How long should it take until you visit the integers with property, "n -> evalb(IsDP(n,2))", 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,2))", for , 1, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 3.7889212907420942040, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 2.5739368738181255622 The skewness is, 1.6181770676330939129 The kurtosis is, 7.1864621230458724328 ----------------------- 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,2))", for , 2, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 7.0989878271546099636, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 3.4801463244250580360 The skewness is, 1.2389137175268941108 The kurtosis is, 5.4471486341573832609 ----------------------- 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,2))", for , 3, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 10.350903238451206271, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 4.2153875741618406754 The skewness is, 1.0496162597705295159 The kurtosis is, 4.6657194578470719587 ----------------------- 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,2))", for , 4, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 13.627452836035317901, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 4.8564537090494077807 The skewness is, 0.88705453258626536652 The kurtosis is, 4.0390305594360281293 ----------------------- 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,2))", for , 5, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 16.938642763368138029, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 5.3776283710317131648 The skewness is, 0.74025007008086048034 The kurtosis is, 3.8058593900897886908 ----------------------- 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,2))", for , 6, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 20.236258803962635905, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 5.7885311855157338135 The skewness is, 0.69502918568531948182 The kurtosis is, 3.8477549423974484228 ----------------------- 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,2))", for , 7, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 23.488870953499059142, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 6.1932126833698816242 The skewness is, 0.70333559386992268754 The kurtosis is, 3.8475699263909053048 ----------------------- 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,2))", for , 8, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 26.725494318175714875, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 6.6371559264032310213 The skewness is, 0.69248371460130774781 The kurtosis is, 3.7319854823562685668 ----------------------- 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,2))", for , 9, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 29.985158163655561798, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 7.0938468137155250089 The skewness is, 0.64981808573303906962 The kurtosis is, 3.6134302853609062808 ----------------------- 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,2))", for , 10, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 33.282982729602373290, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 7.5260728448712921957 The skewness is, 0.60360665763417972342 The kurtosis is, 3.5620926025408637797 ----------------------- 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,2))", for , 11, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 36.612953843528776058, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 7.9254473347290979838 The skewness is, 0.57338922434819859616 The kurtosis is, 3.5107159039660973090 ----------------------- 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,2))", for , 12, times The prob. of reaching that goal in <=, 200, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 39.965177716165597886, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 8.3038789486745129996 The skewness is, 0.54528571544311688543 The kurtosis is, 3.3748063912303329900 ----------------------- 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,2))", for , 13, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999999999997 Conditioned that you are succesful the expected number of rounds for that go\ al is, 43.334442885164970240, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 8.6586851158735021353 The skewness is, 0.49579149771381332156 The kurtosis is, 3.1967817538451368101 ----------------------- 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,2))", for , 14, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999999999972 Conditioned that you are succesful the expected number of rounds for that go\ al is, 46.713453205032554182, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 8.9639253190470050011 The skewness is, 0.42840702057885479096 The kurtosis is, 3.1115350352040540196 ----------------------- 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,2))", for , 15, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999999999770 Conditioned that you are succesful the expected number of rounds for that go\ al is, 50.085367977425228539, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 9.2011595769383761895 The skewness is, 0.37609706206624934813 The kurtosis is, 3.2089222319247230398 ----------------------- 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,2))", for , 16, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999999998276 Conditioned that you are succesful the expected number of rounds for that go\ al is, 53.429573375969652101, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 9.3916449801036987400 The skewness is, 0.37508624626934882364 The kurtosis is, 3.4544889507004676680 ----------------------- 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,2))", for , 17, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999999988052 Conditioned that you are succesful the expected number of rounds for that go\ al is, 56.736925649694603565, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 9.5947380063179592325 The skewness is, 0.43422102963338110925 The kurtosis is, 3.6979568579762664627 ----------------------- 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,2))", for , 18, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999999923137 Conditioned that you are succesful the expected number of rounds for that go\ al is, 60.020073094590620823, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 9.8718542068103340938 The skewness is, 0.51959544852470942493 The kurtosis is, 3.7704070757469442889 ----------------------- 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,2))", for , 19, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999999539359 Conditioned that you are succesful the expected number of rounds for that go\ al is, 63.309538529800188177, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 10.244777606029877880 The skewness is, 0.57627216853887570843 The kurtosis is, 3.6240848815584433831 ----------------------- 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,2))", for , 20, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999997420330 Conditioned that you are succesful the expected number of rounds for that go\ al is, 66.638292111538946332, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 10.681767493656287803 The skewness is, 0.57221782322960670602 The kurtosis is, 3.3627632872239267606 ----------------------- 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,2))", for , 21, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999986462709 Conditioned that you are succesful the expected number of rounds for that go\ al is, 70.025314130022441210, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 11.119272670123700119 The skewness is, 0.51538512534368143193 The kurtosis is, 3.1329664353256265169 ----------------------- 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,2))", for , 22, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999933265007 Conditioned that you are succesful the expected number of rounds for that go\ al is, 73.467781930590916892, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 11.498148572090550018 The skewness is, 0.43631738400311167436 The kurtosis is, 3.0251764391224496842 ----------------------- 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,2))", for , 23, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999999690245562 Conditioned that you are succesful the expected number of rounds for that go\ al is, 76.944834611222316579, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 11.790211182934887982 The skewness is, 0.36679074049715920372 The kurtosis is, 3.0558230223399227117 ----------------------- 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,2))", for , 24, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999998643486502 Conditioned that you are succesful the expected number of rounds for that go\ al is, 80.429014183975995166, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.005310702374862734 The skewness is, 0.32895730865172424157 The kurtosis is, 3.1873412826107095362 ----------------------- 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,2))", for , 25, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999994384375060 Conditioned that you are succesful the expected number of rounds for that go\ al is, 83.898465969918350723, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.181241299900954117 The skewness is, 0.33086222857051373992 The kurtosis is, 3.3515849823692218734 ----------------------- 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,2))", for , 26, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999977986058551 Conditioned that you are succesful the expected number of rounds for that go\ al is, 87.344322379190074665, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.364280808121725601 The skewness is, 0.36499283980112182296 The kurtosis is, 3.4760573889870011590 ----------------------- 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,2))", for , 27, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999918149139404 Conditioned that you are succesful the expected number of rounds for that go\ al is, 90.771336447703738813, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.589757507557440365 The skewness is, 0.41198537937309551922 The kurtosis is, 3.5136214181451780168 ----------------------- 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,2))", for , 28, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999710916758180 Conditioned that you are succesful the expected number of rounds for that go\ al is, 94.193239000432878430, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.870605764394604325 The skewness is, 0.45023529401382331701 The kurtosis is, 3.4603119294388411611 ----------------------- 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,2))", for , 29, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999999028820078019 Conditioned that you are succesful the expected number of rounds for that go\ al is, 97.625980405444900291, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 13.197124082247163846 The skewness is, 0.46580493732981735524 The kurtosis is, 3.3473168174658353899 ----------------------- 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,2))", for , 30, times The prob. of reaching that goal in <=, 200, rounds is, 0.99999996892465337844 Conditioned that you are succesful the expected number of rounds for that go\ al is, 101.08186135336177269, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 13.545209477802760581 The skewness is, 0.45598039658418521405 The kurtosis is, 3.2161558370737906089 ----------------------- To sum up the expected time it takes to visit, "n -> evalb(IsDP(n,2))", k times for k from 1 to , 30, are [3.788921291, 7.098987827, 10.35090324, 13.62745284, 16.93864276, 20.23625880, 23.48887095, 26.72549432, 29.98515816, 33.28298273, 36.61295384, 39.96517772, 43.33444289, 46.71345321, 50.08536798, 53.42957338, 56.73692565, 60.02007309, 63.30953853, 66.63829211, 70.02531413, 73.46778193, 76.94483461, 80.42901418, 83.89846597, 87.34432238, 90.77133645, 94.19323900, 97.62598041, 101.0818614] ---------------- This ends this paper that took, 5581.863, seconds to produce.