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, 10 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, 2.4284979136935042301, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 2.4985553162335780705 The skewness is, 3.3904247398037466440 The kurtosis is, 20.621448464846711904 ----------------------- 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, 4.2393979452582676001 The skewness is, 2.1496467528629638921 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, 9.4988781192189836826, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 5.7679075577335105871 The skewness is, 1.6420771190411456021 The kurtosis is, 7.2098904412910414663 ----------------------- 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, 13.650592713220160044, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 7.1185390567642083297 The skewness is, 1.3892777871372696988 The kurtosis is, 6.1044828267864414313 ----------------------- 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, 5.5085380080315882946 ----------------------- 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, 22.646154021134176350, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 9.5715570828463666903 The skewness is, 1.1503502035655049361 The kurtosis is, 5.0273441353500731526 ----------------------- 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, 27.421159023650740508, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 10.761804643826612997 The skewness is, 1.0474628041080617362 The kurtosis is, 4.6151696546933789142 ----------------------- 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, 32.377528523207407822, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 11.906243772504454946 The skewness is, 0.94877031634226533539 The kurtosis is, 4.2993763433865461239 ----------------------- 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, 37.500299031053022633, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 12.982459558010977471 The skewness is, 0.86252268984492140489 The kurtosis is, 4.0978890145067498816 ----------------------- 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, 42.764718684687791860, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 13.982335862878296967 The skewness is, 0.79744956302958764411 The kurtosis is, 3.9989275441987923326 ----------------------- To sum up the expected time it takes to visit, "isprime", k times for k from 1 to , 10, are [2.428497914, 5.712240468, 9.498878119, 13.65059271, 18.05408931, 22.64615402,27.42115902, 32.37752852, 37.50029903, 42.76471868] ---------------- This ends this paper that took, 5207.631, seconds to produce.