Meepo_

2015-09-16, 06:31 PM

I've always wondered what the best way to roll your D&D/d20 stats was. I've played with the first edition rules; you roll 3 six-sided dice (3d6), add them up, and those are your stats. You used to even roll them order: Str, Dex, Con, Int, Wis, and Cha. It was brutal: I thankfully rolled a 17 for intelligence (and bumped it up to an 18 by sacrificing a few points of strength) to make a bad-ass wizard as my first character. But the variation between power levels of characters was way too much: you were just as likely to be an absolute genius (18) as you were to be only slightly smarter than your average bear (3).

Now I'm all for point-buys. You get exactly what you want, and your stats aren't any better or worse than the next guys'. But after I read 5e, and noticed how it attempts to return to the roots of D&D where fun > optimization, I also realized that rolling your character stats is just about the most D&D-like way to build a character. Later editions attempted to make characters a bit better. All editions from Advanced on have allowed the player to choose which numbers go where. Alternate stat rolling methods began to surface to make adventurers better than normal people. The first alternate way I learned how to roll stats was to roll normally, and re-roll any 1s (3d6r1). The more widely accepted method is to roll 4 six-sided dice, and to add the best 3 (4d6b3). The first removes tiny stats, while the second makes all one's stats better on average. I was faced with a question: which method is the best?

It probably doesn't matter that much, but I just had to know. Mathematically speaking, the average roll on a d6 is 3.5, so the original method yields average stats of 10.5. Re-roll the 1s and the new average per die is 4, so the average stat would be 12. But what is the average stat when you take the 3 best of 4 rolls? If the first 3 rolls are all 6s the method doesn't matter. All 4s and there is a 33% chance of improving the stat with another roll: all 3s and the fourth roll boosts the total 50% of the time. But if you get three 1s, the fourth roll has an 83% chance of boosting the total, whereas if you re-roll your 1s each die has a 100% chance of improving. The 3d6r1 method makes terrible rolls nearly impossible, and the 4d6b1 method makes all your rolls slightly better. But I know no way of mathematically calculating the average of 4d6b1. So I did the only logical thing a D&D nerd would do: I built an experiment.

It takes hundreds of die rolls to produce accurate, average results, and I had no interest in spending my weekends rolling dice [alone and without an awesome story to be a part of], so I coded a nifty program to do it for me. It uses Java (please don't hate), and lets me roll any number of stats, using all three methods, and then calculates the mean of each almost instantaneously. I won't post the code unless anyone cares to see it, and I'll spoiler the results of the first trial so you don't have to scroll past a wall of numbers. But without further ado, here are the results:

Trial # and roll amount

Trial 1: (100)

Trial 2: (100)

Trial 3: (100)

Trial 4: (1000)

Trial 5: (10000)

Trial 6: (1000000)

Trial 7: (1337)

Average of Trials 1 - 3

3d6 (Control)

10.04

10.76

10.29

10.54

10.4957

10.498284

10.477187733732237

10.363333

3d6r1

11.81

12.31

11.77

12.099

12.0322

11.998569

12.051608077786089

11.96333

4d6b1

12.1

12.14

11.73

12.129

12.2332

12.248802

12.100224382946896

11.99

If you really want to see the actual numbers rolled, this spoiler contains the full output of Trial 1:

3d6: 3d6 reroll 1: 4d6 best 3:

11 14 14

11 14 15

8 12 9

9 14 13

11 17 12

9 13 15

11 10 15

12 12 8

6 13 15

9 11 14

16 12 13

16 12 10

14 11 11

11 9 13

10 11 10

8 11 12

6 8 3

4 15 13

7 12 11

8 11 6

7 13 16

10 14 15

10 6 11

9 10 10

6 15 8

11 15 11

9 11 17

11 11 10

11 12 11

7 14 10

8 7 15

9 10 13

9 17 16

12 13 13

11 12 13

11 8 8

17 9 17

10 8 11

18 7 17

9 10 15

10 13 13

7 13 15

9 14 12

10 10 17

9 12 13

9 10 13

8 13 8

13 13 8

12 13 14

14 15 16

9 12 11

8 9 14

13 7 9

8 10 14

5 17 15

6 13 12

7 13 9

10 9 12

9 13 13

10 11 12

14 8 11

4 13 11

9 9 10

8 18 13

13 12 15

12 10 11

11 13 10

7 11 11

8 12 10

11 12 11

11 13 9

12 13 14

13 9 12

11 10 14

11 16 12

17 12 16

7 12 13

8 13 13

7 11 11

7 14 13

14 11 9

7 12 14

8 9 14

10 11 9

9 8 8

15 12 10

10 9 13

10 12 10

13 11 11

11 13 14

9 9 12

11 14 12

8 13 11

11 12 15

10 15 10

11 12 15

11 14 14

12 13 11

11 11 5

14 15 14

Mean:

10.04 11.81 12.1

I started out with three trials of 100 rolls each and obtained the average, but the results were inconclusive. Sometimes 3d6r1 would be better than 4d6b1, and other times it would be the reverse. To get more accurate results I rolled the stats 1,000 and 10,000 times. I then got bored and decided to see what 1,000,000 rolls would be like (took a few minutes, which is a lot by computer standards), and what 1337 would give me (besides more decimal places). It seems to me that the two rolls are not really that different, but the 3d6 had an overall average close to 10.5, the 3d6r1 had an overall average close to 12.0, and the 4d6b1 had an overall average closest to 12.25.

I guess rolling four times and taking the best three is in fact slightly better than rolling three times and re-rolling ones. The superior method is included in the core books, after all. I can't say this wasn't a fun little project- its pretty cool to watch all those numbers scroll by on my screen faster than a CN Rogue to a dragon's hoard. I'll appreciate anything anyone has to say about my experiment: where I screwed up, what I could have done better, whether there are other rolling methods that I should test, or that I should get a new hobby.

Anyways: I bid you all farewell, and may you be blessed with many a powerful character!

Now I'm all for point-buys. You get exactly what you want, and your stats aren't any better or worse than the next guys'. But after I read 5e, and noticed how it attempts to return to the roots of D&D where fun > optimization, I also realized that rolling your character stats is just about the most D&D-like way to build a character. Later editions attempted to make characters a bit better. All editions from Advanced on have allowed the player to choose which numbers go where. Alternate stat rolling methods began to surface to make adventurers better than normal people. The first alternate way I learned how to roll stats was to roll normally, and re-roll any 1s (3d6r1). The more widely accepted method is to roll 4 six-sided dice, and to add the best 3 (4d6b3). The first removes tiny stats, while the second makes all one's stats better on average. I was faced with a question: which method is the best?

It probably doesn't matter that much, but I just had to know. Mathematically speaking, the average roll on a d6 is 3.5, so the original method yields average stats of 10.5. Re-roll the 1s and the new average per die is 4, so the average stat would be 12. But what is the average stat when you take the 3 best of 4 rolls? If the first 3 rolls are all 6s the method doesn't matter. All 4s and there is a 33% chance of improving the stat with another roll: all 3s and the fourth roll boosts the total 50% of the time. But if you get three 1s, the fourth roll has an 83% chance of boosting the total, whereas if you re-roll your 1s each die has a 100% chance of improving. The 3d6r1 method makes terrible rolls nearly impossible, and the 4d6b1 method makes all your rolls slightly better. But I know no way of mathematically calculating the average of 4d6b1. So I did the only logical thing a D&D nerd would do: I built an experiment.

It takes hundreds of die rolls to produce accurate, average results, and I had no interest in spending my weekends rolling dice [alone and without an awesome story to be a part of], so I coded a nifty program to do it for me. It uses Java (please don't hate), and lets me roll any number of stats, using all three methods, and then calculates the mean of each almost instantaneously. I won't post the code unless anyone cares to see it, and I'll spoiler the results of the first trial so you don't have to scroll past a wall of numbers. But without further ado, here are the results:

Trial # and roll amount

Trial 1: (100)

Trial 2: (100)

Trial 3: (100)

Trial 4: (1000)

Trial 5: (10000)

Trial 6: (1000000)

Trial 7: (1337)

Average of Trials 1 - 3

3d6 (Control)

10.04

10.76

10.29

10.54

10.4957

10.498284

10.477187733732237

10.363333

3d6r1

11.81

12.31

11.77

12.099

12.0322

11.998569

12.051608077786089

11.96333

4d6b1

12.1

12.14

11.73

12.129

12.2332

12.248802

12.100224382946896

11.99

If you really want to see the actual numbers rolled, this spoiler contains the full output of Trial 1:

3d6: 3d6 reroll 1: 4d6 best 3:

11 14 14

11 14 15

8 12 9

9 14 13

11 17 12

9 13 15

11 10 15

12 12 8

6 13 15

9 11 14

16 12 13

16 12 10

14 11 11

11 9 13

10 11 10

8 11 12

6 8 3

4 15 13

7 12 11

8 11 6

7 13 16

10 14 15

10 6 11

9 10 10

6 15 8

11 15 11

9 11 17

11 11 10

11 12 11

7 14 10

8 7 15

9 10 13

9 17 16

12 13 13

11 12 13

11 8 8

17 9 17

10 8 11

18 7 17

9 10 15

10 13 13

7 13 15

9 14 12

10 10 17

9 12 13

9 10 13

8 13 8

13 13 8

12 13 14

14 15 16

9 12 11

8 9 14

13 7 9

8 10 14

5 17 15

6 13 12

7 13 9

10 9 12

9 13 13

10 11 12

14 8 11

4 13 11

9 9 10

8 18 13

13 12 15

12 10 11

11 13 10

7 11 11

8 12 10

11 12 11

11 13 9

12 13 14

13 9 12

11 10 14

11 16 12

17 12 16

7 12 13

8 13 13

7 11 11

7 14 13

14 11 9

7 12 14

8 9 14

10 11 9

9 8 8

15 12 10

10 9 13

10 12 10

13 11 11

11 13 14

9 9 12

11 14 12

8 13 11

11 12 15

10 15 10

11 12 15

11 14 14

12 13 11

11 11 5

14 15 14

Mean:

10.04 11.81 12.1

I started out with three trials of 100 rolls each and obtained the average, but the results were inconclusive. Sometimes 3d6r1 would be better than 4d6b1, and other times it would be the reverse. To get more accurate results I rolled the stats 1,000 and 10,000 times. I then got bored and decided to see what 1,000,000 rolls would be like (took a few minutes, which is a lot by computer standards), and what 1337 would give me (besides more decimal places). It seems to me that the two rolls are not really that different, but the 3d6 had an overall average close to 10.5, the 3d6r1 had an overall average close to 12.0, and the 4d6b1 had an overall average closest to 12.25.

I guess rolling four times and taking the best three is in fact slightly better than rolling three times and re-rolling ones. The superior method is included in the core books, after all. I can't say this wasn't a fun little project- its pretty cool to watch all those numbers scroll by on my screen faster than a CN Rogue to a dragon's hoard. I'll appreciate anything anyone has to say about my experiment: where I screwed up, what I could have done better, whether there are other rolling methods that I should test, or that I should get a new hobby.

Anyways: I bid you all farewell, and may you be blessed with many a powerful character!