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<blockquote data-quote="Ainamacar" data-source="post: 6135917" data-attributes="member: 70709">Hmm, fair enough.&nbsp; I would like to note that in the second presentation&nbsp; you do not have to explicitly track how many rolls you have previously made because this is handled&nbsp; implicitly by adding 1 after each roll.&nbsp; An analogy might be tracking&nbsp; hit points, where each &quot;round&quot; you make an &quot;attack&quot; by rolling an&nbsp; ability score: you have regeneration 1 and take &quot;damage&quot; equal to the&nbsp; modifier of the most recent ability score rolled.&nbsp; If you have positive&nbsp; &quot;hit points&quot; you gain that many bonus dice on the next roll.I don't mean to grouse, I am glad you liked how the math works out. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile&nbsp; &nbsp; :)"&nbsp; data-smilie="1"data-shortname=":)" />Out&nbsp; of curiosity I measured the probability of following a known score with&nbsp; a given score.&nbsp; For example, if the previous score rolled is a 15 I&nbsp; determined the average probability of the next one being a 10.&nbsp; From&nbsp; this I can calculate what score is rolled on average after any other&nbsp; roll.&nbsp; Using the Markov chain method they are as follows (to two decimal&nbsp; places, calculated from 100,000 trials):[code]Score&nbsp; Next score (avg)3&nbsp; &nbsp; &nbsp; 15.304-5&nbsp; &nbsp; 14.736-7&nbsp; &nbsp; 14.138-9&nbsp; &nbsp; 13.3910-11&nbsp; 12.6612-13&nbsp; 12.0114-15&nbsp; 11.7716-17&nbsp; 11.4318&nbsp; &nbsp;&nbsp; 11.01[/code]The&nbsp; values are the same for rolls that share modifiers because the number of bonus dice on the next roll only changes if the total modifier changes.For comparison, the averages of nd6 take best 3 (exactly for n=3, approximately for n&gt;3) are[code]n&nbsp;&nbsp; Avg3&nbsp;&nbsp; 10.504&nbsp;&nbsp; 12.245&nbsp;&nbsp; 13.436&nbsp;&nbsp; 14.277&nbsp;&nbsp; 14.908&nbsp;&nbsp; 15.39[/code]We&nbsp; can see that rolling a 10-15 means that, on average, the next roll will&nbsp; be close to that of 4d6 drop lowest, while for 16-18 the next roll will&nbsp; be like that of 3d6.&nbsp; (Even after an 18 the next roll is, on average, a&nbsp; bit better than 3d6 because sometimes an 18 can't make up for earlier&nbsp; terrible rolls.)&nbsp; Rolls as we go below 10 have next-roll averages very&nbsp; near to those of 5d6, 6d6, 7d6 and finally 8d6, so the system is pretty&nbsp; much behaving as expected.From that result it is conceivable we&nbsp; might be able to dispense with the tracking altogether and get a similar&nbsp; outcome.&nbsp; Following the values above, suppose if you roll a 16-18 your&nbsp; next roll is 3d6.&nbsp; If you roll a 10-15 your next roll is 4d6 drop&nbsp; lowest.&nbsp; If 8-9 then 5d6, if 6-7 then 6d6, if 4-5 then 7d6, if 3 then&nbsp; 8d6.&nbsp; Assuming the first roll is 3d6 this non-tracking method gives an&nbsp; average total modifier of 4.64 and a standard deviation of 2.86.&nbsp; If the&nbsp; first roll is 4d6 instead it gives an average total modifier of 5.31 and&nbsp; standard deviation of 2.87.&nbsp; As a reminder, the standard deviations of&nbsp; the 3d6 and 4d6 drop lowest methods are about 3.68 and 3.54,&nbsp; respectively, while for the Markov method it was about 1.71.&nbsp; So&nbsp; changing the number of dice rolled without tracking does reduce the&nbsp; standard deviation, but only by about 20% from the standard methods,&nbsp; compared to about 50% when the total modifier of past rolls is&nbsp; considered.&nbsp; Ahh, statistics.</blockquote>

[QUOTE="Ainamacar, post: 6135917, member: 70709"] Hmm, fair enough. I would like to note that in the second presentation you do not have to explicitly track how many rolls you have previously made because this is handled implicitly by adding 1 after each roll. An analogy might be tracking hit points, where each "round" you make an "attack" by rolling an ability score: you have regeneration 1 and take "damage" equal to the modifier of the most recent ability score rolled. If you have positive "hit points" you gain that many bonus dice on the next roll. I don't mean to grouse, I am glad you liked how the math works out. :) Out of curiosity I measured the probability of following a known score with a given score. For example, if the previous score rolled is a 15 I determined the average probability of the next one being a 10. From this I can calculate what score is rolled on average after any other roll. Using the Markov chain method they are as follows (to two decimal places, calculated from 100,000 trials): [code] Score Next score (avg) 3 15.30 4-5 14.73 6-7 14.13 8-9 13.39 10-11 12.66 12-13 12.01 14-15 11.77 16-17 11.43 18 11.01 [/code] The values are the same for rolls that share modifiers because the number of bonus dice on the next roll only changes if the total modifier changes. For comparison, the averages of nd6 take best 3 (exactly for n=3, approximately for n>3) are [code] n Avg 3 10.50 4 12.24 5 13.43 6 14.27 7 14.90 8 15.39 [/code] We can see that rolling a 10-15 means that, on average, the next roll will be close to that of 4d6 drop lowest, while for 16-18 the next roll will be like that of 3d6. (Even after an 18 the next roll is, on average, a bit better than 3d6 because sometimes an 18 can't make up for earlier terrible rolls.) Rolls as we go below 10 have next-roll averages very near to those of 5d6, 6d6, 7d6 and finally 8d6, so the system is pretty much behaving as expected. From that result it is conceivable we might be able to dispense with the tracking altogether and get a similar outcome. Following the values above, suppose if you roll a 16-18 your next roll is 3d6. If you roll a 10-15 your next roll is 4d6 drop lowest. If 8-9 then 5d6, if 6-7 then 6d6, if 4-5 then 7d6, if 3 then 8d6. Assuming the first roll is 3d6 this non-tracking method gives an average total modifier of 4.64 and a standard deviation of 2.86. If the first roll is 4d6 instead it gives an average total modifier of 5.31 and standard deviation of 2.87. As a reminder, the standard deviations of the 3d6 and 4d6 drop lowest methods are about 3.68 and 3.54, respectively, while for the Markov method it was about 1.71. So changing the number of dice rolled without tracking does reduce the standard deviation, but only by about 20% from the standard methods, compared to about 50% when the total modifier of past rolls is considered. Ahh, statistics. [/QUOTE]

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