> #ok to post 
;
> #Yifan Zhang, 11/6/2020
;
> 
;
> #Q1. 
;
> #Use Wikipedia (or otherwise) to find the final outcome of the electoral votes for the presidential elections for 2000, 2004, 2008, 2012, and 2016, and find the number of ways in which it could have been counted to lead to the final ouctome, by using the appropriate procedures in ComboProject8.txt.
;
> #Year, #RED, #BLUE, TOTAL
;
> #2000, 271, 266, 537+1 FAITHLESS VOTE
;
> #2004, 286, 251, 537+1 FAITHLESS VOTE
;
> #2008, 173, 365, 538
;
> #2012, 206, 332, 538
;
> #2016, 304, 227, 531+7 FAITHLESS VOTE
;
> 
;
> read `ComboProject8.txt`
`This is ComboProject8.txt, a Maple package that is part of Project 8 in Dr. Z.\
's Combinatorics Class at Rutgers University`
`To study and simulate vote counting`
``
`Team Leader: tbd `
``
`Other Team members: tbd `
``
`For a list of all the functions type: Help(); `
`For Help with any of the functions, type  Help(FunctionName):`

;
> coeff(GFv(USEC(), x), x, 271)
16965465344318

;
> coeff(GFv(USEC(), x), x, 286)
16196017263096

;
> coeff(GFv(USEC(), x), x, 365)
3182416524832

;
> coeff(GFv(USEC(), x), x, 332)
8628577597686

;
> coeff(GFv(USEC(), x), x, 304)
13873406885786

;
> 
;
> #Q2.
;
> #Assuming, rather unrealistally, that the probability of winning for either of the candidates is the same for each state, and that they are independent of each other, use the popular vote (also given in Wikipedia), use the appropriate procedure to find the (estimated) probability of the ultimate winner of (i) winning (i.e. scoring at least 270 votes) (ii) scoring that many electoral votes or more.
;
> p:=0.5
Typesetting:-mprintslash([(p := .5)],[.5])

;
> f:=GFvp(USEC(), 1/2, x); 
Typesetting:-mprintslash([(f := 1/2251799813685248*x^538+1/281474976710656*x^
535+5/2251799813685248*x^534+3/2251799813685248*x^533+17/1125899906842624*x^532
+43/2251799813685248*x^531+9/562949953421312*x^530+61/1125899906842624*x^529+
201/2251799813685248*x^528+217/2251799813685248*x^527+427/2251799813685248*x^
526+365/1125899906842624*x^525+115/281474976710656*x^524+717/1125899906842624*x
^523+1165/1125899906842624*x^522+1581/1125899906842624*x^521+1127/
562949953421312*x^520+6821/2251799813685248*x^519+9479/2251799813685248*x^518+
1635/281474976710656*x^517+9341/1125899906842624*x^516+3229/281474976710656*x^
515+35065/2251799813685248*x^514+48207/2251799813685248*x^513+16377/
562949953421312*x^512+87739/2251799813685248*x^511+29393/562949953421312*x^510+
156685/2251799813685248*x^509+206935/2251799813685248*x^508+272173/
2251799813685248*x^507+356345/2251799813685248*x^506+231911/1125899906842624*x^
505+600849/2251799813685248*x^504+387331/1125899906842624*x^503+994337/
2251799813685248*x^502+79423/140737488355328*x^501+1616541/2251799813685248*x^
500+2048369/2251799813685248*x^499+1292799/1125899906842624*x^498+3249925/
2251799813685248*x^497+4069867/2251799813685248*x^496+2539539/1125899906842624*
x^495+6314467/2251799813685248*x^494+7823149/2251799813685248*x^493+9661071/
2251799813685248*x^492+5945021/1125899906842624*x^491+14586067/2251799813685248
*x^490+17839141/2251799813685248*x^489+21749899/2251799813685248*x^488+26438475
/2251799813685248*x^487+32046021/2251799813685248*x^486+38731301/
2251799813685248*x^485+23340503/1125899906842624*x^484+56111173/
2251799813685248*x^483+8408217/281474976710656*x^482+80428483/2251799813685248*
x^481+95923935/2251799813685248*x^480+57058873/1125899906842624*x^479+1058061/
17592186044416*x^478+80172405/1125899906842624*x^477+189392667/2251799813685248
*x^476+6974683/70368744177664*x^475+65607245/562949953421312*x^474+153940465/
1125899906842624*x^473+360421915/2251799813685248*x^472+421035657/
2251799813685248*x^471+490811551/2251799813685248*x^470+285487893/
1125899906842624*x^469+662899405/2251799813685248*x^468+384047447/
1125899906842624*x^467+222062491/562949953421312*x^466+512620199/
1125899906842624*x^465+590565309/1125899906842624*x^464+679103069/
1125899906842624*x^463+194873789/281474976710656*x^462+1786252409/
2251799813685248*x^461+2043040165/2251799813685248*x^460+2332695429/
2251799813685248*x^459+332358821/281474976710656*x^458+1512783735/
1125899906842624*x^457+107410829/70368744177664*x^456+3898350489/
2251799813685248*x^455+4414352627/2251799813685248*x^454+1247690247/
562949953421312*x^453+176051357/70368744177664*x^452+6349587741/
2251799813685248*x^451+7145704429/2251799813685248*x^450+8029647723/
2251799813685248*x^449+4504843269/1125899906842624*x^448+10094708495/
2251799813685248*x^447+11294238537/2251799813685248*x^446+12618513241/
2251799813685248*x^445+7039244329/1125899906842624*x^444+7842928935/
1125899906842624*x^443+17453125685/2251799813685248*x^442+9696808959/
1125899906842624*x^441+21521499793/2251799813685248*x^440+11925924895/
1125899906842624*x^439+26400664613/2251799813685248*x^438+29184882177/
2251799813685248*x^437+2013903353/140737488355328*x^436+35532339797/
2251799813685248*x^435+39134535973/2251799813685248*x^434+43050144947/
2251799813685248*x^433+47301362511/2251799813685248*x^432+51911499571/
2251799813685248*x^431+56905057539/2251799813685248*x^430+62307700923/
2251799813685248*x^429+68146273105/2251799813685248*x^428+18612218211/
562949953421312*x^427+40622408469/1125899906842624*x^426+88564644621/
2251799813685248*x^425+96440188453/2251799813685248*x^424+104904532333/
2251799813685248*x^423+113992004741/2251799813685248*x^422+123738237027/
2251799813685248*x^421+134180113015/2251799813685248*x^420+36338939573/
562949953421312*x^419+39326145049/562949953421312*x^418+85033601175/
1125899906842624*x^417+91842728795/1125899906842624*x^416+99101204021/
1125899906842624*x^415+213662255583/2251799813685248*x^414+230110337445/
2251799813685248*x^413+247593139583/2251799813685248*x^412+133079086477/
1125899906842624*x^411+142926981859/1125899906842624*x^410+9585314627/
70368744177664*x^409+10276153709/70368744177664*x^408+88056450223/
562949953421312*x^407+376948865953/2251799813685248*x^406+403058961273/
2251799813685248*x^405+215304790853/1125899906842624*x^404+229827439823/
1125899906842624*x^403+61281185841/281474976710656*x^402+522448446323/
2251799813685248*x^401+139076801505/562949953421312*x^400+591881433527/
2251799813685248*x^399+629226945869/2251799813685248*x^398+668399676139/
2251799813685248*x^397+709455482123/2251799813685248*x^396+47028129661/
140737488355328*x^395+398719483945/1125899906842624*x^394+844477271443/
2251799813685248*x^393+223404906571/562949953421312*x^392+944920144061/
2251799813685248*x^391+998432176367/2251799813685248*x^390+263552065107/
562949953421312*x^389+278075013417/562949953421312*x^388+586379045449/
1125899906842624*x^387+154453965717/281474976710656*x^386+1300969070333/
2251799813685248*x^385+1368816743171/2251799813685248*x^384+179902476449/
281474976710656*x^383+1512221733293/2251799813685248*x^382+396966026001/
562949953421312*x^381+416546650439/562949953421312*x^380+109201682659/
140737488355328*x^379+457755136173/562949953421312*x^378+1917600677723/
2251799813685248*x^377+1003499094945/1125899906842624*x^376+262405172547/
281474976710656*x^375+2194355954575/2251799813685248*x^374+143272808903/
140737488355328*x^373+598322118139/562949953421312*x^372+2497143781269/
2251799813685248*x^371+2603945128309/2251799813685248*x^370+2713703569031/
2251799813685248*x^369+2826426993933/2251799813685248*x^368+1471060033105/
1125899906842624*x^367+765195995251/562949953421312*x^366+99450516401/
70368744177664*x^365+3307012046945/2251799813685248*x^364+3434561225645/
2251799813685248*x^363+891262782751/562949953421312*x^362+3698465221563/
2251799813685248*x^361+1917391570803/1125899906842624*x^360+3973980781619/
2251799813685248*x^359+4116030292397/2251799813685248*x^358+532612490331/
281474976710656*x^357+2204277044925/1125899906842624*x^356+2279476694427/
1125899906842624*x^355+589006809099/281474976710656*x^354+2433905079487/
1125899906842624*x^353+5026169423561/2251799813685248*x^352+2593538653967/
1125899906842624*x^351+668809383395/281474976710656*x^350+5516300169293/
2251799813685248*x^349+2842243105633/1125899906842624*x^348+5854963107251/
2251799813685248*x^347+3013828551641/1125899906842624*x^346+6202490695643/
2251799813685248*x^345+797422856341/281474976710656*x^344+6558249046909/
2251799813685248*x^343+842375150419/281474976710656*x^342+3460773953287/
1125899906842624*x^341+7105794480365/2251799813685248*x^340+7291642941851/
2251799813685248*x^339+3739496115049/1125899906842624*x^338+1916934572597/
562949953421312*x^337+3928887029809/1125899906842624*x^336+4024494851773/
1125899906842624*x^335+8241272700785/2251799813685248*x^334+8434507865623/
2251799813685248*x^333+4314288798843/1125899906842624*x^332+8823361952383/
2251799813685248*x^331+2254684672229/562949953421312*x^330+2303645887009/
562949953421312*x^329+4705385158993/1125899906842624*x^328+300224089977/
70368744177664*x^327+9803655512445/2251799813685248*x^326+5000046484169/
1125899906842624*x^325+10196350507747/2251799813685248*x^324+10392294222573/
2251799813685248*x^323+5293894562979/1125899906842624*x^322+5391349590887/
1125899906842624*x^321+5488443806749/1125899906842624*x^320+5585108515837/
1125899906842624*x^319+2840637366547/562949953421312*x^318+5776873323005/
1125899906842624*x^317+2935917533811/562949953421312*x^316+5966090700431/
1125899906842624*x^315+12119142062779/2251799813685248*x^314+12304414017695/
2251799813685248*x^313+12487859542917/2251799813685248*x^312+12669341528675/
2251799813685248*x^311+12848723576633/2251799813685248*x^310+6512935062497/
1125899906842624*x^309+6600323348563/1125899906842624*x^308+13372919957601/
2251799813685248*x^307+846409864759/140737488355328*x^306+13709429797337/
2251799813685248*x^305+6936703442893/1125899906842624*x^304+7017180912593/
1125899906842624*x^303+14192169290107/2251799813685248*x^302+896669122687/
140737488355328*x^301+7248925300949/1125899906842624*x^300+228835692305/
35184372088832*x^299+7394745296443/1125899906842624*x^298+14929755436539/
2251799813685248*x^297+7533083756173/1125899906842624*x^296+15198618276161/
2251799813685248*x^295+7663501013473/1125899906842624*x^294+15451216094853/
2251799813685248*x^293+15571160921779/2251799813685248*x^292+7843370065113/
1125899906842624*x^291+15797860694803/2251799813685248*x^290+15904432998027/
2251799813685248*x^289+16006370908045/2251799813685248*x^288+16103591935405/
2251799813685248*x^287+2024502157887/281474976710656*x^286+127215404823/
17592186044416*x^285+8183092214159/1125899906842624*x^284+16443787845527/
2251799813685248*x^283+516134961573/70368744177664*x^282+8291859009999/
1125899906842624*x^281+16645930548169/2251799813685248*x^280+8351452722553/
1125899906842624*x^279+8377298038737/1125899906842624*x^278+8400480062489/
1125899906842624*x^277+4210489888815/562949953421312*x^276+16877560724971/
2251799813685248*x^275+4226933609301/562949953421312*x^274+8466227953609/
1125899906842624*x^273+16951704812511/2251799813685248*x^272+8482732672159/
1125899906842624*x^271+16973726193241/2251799813685248*x^270+8488240282035/
1125899906842624*x^269+16973726193241/2251799813685248*x^268+8482732672159/
1125899906842624*x^267+16951704812511/2251799813685248*x^266+8466227953609/
1125899906842624*x^265+4226933609301/562949953421312*x^264+16877560724971/
2251799813685248*x^263+4210489888815/562949953421312*x^262+8400480062489/
1125899906842624*x^261+8377298038737/1125899906842624*x^260+8351452722553/
1125899906842624*x^259+16645930548169/2251799813685248*x^258+8291859009999/
1125899906842624*x^257+516134961573/70368744177664*x^256+16443787845527/
2251799813685248*x^255+8183092214159/1125899906842624*x^254+127215404823/
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2251799813685248*x^251+16006370908045/2251799813685248*x^250+15904432998027/
2251799813685248*x^249+15797860694803/2251799813685248*x^248+7843370065113/
1125899906842624*x^247+15571160921779/2251799813685248*x^246+15451216094853/
2251799813685248*x^245+7663501013473/1125899906842624*x^244+15198618276161/
2251799813685248*x^243+7533083756173/1125899906842624*x^242+14929755436539/
2251799813685248*x^241+7394745296443/1125899906842624*x^240+228835692305/
35184372088832*x^239+7248925300949/1125899906842624*x^238+896669122687/
140737488355328*x^237+14192169290107/2251799813685248*x^236+7017180912593/
1125899906842624*x^235+6936703442893/1125899906842624*x^234+13709429797337/
2251799813685248*x^233+846409864759/140737488355328*x^232+13372919957601/
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1125899906842624*x^229+12848723576633/2251799813685248*x^228+12669341528675/
2251799813685248*x^227+12487859542917/2251799813685248*x^226+12304414017695/
2251799813685248*x^225+12119142062779/2251799813685248*x^224+5966090700431/
1125899906842624*x^223+2935917533811/562949953421312*x^222+5776873323005/
1125899906842624*x^221+2840637366547/562949953421312*x^220+5585108515837/
1125899906842624*x^219+5488443806749/1125899906842624*x^218+5391349590887/
1125899906842624*x^217+5293894562979/1125899906842624*x^216+10392294222573/
2251799813685248*x^215+10196350507747/2251799813685248*x^214+5000046484169/
1125899906842624*x^213+9803655512445/2251799813685248*x^212+300224089977/
70368744177664*x^211+4705385158993/1125899906842624*x^210+2303645887009/
562949953421312*x^209+2254684672229/562949953421312*x^208+8823361952383/
2251799813685248*x^207+4314288798843/1125899906842624*x^206+8434507865623/
2251799813685248*x^205+8241272700785/2251799813685248*x^204+4024494851773/
1125899906842624*x^203+3928887029809/1125899906842624*x^202+1916934572597/
562949953421312*x^201+3739496115049/1125899906842624*x^200+7291642941851/
2251799813685248*x^199+7105794480365/2251799813685248*x^198+3460773953287/
1125899906842624*x^197+842375150419/281474976710656*x^196+6558249046909/
2251799813685248*x^195+797422856341/281474976710656*x^194+6202490695643/
2251799813685248*x^193+3013828551641/1125899906842624*x^192+5854963107251/
2251799813685248*x^191+2842243105633/1125899906842624*x^190+5516300169293/
2251799813685248*x^189+668809383395/281474976710656*x^188+2593538653967/
1125899906842624*x^187+5026169423561/2251799813685248*x^186+2433905079487/
1125899906842624*x^185+589006809099/281474976710656*x^184+2279476694427/
1125899906842624*x^183+2204277044925/1125899906842624*x^182+532612490331/
281474976710656*x^181+4116030292397/2251799813685248*x^180+3973980781619/
2251799813685248*x^179+1917391570803/1125899906842624*x^178+3698465221563/
2251799813685248*x^177+891262782751/562949953421312*x^176+3434561225645/
2251799813685248*x^175+3307012046945/2251799813685248*x^174+99450516401/
70368744177664*x^173+765195995251/562949953421312*x^172+1471060033105/
1125899906842624*x^171+2826426993933/2251799813685248*x^170+2713703569031/
2251799813685248*x^169+2603945128309/2251799813685248*x^168+2497143781269/
2251799813685248*x^167+598322118139/562949953421312*x^166+143272808903/
140737488355328*x^165+2194355954575/2251799813685248*x^164+262405172547/
281474976710656*x^163+1003499094945/1125899906842624*x^162+1917600677723/
2251799813685248*x^161+457755136173/562949953421312*x^160+109201682659/
140737488355328*x^159+416546650439/562949953421312*x^158+396966026001/
562949953421312*x^157+1512221733293/2251799813685248*x^156+179902476449/
281474976710656*x^155+1368816743171/2251799813685248*x^154+1300969070333/
2251799813685248*x^153+154453965717/281474976710656*x^152+586379045449/
1125899906842624*x^151+278075013417/562949953421312*x^150+263552065107/
562949953421312*x^149+998432176367/2251799813685248*x^148+944920144061/
2251799813685248*x^147+223404906571/562949953421312*x^146+844477271443/
2251799813685248*x^145+398719483945/1125899906842624*x^144+47028129661/
140737488355328*x^143+709455482123/2251799813685248*x^142+668399676139/
2251799813685248*x^141+629226945869/2251799813685248*x^140+591881433527/
2251799813685248*x^139+139076801505/562949953421312*x^138+522448446323/
2251799813685248*x^137+61281185841/281474976710656*x^136+229827439823/
1125899906842624*x^135+215304790853/1125899906842624*x^134+403058961273/
2251799813685248*x^133+376948865953/2251799813685248*x^132+88056450223/
562949953421312*x^131+10276153709/70368744177664*x^130+9585314627/
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1125899906842624*x^203+3928887029809/1125899906842624*x^202+1916934572597/
562949953421312*x^201+3739496115049/1125899906842624*x^200+7291642941851/
2251799813685248*x^199+7105794480365/2251799813685248*x^198+3460773953287/
1125899906842624*x^197+842375150419/281474976710656*x^196+6558249046909/
2251799813685248*x^195+797422856341/281474976710656*x^194+6202490695643/
2251799813685248*x^193+3013828551641/1125899906842624*x^192+5854963107251/
2251799813685248*x^191+2842243105633/1125899906842624*x^190+5516300169293/
2251799813685248*x^189+668809383395/281474976710656*x^188+2593538653967/
1125899906842624*x^187+5026169423561/2251799813685248*x^186+2433905079487/
1125899906842624*x^185+589006809099/281474976710656*x^184+2279476694427/
1125899906842624*x^183+2204277044925/1125899906842624*x^182+532612490331/
281474976710656*x^181+4116030292397/2251799813685248*x^180+3973980781619/
2251799813685248*x^179+1917391570803/1125899906842624*x^178+3698465221563/
2251799813685248*x^177+891262782751/562949953421312*x^176+3434561225645/
2251799813685248*x^175+3307012046945/2251799813685248*x^174+99450516401/
70368744177664*x^173+765195995251/562949953421312*x^172+1471060033105/
1125899906842624*x^171+2826426993933/2251799813685248*x^170+2713703569031/
2251799813685248*x^169+2603945128309/2251799813685248*x^168+2497143781269/
2251799813685248*x^167+598322118139/562949953421312*x^166+143272808903/
140737488355328*x^165+2194355954575/2251799813685248*x^164+262405172547/
281474976710656*x^163+1003499094945/1125899906842624*x^162+1917600677723/
2251799813685248*x^161+457755136173/562949953421312*x^160+109201682659/
140737488355328*x^159+416546650439/562949953421312*x^158+396966026001/
562949953421312*x^157+1512221733293/2251799813685248*x^156+179902476449/
281474976710656*x^155+1368816743171/2251799813685248*x^154+1300969070333/
2251799813685248*x^153+154453965717/281474976710656*x^152+586379045449/
1125899906842624*x^151+278075013417/562949953421312*x^150+263552065107/
562949953421312*x^149+998432176367/2251799813685248*x^148+944920144061/
2251799813685248*x^147+223404906571/562949953421312*x^146+844477271443/
2251799813685248*x^145+398719483945/1125899906842624*x^144+47028129661/
140737488355328*x^143+709455482123/2251799813685248*x^142+668399676139/
2251799813685248*x^141+629226945869/2251799813685248*x^140+591881433527/
2251799813685248*x^139+139076801505/562949953421312*x^138+522448446323/
2251799813685248*x^137+61281185841/281474976710656*x^136+229827439823/
1125899906842624*x^135+215304790853/1125899906842624*x^134+403058961273/
2251799813685248*x^133+376948865953/2251799813685248*x^132+88056450223/
562949953421312*x^131+10276153709/70368744177664*x^130+9585314627/
70368744177664*x^129+142926981859/1125899906842624*x^128+133079086477/
1125899906842624*x^127+247593139583/2251799813685248*x^126+230110337445/
2251799813685248*x^125+213662255583/2251799813685248*x^124+99101204021/
1125899906842624*x^123+91842728795/1125899906842624*x^122+85033601175/
1125899906842624*x^121+39326145049/562949953421312*x^120+36338939573/
562949953421312*x^119+134180113015/2251799813685248*x^118+123738237027/
2251799813685248*x^117+113992004741/2251799813685248*x^116+104904532333/
2251799813685248*x^115+96440188453/2251799813685248*x^114+88564644621/
2251799813685248*x^113+40622408469/1125899906842624*x^112+18612218211/
562949953421312*x^111+68146273105/2251799813685248*x^110+62307700923/
2251799813685248*x^109+56905057539/2251799813685248*x^108+51911499571/
2251799813685248*x^107+47301362511/2251799813685248*x^106+43050144947/
2251799813685248*x^105+39134535973/2251799813685248*x^104+35532339797/
2251799813685248*x^103+2013903353/140737488355328*x^102+29184882177/
2251799813685248*x^101+26400664613/2251799813685248*x^100+11925924895/
1125899906842624*x^99+21521499793/2251799813685248*x^98+9696808959/
1125899906842624*x^97+17453125685/2251799813685248*x^96+7842928935/
1125899906842624*x^95+7039244329/1125899906842624*x^94+12618513241/
2251799813685248*x^93+11294238537/2251799813685248*x^92+10094708495/
2251799813685248*x^91+4504843269/1125899906842624*x^90+8029647723/
2251799813685248*x^89+7145704429/2251799813685248*x^88+6349587741/
2251799813685248*x^87+176051357/70368744177664*x^86+1247690247/562949953421312*
x^85+4414352627/2251799813685248*x^84+3898350489/2251799813685248*x^83+
107410829/70368744177664*x^82+1512783735/1125899906842624*x^81+332358821/
281474976710656*x^80+2332695429/2251799813685248*x^79+2043040165/
2251799813685248*x^78+1786252409/2251799813685248*x^77+194873789/
281474976710656*x^76+679103069/1125899906842624*x^75+590565309/1125899906842624
*x^74+512620199/1125899906842624*x^73+222062491/562949953421312*x^72+384047447/
1125899906842624*x^71+662899405/2251799813685248*x^70+285487893/
1125899906842624*x^69+490811551/2251799813685248*x^68+421035657/
2251799813685248*x^67+360421915/2251799813685248*x^66+153940465/
1125899906842624*x^65+65607245/562949953421312*x^64+6974683/70368744177664*x^63
+189392667/2251799813685248*x^62+80172405/1125899906842624*x^61+1058061/
17592186044416*x^60+57058873/1125899906842624*x^59+95923935/2251799813685248*x^
58+80428483/2251799813685248*x^57+8408217/281474976710656*x^56+56111173/
2251799813685248*x^55+23340503/1125899906842624*x^54+38731301/2251799813685248*
x^53+32046021/2251799813685248*x^52+26438475/2251799813685248*x^51+21749899/
2251799813685248*x^50+17839141/2251799813685248*x^49+14586067/2251799813685248*
x^48+5945021/1125899906842624*x^47+9661071/2251799813685248*x^46+7823149/
2251799813685248*x^45+6314467/2251799813685248*x^44+2539539/1125899906842624*x^
43+4069867/2251799813685248*x^42+3249925/2251799813685248*x^41+1292799/
1125899906842624*x^40+2048369/2251799813685248*x^39+1616541/2251799813685248*x^
38+79423/140737488355328*x^37+994337/2251799813685248*x^36+387331/
1125899906842624*x^35+600849/2251799813685248*x^34+231911/1125899906842624*x^33
+356345/2251799813685248*x^32+272173/2251799813685248*x^31+206935/
2251799813685248*x^30+156685/2251799813685248*x^29+29393/562949953421312*x^28+
87739/2251799813685248*x^27+16377/562949953421312*x^26+48207/2251799813685248*x
^25+35065/2251799813685248*x^24+3229/281474976710656*x^23+9341/1125899906842624
*x^22+1635/281474976710656*x^21+9479/2251799813685248*x^20+6821/
2251799813685248*x^19+1127/562949953421312*x^18+1581/1125899906842624*x^17+1165
/1125899906842624*x^16+717/1125899906842624*x^15+115/281474976710656*x^14+365/
1125899906842624*x^13+427/2251799813685248*x^12+217/2251799813685248*x^11+201/
2251799813685248*x^10+61/1125899906842624*x^9+9/562949953421312*x^8+43/
2251799813685248*x^7+17/1125899906842624*x^6+3/2251799813685248*x^5+5/
2251799813685248*x^4+1/281474976710656*x^3+1/2251799813685248])

;
> evalf(add(coeff(f, x, i), i=270..538))
.4962304641

;
> #The answer is around 0.496
;
> 
;
> #Q3. 
;
> #Using procedure SimCount(L,p,N,K) with N=2000, K=4, four times and with p=3/10, 2/5, 1/2, 3/5, 5/7, and L=USEC(), see whether the first component of the output (that give estimates for the expectation, standard-deviation, and the 3rd, and 4th moments) agree with each other (remember they are only statistical estimates) and how they are close to the true value obtained by using StatAnal(f,x,K) applied to GFvp(USEC(),p,x). I have no clue about the theoreical (exact) value of the probability that such a count is consistent, but the second output of SimCount(L,p,N,K) give estimates. How close, in the above-mentioned four runs are there to each other?
;
> 
;
> #For p=3/10
;
> SimuCount(USEC(),3/10, 2000, 4);
[162.7780000, 47.53066080, .2153509882, 2.747913277], .4260000000

;
> evalf(StatAnal(GFvp(USEC(), 3/10, x), x,4))
[161.4000000, 46.65683230, .2695069204, 2.844567676]

;
> #For p=2/5
;
> SimuCount(USEC(),2/5, 2000, 4);
[215.4825000, 50.48620300, .1823441314, 2.880264282], .2495000000

;
> evalf(StatAnal(GFvp(USEC(), 2/5, x), x,4))
[215.2000000, 49.87825176, .1260503198, 2.769840597]

;
> #For p=1/2
;
> SimuCount(USEC(),1/2, 2000, 4);
[270.4375000, 51.55297368, .6228210294e-2, 2.598034352], .1615000000

;
> evalf(StatAnal(GFvp(USEC(), 1/2, x), x,4))
[269., 50.90677755, 0., 2.748917015]

;
> #For p=3/5
;
> SimuCount(USEC(),3/5, 2000, 4);
[322.0930000, 49.63611942, -.5190689907e-1, 2.706237907], .2395000000

;
> evalf(StatAnal(GFvp(USEC(), 3/5, x), x,4))
[322.8000000, 49.87825176, -.1260503198, 2.769840597]

;
> #For p=5/7
;
> SimuCount(USEC(),5/7, 2000, 4);
[385.6200000, 46.21821718, -.2213024275, 2.788601083], .4715000000

;
> evalf(StatAnal(GFvp(USEC(), 5/7, x), x,4))
[384.2857143, 45.99467583, -.2929144739, 2.861904358]

;
> 
;