The 216 Club

[Ainamacar]

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):

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

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

n   Avg
3   10.50
4   12.24
5   13.43
6   14.27
7   14.90
8   15.39

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.

 

[Shingen]

What do you think of character's ability scores? Where should they fall? Are 18's in the prime ability dandy or do you think they should be much lower?

18 seemed to always lead to issues in most of my games, as getting one is not terribly common. IN fact, that disparity of Attributes causing issues led to me giving out the fixed array for a long time, because 16 is perfectly good, and doesn't lead to such an exaggerated advantage.

 

[howandwhy99]

Each stat has a 1:216 chance of being an 18
Each stat has a 1:54 chance of being an 17 or 18
For every character with 6 stats they have a 1:9 chance of having at least a 17 or 18 in a stat.
For every group of 6 players with 1 PC each they have a 1 1/2 chance of having at least 1 character with a 17 or 18 stat.
For every group where the order of the stats rolled must be kept, meaning there is no assigning of a roll to a desired ability, 1 group of six in 9 will create an initial PC who qualifies for Paladinhood.

For almost every group I've played with the odds of every PC having a score of 18 is 50%.
The odds of them wanting to play a paladin was <1%.

 

[Ainamacar]

Your point is unchanged, @howandwhy99, but your calculation is incorrect. (Tipped off when you said something has a 1 1/2 chance...all probabilities must be between 0 and 1.)

An ability score does have a 1/54 chance of being a 17 or 18 when using the 3d6 method. However, the probability that a character with 6 ability scores has at least one score of 17 or 18 is 1-(1-1/54)^6=2630550167/24794911296~.106092. For a party of 6 players with 1 PC the probability that at least one PC has 17-18 in at least one ability score is 1-(1-2630550167/24794911296)^6=(113809362490648359702243241873878013457817011018044797058180175/
232367482936623516730287712543386283362359123728764547834576896)~.489782. We would expect just about 1 of every 2 groups to have at least one character that could be a Paladin.

If the rolls cannot be rearranged as desired, the probability that at least one of the 6 characters will have a 17-18 in the correct score is also 1-(1-1/54)^6~.106092. Thus, on average it takes about 9.43 groups of 6 players for 1 group to have at least one character that qualifies for Paladinhood.

Assuming we're using the 3d6 method, the probability that *every* PC in a party of 6 has at least one 17-18 is (2630550167/24794911296)^6~1.43*10^-6. Not much better, really, than the proverbial one-in-a-million chance.