Statistics of the Number of Rounds for Reaching an integer with property, "isprime", rolling a fair die with n faces for n from 2 to, 40 By Shalsoh B. Ekhad Suppose you start at, 0, and roll a fair die with n faces, with 1 through n d\ ots, and walk forward according to the number of dots, and stop as soon \ as you hit an integer with property, "isprime" You are allowed up to, 400, die-rolls, but you are interested in the smallest\ number of rolls that would guarantee that you would reach your goal wit\ h prob at least, 0.99999990000000000000 The table below tells you that number of rolls, followed by, conditioned on \ finishing by that number of rolls, the expected duration of such a game,\ followed by the standard-deviation and the scaled moments up to the, 4 with Fair Dice with number of faces from 2 to, 40 Number of faces=, 2, Number of rounds=, 2 [Ave,SD,ThirdScaledMoment,Etc]= , [1.5000000000000000000, 0.50000000000000000000, 0., 1.0000000000000000000] --------------------- Number of faces=, 3, Number of rounds=, 36 [Ave,SD,ThirdScaledMoment,Etc]= , [1.5443477482221103638, 1.0931434233169600653, 4.2364109277655008836, 34.996629514260524236] --------------------- Number of faces=, 4, Number of rounds=, 50 [Ave,SD,ThirdScaledMoment,Etc]= , [2.0435904080208860615, 1.8052051504282744686, 3.7142207037097127297, 25.058286599126621638] --------------------- Number of faces=, 5, Number of rounds=, 49 [Ave,SD,ThirdScaledMoment,Etc]= , [1.9293702992313225124, 1.7724088095275145702, 3.7104239283185695609, 25.153613938887074012] --------------------- Number of faces=, 6, Number of rounds=, 62 [Ave,SD,ThirdScaledMoment,Etc]= , [2.4284921541907827689, 2.4984806654361057029, 3.3891730283230362562, 20.582857692532783494] --------------------- Number of faces=, 7, Number of rounds=, 59 [Ave,SD,ThirdScaledMoment,Etc]= , [2.1364797833761478522, 2.2036099858946277276, 3.7824698255013654264, 24.994140519331001116] --------------------- Number of faces=, 8, Number of rounds=, 64 [Ave,SD,ThirdScaledMoment,Etc]= , [2.5684640070550435904, 2.6855330511508875157, 3.2101842352053234977, 18.918691831782846108] --------------------- Number of faces=, 9, Number of rounds=, 65 [Ave,SD,ThirdScaledMoment,Etc]= , [2.7287947888759995653, 2.7515913114572761183, 3.0604287587374549063, 17.681134215368002342] --------------------- Number of faces=, 10, Number of rounds=, 70 [Ave,SD,ThirdScaledMoment,Etc]= , [2.9713505927530138576, 3.0272614089633627320, 2.9846434278457894323, 16.965989768576117365] --------------------- Number of faces=, 11, Number of rounds=, 69 [Ave,SD,ThirdScaledMoment,Etc]= , [2.7823651109352962612, 2.9093476825987092580, 3.1120990551441103335, 18.164816128301934824] --------------------- Number of faces=, 12, Number of rounds=, 75 [Ave,SD,ThirdScaledMoment,Etc]= , [3.0814151699645665008, 3.2629425602826572008, 2.9881564513330378797, 16.956598864320083578] --------------------- Number of faces=, 13, Number of rounds=, 74 [Ave,SD,ThirdScaledMoment,Etc]= , [2.8871959185173323636, 3.1276616908867724102, 3.1265975596091342608, 18.228547167147828658] --------------------- Number of faces=, 14, Number of rounds=, 77 [Ave,SD,ThirdScaledMoment,Etc]= , [3.1332476846631770828, 3.3636402274387133517, 2.9756287175063523946, 16.864061097778595698] --------------------- Number of faces=, 15, Number of rounds=, 77 [Ave,SD,ThirdScaledMoment,Etc]= , [3.2365862383458321184, 3.3983283931269076156, 2.9183470706683512110, 16.445994929559074101] --------------------- Number of faces=, 16, Number of rounds=, 79 [Ave,SD,ThirdScaledMoment,Etc]= , [3.3900174193564578494, 3.5388514442731559627, 2.8725833875585385771, 16.045188682216492365] --------------------- Number of faces=, 17, Number of rounds=, 79 [Ave,SD,ThirdScaledMoment,Etc]= , [3.2348612400956163751, 3.4544948119081634839, 2.9626690489327396561, 16.821246907160289013] --------------------- Number of faces=, 18, Number of rounds=, 83 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4533733280528454794, 3.6927232697601738466, 2.8922304547809610555, 16.144576695190755577] --------------------- Number of faces=, 19, Number of rounds=, 82 [Ave,SD,ThirdScaledMoment,Etc]= , [3.2970451244721922078, 3.5999583898455054947, 2.9827352394167096014, 16.912726920012296473] --------------------- Number of faces=, 20, Number of rounds=, 84 [Ave,SD,ThirdScaledMoment,Etc]= , [3.4723329691280602313, 3.7650362823532845429, 2.9041193381554332771, 16.208137890402385701] --------------------- Number of faces=, 21, Number of rounds=, 84 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5426354540227691912, 3.7848698903265788123, 2.8719586434112609274, 15.970226937003743643] --------------------- Number of faces=, 22, Number of rounds=, 86 [Ave,SD,ThirdScaledMoment,Etc]= , [3.6471048235830236523, 3.8792881625411180864, 2.8474429571342715084, 15.759598333118914157] --------------------- Number of faces=, 23, Number of rounds=, 85 [Ave,SD,ThirdScaledMoment,Etc]= , [3.5122413296961164493, 3.8092232671617927215, 2.9165793353120142394, 16.328967814323812053] --------------------- Number of faces=, 24, Number of rounds=, 88 [Ave,SD,ThirdScaledMoment,Etc]= , [3.6900487996846355800, 3.9986602385576134167, 2.8542402992219040493, 15.744080606066633573] --------------------- Number of faces=, 25, Number of rounds=, 88 [Ave,SD,ThirdScaledMoment,Etc]= , [3.7372894496538530918, 4.0057902646417855415, 2.8375363422552740957, 15.625249986487685410] --------------------- Number of faces=, 26, Number of rounds=, 89 [Ave,SD,ThirdScaledMoment,Etc]= , [3.8538176349117030824, 4.1092135121576991657, 2.7974945080105150211, 15.253944899638809237] --------------------- Number of faces=, 27, Number of rounds=, 89 [Ave,SD,ThirdScaledMoment,Etc]= , [3.8937536620461898682, 4.1133686249016197131, 2.7855785823895886482, 15.173773091386932987] --------------------- Number of faces=, 28, Number of rounds=, 90 [Ave,SD,ThirdScaledMoment,Etc]= , [3.9877959183545914708, 4.1997051421816924389, 2.7621406803111528413, 14.948769977934996267] --------------------- Number of faces=, 29, Number of rounds=, 90 [Ave,SD,ThirdScaledMoment,Etc]= , [3.8745963279823162782, 4.1474860447758304078, 2.8054717787501605592, 15.292802341331785663] --------------------- Number of faces=, 30, Number of rounds=, 93 [Ave,SD,ThirdScaledMoment,Etc]= , [4.0320751278272932471, 4.3259618368437143581, 2.7693156648598876315, 14.954287355533361829] --------------------- Number of faces=, 31, Number of rounds=, 92 [Ave,SD,ThirdScaledMoment,Etc]= , [3.9125212323009624654, 4.2640753202258797785, 2.8179676646433564077, 15.339652052928019823] --------------------- Number of faces=, 32, Number of rounds=, 93 [Ave,SD,ThirdScaledMoment,Etc]= , [3.9962512542383982273, 4.3319481197661970373, 2.7945136217458470871, 15.144641138640006427] --------------------- Number of faces=, 33, Number of rounds=, 93 [Ave,SD,ThirdScaledMoment,Etc]= , [4.0378933842760971485, 4.3406736054033755298, 2.7776202489398739767, 15.021079228600899080] --------------------- Number of faces=, 34, Number of rounds=, 94 [Ave,SD,ThirdScaledMoment,Etc]= , [4.1192137759029985824, 4.4123898970545046720, 2.7565431385638821225, 14.834185886562422785] --------------------- Number of faces=, 35, Number of rounds=, 94 [Ave,SD,ThirdScaledMoment,Etc]= , [4.1569729520476639727, 4.4200738274903373341, 2.7423541285422096379, 14.732550503637306717] --------------------- Number of faces=, 36, Number of rounds=, 96 [Ave,SD,ThirdScaledMoment,Etc]= , [4.2663501001264788985, 4.5350785118739938742, 2.7189882258220501125, 14.520076898967678927] --------------------- Number of faces=, 37, Number of rounds=, 95 [Ave,SD,ThirdScaledMoment,Etc]= , [4.1591456626547796819, 4.4846436115665899511, 2.7566296955941220803, 14.808241228371602608] --------------------- Number of faces=, 38, Number of rounds=, 96 [Ave,SD,ThirdScaledMoment,Etc]= , [4.2276783644270570961, 4.5391020069676079519, 2.7387348512757772122, 14.668981345650383746] --------------------- Number of faces=, 39, Number of rounds=, 97 [Ave,SD,ThirdScaledMoment,Etc]= , [4.2589096540563941024, 4.5435219288488050977, 2.7284101273912616325, 14.598236778482318640] --------------------- Number of faces=, 40, Number of rounds=, 98 [Ave,SD,ThirdScaledMoment,Etc]= , [4.3321864362714841593, 4.6131173627923874932, 2.7124337674151330232, 14.447944592160975488] --------------------- This ends this paper that took, 202.241, seconds to produce.