Introduction to Statistics Chapter 1 Web

8/15/2019 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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=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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)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 arri&ethat 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%


39/54

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


44/54

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.


45/54

$he *ational

5ottery


46/54

*+ 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


47/54

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


49/54

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


50/54

$ossing a fair coin


51/54

$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


52/54

$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.


53/54

8osencrantMand

uildensternare 0ead


54/54

#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

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