9.6.EXERCISES 153
Writea programthatsimulatesmultiplegamesofblackjackandestimatestheprobabilitythatthedealer
willbust.
- Ablackjackdealeralwaysstartswithonecardshowing.It wouldbeusefulfora playertoknowthe
dealer’s bustprobability(seepreviousproblem)foreachpossiblestartingvalue.Writea simulation
programthatrunsmultiplehandsofblackjackforeachpossiblestartingvalue(ace–10)andestimates
theprobabilitythatthedealerbustsforeachstartingvalue.
- MonteCarlotechniquescanbeusedtoestimatethevalueofpi.Supposeyouhave a rounddartboard
thatjustfitsinsideofa squarecabinet. If youthrowdartsrandomly, theproportionthathitthedart
boardvs.thosethathitthecabinet(inthecornersnotcoveredbytheboard)willbedeterminedbythe
relative areaofthedartboardandthecabinet.Ifnis thetotalnumberofdartsrandomlythrown(that
landwithintheconfinesofthecabinet),andhis thenumberthathittheboard,it is easytoshow that
π 4
h
n
Writea programthatacceptsthe“numberofdarts”asaninputandthenperformsa simulationto
estimateπ. Hint:youcanuse2*random() - 1togeneratethexandycoordinatesofa random
pointinsidea 2x2squarecenteredat
0 0 . Thepointliesinsidetheinscribedcircleifx^2 y^2 1.
- Writea programthatperformsa simulationtoestimatetheprobabilityofrollingfive-of-a-kindina
singlerolloffive six-sideddice.
- Arandomwalkis a particularkindofprobablisticsimulationthatmodelscertainstatisticalsystems
suchastheBrownianmotionofmolecules.Youcanthinkofa one-dimensionalrandomwalkinterms
ofcoinflipping.Supposeyouarestandingona verylongstraightsidewalkthatextendsbothinfront
ofandbehindyou.Youflipa coin.If it comesupheads,youtake a stepforward;tailsmeanstotake a
stepbackward.
Supposeyoutake a randomwalkofnsteps.Onaverage,how many stepsawayfromthestartingpoint
willyouendup?Writea programtohelpyouinvestigatethisquestion.
- Supposeyouaredoinga randomwalk(seepreviousproblem)ontheblocksofa citystreet.Ateach
“step”youchoosetowalkoneblock(atrandom)eitherforward,backward,leftorright. Innsteps,
how fardoyouexpecttobefromyourstartingpoint?Writea programtohelpanswerthisquestion.
- Writea graphicalprogramtotracea randomwalk(seeprevioustwo problems)intwo dimensions.
Inthissimulationyoushouldallow thesteptobetakeninanydirection.Youcangeneratea random
directionasanangleoff ofthexaxis.
angle = random()* 2 * math.pi
Thenewxandypositionsarethengivenbytheseformulas.
x = x + cos(angle)
y = y + sin(angle)
Theprogramshouldtake thenumberofstepsasaninput.Startyourwalkerat thecenterofa 100x100
gridanddraw a linethattracesthewalkasit progresses.
- (Advanced)Hereisa puzzleproblemthatcanbesolvedwitheithersomefancy analyticgeometry
(calculus)ora (relatively)simplesimulation.
Supposeyouarelocatedattheexactcenterofa cube. Ifyoucouldlookallaroundyouinevery
direction,eachwallofthecubewouldoccupy^16 ofyourfieldofvision.Supposeyoumove towardone
ofthewallssothatyouarenow half-waybetweenit andthecenterofthecube.Whatfractionofyour
fieldofvisionis now takenupbytheclosestwall?Hint:usea MonteCarlosimulationthatrepeatedly
“looks”ina randomdirectionandcountshow many timesit seesthewall.
