How long should it take until you visit the integers with property, "n -> evalb(IsDP(n,3))", for the k-th time if you roll a fair die with, 6, faces for k from 1 to, 30 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, "n -> evalb(IsDP(n,3))", for , 1, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999999501926 Conditioned that you are succesful the expected number of rounds for that go\ al is, 17.616897335828798542, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 9.3261053316121849939 The skewness is, 1.2567896999400209166 The kurtosis is, 5.0418472845923545052 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 2, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999978153782 Conditioned that you are succesful the expected number of rounds for that go\ al is, 28.596252324281988294, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 11.157841314911990922 The skewness is, 0.93776622882399139460 The kurtosis is, 4.2641746771809241468 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 3, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999528115220 Conditioned that you are succesful the expected number of rounds for that go\ al is, 38.020325322186092566, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.479530793511262501 The skewness is, 0.81175232398970259173 The kurtosis is, 3.8341669734937284478 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 4, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999993307651009 Conditioned that you are succesful the expected number of rounds for that go\ al is, 46.693062385961774643, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 13.580572167857577590 The skewness is, 0.69280890864989575615 The kurtosis is, 3.7249605106929817751 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 5, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999929898388472 Conditioned that you are succesful the expected number of rounds for that go\ al is, 54.892802766346504432, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 14.486431472472948643 The skewness is, 0.66698961387699746776 The kurtosis is, 3.7835629496742538917 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 6, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999421518287946 Conditioned that you are succesful the expected number of rounds for that go\ al is, 62.747776584480892247, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 15.413664961099616259 The skewness is, 0.67330531522173540417 The kurtosis is, 3.7768138592611217017 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 7, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999996082806352221 Conditioned that you are succesful the expected number of rounds for that go\ al is, 70.415107984375332733, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 16.409414739222399534 The skewness is, 0.65934957002666741191 The kurtosis is, 3.6408242540541859093 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 8, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999977612345969817 Conditioned that you are succesful the expected number of rounds for that go\ al is, 78.007703050482131929, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 17.401844643022972710 The skewness is, 0.60893591945564359309 The kurtosis is, 3.3799557748592482485 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 9, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999889758964619125 Conditioned that you are succesful the expected number of rounds for that go\ al is, 85.566052039471742242, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 18.286627576699349714 The skewness is, 0.51812641675753146170 The kurtosis is, 3.0929646333557157057 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 10, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999524854781218845 Conditioned that you are succesful the expected number of rounds for that go\ al is, 93.068614917023226305, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 18.954867124736557907 The skewness is, 0.40609986996624998563 The kurtosis is, 2.9655049955215000672 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 11, times The prob. of reaching that goal in <=, 200, rounds is, 0.99998185019623807341 Conditioned that you are succesful the expected number of rounds for that go\ al is, 100.44867665507477337, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 19.358065081679498993 The skewness is, 0.32177981658086591250 The kurtosis is, 3.1033378742492618358 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 12, times The prob. of reaching that goal in <=, 200, rounds is, 0.99993793066532649655 Conditioned that you are succesful the expected number of rounds for that go\ al is, 107.63165291222949199, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 19.581792293887666783 The skewness is, 0.31432089800494074178 The kurtosis is, 3.4316632267480594167 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 13, times The prob. of reaching that goal in <=, 200, rounds is, 0.99980835663058790566 Conditioned that you are succesful the expected number of rounds for that go\ al is, 114.58624156405740886, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 19.818419044959473709 The skewness is, 0.39086178814383103493 The kurtosis is, 3.7110771025182910199 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 14, times The prob. of reaching that goal in <=, 200, rounds is, 0.99946195982139212444 Conditioned that you are succesful the expected number of rounds for that go\ al is, 121.34803343510340617, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 20.238539090550529408 The skewness is, 0.49339267438346554580 The kurtosis is, 3.6927117013269388331 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 15, times The prob. of reaching that goal in <=, 200, rounds is, 0.99861800913473429950 Conditioned that you are succesful the expected number of rounds for that go\ al is, 127.99410741018916009, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 20.865325980504410801 The skewness is, 0.53763163088090580428 The kurtosis is, 3.3393364634314455818 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 16, times The prob. of reaching that goal in <=, 200, rounds is, 0.99673487076618097434 Conditioned that you are succesful the expected number of rounds for that go\ al is, 134.58754815908014700, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 21.554224681795759178 The skewness is, 0.48164094914753855956 The kurtosis is, 2.8347375283412575353 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 17, times The prob. of reaching that goal in <=, 200, rounds is, 0.99287016914253043976 Conditioned that you are succesful the expected number of rounds for that go\ al is, 141.12855046429820652, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 22.077293057473103399 The skewness is, 0.33896142478789228686 The kurtosis is, 2.3874714207707270566 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 18, times The prob. of reaching that goal in <=, 200, rounds is, 0.98554818224841369364 Conditioned that you are succesful the expected number of rounds for that go\ al is, 147.53841031899024178, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 22.231246750216374178 The skewness is, 0.14420589081126417146 The kurtosis is, 2.1105866694030594846 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 19, times The prob. of reaching that goal in <=, 200, rounds is, 0.97270016628208622753 Conditioned that you are succesful the expected number of rounds for that go\ al is, 153.68022258144238926, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 21.906263433568753169 The skewness is, -0.073038951341242347456 The kurtosis is, 2.0345007527630789316 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 20, times The prob. of reaching that goal in <=, 200, rounds is, 0.95175962404272931005 Conditioned that you are succesful the expected number of rounds for that go\ al is, 159.40139518228206271, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 21.102741863290863133 The skewness is, -0.29335163596435395817 The kurtosis is, 2.1508732739726919799 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 21, times The prob. of reaching that goal in <=, 200, rounds is, 0.91997656031579815767 Conditioned that you are succesful the expected number of rounds for that go\ al is, 164.57630574507237735, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 19.909475368506151624 The skewness is, -0.50390340536439600579 The kurtosis is, 2.4362306264237661090 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 22, times The prob. of reaching that goal in <=, 200, rounds is, 0.87495178748678328154 Conditioned that you are succesful the expected number of rounds for that go\ al is, 169.13271145623126262, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 18.463206212677805598 The skewness is, -0.69464634847330154157 The kurtosis is, 2.8564041798811609755 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 23, times The prob. of reaching that goal in <=, 200, rounds is, 0.81529715131338585718 Conditioned that you are succesful the expected number of rounds for that go\ al is, 173.05698124270697065, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 16.907810230960505993 The skewness is, -0.85660268572948555364 The kurtosis is, 3.3633626202277299086 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 24, times The prob. of reaching that goal in <=, 200, rounds is, 0.74123895835943136905 Conditioned that you are succesful the expected number of rounds for that go\ al is, 176.38320382753781205, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 15.364974023302158320 The skewness is, -0.98223785305584703082 The kurtosis is, 3.8938717667381115798 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 25, times The prob. of reaching that goal in <=, 200, rounds is, 0.65494924276584592574 Conditioned that you are succesful the expected number of rounds for that go\ al is, 179.17527966285063983, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 13.920153200608663991 The skewness is, -1.0671554282378990701 The kurtosis is, 4.3761623044854038636 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 26, times The prob. of reaching that goal in <=, 200, rounds is, 0.56044453826760072115 Conditioned that you are succesful the expected number of rounds for that go\ al is, 181.50969061470808746, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.621407471798517963 The skewness is, -1.1118767919109032806 The kurtosis is, 4.7455610888977039462 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 27, times The prob. of reaching that goal in <=, 200, rounds is, 0.46302498083584367198 Conditioned that you are succesful the expected number of rounds for that go\ al is, 183.46291135419050648, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 11.486046555909624076 The skewness is, -1.1223625016714526745 The kurtosis is, 4.9628262659996128477 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 28, times The prob. of reaching that goal in <=, 200, rounds is, 0.36838556430855488597 Conditioned that you are succesful the expected number of rounds for that go\ al is, 185.10414366946446739, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 10.510217555086763284 The skewness is, -1.1086206769742823297 The kurtosis is, 5.0247600309391434179 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 29, times The prob. of reaching that goal in <=, 200, rounds is, 0.28164614126501818734 Conditioned that you are succesful the expected number of rounds for that go\ al is, 186.49228754649899455, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 9.6780846112219684396 The skewness is, -1.0819670696943791252 The kurtosis is, 4.9606168153946954904 ----------------------- 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, "n -> evalb(IsDP(n,3))", for , 30, times The prob. of reaching that goal in <=, 200, rounds is, 0.20656682901179501213 Conditioned that you are succesful the expected number of rounds for that go\ al is, 187.67560395267285913, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 8.9689211152417028609 The skewness is, -1.0522942654880194060 The kurtosis is, 4.8172187814004230610 ----------------------- To sum up the expected time it takes to visit, "n -> evalb(IsDP(n,3))", k times for k from 1 to , 30, are [17.61689734, 28.59625232, 38.02032532, 46.69306239, 54.89280277, 62.74777658, 70.41510798, 78.00770305, 85.56605204, 93.06861492, 100.4486767, 107.6316529, 114.5862416, 121.3480334, 127.9941074, 134.5875482, 141.1285505, 147.5384103, 153.6802226, 159.4013952, 164.5763057, 169.1327115, 173.0569812, 176.3832038, 179.1752797, 181.5096906, 183.4629114, 185.1041437, 186.4922875, 187.6756040] ---------------- This ends this paper that took, 11253.702, seconds to produce.