Rolling Method I-V probabilities

[SBLaxman]

This is a repost (sorry mods) of the conclusion of a thread in the general discussion tab. I know a lot of 1e posters don't view GD, and this is a 1e specific topic. After several dice-rolling-method threads, I was curious on what the results of the different methods actually were, so I busted out Matlab (a math programming software package) and wrote up a script to find out. For methods 0 - II, actual probabilities were mathematically determined. For methods III - V, The sheer volume of numbers prevented my from calculating all possible outcomes (9d6 has 10,077,696 possible outcomes!!!!). For these methods, I generated instead 100,000 characters using each method. For Method IV and V, this was done for each class, resulting in 1,000,000 character generations for each method.

The methods were determined as follows:

• Method 0: 3d6 in Order clearly
  • non standard, but used by some
• Method I: 4d6, Arrange to Taste. Lowest die dropped.
• Method II: 3d6 is rolled 12 times, top 6 retained, arrange to taste
• Method III: 3d6 is rolled 6 times for each score, best roll taken.
  • For this method, the player cannot arrange the scores as desired.
• Method IV: 6 characters are arranged by 3d6 in order. Best character chosen.
  • For this method, user input is needed to pick the "best" character. Because of this, it was repeated 10 times (once for each class) with criteria of choosing as follows
    • First: Does the Class Qualify
    • Second: Is the Prime Requisite High
    • Third: Is the secondary prime high (for assassin, dex was chosen first, then strength, for example)
• Method V: xd6, Pick best 3 for each score, based on class
  • From UA, repeated for each class

Summary Part 1: The "average highest score" table. In this table, Method I-V are presented, as well as "Method 0" (3d6) and Method Ib (4d6, reroll 1s, keep 1 on re-roll). The numbers presented are the change a character has of having X as their highest score. For example, with method 1, a character has a 9.34% having an 18 as their highest score, while a 20.83% chance of having a 17 as their highest score, while 1.49% of characters had no score higher than 12.

Ability Score	Method 0	Method I	Method Ib	Method II	Method III	Method IV	Method V
3	0.000	0.000	0.000	0.000	0.000	0.000	0.000
4	0.000	0.000	0.000	0.000	0.000	0.000	0.000
5	0.000	0.000	0.000	0.000	0.000	0.000	0.000
6	0.000	0.000	0.000	0.000	0.000	0.000	0.000
7	0.000	0.000	0.000	0.000	0.000	0.000	0.000
8	0.020	0.000	0.000	0.000	0.000	0.004	0.000
9	0.190	0.003	0.000	0.000	0.000	0.050	0.000
10	1.490	0.004	0.001	0.028	0.000	0.316	0.000
11	4.370	0.280	0.023	0.356	0.001	1.365	0.000
12	10.810	1.490	0.260	2.331	0.009	4.239	0.005
13	17.340	5.400	1.670	9.293	0.165	10.194	0.068
14	20.980	13.400	6.900	19.244	1.214	17.769	0.696
15	20.230	22.640	17.740	25.407	6.745	23.706	4.163
16	13.790	26.690	28.820	23.242	22.052	22.352	17.218
17	7.810	20.830	26.860	14.727	37.931	14.532	36.416
18	2.970	9.340	17.730	5.392	31.882	5.472	41.435

So, what does this say? Method V clearly gives quite a strong chance of an 18, as does method III. Method IV has quite a low chance of an 18, as does Method II.

Summary Part 2: Average scores...

Method	Average	StD
Method 0	10.50	2.97
Method I	12.24	2.85
Method Ib	13.20	2.57
Method II	12.74	0.98
Method III	14.28	2.36
Method IV	11.21	1.21
Method V	14.12	0.96

So, what does this show? Method III provides the highest average score, only slightly better than Method V. Method IV again is the lowest of all possible scores, outside of 3d6. Method II is only slightly better than Method I.

Summary Part 3: the summary of class availability by method...

Class	Method 0	Method I	Method II	Method III	Method IV	Method V
Cleric	74.07	100.00	99.99	99.99	99.99	99.91
Druid	3.47	78.33	68.13	44.23	24.87	75.10
Fighter	67.21	100.00	99.99	99.99	97.88	99.88
Paladin	0.10	25.06	18.77	11.95	0.88	31.91
Ranger	0.18	30.56	34.09	26.56	1.78	49.69
MU	74.07	99.99	99.99	99.99	99.99	99.91
Illusionist	1.50	35.66	24.29	17.45	4.17	44.24
Thief	74.07	99.99	99.99	99.99	99.99	99.91
Assassin	7.03	99.18	99.40	80.85	53.67	86.41
Monk	0.04	13.65	9.16	12.11	0.37	31.05

The results? Method V far and away provides the best class availability. Interestingly, of all the other methods, Method I reigns supreme, far better than Method II, III, or IV (which is barely better than 3d6 in order).

Method III increases the average score by a good bit, but actually decreases the availability of classes compared to method I, interestingly enough

100% is really 99.99.., just got rounded to 100 by the spreadsheet

Also of interest, a histogram (distribution) of dice results. For 4d6 through 9d6, it was "pick best 3". These are distributions of a single dice set, not of a characters ability scores for any given method (though a Method 0, I, or III character would match the appropriate curve.

[SBLaxman]

I need to go back and verify my numbers for Method III. Looking at my script, I may have accidentally calculated 6d6 pick 3, instead of 3d6 x6, pick best. I may have done it correctly, but I need to double check.

[SBLaxman]

Double Post

[Justisaur]

You left out throwing out characters without 2 15 or better scores, or with 2 of 5 or less (impossible to qualify for any class).

[SBLaxman]

You left out throwing out characters without 2 15 or better scores

A good point, though secondary. This was simply to show the base distributions of the methods. Table 1 and 2 do not take into account if the character is valid, only what the numbers rolled are. Table 3 does take into account character validity, however it only considered the minimum scores listed. The "two 15+" recommendation, while followed by myself, is a recommendation and not a hard rule, so it was omitted.

or with 2 of 5 or less (impossible to qualify for any class).

I failed to include the sub-5 score in accounting for class availability, though I believe this did not occur in most methods (though clearly it will occur in method 0 fairly often). When I get time I'll add a post with that information.

I also negate to add an extra category for "fail to achieve any class" due to not reaching mimumums. For methods I-V, this is less than 0.1%, as 99.9% or better of all rolled characters qualified to be a thief, cleric, or MU (fighters are slightly lower due to the 7 con required). For method 0 (3d6 in order) there is certainly a chance of a null character. While I had calculated that, I can't find the post at the moment. Simulating with 100,000 characters, one may expect less than 100 would fail to qualify for any class using any BtB method.

[prespos]

Here are the ability score generation articles from Dragon, if you are interested:

GENERATION OF ABILITY SCORES
Combined Generation : Character-creation tables all in one place (114.40)
What are the Odds? : Unearthed Arcana's "Method V" analyzed (117.14)
Roll On! : Expanding Unearthed Arcana's character-generation Method V (140.44)
Give Demi-Humans an Even Break! : More dice are nice when rolling ability scores (129.24)
Let the Good Dice Roll : Know the odds before you create your character (132.38)

[SBLaxman]

Here are the ability score generation articles from Dragon, if you are interested:

GENERATION OF ABILITY SCORES
Combined Generation : Character-creation tables all in one place (114.40)
What are the Odds? : Unearthed Arcana's "Method V" analyzed (117.14)
Roll On! : Expanding Unearthed Arcana's character-generation Method V (140.44)
Give Demi-Humans an Even Break! : More dice are nice when rolling ability scores (129.24)
Let the Good Dice Roll : Know the odds before you create your character (132.38)

Thanks, I'll have to dig them up and see if I wasted my time

Looking them over...

114.40 has a nice table of class minumums
117.14 is the exact same data as in the chart in the OP, but in numeric (vs graphical) form
140.44 adds a table for Method V for multi-class characters and for classes in OA
129.24 is a modified Method V for demi-humans
132.38 presents the similar data as in my table 1, though it a different format (it presents the chance of rolling X or higher on a single roll, while I'm displaying the chance of X being the max score of a character.

So no, I didn't waste me time, especially since 132.38 only covered 3d6 and Method I/II/III, not IV or V, which I did.

Thanks prespos, a fountain of references as always!