How long should it take until you visit the integers with property, "isprime", for the , 20, -th time if you roll a fair die with, 6, faces By Shalosh B. Ekhad If you roll a fair die with, 6, faces starting at, 0, and look at the runnin\ g total and keep going until that running total has the property, "isprime", for , 20, times The prob. of reaching that goal in <=, 500, rounds is, 0.99999999999999999972 Conditioned that you are succesful the expected number of rounds for that go\ al is, 100.26480683975818340, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 22.677184601850379544 The skewness is, 0.53510980922322583233 The kurtosis is, 3.4359882942130068387 The prob. generating function, in t, for the r.v. reaching property, isprime, for the, 20 -h time is .1681943355e-16*t^451+.5650595173e-8*t^286+.2572227035e-16*t^448+.3051076787e-\ 10*t^329+.1632631682e-14*t^417+.3298264141e-11*t^350+.1208458614e-14*t^419+.\ 6168420988e-6*t^241+.5490171932e-6*t^242+.3655169920e-13*t^388+.1545538247e-1*t ^87+.6134044331e-2*t^66+.1748589033e-1*t^99+.4148054882e-11*t^348+.8239008074e-\ 18*t^473+.1770235523e-7*t^275+.3100640823e-4*t^198+.4907426965e-3*t^171+.\ 5089469751e-2*t^134+.1754843594e-9*t^313+.4156382095e-18*t^479+.2759909144e-8*t ^292+.4894057277e-13*t^384+.2963490385e-9*t^309+.2547390136e-14*t^414+.\ 1068836736e-11*t^360+.3377666407e-9*t^308+.1583882537e-1*t^80+.3034937569e-6*t^ 247+.2001712363e-12*t^372+.3889635542e-7*t^28+.6904332675e-4*t^43+.4495062385e-\ 8*t^288+.1026117541e-1*t^72+.1581649198e-13*t^399+.2388795420e-7*t^272+.\ 1723663366e-12*t^373+.4881895127e-6*t^243+.4040549982e-2*t^137+.2063647176e-18* t^485+.1194995342e-9*t^316+.2133987062e-8*t^294+.8251874403e-11*t^342+.\ 1111017346e-6*t^256+.7808030418e-13*t^379+.1109371768e-4*t^211+.6579767588e-11* t^344+.1202196545e-11*t^359+.3129211093e-8*t^291+.8205742227e-5*t^214+.\ 2944864538e-14*t^413+.6658965743e-10*t^321+.7902846054e-2*t^128+.2106406877e-4* t^203+.1701067108e-1*t^105+.1566996290e-1*t^82+.1575380953e-1*t^89+.1306359970e\ -2*t^54+.3055588443e-3*t^176+.3851791701e-6*t^245+.3999690847e-5*t^221+.\ 1192918718e-1*t^116+.3993968680e-8*t^289+.1925254406e-5*t^230+.3749846791e-2*t^ 138+.1028413364e-1*t^122+.5630691713e-12*t^365+.5952651681e-2*t^132+.1160493528\ e-1*t^117+.1568346382e-4*t^207+.1006754305e-4*t^212+.6424944866e-2*t^131+.\ 2236717155e-16*t^449+.1660648859e-1*t^96+.1691611619e-1*t^97+.1862368133e-3*t^ 181+.4344347212e-3*t^49+.4384761579e-9*t^306+.8324682975e-10*t^319+.1523163202e\ -3*t^183+.6123347411e-19*t^495+.1466616480e-5*t^233+.9482851032e-14*t^404+.\ 3347117419e-10*t^328+.8322142939e-3*t^164+.1561243309e-1*t^88+.6914246521e-2*t^ 130+.6922691261e-6*t^240+.2130660987e-6*t^250+.1329076462e-5*t^234+.1439873328e\ -13*t^400+.1459424540e-1*t^77+.2355052757e-2*t^144+.5352173691e-3*t^170+.\ 1421380485e-1*t^111+.2932478571e-2*t^61+.2913196651e-7*t^270+.9809858880e-3*t^ 161+.7808411748e-3*t^165+.3096201531e-5*t^224+.7345375924e-18*t^474+.6451478299\ e-12*t^364+.8380380252e-2*t^127+.1055425355e-8*t^25+.2107113088e-11*t^354+.\ 1685205129e-3*t^182+.6919581939e-19*t^494+.4482770439e-14*t^410+.1544808213e-5* t^34+.1272268888e-1*t^114+.6656346255e-15*t^423+.5754989511e-15*t^424+.\ 4484715594e-3*t^172+.2769414024e-2*t^142+.7410521326e-2*t^129+.1080201945e-5*t^ 236+.3371024630e-3*t^175+.2507155145e-3*t^178+.1766444920e-1*t^100+.1059409157e\ -13*t^403+.3848741325e-9*t^307+.1272685580e-10*t^338+.2279835860e-9*t^311+.\ 7373244920e-11*t^343+.2882649670e-15*t^429+.1941307959e-16*t^450+.9221012213e-\ 11*t^341+.1178508413e-13*t^402+.8440908010e-9*t^301+.1330207305e-4*t^209+.\ 1497307200e-17*t^468+.1006316874e-6*t^257+.5091499000e-4*t^193+.1585251440e-1*t ^90+.7397975389e-9*t^302+.5419945772e-19*t^496+.3214847672e-7*t^269+.2267457296\ e-5*t^228+.5202896205e-7*t^264+.3341716053e-3*t^48+.5816285841e-3*t^169+.\ 1548120249e-1*t^83+.4086504721e-3*t^173+.1084547259e-2*t^53+.6940303538e-17*t^ 457+.2631897879e-11*t^352+.2185219645e-4*t^40+.2048633315e-13*t^396+.5691773586\ e-9*t^304+.2533576995e-10*t^331+.4119021181e-4*t^195+.3718600333e-6*t^31+.\ 1601586576e-1*t^93+.7056493567e-3*t^51+.2394650552e-6*t^249+.7982022673e-4*t^ 189+.1995526258e-2*t^146+.8903960127e-2*t^70+.5964248416e-5*t^217+.8452360509e-\ 14*t^405+.5226619018e-11*t^346+.1705438598e-2*t^148+.1286227270e-1*t^75+.\ 2161623240e-2*t^58+.1575471667e-1*t^108+.1956539985e-7*t^274+.9636161424e-9*t^ 300+.1523851019e-1*t^78+.7758781259e-6*t^239+.6478905054e-16*t^441+.2323792599e\ -18*t^484+.4110873522e-8*t^26+.1590959372e-1*t^91+.2092442772e-5*t^229+.\ 9636111446e-2*t^124+.4733179390e-7*t^265+.1640359260e-8*t^296+.3368013811e-2*t^ 62+.1533400164e-1*t^84+.8681772852e-6*t^238+.2225441848e-15*t^431+.2529899786e-\ 15*t^430+.2944888137e-11*t^351+.4655857089e-11*t^347+.1773770900e-1*t^101+.\ 2056777729e-3*t^180+.6350976018e-13*t^381+.5866053872e-11*t^345+.3919111783e-13 *t^387+.1144419588e-2*t^157+.1214329994e-2*t^155+.2167494026e-2*t^145+.\ 2556215942e-2*t^143+.1565702115e-1*t^79+.3185157418e-13*t^390+.5942303031e-5*t^ 37+.5308841119e-13*t^383+.1370963408e-7*t^27+.2629827922e-4*t^200+.1252040390e-\ 2*t^154+.1183642142e-7*t^279+.1372239079e-3*t^45+.1193334556e-1*t^74+.\ 1613885701e-1*t^94+.1369125674e-1*t^112+.3657396574e-5*t^222+.1448411928e-7*t^ 277+.1612172900e-5*t^232+.2429060057e-8*t^293+.5715087395e-7*t^263+.3853211188e\ -17*t^461+.1171913361e-17*t^470+.1899159814e-6*t^251+.1992873701e-6*t^30+.\ 9987851218e-19*t^491+.1274582294e-18*t^489+.4707053379e-2*t^135+.1871993151e-8* t^295+.6312708594e-8*t^285+.5403959802e-10*t^323+.4798940432e-19*t^497+.\ 5370194878e-5*t^218+.1068740055e-2*t^159+.5046085469e-8*t^287+.7820265103e-19*t ^493+.1999728491e-9*t^312+.2648640225e-5*t^226+.3545486304e-7*t^268+.2578034606\ e-13*t^393+.2354200224e-11*t^353+.1696620559e-6*t^252+.1731891192e-15*t^433+.\ 1663913519e-1*t^106+.1230107950e-1*t^115+.3339444106e-17*t^462+.8269000959e-2*t ^69+.2608886617e-2*t^60+.1377636136e-1*t^76+.8819878801e-3*t^52+.1769934055e-1* t^102+.6553366453e-18*t^475+.1305406936e-13*t^401+.1961578282e-15*t^432+.\ 8931071628e-4*t^188+.3714824735e-3*t^174+.4995315309e-9*t^305+.7607284186e-2*t^ 68+.8838278270e-19*t^492+.4603091942e-2*t^64+.2392455428e-5*t^35+.1831212154e-\ 18*t^486+.2638507473e-7*t^271+.4394137482e-5*t^220+.1933716080e-17*t^466+.\ 1127003509e-12*t^376+.1291718379e-12*t^375+.5220770553e-18*t^477+.1447256236e-4 *t^208+.2780983764e-10*t^330+.1145270440e-10*t^339+.7431439649e-10*t^320+.\ 3673235727e-10*t^327+.6887021188e-7*t^261+.9648313850e-8*t^281+.1039695738e-14* t^420+.2380382779e-9*t^24+.1535378813e-2*t^55+.1581900450e-1*t^81+.1112608343e-\ 3*t^186+.3539649192e-8*t^290+.2694796285e-6*t^248+.1404827653e-14*t^418+.\ 4659193572e-18*t^478+.3660691796e-12*t^368+.1722420341e-1*t^98+.2526858302e-17* t^464+.1309636921e-7*t^278+.1454898698e-16*t^452+.7341816658e-16*t^440+.\ 1541359448e-9*t^314+.1348731186e-11*t^358+.1318950034e-1*t^113+.3391273244e-4*t ^197+.4239847445e-12*t^367+.3928941994e-2*t^63+.9977966940e-4*t^187+.3907212703\ e-7*t^267+.4330961800e-15*t^426+.9978878605e-2*t^123+.1200577073e-11*t^21+.\ 1960893863e-4*t^204+.1824059476e-4*t^205+.1105464851e-1*t^73+.1492001579e-2*t^ 150+.5354531542e-2*t^65+.1412271033e-2*t^151+.3728249063e-4*t^196+.1633784472e-\ 1*t^95+.8807058153e-13*t^20+.9473468226e-12*t^361+.8367241489e-12*t^362+.\ 4530981925e-13*t^385+.1411504752e-10*t^337+.3697441142e-11*t^349+.1069103597e-7 *t^280+.3396699500e-14*t^412+.1488849898e-12*t^374+.4358247830e-2*t^136+.\ 6297017358e-3*t^168+.5505594854e-2*t^133+.1441587702e-4*t^39+.9449667751e-7*t^ 29+.8833294892e-2*t^126+.1732095159e-1*t^104+.1965706623e-2*t^57+.1519656743e-6 *t^253+.5788876008e-13*t^382+.1106121073e-1*t^119+.1081274502e-1*t^120+.\ 4034645017e-10*t^326+.4208833422e-13*t^386+.7037269163e-8*t^284+.7712988837e-15 *t^422+.2770252877e-13*t^392+.6633126649e-14*t^407+.3419143797e-6*t^246+.\ 1257306650e-8*t^298+.5988587729e-10*t^322+.8055228240e-17*t^456+.2394187200e-13 *t^394+.2217887460e-13*t^395+.4410511993e-16*t^444+.7361953689e-12*t^363+.\ 4302272681e-7*t^266+.1726357525e-10*t^335+.1229758733e-6*t^255+.2207219004e-17* t^465+.2939319178e-18*t^482+.3767726152e-19*t^499+.1623815021e-18*t^487+.\ 3301641681e-18*t^481+.5981548673e-17*t^458+.6297262180e-6*t^32+.5709077091e-16* t^442+.2710468126e-12*t^370+.9348270908e-17*t^455+.1028608180e-10*t^340+.\ 6791511323e-3*t^167+.7560729413e-7*t^260+.2952263426e-16*t^447+.1128464274e-18* t^490+.1354703329e-15*t^435+.7502729366e-14*t^406+.9695497924e-6*t^237+.\ 4891019257e-10*t^324+.1525933150e-1*t^109+.9553605682e-2*t^71+.3255576155e-4*t^ 41+.9102699598e-5*t^213+.4337643001e-6*t^244+.3771386613e-15*t^427+.1595114127e\ -1*t^92+.5571285449e-3*t^50+.1730484747e-13*t^398+.1323029608e-17*t^469+.\ 1527965453e-1*t^85+.1026387828e-2*t^160+.2768068774e-3*t^177+.2861615428e-5*t^ 225+.6487495663e-9*t^303+.1510120175e-11*t^357+.8761541344e-11*t^22+.9253632288\ e-2*t^125+.4776508607e-4*t^42+.1055964336e-9*t^317+.9359677318e-10*t^318+.\ 1040294431e-17*t^471+.1084355732e-16*t^454+.2199377889e-14*t^415+.4861483185e-\ 10*t^23+.2097027185e-10*t^333+.1886434012e-11*t^355+.7830011033e-8*t^283+.\ 3907764227e-14*t^411+.1688483372e-11*t^356+.4894976944e-12*t^366+.1601420250e-7 *t^276+.1061632910e-15*t^437+.1531061620e-15*t^434+.8305677877e-7*t^259+.\ 1237974261e-3*t^185+.8831756321e-3*t^163+.7296588546e-3*t^166+.9330175177e-3*t^ 162+.1199212336e-15*t^436+.1755579960e-1*t^103+.1131999556e-1*t^118+.1347725254\ e-2*t^152+.2849106646e-4*t^199+.6634365788e-5*t^216+.1904172623e-10*t^334+.\ 1699131258e-17*t^467+.3358950996e-5*t^223+.2451933694e-5*t^227+.8757370826e-13* t^378+.1217411384e-4*t^210+.5022455767e-16*t^443+.2306276530e-10*t^332+.\ 2328723627e-12*t^371+.2270859750e-3*t^179+.9395402816e-16*t^438+.7127585822e-4* t^190+.1841048667e-2*t^147+.2993990471e-2*t^141+.1881197117e-3*t^46+.4454990010\ e-17*t^460+.3381672106e-16*t^446+.1100564346e-8*t^299+.7015740083e-13*t^380+.\ 1000001121e-5*t^33+.2529953700e-3*t^47+.9898215076e-13*t^377+.1365044284e-6*t^ 254+.1765178204e-5*t^231+.8950346458e-15*t^421+.1562453644e-10*t^336+.\ 1621750937e-1*t^107+.3481366323e-2*t^139+.2614593024e-18*t^483+.1439016403e-18* t^488+.1758040778e-2*t^56+.1693691148e-4*t^206+.5687160969e-4*t^192+.2263647641\ e-4*t^202+.2599453796e-9*t^310+.9251289914e-18*t^472+.2901033204e-17*t^463+.\ 4571034696e-4*t^194+.1200123933e-5*t^235+.5841796514e-14*t^408+.4848651212e-5*t ^219+.3411845685e-13*t^389+.3705729906e-18*t^480+.1256751067e-16*t^453+.\ 1055847415e-1*t^121+.2364939243e-2*t^59+.5848997529e-18*t^476+.8309397815e-16*t ^439+.4437876044e-10*t^325+.1355807304e-9*t^315+.5159083280e-17*t^459+.\ 3757617493e-5*t^36+.8697710403e-8*t^282+.6274436263e-7*t^262+.4986436635e-15*t^ 425+.1295409364e-2*t^153+.9331849356e-5*t^38+.1374519362e-3*t^184+.3341721362e-\ 19*t^500+.9134601954e-7*t^258+.2436531427e-4*t^201+.1532882716e-1*t^86+.\ 3230593849e-2*t^140+.4250983996e-19*t^498+.6364101062e-4*t^191+.1896008982e-14* t^416+.1886180412e-13*t^397+.3865785500e-16*t^445+.3152630364e-12*t^369+.\ 2972059809e-13*t^391+.7381871787e-5*t^215+.1474147160e-1*t^110+.5126128750e-14* t^409+.9820796323e-4*t^44+.6894751837e-2*t^67+.1436362238e-8*t^297+.3292862383e\ -15*t^428+.1179280965e-2*t^156+.1107938461e-2*t^158+.1589239881e-2*t^149+.\ 2162109379e-7*t^273 Here is a plot of the prob. distribution 0.4H*H HHHHHHHHH+HHHHH HHHH 0.2 + HHHHHH HHHHHHH + HHHHHHHHHHHHHHH -******************-+--+---+--+--+--+--+--+--+--+--+---+--+--***************- -4 -2 0 2 4 To sum up the expectation, s.d., and the scaled moments up to the, 4, -th are [100.26480683975818340, 22.677184601850379544, 0.53510980922322583233, 3.4359882942130068387] ----------------------- ---------------- This ends this paper that took, 4216.398, seconds to produce. How long should it take until you visit the integers with property, "isprime", for the , 40, -th time if you roll a fair die with, 6, faces By Shalosh B. Ekhad If you roll a fair die with, 6, faces starting at, 0, and look at the runnin\ g total and keep going until that running total has the property, "isprime", for , 40, times The prob. of reaching that goal in <=, 500, rounds is, 0.99999998164326044596 Conditioned that you are succesful the expected number of rounds for that go\ al is, 229.89037827865298072, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 36.127190160746335948 The skewness is, 0.37779489989973806053 The kurtosis is, 3.1278526033881428377 The prob. generating function, in t, for the r.v. reaching property, isprime, for the, 40, -h time is .2877862856e-5*t^413+.2725831065e-7*t^98+.1804957998e-3*t^139+.3010698175e-2*t^ 290+.4180439459e-3*t^149+.1056288788e-1*t^211+.1039062966e-1*t^243+.7079137661e\ -2*t^187+.2188304772e-5*t^114+.5969498207e-2*t^263+.3181055761e-3*t^145+.\ 8523210026e-3*t^314+.1527056264e-2*t^304+.2346008587e-3*t^339+.5084630766e-11*t ^76+.3222573075e-26*t^46+.5671029829e-4*t^130+.1217234524e-4*t^122+.1449785074e\ -6*t^448+.3830517226e-7*t^462+.3225072607e-2*t^288+.3109625702e-5*t^412+.\ 6380252756e-2*t^184+.7485814288e-5*t^393+.2665934044e-4*t^374+.6359086449e-7*t^ 457+.1367689623e-7*t^96+.7240806529e-6*t^428+.2044197607e-3*t^342+.6379648101e-\ 4*t^363+.7848630739e-8*t^480+.1103642095e-4*t^384+.5746043167e-2*t^265+.\ 5637920171e-2*t^266+.2417374528e-4*t^375+.4590120068e-2*t^177+.1367714777e-2*t^ 306+.2861078767e-9*t^85+.6219099913e-2*t^261+.5862677556e-2*t^182+.1272187574e-\ 4*t^382+.2507627587e-6*t^441+.5362363410e-8*t^485+.2584748315e-8*t^494+.\ 3765537435e-10*t^80+.4114754607e-5*t^117+.1117930321e-1*t^236+.1557444643e-13*t ^68+.6002258890e-4*t^364+.2244598979e-3*t^340+.6242282188e-5*t^399+.1042688136e\ -1*t^228+.1112096317e-1*t^234+.8094725383e-2*t^252+.4817773974e-4*t^129+.\ 2171811653e-16*t^61+.1037494176e-4*t^385+.6089645792e-2*t^262+.8455717360e-2*t^ 202+.4844854111e-2*t^273+.4748997805e-3*t^323+.1752808108e-3*t^345+.2908437018e\ -2*t^170+.1153305887e-7*t^475+.1249830412e-7*t^474+.6253011063e-8*t^483+.\ 7987409004e-3*t^315+.6690286575e-2*t^258+.1044183250e-1*t^221+.5342889526e-2*t^ 180+.2324374669e-7*t^467+.6856665411e-5*t^396+.7217027932e-3*t^156+.5601594776e\ -2*t^181+.3860601433e-2*t^174+.9342587001e-7*t^453+.1104302003e-1*t^233+.\ 3141772893e-2*t^171+.5261218128e-4*t^366+.1047052187e-3*t^354+.1642448150e-4*t^ 379+.1482090789e-3*t^348+.5095281495e-5*t^118+.9091540841e-6*t^110+.2871900667e\ -3*t^334+.7968263743e-5*t^391+.2984945958e-3*t^333+.3379272631e-2*t^172+.\ 7565641103e-2*t^254+.3459285108e-7*t^463+.6324745305e-5*t^119+.1054863927e-1*t^ 229+.1115616391e-1*t^237+.7178653512e-17*t^60+.1174031528e-3*t^352+.8507695354e\ -7*t^454+.2655296837e-5*t^414+.1686357016e-8*t^499+.1943093356e-9*t^84+.\ 6867464947e-8*t^94+.7404789139e-7*t^101+.7717081371e-5*t^392+.1167231138e-5*t^ 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^78+.2692935403e-3*t^143+.7165599992e-6*t^109+.1761544305e-19*t^55+.6360700461e\ -2*t^260+.6654297507e-5*t^397+.5202063295e-7*t^459+.3962279985e-6*t^435+.\ 1108280439e-3*t^353+.9330163027e-5*t^387+.1285508143e-5*t^422+.2702990965e-6*t^ 440+.5855830965e-2*t^264+.4632471268e-25*t^47+.7951303650e-6*t^427+.7781339768e\ -2*t^192+.2226432958e-3*t^141+.1109890059e-1*t^238+.6628404673e-2*t^185+.\ 1059670459e-5*t^424+.1475953901e-7*t^472+.1398660596e-3*t^349+.1293687214e-2*t^ 307+.2331093204e-2*t^296+.8827677212e-11*t^77+.9622351750e-6*t^425+.1135543776e\ -3*t^135+.1523899021e-2*t^163+.1074692598e-1*t^241+.3329273108e-2*t^287+.\ 9077385230e-2*t^205+.2008900319e-3*t^140+.1223097271e-6*t^450+.7865453729e-5*t^ 120+.2811424435e-8*t^493+.9681416799e-8*t^95+.1093185558e-1*t^214+.5069058035e-\ 6*t^432+.1574172512e-6*t^447+.4543571324e-24*t^48+.1119803583e-14*t^65+.\ 7930086059e-2*t^195+.7733492141e-7*t^455+.7268540478e-5*t^394+.1032646691e-1*t^ 227+.5344017487e-7*t^100+.9371641901e-4*t^356+.6762906818e-4*t^362+.7477337872e\ -3*t^316+.8095717644e-13*t^70+.3836878937e-3*t^327+.7996951441e-2*t^197+.\ 6994842352e-3*t^317+.3505595808e-3*t^329+.9134415062e-8*t^478+.6746894581e-8*t^ 482+.3381153486e-4*t^127+.3301325686e-6*t^106+.1513793574e-20*t^53+.3349287345e\ -5*t^411+.1017494433e-6*t^102+.8308873575e-2*t^201+.7277401276e-8*t^481+.\ 6227139890e-16*t^62+.5581148136e-5*t^402+.1232701453e-2*t^161+.5346668033e-5*t^ 403+.4492715543e-2*t^276+.5088002372e-2*t^179+.1798379629e-2*t^301+.5629519585e\ -4*t^365+.1089903886e-2*t^310+.7092053385e-2*t^256+.6881790779e-2*t^257+.\ 1025587254e-1*t^226+.1385643284e-6*t^103+.1089136564e-1*t^240+.5104483229e-3*t^ 152+.1088094592e-1*t^217+.3673248695e-3*t^147+.3118944492e-2*t^289+.9958930701e\ -2*t^245+.4541472923e-4*t^368+.4607937648e-5*t^406+.1986353293e-4*t^377+.\ 1267973871e-27*t^45+.5754570568e-7*t^458+.1441612309e-3*t^137+.3481971946e-8*t^ 92+.7440518301e-2*t^189+.1662176569e-29*t^44+.1865281293e-4*t^124+.1246592240e-\ 8*t^89+.3634681138e-2*t^284+.1501050110e-4*t^380+.9466136657e-2*t^247+.\ 3126088761e-7*t^464+.1181279059e-4*t^383+.7640271151e-12*t^73+.2442991196e-5*t^ 415+.9083096293e-3*t^313+.9862447215e-8*t^477+.1547132647e-8*t^500+.4758677764e\ -3*t^151+.6541707049e-3*t^318+.1024010848e-6*t^452+.8246320368e-5*t^390+.\ 1426905168e-5*t^112+.3103071668e-3*t^332+.2935787375e-3*t^144+.5079860885e-2*t^ 271+.8106030390e-2*t^199+.1615787148e-3*t^138+.3736063680e-2*t^283+.3922618711e\ -3*t^148+.2496861183e-6*t^105+.2002535714e-8*t^497+.2113173332e-7*t^468+.\ 1095903206e-1*t^215+.2291167964e-4*t^125+.5308910801e-2*t^269+.2392017129e-10*t ^79+.4319505614e-6*t^107+.2750251837e-14*t^66+.3856484348e-4*t^370+.1848257231e\ -3*t^344+.5728970741e-3*t^320+.6119417905e-3*t^319+.2325730124e-6*t^442+.\ 1332864477e-6*t^449+.3603265880e-8*t^490+.2052215467e-5*t^417+.1758020803e-7*t^ 470+.2253356017e-17*t^59+.9788566924e-5*t^121+.1116733059e-1*t^235+.1319331624e\ -3*t^350+.3056053533e-8*t^492+.3662854339e-6*t^436+.5898964281e-19*t^56+.\ 9912218304e-3*t^159+.4960630824e-8*t^486+.2455411023e-3*t^142+.9811412956e-5*t^ 386+.8892193115e-3*t^158+.3533516017e-2*t^285+.1100969138e-1*t^239+.5420291695e\ -2*t^268+.5985066383e-3*t^154+.1965788723e-22*t^50+.3663207333e-3*t^328+.\ 1105710685e-2*t^160+.1244396724e-3*t^351+.4585345092e-8*t^487+.8559403868e-5*t^ 389+.3227890308e-3*t^331+.1613009530e-2*t^303+.2681126784e-2*t^169+.1011889093e\ -1*t^209+.1569234333e-3*t^347+.3361323443e-3*t^330+.2108361544e-2*t^298+.\ 1837865998e-8*t^498+.1073169556e-1*t^212+.8768027493e-10*t^82+.6517055622e-2*t^ 259+.7961942854e-2*t^196+.7893888445e-2*t^194+.6034623392e-6*t^430+.4962808832e\ -2*t^272+.4345603505e-2*t^176+.8650893503e-2*t^250+.9331177553e-2*t^206+.\ 4047552287e-2*t^280+.6609346108e-14*t^67+.6123484197e-2*t^183+.1496967493e-11*t ^74+.3532933223e-4*t^371+.1034111099e-1*t^222+.8928926879e-2*t^249+.2935670046e\ -4*t^373+.4482865742e-3*t^324+.2445131712e-2*t^295+.4884990023e-8*t^93+.\ 2674409503e-2*t^293+.6862978211e-2*t^186+.1281880594e-3*t^136+.1996517205e-6*t^ 444+.3347377666e-23*t^49+.2481907380e-8*t^91+.2815654983e-11*t^75+.1065851501e-\ 7*t^476+.4378056483e-2*t^277+.4243870087e-3*t^325+.7995216574e-3*t^157+.\ 1945521817e-3*t^343+.1356840198e-7*t^473+.2762621632e-3*t^335+.2559841025e-2*t^ 294+.2900345848e-2*t^291+.2001214373e-2*t^299+.2788109244e-2*t^292+.4155352107e\ -2*t^279+.7650972376e-4*t^132+.1027107835e-2*t^311+.5592363261e-6*t^108+.\ 2375026182e-8*t^495+.1081508642e-1*t^231+.1509938141e-4*t^123+.7320533065e-2*t^ 255+.9201775730e-2*t^248+.1869236644e-6*t^104+.1932728361e-7*t^97+.8781867861e-\ 4*t^133+.1925263373e-7*t^469+.6450255243e-5*t^398+.5808512551e-5*t^401+.\ 8634344956e-2*t^203+.2155770169e-6*t^443+.4193496095e-4*t^369+.8465600179e-8*t^ 479+.4858917994e-5*t^405+.9595997717e-2*t^207 Here is a plot of the prob. distribution | HHHHHHHH*HHHHH HHHH + HHHHHHH +***************************-+--+--+--+--+--+--+-***************************- | -4 -2 2 4 To sum up the expectation, s.d., and the scaled moments up to the, 4, -th are [229.89037827865298072, 36.127190160746335948, 0.37779489989973806053, 3.1278526033881428377] ----------------------- ---------------- This ends this paper that took, 9243.026, seconds to produce. How long should it take until you visit the integers with property, "isprime", for the , 60, -th time if you roll a fair die with, 6, faces By Shalosh B. Ekhad If you roll a fair die with, 6, faces starting at, 0, and look at the runnin\ g total and keep going until that running total has the property, "isprime", for , 60, times The prob. of reaching that goal in <=, 500, rounds is, 0.99394787259690656644 Conditioned that you are succesful the expected number of rounds for that go\ al is, 370.52415776389690307, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 46.024513536313478563 The skewness is, 0.14067634057220270295 The kurtosis is, 2.6164507340399231243 The prob. generating function, in t, for the r.v. reaching property, isprime, for the, 60, -h time is .4214452058e-2*t^428+.2471225385e-4*t^233+.1620367217e-3*t^252+.1464674787e-2*t ^461+.2252244834e-2*t^450+.3395753179e-2*t^434+.9295734309e-2*t^362+.6220505261\ e-2*t^403+.5610272340e-2*t^329+.1257806587e-2*t^464+.1537195225e-2*t^460+.\ 2378956077e-2*t^448+.2440188794e-2*t^447+.1906406191e-2*t^455+.2049477977e-2*t^ 453+.1576619399e-7*t^183+.6254034955e-3*t^479+.5312870209e-2*t^391+.5445986301e\ -2*t^395+.1213056499e-2*t^282+.2186414143e-8*t^175+.6149904609e-2*t^334+.\ 2619851811e-2*t^444+.1742549013e-5*t^211+.5024115072e-4*t^239+.1371024235e-30*t ^87+.4124629519e-3*t^266+.2560132298e-2*t^445+.2884340343e-22*t^104+.1153821521\ e-11*t^151+.8408012637e-4*t^244+.1457125437e-15*t^127+.8444204571e-14*t^136+.\ 3726456349e-12*t^147+.4146479706e-2*t^307+.5377716929e-2*t^389+.8854394576e-24* t^100+.9065558702e-2*t^360+.5804482837e-2*t^399+.1499135693e-25*t^96+.\ 2714650084e-13*t^139+.1248472741e-24*t^98+.3068985165e-5*t^215+.4069957339e-19* t^114+.4056956964e-2*t^429+.8028371125e-2*t^347+.1563398019e-14*t^132+.\ 7602251761e-5*t^223+.5843731103e-2*t^385+.1258486236e-13*t^137+.6840140168e-5*t ^222+.1293541275e-2*t^283+.6228686779e-4*t^241+.4901642861e-3*t^486+.7313920766\ e-3*t^475+.1117850592e-9*t^165+.1699236580e-4*t^230+.6552703369e-2*t^408+.\ 3290322096e-2*t^435+.2722424978e-3*t^260+.1299044312e-3*t^249+.9459686872e-29*t ^90+.8339670282e-2*t^351+.1699092686e-38*t^76+.6769263590e-2*t^381+.2550476812e\ -3*t^259+.5423153598e-2*t^421+.5107874377e-3*t^269+.6034827299e-2*t^384+.\ 8458848555e-5*t^224+.3490664495e-2*t^302+.4492633654e-4*t^238+.2606877305e-2*t^ 295+.5385481907e-2*t^394+.7634831277e-3*t^474+.9405634306e-2*t^363+.4660576735e\ -2*t^311+.3060316292e-3*t^498+.2476473883e-20*t^110+.4539912607e-7*t^188+.\ 5875406250e-3*t^271+.1198989432e-3*t^248+.1066418002e-2*t^280+.9997219562e-3*t^ 279+.6154336043e-5*t^221+.5823099066e-3*t^481+.6733325788e-6*t^205+.8019525076e\ -10*t^164+.9689924799e-3*t^469+.5602250804e-4*t^240+.1003983989e-19*t^112+.\ 2704604762e-6*t^199+.5372830991e-2*t^323+.2732502604e-2*t^296+.7356991971e-2*t^ 379+.9418898021e-2*t^371+.7983138961e-3*t^473+.1904055572e-39*t^75+.6032294765e\ -3*t^480+.3338244260e-3*t^496+.6393674923e-2*t^405+.2926817260e-3*t^499+.\ 9625737980e-2*t^369+.3486783230e-5*t^216+.1522893844e-26*t^94+.4957956236e-5*t^ 219+.4732971957e-3*t^487+.3779341853e-3*t^493+.5363947842e-36*t^79+.1924663632e\ -4*t^231+.7925256677e-2*t^346+.5434338100e-3*t^483+.9322178329e-7*t^192+.\ 6254484895e-2*t^383+.7205073012e-3*t^274+.2186389821e-2*t^451+.2023140003e-5*t^ 212+.3578021065e-3*t^264+.2316387841e-2*t^449+.1590827391e-16*t^123+.1268470064\ e-22*t^103+.6983969348e-32*t^85+.4279121674e-2*t^308+.2067649727e-12*t^145+.\ 1324565919e-2*t^463+.1857817552e-13*t^138+.2842775037e-8*t^176+.1996829938e-2*t ^290+.2234977472e-2*t^292+.2680249053e-2*t^443+.8627392913e-16*t^126+.\ 4554617238e-27*t^93+.6452203845e-2*t^413+.2391821790e-3*t^258+.1518362476e-12*t ^144+.5075087962e-2*t^423+.1054636838e-4*t^226+.1882592174e-2*t^289+.1469779719\ e-2*t^285+.6073982408e-45*t^70+.6926555202e-2*t^339+.2500430447e-2*t^446+.\ 6515852166e-2*t^407+.1041877234e-10*t^158+.1314608010e-37*t^77+.5786196801e-2*t 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298+.9668508132e-2*t^368+.9673458167e-2*t^367+.8401800366e-2*t^352+.3938298855e\ -15*t^129+.2796808224e-4*t^234+.1273786888e-17*t^119+.9431826817e-5*t^225+.\ 2892548152e-21*t^107+.6373075349e-22*t^105+.8595306312e-2*t^355+.2181215252e-4* t^232+.6016614777e-2*t^333+.2851790319e-16*t^124+.4722078215e-2*t^425+.\ 6559488020e-12*t^149+.2818890168e-11*t^154+.3103829209e-2*t^437+.9370363818e-3* t^278+.8361082992e-3*t^472+.7088627495e-2*t^340+.2394668015e-29*t^89+.\ 7908644952e-13*t^142+.4081468638e-10*t^162+.6602252953e-2*t^337+.3335958410e-3* t^263+.4990169958e-20*t^111+.1759180278e-2*t^457+.1502364973e-4*t^229+.\ 2685917356e-5*t^214+.5315538148e-2*t^320+.4919127613e-6*t^203+.7441948964e-11*t ^157+.4537832163e-2*t^310+.3931999740e-3*t^492+.1097957131e-6*t^193+.5708266346\ e-48*t^68+.1536302076e-32*t^84+.4880243170e-26*t^95+.1494537090e-5*t^210+.\ 5355770597e-2*t^322+.7247534409e-2*t^341+.1183021174e-4*t^227+.5332557185e-2*t^ 390+.6736791837e-3*t^273+.1000183277e-14*t^131+.3725804564e-7*t^187+.6279033828\ e-2*t^415+.8194549129e-19*t^115+.2417229082e-14*t^133+.4378654614e-2*t^427+.\ 3840685748e-3*t^265+.2357607475e-2*t^293+.7810373365e-2*t^345+.1277513483e-5*t^ 209+.7401050776e-2*t^342+.1833145758e-2*t^456+.8698874643e-17*t^122+.1251922978\ e-7*t^182+.4429293961e-5*t^218+.5387041148e-9*t^170+.2457979511e-17*t^120+.\ 1402707854e-3*t^250+.5074163485e-3*t^485+.4757975615e-3*t^268+.2742079224e-2*t^ 442+.3619390930e-2*t^303+.1974056530e-7*t^184+.1299852984e-42*t^72+.1223154972e\ -20*t^109+.1856281894e-40*t^74+.8224771653e-3*t^276+.5590398866e-2*t^420+.\ 5248742977e-2*t^318+.2337736333e-6*t^198+.2106773887e-3*t^256+.3939793562e-5*t^ 217+.3193230813e-2*t^436+.6374440270e-2*t^414+.3562367406e-4*t^236+.6507922700e\ -18*t^118+.2060410687e-10*t^160+.6571411394e-2*t^409+.8555572985e-2*t^375+.\ 2806180502e-2*t^441+.6511293021e-2*t^412+.7900678062e-6*t^206+.5415971699e-2*t^ 325+.1379078995e-2*t^284+.3035742883e-7*t^186+.3482263770e-3*t^495+.3765791958e\ -2*t^431+.3021331040e-2*t^438+.4898249172e-2*t^424+.3236368069e-2*t^300+.\ 5480231266e-3*t^270+.6293972391e-3*t^272+.3363091463e-2*t^301+.5543316071e-2*t^ 328+.6121002230e-2*t^402+.8780528361e-3*t^277+.5337665351e-2*t^321+.2020682247e\ -19*t^113+.1463802620e-10*t^159+.5450174649e-2*t^388+.5344109590e-11*t^156+.\ 8120043994e-2*t^348+.3286675921e-18*t^117+.3700927429e-14*t^134+.1331269105e-4* t^228+.1486246695e-34*t^81+.3342195467e-33*t^83+.5139276125e-2*t^316+.\ 6167297497e-2*t^416+.5831830126e-30*t^88+.9690972643e-44*t^71+.4952080561e-12*t ^148+.9053513917e-2*t^373+.2138285934e-9*t^167+.6040775454e-2*t^417+.1978612998\ e-2*t^454+.3907157999e-2*t^430+.9213198655e-3*t^470+.3923270643e-13*t^140+.\ 7971180090e-2*t^377+.1102823339e-3*t^247+.2901250138e-10*t^161+.5750416361e-2*t ^419+.2114501170e-2*t^291+.3159907743e-4*t^235+.4416242235e-25*t^97+.2797208365\ e-3*t^500+.5405866662e-23*t^102+.2227325485e-23*t^101+.8850989742e-2*t^358+.\ 3749467500e-2*t^304+.3633334610e-2*t^432+.2931510641e-35*t^80+.4883697214e-2*t^ 313+.7664161065e-2*t^378+.9586128824e-2*t^365+.2118783254e-2*t^452+.1286238224e\ -6*t^194+.1010870921e-3*t^246+.1272758194e-8*t^173+.2416124750e-15*t^128+.\ 9278322255e-6*t^207+.2075000889e-11*t^153+.3588212710e-28*t^91+.8687050514e-12* t^150+.1550500932e-9*t^166+.2787362008e-12*t^146+.1610760362e-2*t^459+.\ 2873456568e-2*t^440+.3510000663e-2*t^433+.3110166618e-2*t^299+.5901194150e-2*t^ 418+.5286667315e-2*t^319+.5551419925e-2*t^387+.5682472072e-2*t^386+.5894958572e\ -2*t^332+.4567532371e-3*t^488+.9542790333e-2*t^370+.1137432751e-2*t^281+.\ 1853154039e-3*t^254+.3197524688e-3*t^497+.7547137555e-2*t^343+.5447747951e-2*t^ 326+.2944853482e-2*t^439+.5910109519e-2*t^400+.2481761478e-2*t^294+.3987286818e\ -9*t^169+.5530499266e-5*t^220+.5607045893e-2*t^397+.7867297268e-7*t^191+.\ 4665985338e-17*t^121+.5489577200e-2*t^327+.1132610719e-2*t^466+.2019976364e-6*t ^197+.5391907345e-2*t^324+.8273735116e-2*t^350+.1645636280e-18*t^116+.\ 5987950964e-21*t^108+.6500416425e-2*t^382+.7155905439e-34*t^82+.4740558092e-8*t ^178+.8671340049e-2*t^356+.7684036768e-2*t^344+.5729989790e-10*t^163+.\ 5250711485e-2*t^422+.3864890315e-11*t^155+.7056598112e-2*t^380+.1772353940e-2*t ^288+.7017957989e-3*t^476+.7225392267e-9*t^171+.2858328491e-2*t^297+.3678958399\ e-8*t^177+.1509811439e-3*t^251+.5010040776e-16*t^125+.5199994282e-2*t^317 Here is a plot of the prob. distribution H* HHHHH*HH HHH HHHHHH + HHHH HHHH HHHHHHH + HHHHHHHHHH +*********************-+--+--+--+--+--+--+--+--+--+--+--+-******************- -4 -2 0 2 4 To sum up the expectation, s.d., and the scaled moments up to the, 4, -th are [370.52415776389690307, 46.024513536313478563, 0.14067634057220270295, 2.6164507340399231243] ----------------------- ---------------- This ends this paper that took, 14719.254, seconds to produce.