
=im=im E,P E,P∞∞ P( +6 * +! ) * 1.P( +6 * +! ) * 1.
In the limit as E tends to infinityIn the limit as E tends to infinity
the probability of matchingthe probability of matching
numbers of heads and tailsnumbers of heads and tails
approaches 1.approaches 1.
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8osencrantMand
uildensternare 0ead
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#rob of e3ual
numbers of ) and $
$f $ft$ssest$sses
22 44 66 ;; 1010
?? 3!;3!; 5!165!16 35!12;35!12; 63!25663!256
P"$.P"$. 0&50&5 0&3'50&3'5 0&31250&3125 0&2'30&2'3 0&2460&246