How long should it take until you  visit the integers with property, "isprime",

    for the , 100,  -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 , 100, times



 The prob. of reaching that goal in <=, 1200, rounds is, 0.99999999990782734390





Conditioned that you are succesful the expected number of rounds for that go\

    al is, 676.31537634061272787,  (and this is a good estimate for unconditi\

    onal expectation) and the standard deviation is, 65.276593284568584017



                    The skewness is, 0.22304111567227178273

                     The kurtosis is, 3.0704308328514590521

The prob. generating function, in t,  for the r.v. reaching property, conver\

    t(P,string), for the, k, -h time is



.1840279175e-2*t^782+.7774296589e-8*t^399+.1408493503e-4*t^467+.2164703648e-7*t
^406+.4033026397e-31*t^208+.2230303711e-22*t^251+.1157056295e-3*t^869+.\
1079780132e-4*t^940+.4193812492e-2*t^708+.4070655702e-4*t^902+.1335401009e-65*t
^137+.2268445295e-2*t^776+.4140622459e-8*t^1092+.9659897931e-4*t^874+.\
1635179009e-23*t^244+.3268289279e-2*t^744+.2054202588e-3*t^851+.1686805828e-4*t
^470+.3106586299e-8*t^391+.5509062702e-3*t^818+.1512575512e-3*t^509+.4034988106\
e-2*t^615+.6107590075e-9*t^374+.6107172701e-2*t^649+.1377855795e-54*t^150+.\
4681749504e-2*t^634+.1154231252e-3*t^503+.2592278720e-8*t^389+.1322021384e-4*t^
466+.7513155323e-4*t^495+.3389371425e-2*t^736+.1587526767e-2*t^786+.7200052609e\
-11*t^1198+.5338794011e-3*t^819+.8115050596e-7*t^1038+.1118343374e-15*t^297+.\
3145734531e-42*t^173+.2443535982e-19*t^271+.1474527718e-2*t^567+.1345926972e-14
*t^306+.3300396015e-2*t^741+.4417112403e-10*t^1168+.5006122829e-10*t^1166+.\
2707546227e-33*t^200+.9836518392e-3*t^557+.2505265625e-9*t^368+.2495458243e-2*t
^584+.3564054156e-12*t^333+.4228433117e-20*t^266+.1488538140e-15*t^298+.\
4501704498e-2*t^632+.1834039227e-4*t^926+.4514842892e-4*t^898+.5398307314e-7*t^
1045+.6427640284e-10*t^1162+.3563409056e-2*t^728+.1415297720e-2*t^566+.\
1999545560e-3*t^516+.7279416288e-10*t^1160+.2002247475e-4*t^473+.5302253194e-7*
t^412+.2519708346e-7*t^407+.1596377741e-2*t^569+.4019154268e-2*t^621+.\
1810146019e-9*t^1145+.9320474086e-10*t^1156+.3575905755e-23*t^246+.1782512033e-\
2*t^572+.1098666025e-3*t^502+.1785228377e-21*t^257+.6018069109e-20*t^267+.\
1511526381e-62*t^140+.2869005740e-6*t^1017+.7429136606e-6*t^999+.3217236064e-2*
t^756+.3827372403e-2*t^715+.7020376651e-2*t^676+.3166770383e-22*t^252+.\
1996913970e-24*t^239+.5352019219e-18*t^280+.8356710068e-3*t^552+.2013391459e-8*
t^386+.3819665049e-4*t^484+.1113834931e-2*t^796+.3078040764e-9*t^1136+.\
1039077196e-3*t^872+.3246268062e-2*t^595+.4299214057e-3*t^826+.1032183837e-10*t
^1192+.3796049456e-2*t^716+.2377958608e-3*t^846+.9737741020e-19*t^275+.\
5888210511e-10*t^360+.2651521759e-4*t^916+.5095487184e-7*t^1046+.6442060815e-11
*t^348+.1242160572e-9*t^364+.5584330839e-5*t^455+.2073905683e-51*t^155+.\
1215485937e-19*t^269+.2104180761e-10*t^1180+.2468782168e-4*t^918+.2182274813e-\
36*t^190+.8573791180e-72*t^132+.1048151206e-17*t^282+.1053528346e-9*t^1154+.\
1474264599e-5*t^442+.9768866937e-9*t^378+.1818697180e-5*t^444+.2431525044e-9*t^
1140+.2512386409e-4*t^477+.3500454335e-2*t^731+.5958256857e-2*t^647+.3910024728\
e-27*t^226+.4822836744e-13*t^322+.1044437108e-3*t^501+.4072650615e-6*t^429+.\
5229198383e-12*t^335+.3722961570e-2*t^719+.3543230064e-2*t^729+.1323726209e-5*t
^985+.3685614577e-2*t^721+.5673238377e-10*t^1164+.1787101885e-4*t^471+.\
1027771597e-7*t^401+.1277694326e-2*t^792+.1419825802e-3*t^863+.8764176610e-10*t
^1157+.1418747192e-8*t^1111+.1993585431e-41*t^175+.2569758665e-3*t^522+.\
3410949322e-2*t^735+.5024596509e-6*t^1007+.2620322555e-29*t^216+.1199730034e-6*
t^418+.3903830190e-2*t^713+.1018437656e-16*t^289+.5452137288e-6*t^432+.\
3442615187e-4*t^908+.2745787410e-4*t^915+.1077327796e-3*t^871+.1769643992e-3*t^
856+.5193382727e-2*t^639+.3781308041e-5*t^451+.1389230134e-6*t^1029+.1549367420\
e-7*t^1067+.3897854895e-10*t^1170+.3504595300e-30*t^212+.2036128921e-17*t^284+.\
1252194133e-25*t^233+.3877159598e-2*t^605+.2045116496e-7*t^1062+.4125188505e-2*
t^626+.6660098542e-3*t^812+.1231871864e-10*t^1189+.3116826703e-28*t^221+.\
5402089160e-17*t^287+.3137809212e-4*t^911+.2288711624e-7*t^1060+.5308384042e-2*
t^697+.6752686550e-2*t^663+.5225269687e-23*t^247+.6060852867e-7*t^1043+.\
1927227113e-10*t^354+.4702395013e-10*t^1167+.7496357144e-18*t^281+.4805502531e-\
3*t^535+.5146667847e-4*t^893+.7096586420e-3*t^810+.3582668835e-2*t^727+.\
6945789262e-2*t^669+.4382911696e-3*t^533+.1973395715e-32*t^203+.5289734990e-4*t
^892+.4288524845e-4*t^900+.2827190699e-17*t^285+.2376101043e-10*t^1178+.\
1933783323e-7*t^1063+.1280065361e-24*t^238+.2260392175e-2*t^580+.3094398096e-3*
t^837+.2703815835e-6*t^1018+.1159213232e-6*t^1032+.2318443843e-2*t^581+.\
8344093560e-4*t^878+.7652302651e-7*t^1039+.2954162655e-2*t^591+.1006758582e-7*t
^1075+.1977073795e-4*t^924+.6174149312e-2*t^650+.4684109290e-5*t^958+.\
6362060725e-11*t^1200+.5855910021e-6*t^1004+.3186937197e-3*t^836+.2118523012e-4
*t^474+.3601212057e-2*t^600+.4015729537e-2*t^620+.1335291883e-8*t^1112+.\
1264277885e-37*t^186+.4879680113e-2*t^636+.4365006951e-8*t^1091+.4747434051e-24
*t^241+.2052791186e-47*t^162+.2585629470e-37*t^187+.3025928001e-2*t^592+.\
1466766565e-7*t^1068+.1464848369e-4*t^932+.7276893209e-3*t^547+.1256308005e-8*t
^1113+.1261705493e-31*t^206+.2955072811e-38*t^184+.3407956141e-7*t^409+.\
1994491578e-3*t^852+.7641362805e-14*t^313+.1828903415e-7*t^1064+.6128954227e-4*
t^887+.6347542116e-2*t^653+.6988560134e-4*t^883+.1003339594e-4*t^462+.\
9171869284e-5*t^944+.1730106828e-7*t^1065+.7084442221e-4*t^494+.2080626485e-3*t
^517+.7348653599e-5*t^949+.4403354692e-9*t^1130+.8946830452e-22*t^255+.\
3829459171e-6*t^1012+.1936260220e-3*t^853+.8865767903e-3*t^803+.6055992317e-3*t
^815+.3978661590e-3*t^531+.4219783952e-2*t^628+.6624999823e-8*t^1083+.\
2556260706e-2*t^772+.1578629122e-4*t^930+.1284203266e-3*t^866+.1027149009e-20*t
^262+.6294016294e-2*t^652+.8080070663e-7*t^415+.5987110689e-8*t^397+.9916216837\
e-4*t^500+.1576426355e-3*t^510+.1411028577e-4*t^933+.1905523743e-2*t^574+.\
1481360814e-13*t^316+.3949909321e-2*t^712+.4059552044e-2*t^710+.1248429313e-2*t
^563+.4848083571e-8*t^1089+.3590856379e-4*t^483+.3232716274e-25*t^235+.\
3744444826e-2*t^718+.5220748973e-4*t^489+.5598466178e-4*t^890+.3665061300e-2*t^
601+.1844269367e-2*t^573+.4565955846e-15*t^302+.6887590790e-2*t^667+.2685912811\
e-10*t^1176+.3815330488e-18*t^279+.5152539344e-28*t^222+.6964588536e-68*t^135+.\
1405519104e-38*t^183+.3780784769e-2*t^603+.3951269769e-2*t^607+.2556347040e-2*t
^585+.1177235908e-5*t^988+.3261298047e-2*t^745+.3578923274e-21*t^259+.\
1466870174e-3*t^862+.1641687848e-3*t^511+.5504089945e-2*t^642+.6443881052e-2*t^
655+.2826973186e-53*t^152+.1044507582e-8*t^1116+.5292227365e-6*t^1006+.\
1260290205e-4*t^936+.8057743394e-3*t^806+.8485771647e-6*t^996+.5765608585e-4*t^
889+.4092897253e-69*t^134+.2618589098e-2*t^586+.3426421956e-3*t^528+.4001726955\
e-2*t^711+.4753714136e-4*t^896+.3822159354e-7*t^1051+.2841739131e-4*t^914+.\
3282452802e-3*t^835+.8882174906e-3*t^554+.7230544284e-24*t^242+.4778938942e-2*t
^635+.4039390746e-2*t^613+.2814946247e-2*t^589+.2753632181e-2*t^769+.5402356438\
e-2*t^641+.8031939084e-6*t^436+.2955738008e-3*t^525+.5450812887e-3*t^538+.\
2282602089e-46*t^164+.5789544431e-2*t^645+.4420502082e-2*t^631+.3703547263e-2*t
^720+.1072111753e-26*t^228+.4312785269e-12*t^334+.5189698942e-2*t^698+.\
6874856219e-3*t^811+.1705801778e-9*t^1146+.2883770013e-2*t^590+.6335355611e-2*t
^688+.5148195093e-25*t^236+.2021377944e-5*t^445+.6607192606e-2*t^659+.\
9453304506e-3*t^801+.9312248857e-4*t^875+.2044826078e-44*t^168+.9728316883e-11*
t^1193+.4178750918e-4*t^901+.1599766151e-8*t^1109+.1082852634e-5*t^439+.\
1564602354e-22*t^250+.1613188370e-5*t^980+.1307666224e-6*t^1030+.3319961170e-2*
t^596+.2178403422e-3*t^849+.1194898466e-8*t^380+.2707822692e-6*t^425+.\
3353030255e-10*t^357+.1865369790e-10*t^1182+.4904467804e-4*t^488+.6938708308e-\
13*t^324+.1991425634e-5*t^975+.6452138650e-3*t^813+.7218628561e-21*t^261+.\
6140788474e-38*t^185+.5667512264e-2*t^694+.4678047156e-9*t^1129+.1825849993e-13
*t^317+.4021568348e-9*t^371+.2177895653e-6*t^423+.8047023612e-4*t^879+.\
1470228305e-10*t^1186+.3787940697e-5*t^962+.5867517749e-3*t^816+.7941909630e-35
*t^195+.3668613407e-2*t^722+.1848506911e-3*t^514+.3040239485e-7*t^1055+.\
1360779205e-33*t^199+.3035094103e-11*t^344+.1777393415e-3*t^513+.6761072123e-9*
t^1123+.1240853796e-3*t^867+.1701414962e-4*t^928+.4455376959e-36*t^191+.\
2037504658e-9*t^1143+.1388561137e-3*t^507+.3377584446e-4*t^482+.6487656572e-2*t
^656+.5010133437e-4*t^894+.9131633002e-7*t^1036+.6007995728e-15*t^303+.\
4289649895e-29*t^217+.6674116571e-4*t^493+.1338408647e-10*t^352+.9923343634e-13
*t^326+.3392905396e-2*t^597+.8609991919e-3*t^553+.4483519665e-22*t^253+.\
2294849200e-4*t^920+.2690489856e-2*t^770+.4936771560e-5*t^957+.6916748348e-2*t^
680+.3381069617e-3*t^834+.5698285210e-2*t^644+.1225786688e-49*t^158+.3313061894\
e-13*t^320+.9493298126e-3*t^556+.1301941148e-2*t^564+.3331268986e-2*t^739+.\
3545576944e-4*t^907+.4027446401e-2*t^622+.1390841054e-16*t^290+.1328827554e-5*t
^441+.1054871541e-6*t^417+.5235888408e-3*t^537+.4589047739e-2*t^633+.3994864852\
e-5*t^961+.7325488068e-3*t^809+.6431989007e-2*t^687+.6482368190e-27*t^227+.\
2115477770e-3*t^850+.4149256279e-10*t^1169+.6770073477e-11*t^1199+.9166726496e-\
11*t^1194+.1238327605e-4*t^465+.3079242500e-5*t^449+.1462575810e-17*t^283+.\
1716552998e-3*t^857+.6681094557e-2*t^661+.2864847796e-55*t^149+.3148621130e-2*t
^760+.1312534669e-8*t^381+.1027592063e-60*t^142+.1073665008e-56*t^147+.\
1323633090e-2*t^791+.1908551517e-2*t^781+.2503502346e-11*t^343+.2264233421e-14*
t^308+.1115799542e-10*t^351+.1315573314e-7*t^1070+.2378838412e-8*t^388+.\
7692425547e-75*t^130+.3265709548e-9*t^1135+.2563021500e-7*t^1058+.2017675511e-\
25*t^234+.3233316698e-10*t^1173+.2951772802e-12*t^332+.3461774989e-15*t^301+.\
1233854285e-2*t^793+.6326436913e-4*t^886+.2510885034e-5*t^970+.1131742681e-5*t^
989+.2258391115e-6*t^1021+.8050781973e-5*t^947+.9631415627e-6*t^993+.6488397568\
e-3*t^543+.2486042251e-2*t^773+.4632415542e-4*t^897+.2916176395e-5*t^967+.\
7095830163e-6*t^1000+.6529160370e-2*t^657+.2637174937e-5*t^969+.3687650426e-6*t
^428+.9059945724e-8*t^1077+.2143344733e-8*t^1104+.6805536275e-8*t^398+.\
2624558331e-2*t^771+.2917787035e-3*t^839+.6611886228e-6*t^434+.1077278034e-2*t^
797+.6423182194e-7*t^1042+.3917116478e-2*t^606+.6263204046e-16*t^295+.\
2163290514e-7*t^1061+.2078091739e-30*t^211+.3218925394e-7*t^1054+.2376781427e-2
*t^582+.8145866119e-25*t^237+.3343982403e-8*t^1096+.6357499045e-54*t^151+.\
8123986577e-6*t^997+.6644509823e-2*t^660+.3478194368e-2*t^732+.1976576842e-15*t
^299+.6059086764e-5*t^953+.2450240412e-12*t^331+.1655344088e-10*t^1184+.\
6984425721e-2*t^678+.8921091794e-8*t^400+.1767555036e-26*t^229+.1699265044e-12*
t^329+.1060584575e-2*t^559+.1597402229e-29*t^215+.7805770274e-3*t^807+.\
4019439215e-2*t^618+.9402970765e-4*t^499+.6686945317e-3*t^544+.1535005445e-2*t^
568+.3190073934e-2*t^758+.1083061764e-8*t^379+.4437444997e-11*t^346+.1091747599\
e-6*t^1033+.1647379816e-2*t^785+.4170364073e-3*t^827+.4041499623e-2*t^623+.\
5791299165e-13*t^323+.4592564713e-3*t^534+.7884955631e-15*t^304+.3178992856e-4*
t^481+.1161432878e-10*t^1190+.3421224841e-8*t^392+.3618564168e-2*t^725+.\
5088034752e-52*t^154+.5887105127e-44*t^169+.3255936022e-2*t^746+.5081877243e-5*
t^454+.4145885114e-9*t^1131+.1342762791e-9*t^1150+.6117703601e-7*t^413+.\
5284111032e-9*t^1127+.1506846321e-8*t^1110+.1431521092e-5*t^983+.6883643066e-3*
t^545+.1008656129e-2*t^799+.4958031806e-2*t^700+.6787726380e-2*t^664+.\
1543922665e-64*t^138+.4614164220e-5*t^453+.1590568048e-4*t^469+.1111081310e-8*t
^1115+.2130740143e-4*t^922+.5087892777e-2*t^638+.1094964613e-10*t^1191+.\
4982996050e-2*t^637+.2661868037e-4*t^478+.3414518007e-5*t^450+.5785867215e-2*t^
693+.4877384650e-10*t^359+.3227431770e-58*t^145+.1224294217e-5*t^987+.\
2751757388e-3*t^841+.8380776759e-16*t^296+.2817977482e-3*t^524+.4004510693e-13*
t^321+.5079129554e-21*t^260+.3121833460e-2*t^761+.2085381152e-2*t^577+.\
6236338469e-2*t^651+.4879389606e-4*t^895+.4123556722e-2*t^709+.1206834089e-42*t
^172+.1300110606e-61*t^141+.3408377806e-7*t^1053+.4846370191e-2*t^701+.\
1328489955e-3*t^865+.3454924250e-9*t^370+.3956231765e-7*t^410+.2315277401e-10*t
^355+.3466135858e-16*t^293+.4213150603e-5*t^960+.4210627805e-8*t^394+.\
1061591994e-7*t^1074+.3618921504e-6*t^1013+.5440085429e-4*t^891+.9299202005e-11
*t^350+.7040458524e-7*t^414+.4062083927e-2*t^624+.2672463809e-3*t^842+.\
2595526770e-3*t^843+.4652135636e-9*t^372+.4738319779e-2*t^702+.3464508988e-2*t^
598+.1904232291e-4*t^925+.9549572434e-8*t^1076+.2421806029e-7*t^1059+.\
4016179341e-2*t^619+.6970724431e-2*t^670+.1001952218e-3*t^873+.1567027813e-8*t^
383+.7011552755e-5*t^950+.1605626658e-10*t^353+.2774004264e-5*t^448+.1198462546\
e-3*t^868+.1800805293e-8*t^1107+.2354896588e-27*t^225+.9125835164e-36*t^192+.\
6840688474e-10*t^1161+.4089825043e-2*t^625+.7430446886e-17*t^288+.4441296524e-2
*t^705+.4049435424e-6*t^1011+.4047893500e-7*t^1050+.7744277102e-11*t^349+.\
4177863620e-3*t^532+.7108402591e-10*t^361+.6397340503e-2*t^654+.6289700829e-8*t
^1084+.7006242035e-2*t^677+.8863543055e-6*t^437+.4569195909e-49*t^159+.\
6717128877e-2*t^662+.6992014140e-2*t^671+.1892473179e-16*t^291+.2049744035e-2*t
^779+.2435691520e-2*t^583+.1186230863e-12*t^327+.3858745834e-35*t^194+.\
6522716835e-2*t^686+.7817636169e-9*t^376+.1042254748e-2*t^798+.1373699653e-3*t^
864+.5295053697e-8*t^396+.2525953297e-10*t^1177+.4810063018e-7*t^1047+.\
3225785265e-6*t^1015+.6936774162e-47*t^163+.1651918365e-43*t^170+.1757011433e-\
10*t^1183+.2414360737e-2*t^774+.1264147655e-9*t^1151+.3045526939e-39*t^181+.\
2238162668e-13*t^318+.1269086602e-3*t^505+.2917800825e-14*t^309+.2931704933e-7*
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2122075871e-2*t^778+.1044959120e-5*t^991+.3785674681e-8*t^393+.6755789989e-4*t^
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9175838168e-3*t^555+.3237855869e-2*t^753+.6855504968e-2*t^666+.6035230941e-2*t^
648+.9563755943e-5*t^943+.1188039486e-7*t^402+.8539390634e-28*t^223+.2903170887\
e-26*t^230+.8009998255e-42*t^174+.4018746831e-2*t^610+.5548001186e-2*t^695+.\
1613663870e-3*t^859+.1978464216e-2*t^780+.3718194210e-32*t^204+.2813593895e-2*t
^768+.5602951734e-2*t^643




                    Here is a plot of the prob. distribution




                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                      HH*H                                      
                                   HHHH +HHH                                    
                              HHHHHH    +  HHHHHHHHH                            
                        HHHHHHH         +          HHHHHHH                      
  +**********************+--+--+--+--+--+--+--+--+--+--+-*********************- 
          -4            -2             0               2             4          
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                
                                                                                


 To sum up the expectation, s.d., and the scaled moments up to the, 4, -th are



[676.31537634061272787, 65.276593284568584017, 0.22304111567227178273,

    3.0704308328514590521]



                            -----------------------



                                ----------------

                                                     7
    This ends this paper that took, 0.1614644849 x 10 , seconds to produce.