How long should it take until you visit the integers with property, "isprime", 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, "isprime", for , 1, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 2.4284979136935042302, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 2.4985553162335780705 The skewness is, 3.3904247398037466438 The kurtosis is, 20.621448464846711903 ----------------------- 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 , 2, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 5.7122404679510382653, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 4.2393979452582675998 The skewness is, 2.1496467528629638934 The kurtosis is, 10.047545228613741860 ----------------------- 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 , 3, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 9.4988781192189836827, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 5.7679075577335105845 The skewness is, 1.6420771190411456091 The kurtosis is, 7.2098904412910414566 ----------------------- 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 , 4, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 13.650592713220160043, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 7.1185390567642083354 The skewness is, 1.3892777871372696888 The kurtosis is, 6.1044828267864414167 ----------------------- 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 , 5, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 18.054089314992686022, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 8.3598783694304875694 The skewness is, 1.2554075652090381430 The kurtosis is, 5.5085380080315882762 ----------------------- 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 , 6, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 22.646154021134176348, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 9.5715570828463666913 The skewness is, 1.1503502035655049337 The kurtosis is, 5.0273441353500731448 ----------------------- 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 , 7, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 27.421159023650740516, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 10.761804643826612982 The skewness is, 1.0474628041080617518 The kurtosis is, 4.6151696546933789158 ----------------------- 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 , 8, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 32.377528523207407821, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 11.906243772504454950 The skewness is, 0.94877031634226532590 The kurtosis is, 4.2993763433865461398 ----------------------- 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 , 9, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 37.500299031053022628, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.982459558010977463 The skewness is, 0.86252268984492142991 The kurtosis is, 4.0978890145067497547 ----------------------- 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 , 10, times The prob. of reaching that goal in <=, 400, rounds is, 1.0000000000000000000 Conditioned that you are succesful the expected number of rounds for that go\ al is, 42.764718684687791856, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 13.982335862878296970 The skewness is, 0.79744956302958764618 The kurtosis is, 3.9989275441987923017 ----------------------- 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 , 11, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999999999999 Conditioned that you are succesful the expected number of rounds for that go\ al is, 48.143205551549412331, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 14.918414655801896848 The skewness is, 0.75694284460840510338 The kurtosis is, 3.9630488524371691534 ----------------------- 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 , 12, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999999999989 Conditioned that you are succesful the expected number of rounds for that go\ al is, 53.613514592847917353, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 15.818543482904486561 The skewness is, 0.73622628499502110209 The kurtosis is, 3.9427895560177179238 ----------------------- 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 , 13, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999999999919 Conditioned that you are succesful the expected number of rounds for that go\ al is, 59.164066546733484968, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 16.710983956740748499 The skewness is, 0.72507157169726602326 The kurtosis is, 3.9031802915767648959 ----------------------- 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 , 14, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999999999455 Conditioned that you are succesful the expected number of rounds for that go\ al is, 64.793373495424123028, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 17.611557422297335808 The skewness is, 0.71313873728038838584 The kurtosis is, 3.8308430747532868822 ----------------------- 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 , 15, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999999996613 Conditioned that you are succesful the expected number of rounds for that go\ al is, 70.505171273495329962, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 18.519767782622361469 The skewness is, 0.69392893821914855815 The kurtosis is, 3.7314241300193262998 ----------------------- 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 , 16, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999999980459 Conditioned that you are succesful the expected number of rounds for that go\ al is, 76.302841609017492569, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 19.422732435497725257 The skewness is, 0.66573440994627524427 The kurtosis is, 3.6223695206843828422 ----------------------- 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 , 17, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999999894832 Conditioned that you are succesful the expected number of rounds for that go\ al is, 82.185662134333369730, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 20.302274811336201458 The skewness is, 0.63073738937313140794 The kurtosis is, 3.5254482822337134204 ----------------------- 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 , 18, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999999469941 Conditioned that you are succesful the expected number of rounds for that go\ al is, 88.147576263973941615, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 21.141969703094017470 The skewness is, 0.59365501263113232943 The kurtosis is, 3.4590868939862262302 ----------------------- 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 , 19, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999997489446 Conditioned that you are succesful the expected number of rounds for that go\ al is, 94.178112559489133288, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 21.932923954888551603 The skewness is, 0.56018120696339862300 The kurtosis is, 3.4312822946271956163 ----------------------- 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 <=, 400, rounds is, 0.99999999999988791100 Conditioned that you are succesful the expected number of rounds for that go\ al is, 100.26480683972369563, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 22.677184601617557455 The skewness is, 0.53510980896389320059 The kurtosis is, 3.4359882905461904816 ----------------------- 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 , 21, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999952694857 Conditioned that you are succesful the expected number of rounds for that go\ al is, 106.39629965563239809, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 23.387307049413526011 The skewness is, 0.52051734601875814000 The kurtosis is, 3.4553514431666884159 ----------------------- 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 , 22, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999810812432 Conditioned that you are succesful the expected number of rounds for that go\ al is, 112.56502065861703585, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 24.081633935216226934 The skewness is, 0.51482837931528280962 The kurtosis is, 3.4675148895681444726 ----------------------- 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 , 23, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999999281375918 Conditioned that you are succesful the expected number of rounds for that go\ al is, 118.76840919959805095, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 24.776998119083176134 The skewness is, 0.51344092276361256292 The kurtosis is, 3.4566369453354043959 ----------------------- 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 , 24, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999997402027280 Conditioned that you are succesful the expected number of rounds for that go\ al is, 125.00819936185045282, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 25.482183442228054018 The skewness is, 0.51080479882600115968 The kurtosis is, 3.4199435496604234004 ----------------------- 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 , 25, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999991044008558 Conditioned that you are succesful the expected number of rounds for that go\ al is, 131.28816827897860826, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 26.195216562248105797 The skewness is, 0.50290532426453145250 The kurtosis is, 3.3679599201701849571 ----------------------- 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 , 26, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999970508630159 Conditioned that you are succesful the expected number of rounds for that go\ al is, 137.61140973861670040, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 26.905543028897960063 The skewness is, 0.48883189693992319466 The kurtosis is, 3.3183350285303488233 ----------------------- 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 , 27, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999907087188270 Conditioned that you are succesful the expected number of rounds for that go\ al is, 143.97831102445874480, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 27.599719516892873218 The skewness is, 0.47078413796541234234 The kurtosis is, 3.2873676776291046703 ----------------------- 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 , 28, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999719520674878 Conditioned that you are succesful the expected number of rounds for that go\ al is, 150.38598811755200515, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 28.267848171166140569 The skewness is, 0.45281977256005170482 The kurtosis is, 3.2835481170478992542 ----------------------- 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 , 29, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999999187600358805 Conditioned that you are succesful the expected number of rounds for that go\ al is, 156.82924624679348964, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 28.908071859042350423 The skewness is, 0.43911448406748274968 The kurtosis is, 3.3051185719588134719 ----------------------- 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 , 30, times The prob. of reaching that goal in <=, 400, rounds is, 0.99999997739315056535 Conditioned that you are succesful the expected number of rounds for that go\ al is, 163.30251726839510113, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 29.527602103131645560 The skewness is, 0.43242041420746583182 The kurtosis is, 3.3414987535190792816 ----------------------- To sum up the expected time it takes to visit, "isprime", k times for k from 1 to , 30, are [2.428497914, 5.712240468, 9.498878119, 13.65059271, 18.05408931, 22.64615402, 27.42115902, 32.37752852, 37.50029903, 42.76471868, 48.14320555, 53.61351459, 59.16406655, 64.79337350, 70.50517127, 76.30284161, 82.18566213, 88.14757626, 94.17811256, 100.2648068, 106.3962997, 112.5650207, 118.7684092, 125.0081994, 131.2881683, 137.6114097, 143.9783110, 150.3859881, 156.8292462, 163.3025173] ---------------- This ends this paper that took, 78547.247, seconds to produce.