SecondeeuropéenneExercicesdemathématiques Chapitre6 SimulationetéchantillonageSimulatingandsampling

Seconde européenne

Exercices de mathématiques

Chapitre 6

Simulation et échantillonage Simulating and sampling

Yucca Muffin , by Milo Beckman

At the end of this chapter, you should be able to :

• compute the margin of error and fluctuation interval at 95% confi- dence for a known probability ;

• use a fluctuation interval to accept or reject an assumption ;

• compute the confidence interval at 95% for a sample ;

• use a confidence interval to estimate a probability ;

• use the calculator to simulate a random experiment.

Aymar de Saint-Seine et Mickaël Védrine

Année scolaire 2010/2011

Margin of error

6.1 Estimation and sampling : an election

In 2008, the Amerian eletors had to hoose between the Republian John MCain and the

Demorat Barak Obama. OnEletion day,Obama won with53% of thevotes.Of ourse,this

wasnot knownto the andidatesbeforetheeletion. Surveys wereorganised byboth partiesto

estimate the proportion of eletors who wanted to vote for eah andidate. As it's impossible

to gather theopinionsofall the eletors,surveys arearriedoversmall partsof thepopulation,

alledsamples. We will onsiderthat samplesarebuilt randomly.

Part A – Point estimate

1

^{. Overasampleof900eletors,497delaredthattheywantedto votefor Obama.Compute}

theperentage ofpotential Obama eletors inthissample.

2

^{. Theperentageomputedfor thesampleisapointestimate oftheatualperentageinthe}

whole population. Compute the dierene between this point estimateand the truevalue

(knownonly afterthe eletion). Whatdo you think ofthis estimation?

3

^{. Ten other surveys wereorganized over the same period. The size of eah sample and the}

numberof potential Obama eletorsaregiven inthetable below.

Survey 1 2 3 4 5 6 7 8 9 10

Size 895 873 900 885 899 842 878 900 897 892

Obama eletors 462 493 501 437 467 447 468 495 488 478

a

^{. Compute the point estimatefor eah survey.Round theanswersto 2DP.}

b

^{. Howmany surveys gave apoint estimateequal to thetrueperentage?}

c

^{. IfMCain had only knownabout the4th survey,what ouldhe havededued?}

d

^{. What doyouthink of thepoint estimatemethod?}

Part B – Margin of error

Estimationbypoint estimateisnot averygoodmethod.Thehanesthatthesamplewill yield

the true value in the whole population are very small. Furthermore, There may be important

dierenes between the point estimates obtained from dierent samples. This phenomenon is

knownassampling utuation.

To illustrate this, one hundred surveys were simulated with a omputer, eah one over a po-

pulation of 900 people. (To do so, the perentage of Obama eletors in the whole population,

p = 0.53

^{,wasused). The satterplotbelowshows theperentage ofpotential Obamaeletors in}

eah survey.

0.48 0.50 0.52 0.54 0.56 0.58

0 10 20 30 40 50 60 70 80 90 100

b b b b b b

b b

b b b b

b b b

b b

b b b b b b

b b

b b

b b

b b

b b

b b b

b b b

b b

b b b

b b b b b b b b b b

b b

b b b b

b b

b b b b b b b

b b

b b

b b

b b

b b

b b

b b b

b b b b b b b b b

b b

b b b b b

1

^{. Onthe}satterplot, showtheperentage

p

^{ofObamaeletors inthewholepopulationwith}

a horizontal redline.How manysimulated surveys gave thatexat value?

2

^.

a

^{. The value}

m = √ ¹ n

,where

n

^{is the size of a sample, is alledthe margin of error at}

95% ondene for thatsample. Computethis valueto 3DP.

b

^{. Onthe graph, showthevalues}

p − m

^and

p + m

^{withtwohorizontal bluelines.}

c

^{. Howmanysurveysgaveapointestimateinluded intheinterval}

[p −m; p + m]

^,alled

utuation interval at

95%

^ondene?

d

^{. Is the answerto thepreviousquestion onsistent withthenameof theinterval?}

3

^{. Computethemarginoferrorandutuation intervalat95%ondeneforsamplesofsize}

n = 25

^{, then}

n = 100

^and

n = 500

^{. What do you notie about the marginof error when}

thesize of the sampleinreases?

6.2 The French lottery and odd numbers

The priniples of the Frenh National Lottery (Loto) are fairly simple. Eah player piks six

numbers (plus one, that we won't onsider in this exerise) between 1 and 49. On lottery day,

49 ballswith numbersfrom 1 to 49 arerandomly drawnfrom a mahine. The ballsarenot put

bakinthemahine,sothe samenumberannotappeartwieinadrawing.The orderinwhih

theballsaredrawn isirrelevant.

Among thenumbers from 1to 49,there are25 oddnumbers and 24 even numbers. So it seems

thatthe Frenh lottery favours odd numbers. Thisiswhat we will study inthisexerise.

Part A – Drawing a single number

In this part,weonsider the random experiment that onsistsindrawing asingle ball fromthe

49 inthemahine.

1

^{. What isthe}probabilityofthedrawnnumberbeingodd?Give theresultasan irreduible frationand asan approximate value to2DP.

2

^{. Fifty samples,eah made of}

n = 100

independant drawings of a ball were simulated with a omputer. For eah sample, theproportion of odd numbers was omputed. The results

of theseftysamples of size

100

^{are given below.}

0.44 0.52 0.50 0.44 0.51 0.41 0.44 0.40 0.57 0.50

0.51 0.43 0.59 0.46 0.55 0.35 0.55 0.43 0.53 0.53

0.45 0.42 0.47 0.48 0.50 0.45 0.48 0.47 0.46 0.57

0.52 0.55 0.53 0.46 0.45 0.44 0.45 0.48 0.51 0.46

0.55 0.48 0.43 0.51 0.49 0.38 0.52 0.40 0.50 0.46

a

^{. Howmany samplesshowed a proportion equal to thetheoretialvalueto 2DP?}

b

^{. Compute the utuationinterval at}

95%

^ondene.

c

^{. Howmany samplesshowed a proportion insidethemargin of error?}

d

^{. Can yound amargin oferror at}

99%

^ondene?

Part B – Drawing six numbers

In this seond part, we onsider the random experiment that onsists of drawing suessively

six balls, without putting them bak inthe mahine. It an be proven that in eah drawing of

six numbers, there is an average of

3.0612

^{odd numbers, so a proportion}

q = ^3.0612 ₆ ≈ 0.51

^{, or}

approximately

51%

^.

Fiftysamples,eahmade of

n = 100

independant drawingsof sixsuesiveballsweresimulated withaomputer. Foreah sample,the proportionofoddnumberswasomputed.Theresultsof

these ftysamples ofsize

100

^{aregivenbelow.}

0.515 0.468 0.517 0.485 0.498 0.473 0.505 0.507 0.492 0.498

0.508 0.408 0.563 0.612 0.542 0.497 0.508 0.498 0.500 0.535

0.498 0.508 0.525 0.478 0.517 0.528 0.492 0.487 0.535 0.523

0.512 0.543 0.522 0.482 0.530 0.478 0.508 0.532 0.528 0.527

1

^{. Compute the utuationinterval at}

95%

^ondene.

2

^{. Howmany samplesshowed a proportion insidetheinterval?}

Part C – Probabilities on the number of odd numbers

The table belowshows theprobabilities of drawing

k

^{odd numbers amongthesix, for}

k

^from

0

to

6

^{.Values havebeen roundedto 3DP.}

Oddnumbers

0 1 2 3 4 5 6

Probability

0.010 0.076 0.228 0.333 0.250 0.091 0.013

Foreahofthefollowingsentenes,sayifit'strueorfalse.Justifyeahanswerwithaomputation

or an explanation.

1

^{. There aremore hanesto draw4 oddnumbers or morethan 2 odd numbers orless.}

2

^{. There areasmany hanes to drawat least3odd numbersthan at least3 even numbers.}

3

^{. There aremore than}

90%

^{hanes to draw at least2oddnumbers.}

4

^{. There areasmany hanes to drawexatly3 odd numbersthan exatly3 even numbers.}

5

^{. There are}

50%

^{hanes to drawasmanyoddnumbers and even numbers.}

6

^{. There aremore hanesto drawno even numberthan to drawno oddnumber.}

6.3 Using margins of error to make decisions

Part A – Accepting or rejecting an assumption

It isknow thatintheFrenhpopulation,

26%

^{areallergi to pollen.In onepartiular sampleof}

400 people,120 suerfrom thatallergy.

1

^{. Compute the utuationinterval at}

95%

^ondene.

2

^{. What isthe frequeny ofallergi} individualsinthis sample?

3

^{. Wouldyouonsider this sample}representative of theFrenh population?

Part B – Parity in French Region councils

After the 2004 regional eletions in Frane, the repartition between women and men in four

regionalounilswasasfollows.Weonsiderthattheseounils arerandomsamplesoftheloal

politiianpopulationinFrane.

Men Women Total

Burgundy 32 25 57

Brittany 38 47 85

Rhne-Alpes 81 76 57

Île-de-Frane 103 106 209

1

^{. Supposing that parity between men andwomen isreal in a regionalounil, what should}

betheperentage ofwomeninthatounil?

2

^{. Find out the utuation interval at}

95%

^{ondene for the proportion of women ineah}

ounil.

3

^{. Whatdoyouthinkoftheparitybetweenmenandwomenintheloalpolitiianpopulation}

inFrane.

Part C – A car factory

Inaarfatory,aontrolisdoneforawsofthetypegrainyspotsonthehood.Normally,

20%

of thevehilespresent thiskind ofaws.Whileontrolling a randomsampleof 50vehiles, itis

seen that13 vehileshave it. Shoulditbea matterof onern?

Part D – Rodrigo Partida’s case

In 1970, theMexian-Amerian RodrigoPartida was sentened to eight yearsof prison. He ap-

pealed to thejudgment ontending thathe was denied due proess and equal protetion of law

beausethegrandjuryofHidalgoCounty,Texas,whihinditedhim,wasunonstitutionallyun-

derrepresentedbyMexian-Amerians.Heintroduedevidenethatin1970,thetotalpopulation

of HidalgoCountywas181,535 personsof whih143,611, or approximately79.2% werepersons

of Spanish language or Spanish surname. Next, he presented evidene showing theomposition

ofthegrandjurylistsoveraperiodoftenyearspriorto andinludingthetermofourtinwhih

theinditment againsthim wasreturned. Ofthe870 personsseleted for grandjury duty,only

39.0%wereMexian-Amerians.Ifyouwereajudgeintheourtofappeals,howwouldyoureat

to these allegations?

Confidence interval

6.4

^{In thisexerise, we lookagainatthe US 2008 eletion.Wewill introdue a better method}

of estimation, based on the onept of margin of error. Instead of a simple point estimate, we

will build for eah samplea ondene interval whose diameter dependson themargin of error

weallow.

We still note

p

^{the perentage of Obama eletors inthe whole population (so}

p = 0.53

^{). Now,}

onsiderasamplefosize

n

^{yielding apoint estimate}

f

^of

p

^{.We'veseeninthepreviouspartthat}

themargin of error at 95%ondene is

m = √ ¹ n

.Indeed, theprobabilityof thepoint estimate

f

^{beingintheinterval}

h

p − √ ¹ n , p + √ ¹ n

i

is approximately equal to

95%

^.

1

^{. Translatethe fatthat}

f

^{belongs to thatintervalwithtwo} inequalities.

2

^{. Prove thatthe fatthat}

f

^{belongs to thatintervalisequivalent to thefatthat}

p

^belongs

to theinterval

h f − √ ¹

n , f + ^{√ 1} _n i

.

The interval

h f − √ ¹

n , f + √ ¹ n

i

is alled a

95%

^{ondene interval.} Intuitively, this means that, knowing

f

^{and not}

p

^{, we have a}

5%

^{riskof beingwrong ifwe onsiderthat}

p

^{isin theinterval.}

But, as

p

^{is xed,it's notreally orretto talkabout} probability.One theondeninterval is determined,

p

^{iseitherinitor not!}

1

^{. Find the}

95%

^{ondene intervals for the surveys ofexerie 1 partA.}

2

^{. Howmany surveys gave aondene interval inludingthereal value?}

6.5 The referendum on the European constitution

1

^{. The Frenh referendum on the Treaty} establishing a Constitution for Europe was held on 29 May 2005 to deide whetherFrane should ratifythe proposed Constitution of the

EuropeanUnion. Thequestionputtovoters was:Doyouapprovethebillauthorisingthe

ratiation of the treatyestablishing aConstitution for Europe?

Belowaregiven the results ofsome surveys arriedout before thereferendum.

18 and 19Marh 2005 Ipsos 860 0.52

25 and 26Marh 2005 Ipsos 944 0.54

1erand 2April2005 Ipsos 947 0.52

16 and 17Marh 2005 CSA 802 0.51

23 Marh 2005 CSA 856 0.55

1and 2April2005 Louis Harris 1004 0.54

31 Marh and1 April2005 IFOP 868 0.55

24 Marh 2005 IFOP 817 0.53

a

^{. Find the}

95%

^{ondene interval for eahsurvey.}

b

^{. The result was a vitoryfor the}"No"ampaign, with

54.67%

^{. A ommentator then}

saidthatnotmanysurveyshadantiipated suhadeisiveresult.Whatdoyouthink

of thatopinion?

2

^{. TheUnitedKingdomreferendumwasexpetedtotakeplaein2006.Followingtherejetion}

of theConstitution byvoters inFrane inMay2005 andintheNetherlands inJune 2005,

thereferendum waspostponed indenitely.

ICMresearhasked1,000votersinthethirdweekofMay2005Iftherewereareferendum

tomorrow, wouldyou vote for Britain to signup to theEuropean Constitution or not? :

57%saidno.Findthe

95%

^{ondeneintervalforthissurvey.Ifyouwereapolitiian,what}

would youdedue fromthis?

6.6 M&Ms

JoshMadisonisa30-somethingNewYorkerwhorunsapersonalwebsiteontheinternet.Inone

of hisartiles, he statesthefollowing ruialproblem.

I love M&M's. I'm partial to the plain Milk Choolate variety, but I've been known to have a

Peanut from time to time inorder to remind myself why I don't like them that muh. Often,

while eating apak, I'llwonderhowthey're madeand how theolorsaredistributed.

After wondering about it a little more, I heked out M&M's web site. Aording to it, eah

pakageofMilkChoolateM&M'sshouldontain30%blue,20%brown,10%green,10%orange,

10%red,and20%yellowM&M's.IhekedthenextfewpakagesofM&M'sthatIateandfound

that their perentages were not even lose to the stated distribution. In my mind, this sort of

onrmed my thoughts about how they produe M&M's : When they make M&M's, in any

prodution run,theyprodue thestated perentageofeah olorand thenjustllthepakso

a onveyor line or some other weight based method. This would mean that anysingle pakage

ould be way o from the stated perentage; but analyze the ounts over a large number of

pakages, and theyshould onvergetowards thestated perentages.

Today,we willstudythis essentialproblemon afewbagsofM&Ms. First,we willonsidereah

bag asa sample.

1

^{. Open your bag of M&Ms and ount the number of andies of eah olor and the total}

number of andies. Give the results in an absolute frequeny table. As soon as you have

theresults,give them tothe teaherwhowill gather thedataforthe wholelass.

2

^.

a

^{. Compute the ondeneintervalat}

95%

^{for eah olor ofyour sample.}

b

^{. Consider the blue M&Ms. For how many bags was the expeted perentage in the}

ondene interval?What wouldyou onlude fromthis result?

c

^{. Answerthe previous question for theother olors.}

3

^{. To get a larger sample, we will now usethe total numbers of andies of eah olor inall}

the bagsinthe lass. Compute theondene interval withthatsample and ompare the

experimentalresults withthe expetedvalues.What wouldyou onlude fromthat?

Simulations

6.7 Random walks on an axis

0 1 2 3

−1

−2

−3

A ea ismoving along an axis.It startsfrom theorigin and,after eahjump, lands one unitto

theright orone unit to theleft,randomlyand withthe same probability.

A sequeneofjumpsisalledawalk. For example,ifthe ea isalways jumpingto theright,the

walkwill benotedRRRR.Ifitalternates between rightand left,thewalk willbe notedRLRL.

Part A – Simulations of 4-jumps walks

TheRandom orAlea funtiononyouralulator deliversarandomdeimal number between

0

^and

1

^.

1

^{. Devise a method to simulatea 4-jumpswalk using theRandom funtion.}

2

^{. Simulate 25walks andnotethe nal position oftheea at theend ofeah walk.}

3

^{. What arethe possiblenalpositions ontheaxis?Explainwhy someare}impossible.

4

^{. Count thenumber ofwalks for eah nalpositionand showtheountsina table.}

5

^{. Add arowto theprevious table withtheabsolutefrequeniesfor thewholelass.}

6

^{. Compute the relative frequeniesfor thewholelass.}

7

^{. Compute the average nalpositionof theea at theend of a4-jumpswalk.}

Part B – An algorithm

A random walk an be desribed bythe algo-

rithmshownontheright-handside,wherethe

aleafuntion deliversarandomnumberinthe

interval

[0, 1[

^{.Partsofthealgorithmhavebeen}

omitted on purpose.

begin

0 → x

^;

1 → i

^;

while

i 6 4

^do

if alea <

0.5

^then

. . . . → x

^;

else

. . . . → x

^;

end if

i + 1 → i

^;

end while

Output:

x

end

1

^{. Explain the funtionsoftheintegers}

x

^and

i

^{inthis algorithm.}

2

^{. Fill thetwo inompletelines.}

3

^{. Here are the results of applying the algo-}

rithm one. What is the nal position of

theea at the end ofthis walk?

i 1 2 3 4

alea

0.37 0.01 0.93 0.11

x 0 1 2 1 2

4

^{. Applythealgorithm toreatevenewwalks,usingtherandomfuntionofyouralulator}

and displaying allthe steps of thealgorithm like intheexample oftheprevious question.

Part C – Probabilistic study

In thispart, we will useprobabilities to study thesituation and ompare thetheoretialresults

to thefrequenieswefound inpartA.

1

^{. Drawa tree to showall the possible 4-jumps walks. At theend of eah branh, write the}

nalpositionof the ea.

2

^{. Use the tree to ompute the} probability of eah nal position and give the results in a probabilitytable.

3

^{. Compute the marginoferror at}

95%

^{ondenefor your sampleof 25 randomwalks.}

4

^{. For eah} probability, ount in the lass how many samples of 25 walks gave a frequeny within the marginof error.

6.8 A birth policy

A government has deided to impose a strit

birth poliy. Births in a family must stop as

soon asa boyis born or after thebirth ofthe

fourth hild.

Weonsiderinthisexerisethattheprobabili-

ties ofgivingbirth to agirl ora boyareequal

and that eah birth is independant from the

previous birthsinthesame family.

Thisbirthpoliyan berepresentedasatree,

where thepossiblefamilies areboxed.

∅

B G

GB GG

GGB GGG

GGGB GGGG

Part A – Simulation and statistical view

1

^{. Doyou think thatthis poliy will favour boys or girls?No justiationis expeted.}

2

^{. Devise a method to simulatetheompositionof afamily withthealulator.}

3

^{. Simulate and write down the omposition of 100 families. Count the number of hildren}

perfamilyand showthe results ina tablewithabsolute andrelative frequenies.

4

^{. Computethearithmetimean}

m 4

^andthemedian

d 4

^{forthenumberofhildrenperfamily}

inyour sample of

100

^families.

5

^{. Compute the arithmetimean}

M 4

^{and themedian}

D 4

^{for allthefamilies inthelass.}

Part B – An algorithm

Thisproessanbedesribedasanalgorithm.

The output is then a list of digits, with

0

^re-

presenting agirl and

1

representing a boy.

1

^{. Explainthe funtionsofthe wholenum-}

bers

x

^and

i

^{inthis algorithm.}

2

^{. Explainthe ondition}

x 6= 1

^and

i 6 4

^.

Doesitensurethatthealgorithmwillal-

ways stop?

3

^{. Whatisthe funtionof thelist}

L

^inthis

algorithm?

4

^{. Explain the notation}

L(i)

^.

begin

Clearlist

L

^;

0 → x

^;

0 → i

^;

while

x 6= 1

^and

i 6 4

^do

i + 1 → i

^;

if alea <

0.5

^then

0 → x

^;

else

1 → x

^;

end if

x → L(i)

^;

end while

Output:

L

end

5

^{. Herearetheresultsofapplyingthealgo-}

rithm one. Apply the algorithm to get

5 families,displaying allthesteps ofthe

algorithm likeinthe example.

i 1 2 3

alea

0.37 0.01 0.93

x 0 0 0 1

L () (0) (0, 0) (0, 0, 1) Part C – Probabilistic study

1

^{. Copythetreeat thebeginning oftheexerise andadd the}probabilities.

2

^{. Compute the} probability ofeah type offamily.

3

^{. Showina table the possible numbersof hildrenand their} probabilities. Arethese proba- bilities onsistentwiththe frequeniesfoundat theend of partA?

4

^{. Use thetable to ompute theexpeted valuefor thenumberof hildrenina family.}

Part D – The percentage of girls

The aim of this part is to study the perentage of girls

g

^{indued by this birth poliy, and}

therefore answerthe rstquestion ofpart A. To do so, we will rst use thesimulations of part

A, andthenthe probabilities of partC.

1

^{. Count thenumber andperentage ofgirls inyour 100simulated families.}

2

^{. From the various values found in the lass, what ould be the theoretial value of this}

perentage?

3

^{. Compute the}

95%

^{ondene intervalfor theperentage ofgirls inyour sample.}

4

^{. Ofall the ondene intervals inthelass, how manyinlude thepossible valuewefound}

inquestion 2?Doesitvalidateor invalidatethis hypothesis?

5

^{. Use partCto ndthe true valueoftheperentage}

g

^.

6.9 Random walks on a tetrahedron

An ant is walking on the edges of a tetrahedron

ABCD

^,

starting from vertex

A

^{.When itgets to a vertex,it hooses}

randomly the next edge it will walk on. The aim of this

exerise is to study the time it will take for the ant to go

bak to vertex

A

^{, assuming that itwalks along one edge in}

exatly1 minute.

A

B C D

A walk will be notedasa suessionof verties,asintheexample below:

A → D → B → C → A.

A walk will always start from

A

^{andstop assoonasthe ant omes bakto}

A

^.

Part A – Simulations

1

^{. Devise a method to simulatea random walk.}

2

^{. Simulate25 random walksandount the duration ofeah one.Gatherthedata inatable}

withtheabsolute frequenyof eahduration (from 1 to 20minutes).

3

^{. Explain the value inthe olumn for1 minute.}

4

^{. Is theduration neessarilylessthan 20 minutes?}

5

^{. Find out the minimum, maximum,range, mean andmedian of this data.}

6

^{. Carry out the previous}omputations for allthesimulated walksinthelass.

Part B – Probabilistic study

In this part, we will gather the verties B, C, D, for whih the walk doesn't end in a single

outome notedBCD. Therefore,from vertexA,theonly possibilityisto goto BCD,while from

BCDit's possible to goto A or stayinBCD.

A

BCD

1

^{. Illustratethis new way ofseeing theproblemwitha more simplegraph.}

2

^{. Starting from BCD, ompute the} probabilities to go to A and to stay inBCD. Usethese results to put the right probabilities on thearrows ofthe previousgraph.

3

^{. Showina} probability treetherst four steps ofthis proess.

4

^{. Compute the} probabilities ofa 2-minutes, a 3-minutesand a 4-minuteswalk.

5

^{. Without adding a level to the tree, onjeture a value for the} probability of a 5-minutes walk. Deduethe probability ofa walk lasting5 minutes or less.

Part C – Estimation of a percentage

In thispart, we will ndan estimation oftheperentage

p

^{of walkslasting 5minutesor less.}

1

^{. Compute the perentageof walkslasting 5minutesor lessinyour 25 simulated walks.}

2

^{. Compute the}

95%

^{ondene intervalfor thisperentage inyour sample.}

3

^{. Doesthis intervalvalidate the valueonjeturedinpartB?}

4

^{. Compute the}

95%

^{ondene interval for this perentage in the sample made of all the}

simulated walksinthe lass.

5

^{. Doesthis intervalvalidate the valueonjeturedinpartB?}

Homework

6.10

^{Galileo Galilei (15 February 1564 - 8 January 1642)was an Italian physiist, mathema-}

tiian, astronomer and philosopher, and withouth any doubt one of the greatest minds of all

times. He wasborn, thesame yearas Shakespeare, in theItalian townof Pisa.He was sent to

the Universityof Pisa to study mediine, but in1589 beame professor of mathematis (at the

age of 25) through the favours of Ferdinando dei Medii, the Grand Duke of Tusany. During

this period,Galileowasordered bytheGrandDuke ofTusanytoexplainaparadoxarisingin

theexperiment oftossing threedie.

Why, although there were an equal number of 6 partitions of the numbers 9 and 10. did experience

state that the chance of throwing a total 9 with three fair dice was less than that of throwing a

total of 10 ?

Part A – Simulating 1000 throws

Useaspreadsheet (OpenOeCalor MirosoftExel,for example)to simulate1000 throwsof

afairdie,andountthefrequenyofeahresult.To doso,youmayusethefollowing funtions:

=ALEA.ENTRE.BORNES(1;N) : Produes arandom integer between 1and N.

=SOMME(A1:A7) : Computes the sum of the numbers in ells A1 to A7, inluding all ells in

between.

=NB.SI(A1:A7;k) : Countshowmany timesthevalue

k

^{ours intheellsA1 to A7.}

Give the results of your 1000 simulations in an absolute frequeny table. Does it onrm the

Duke'sstatement?

Part B – Galileo’s solution

Below is an extrat from Galileo's answer to the Great Duke of Tusany, translated by E. H.

Thorne.

The fact that in a dice-game certain numbers are more advantageous than others has a very obvious reason, i.e. that some are more easily and more frequently made than others, which depends on their being able to be made up with more variety of numbers. Thus a 3 and an 18, which are throws which can only be made in one way with 3 numbers (that is, the latter with 6.6.6 and the former with 1.1.1, and in no other way), are more difficult to make than, e.g. 6 or 7, which can be made up in several ways, that is, a 6 with 1.2.3 and with 2.2.2 and with 1.1.4, and a 7 with 1.1.5, 1.2.4, 1.3.3, and 2.2.3. Nevertheless, although 9 and 12 can be made up in as many ways as 10 and 11, and therefore they should be considered as being of equal utility to these, yet it is known that long observation has made dice-players consider 10 and 11 to be more advantageous than 9 and 12. And it is clear that 9 and 10 can be made up by an equal diversity of numbers (and this is also true of 12 and 11). [...]

Three special points must be noted for a clear understanding of what follows. The first is that that sum of the points of 3 dice, which is composed of 3 equal numbers, can only be produced by one single throw of the dice : and thus a 3 can only be produced by the three ace-faces, and a 6, if it is to be made up of 3 twos, can only be made by a single throw. Secondly : the sum which is made up of 3 numbers, of which two are the same and the third different, can be produced by three throws : as e.g., a 4 which is made up of a 2 and of two aces, can be produced by three different throws ; that is, when the first die shows 2 and the second and third show the ace, or the second die a 2 and the first and third the ace ; or the third a 2 and the first and second the ace.

And so e.g., an 8, when it is made up of 3.3.2, can be produced also in three ways : i.e. when the first die shows 2 and the others 3 each, or when the second die shows 2 and the first and third 3, or finally when the third shows 2 and the first and second 3. Thirdly the sum of points which is made up of three different numbers can be produced in six ways. As for example, an 8 which is made up of 1.3.4. can be made with six different throws : first, when the first die shows 1, the second 3 and the third 4 ; second, when the first die still shows 1, but the second 4 and the third 3 ; third, when the second die shows 1, and the first 3 and the third 4 ; fourth, when the second still shows 1, and the first 4 and the third 3 ; fifth, when the third die shows 1, the first 3, and the second 4 ; sixth, when the third shows 1, the first 4 and the second 3. Therefore, we have so far declared these three fundamental points ; first, that the triples, that is the sum of three-dice throws, which are made up of three equal numbers, can only be produced in one way ; second, that the triples which are made up of two equal numbers and the third different, are produced in three ways ; third, that those triples which are made up of three different numbers are produced in six ways.

1

^{. Intherstparagraph,Galileoexplainsthatthereisonly onewaytomakea3,but several}

to make a6.

b

^{. In the samemanner, ndout all theways to make thenumbers9 and10.}

c

^{. What statement inGalileo's text isthenproved?}

2

^{. Explain the seond paragraph with the example of the dierent ways to make a 6, and}

deduehowmany dierent throws atually make this value.

3

^{. Compute in the same way the number of dierent throws that make a 9 and those that}

make a10.

4

^{. Write the onlusion of Galileo's paper, explaining to the Great Duke of Tusany the}

solution to thisproblem. (Minimum 50words,maximum150 words.)

Part C – The Passe-Dix game

Passe-dix, alsoalled passage inEnglish, isa gameof haneusing die. It isplayed withthree

die. There is always a banker, and the number of players is unlimited. To win, a player must

throwa point above ten(orpass tenwhene thename ofthegame).

1

^{. Use Galileo's method to ompute thenumber of throws that an give eah possible sum.}

Givetheresults inan absolutefrequeny table.

2

^{. Add to the previous table the} probabilities of eah sum, given as an irreduible fration and anapproximate value to3 DP.

3

^{. Computetherelativefrequeniesinthe1000simulatedthrowsofpartAandomparethem}

to the probabilities. Arethevalues verydierent?

4

^{. Compute the} probability ofpassing ten. Is ita fairgame?

Last year’s test

A random raeisgoing onbetween a hare 1

and a tortoise 2

.Itis played byrolling repeatedlya

four-sided fairdie.

•

^{Ifa4 turns up,the harediretlyreahes the nishline and wins.}

•

^{Ifa1,a2ora3turnsup,thetortoisemovestowardsthenishlineandthedieisrolledagain.}

Thetortoise winsafter movingfour times.

Part A – Simulation and statistical study 1

^{. Devise a method to simulatethegame witha alulator.}

2

^{. A sampleof 140 raes have been simulated. Below aregiven thewinners for eah ofthese}

raes,where theletter Hstands for the hareand Tfor thetortoise.

H H H T H H H T T H H T H H H H H T T T

T T H T T T T H H H T T T H T TH H H T

T H H H H T T T T H H H T H H H H H H H

T T T H H H H H T T T T H T H H H T T H

T T H T H H T H T H T T T H T H H H T T

T H H T H H T T H T T H H H H T T H H H

T T H T H H H H H T H H H H H H H H H H

3

^{. Copythetablebelow andll itout withtheabsoluteand relative} frequenies.

1. hare:lièvre

2. tortoise :tortue

Absolutefrequenies

Relativefrequenies

4

^{. Compute the ondeneintervalat 95%for theperentage ofraes won bythehare.}

5

^{. Explain inafewwordswhat aondene interval at 95%ondene is.}

6

^{. What isthe range3 of this ondeneinterval?Howould we lower it?}

7

^{. Aording to this sample, do you think that these rules are to the advantage of the hare}

or the tortoise?

Part B – An algorithm

Thisgame an be representedasthe following algorithm.

Input:

0 → T

^;

O → H

^;

begin

while

H 6= 1

^and

T 6= 4

^do

if alea <

0.25

^then

1 → H

^;

else

T + 1 → T

^;

end if

end while

if ... then

Output :Thehare wins.

else

Output :Thetortoise wins.

end if

end

8

^{. Explain the funtionsofthenumbers}

H

^and

T

^{inthis algorithm.}

9

^{. Explain the ondition}

H 6= 1

^and

T 6= 4

^{. Doesit ensure thatthe algorithm will always}

stop?

10

^{. What shouldbe entered inthelast If instrution sothattheoutput willbeorret?}

Part C – A probabilistic view

11

^{. Copy andll out thefollowing}probability tree:

∅

. . .

H

T

. . .

. . .

H

T

. . .

. . . . . .

. . . . . .

12

^{. At the end of eah branh of the tree, write down thewinner and the}probability of this situation.

13

^{. Dedue thatthe}probabilityof theharewinning is

175 256

^.

14

^{. Aording tothe followingresultdo youthink thattheserules areto theadvantageofthe}

hareor the tortoise?

15

^{. The} probability omputed in question 13 is not inluded in the ondene interval from part A.Doesitmeanthat somethingis wrong withone ofthese omputations?

Glossary

English Frenh Explanation

Survey Sondage A method for olleting quantitative

information about items in a popula-

tion.

Sample Éhantillon A subset of a population seleted for

measurement, observation or questio-

ning,toprovidestatistialinformation

about thepopulation.

Sampling Éhantillonage The proess or tehnique of obtaining

a representative sample.

Margin oferror Marge d'erreur An expression of thelakof preision

intheresultsobtained fromasample.

Flutuation interval Intervalle de utuation For a ertain proportion of samples,

the interval where theparameter stu-

died shouldbe.

Estimate (verb) Estimer To alulate roughly, often from im-

perfet data.

Estimate Estimation A rough alulation or guess.

Estimation Estimation The proessof makingan estimate.

Point estimate Estimation pontuelle A single value omputed from sample

data, used as a "best guess" for an

unknownpopulation parameter.

Condene interval Intervalle de onane A partiular kind of interval estimate

of apopulationparameter.

Simulate (verb) Simuler Tomodel, repliate, dupliatethe be-

havior, appearane or properties of a

systemor environment

Simulation Simulation Something whih simulates a system

or environment inorderto predita-

tual behaviour.

Aw,people an ome up withstatistis toprove anything,Kent. Forfty perent of all

people know that. (Homer Simpson)

Lottery :A tax on people who are badat math. (Anonymous)

Do not put your faith in what statistis sayuntil you have arefully onsidered what

they donot say. (William W.Watt)

He uses statistis as a drunken man uses lampposts - for support rather than for

illumination. (Andrew Lang)