Probability help?

[Jeff Wilder]

My knowledge of probability is too basic to handle this, so I was hoping for help.

(1) Assume 4d6, drop lowest, for each roll.

(2) Assume six rolls.

(3) Assume X sets of rolls. (In my case, X would be 4, 5, or 6.)

What will be the average value of a roll for the best (highest) set of rolls?

(4) Same question, but for 3d6 for each roll.

E.g., if one set of rolls generates 9, 12, 13, 13, 15, 16 and another set generates 8, 14, 14, 15, 17, 18, the average value of a roll for the best set (second, in this case) is 14.33.

While the equation would be great, just an answer would be great, too. Brute-force computations would be plenty close enough for my purposes.

BTW, I'm asking because I'm considering moving back to random generation of abilities, combining with boosting players who roll worse up to the level of the player who rolls best. (With limits on how the "boost" is spent.) I truly believe that point-buy is the fairest way to do things, but I like the surprises that can happen with random rolling. So I'm working on a solution to keep randomness but smooth out the disparity in results between players.

 

[blargney the second]

Have you considered using the Three Dragon Ante method from Dragon magazine to generate stats? It's both fair and random.

 

[Mustrum_Ridcully]

There must be some formulas to express these dice rolls.

One thing I wonder is: What do you like about the random distribution in particular?

I have always considered introducing elements of randomness without necessarily "breaking" the game or balance.

Maybe define 3 stat sets and roll 1d6:
1-2) 15, 14, 13, 12, 10, 8
3-4) 16, 14, 11, 12, 10, 8
4-5) 14, 14, 14, 11, 10, 9 (note: one bonus point compared to the rest)
5-6) 17, 14, 12, 10, 10, 6 (note: 6 is lower then point buy normally allows)
That's your starting set, you can distribute it as you see fit.

Now, roll 1d6 (1: Str, 2: Dex, 3:Con; 4:Int; 5:Wis; 6:Cha). Spend 3 points on that ability score (normal point buy method). If you can't buy a further advancement but have some points left, roll again to spend the remaining points (or just chose one ability).
Repeat this a second time, but reroll if you end up with a stat you have already have increased.

In addition, you may allow one reroll in the entire process. (Allow the decision to reroll something at the end)

You'd end up with ~31 point buy characters. The players still had a major choice in their ability score distribution by chosing their preferences, but the rolls might end up with some surprises (like a melee-type character that ends up spending his extra points on Charisma and Intelligence, originally his lowest statistics, or a wizard-type character that gets some boosts to strength or dex.)

At best, the 3 extra points on your favored ability score can only get you a net +1 bonus to it, which should still result in fairly effective characters, even if you end up with extra points in ability scores you didn't want.

---
Alternatively, you might use 3d6 roll in order (or 4d6 drop lowest) to define priorities. Arrange the stats according to these priorities. You can only spend any points on a ability if you have spend at least 3 points on the next higher stat, and the rest is 25 point buy.
So, a character with 3d6 results like Str 3, Dex 9, Con 12, Int5, Wis 16, Cha 15 that shall become a fighter would need to have at least an 11 in all attributes other then Strength. Even with 25 point buy, you are guaranteed to have 10 points remaining to spend on your favored ability score (so this fighter could reach a Strength of 16, though more likely end up with Strength 15 and Con and Dex of 12. You are not promised to get a super-optimized character, but you will get one that might take you to unexpected places.

 

[Mustrum_Ridcully]

Have you considered using the Three Dragon Ante method from Dragon magazine to generate stats? It's both fair and random.

Hmm. Makes me wish I had that (or any) dragons (I assume it wasn't the online version?)

 

[Jeff Wilder]

Have you considered using the Three Dragon Ante method from Dragon magazine to generate stats? It's both fair and random.

I vaguely remember liking that system. It would present some logistical problems in generating stats for everyone -- i.e., I'd actually have to meet with each player individually -- but on the other hand it could also make it fun to actually get the group together to generate PCs, which we just don't seem to do anymore.

Leading to another question about the Three Dragon Ante (TDA) method, one of the arguments in favor of random generations that I feel is valid -- probably the only one I feel is valid -- is that if you limit players' ability to swap scores into stats (limit, not eliminate), you can get some interesting results that just don't happen in point-buy. Things like a low-DEX wizard, or a high-STR sorcerer. I find that possibility appealing.

So, does TDA allow limiting score-to-stat assignment? I don't remember the article well enough to say. If it doesn't, though, it doesn't give me the only thing I'm really looking for from random stats.

 

[Mustrum_Ridcully]

Leading to another question about the Three Dragon Ante (TDA) method, one of the arguments in favor of random generations that I feel is valid -- probably the only one I feel is valid -- is that if you limit players' ability to swap scores into stats (limit, not eliminate), you can get some interesting results that just don't happen in point-buy. Things like a low-DEX wizard, or a high-STR sorcerer. I find that possibility appealing.

I knew that's what you'd like about it. That is the only part of random generation that I like, while I hate all the balancing woes...

 

[Umbran]

(1) Assume 4d6, drop lowest, for each roll.

(2) Assume six rolls.

(3) Assume X sets of rolls. (In my case, X would be 4, 5, or 6.)

What will be the average value of a roll for the best (highest) set of rolls?

While you imply later that it looks like you're thinking in terms of average stat, I note that "best set of rolls" is not well defined - highest average stat, highest sum of stats, highest maximum stat, highest average modifier, highest sum of modifiers, and highest maximum modifier all will give slightly different answers.

Also, how you choose to resolve ties will have an impact on the final result - If two sets have the same average of stats, keeping the one with the highest maximum stat and keeping the one with the highest sum of modifiers won't give you the same results.

BTW, I'm asking because I'm considering moving back to random generation of abilities, combining with boosting players who roll worse up to the level of the player who rolls best. (With limits on how the "boost" is spent.) I truly believe that point-buy is the fairest way to do things, but I like the surprises that can happen with random rolling. So I'm working on a solution to keep randomness but smooth out the disparity in results between players.

There's reasonable ways of doing this without knowing the averages. For example:

Reroll any set of stats that is under a particular total modifier threshhold.

 

[Jeff Wilder]

While you imply later that it looks like you're thinking in terms of average stat, I note that "best set of rolls" is not well defined

You're right. I am just talking about average stat, and I don't care about tie-breakers, really. I'll go with "highest modifier total," but it doesn't matter much, because I'm kinda just trying to get an idea of the stats I can expect if I raise everybody else up to the level of the highest roller of, say, five players.

(My gut says "way too powerful," but I do know enough probability to know that it's often unintuitive.)

Reroll any set of stats that is under a particular total modifier threshhold.

That's what I used before switching to point-buy. Getting relatively screwed is not much more fun than getting absolutely screwed. I want PCs to start on reasonably even footing.

 

[Umbran]

... I'm kinda just trying to get an idea of the stats I can expect if I raise everybody else up to the level of the highest roller of, say, five players.

Ah, for that, what's more important than the average is the expected deviation from the average. If the expected deviation is large, knowing the average isn't so informative.

That's what I used before switching to point-buy. Getting relatively screwed is not much more fun than getting absolutely screwed. I want PCs to start on reasonably even footing.

Well, then say "reroll until you are within X total modifier of the leader". Or, if you want to avoid the way-too-powerful, "Roll until you have a total modifier between X and Y" - so someone who rolls too well needs to reroll.

 

[Krensky]

Hmm. Makes me wish I had that (or any) dragons (I assume it wasn't the online version?)

No, it was in the print ones (#346), and it's a later one so Paizo lost the rights to make a PDF from it before they could make a PDF of it. The still, apparently, have back issues of it.

Essentially it involves shuffling the deck, playing them out in a pattern and placing tokens (representing point buy points) on the cards and then moving them based on what the cards said. Characters tend to be middle of the road for their PB, but there can be some severe swings. It sounds hokey, but it gives a good (and usually non-optimized) result from when I've used it and the notes in the article also briefly discuss converting the "meanings" of the cards into characterization and backstory.

So, does TDA allow limiting score-to-stat assignment? I don't remember the article well enough to say. If it doesn't, though, it doesn't give me the only thing I'm really looking for from random stats.

Well, if I remember right the default is that you get whatever value you can buy for the number of tokens on the Stat's card. A sidebar talked about allowing swapping cards or point values though.