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Author Topic: Probability Theory and You  (Read 6107 times)

Jamfke

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Probability Theory and You
« on: May 09, 2020, 10:31:50 am »
I'm not much of a probability genius. I understand a little bit about it, but nothing in depth. My question here is primarily one to determine which type of resolution methods are your favorite for certain activities performed in a game and why.

Like D20 vs percentiles for combat resolution or skill usage. Which is your favorite and why? Does probability factor into your reasoning at all or is it more of a sentimental thing (cuz it is the way!)?
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finarvyn

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« Reply #1 on: May 09, 2020, 11:46:18 am »
I had a friend in high school who wanted to write his own D&D variant, and one thing he used is 1d20 to roll character attributes. His reason was because he thought that a scale that went from 3 up was stupid, which in a way it is. Anyway, his problem was a total lack of understanding of probability. While 3d6 gives most scores in the middle and few on the extremes (the bell curve) a single d20 gives the exact same average but also an identical chance of getting an extreme number as an average number. Probably everyone reading this board understands this, but he didn't. From that standpoint, I feel like a basic understanding of probability is essential for anyone who wants to design a game.

I find it interesting when I see a percentile-based RPG where every single probability or bonus is divisible by five. Those folks are playing a d20 game but they don't seem to even realize it or for some reason think that rolling percentile dice instead of a d20 is more fun somehow. Again, a basic understanding of probability is valuable.

Just some observations. A lot of folks don't like math, I get that.
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HappyDaze

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« Reply #2 on: May 09, 2020, 12:58:24 pm »
I'm not a fan of card draw resolutions, especially when the deck is not reshuffled between draws. This is because previous draws will influence the outcome of later draws--if I did great last turn with good draws, then this turn I'm not as likely to get draws as good since those cards are no longer in the deck. Likewise the reverse is getting the "you suck" hand and knowing that your chance of doing better next turn is higher. In both cases, I'd prefer that each outcome was independent of the last, so rolling dice has always been my preference.

VisionStorm

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« Reply #3 on: May 09, 2020, 01:12:56 pm »
I have tried and enjoyed (or hated) many types of task resolution mechanics, including d20 (roll under or over), d100 (roll under or over, or compare to table FASERIP-style), Dice Pools (add together or count successes), and more. But the one I find most effective and enjoy most is d20+Modifier (roll over).

d20+Mod IMO is the most simple and straightforward mechanic and the easiest to teach. Rolling "high" to succeed is intuitive and a d20 has a decent range of variables without needing to roll a bunch of dice or explaining to people how to count the "1s" and "10s". It has the issue of lacking a bellcurve, but makes up for it with speed of play. Almost every other task resolution mechanic (other than d10+Mod, which is identical, but with a d10) takes longer to resolve and more effort from the players, and attention away from the game.

SHARK

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« Reply #4 on: May 09, 2020, 01:27:12 pm »
Quote from: VisionStorm;1129384
I have tried and enjoyed (or hated) many types of task resolution mechanics, including d20 (roll under or over), d100 (roll under or over, or compare to table FASERIP-style), Dice Pools (add together or count successes), and more. But the one I find most effective and enjoy most is d20+Modifier (roll over).

d20+Mod IMO is the most simple and straightforward mechanic and the easiest to teach. Rolling "high" to succeed is intuitive and a d20 has a decent range of variables without needing to roll a bunch of dice or explaining to people how to count the "1s" and "10s". It has the issue of lacking a bellcurve, but makes up for it with speed of play. Almost every other task resolution mechanic (other than d10+Mod, which is identical, but with a d10) takes longer to resolve and more effort from the players, and attention away from the game.

Greetings!

Good points, VisionStorm! For a number of years, I played Rolemaster. Yes, sometimes called *Chartmaster*:D The game system provided probably unparalleled depth, and richness of detail. My players deeply enjoyed many aspects and features of the Rolemaster system, but ultimately grew weary from the charts, the math, the general complexity, and the bookkeeping involved.

So, yeah. Adult gamers for the most part--in my experience--hugely demand speed, simplicity/ease of use, and fun. That is the "holy trinity.":D

Semper Fidelis,

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Omega

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« Reply #5 on: May 09, 2020, 06:01:22 pm »
Over on BGG we've had a few of these threads and/or threads where someone either steadfastly disbelieves the bell curve or fanatically keeps harping that "a d20 is not balanced because its average is 10.5" no. I kid you not. That ones been spouted a few times.

Nobby-W

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« Reply #6 on: May 09, 2020, 06:09:15 pm »
It's complicated.

The difference between different dice mechanics is largely about the kurtosis, although some such as D20 and percentile dice behave essentially the same in this regard.

[ATTACH=CONFIG]4452[/ATTACH]

Fig 1. Random chart off the interwebs showing distributions with different kurtosis.

Figure 1 shows some probability density functions with differing kurtosis.  Kurtosis drives the likelihood that a roll will produce extreme results in the tails.  Dice with high kurtosis are more swingy.  The most swingy mechanics are D20 and percentile dice (which behave as if they were a single 100-sided die).  These have a flat distribution where every possible outcome has the same probability.  

[ATTACH=CONFIG]4453[/ATTACH]
Fig. 2. A flat distribution such as a D20 or D100.

Different dice mechanics have different probability distributions.  4DF, for example, has relatively low probability of rolling anything outside the range of +2 to -2.  There is about 6% chance of rolling higher than +2.  This makes the game dependent of the FATE point economy as the dice roll outcomes are largely clustered around the middle in the range +2 to -2.  D&D's D20 mechanic, however, is just as likely to roll a 1, 10 or 20, giving it a much higher probability of rolling in the extremes.  Traveller's 2D6 mechanic sits somewhere in the middle.  Exploding dice mechanics complicate the calculation - Savage Worlds is famous for being fiddly to calculate the odds for dice rolls.

Generally the more dice involved the lower your kurtosis.  One issue with buckets-of-dice systems is that the kurtosis changes with the number of dice being rolled.

Having said all that I have a soft spot for 2D6, as it lends itself to quick mental arithmetic and bonuses of (say) +1 to +3 have more-or-less sensible effects across a wide range of target rolls, which is a nice happenstance.  It's more swingy than 4DF and less swingy than D20.

I'll also add some observations about dice mechanics on systems for PbP use.

I'm running a Scum and Villainy campaign on a forum at the moment, and the dice mechanic is slightly clumsy as you have to have some back-and-forth between the DM and players to even determine how many dice to roll.  For a game to work nicely in PbP I would suggest a dice mechanic with the following attributes.

  • Always rolling the same dice - the difference is the target.
  • Bonuses can be added after the fact.  If you have metacurrency then it can just be added later.
  • Avoid contested rolls.  Keep it to the player making one dice roll - through a roller app or otherwise.  The DM can subtract the opponent's bonus from the roll.
This approach means that the player can roll straightaway and the GM can frig the roll with whatever bonuses apply.  On a PbP game that can save a day or two of turnaround time for a single roll.

2D6 + bonuses would work fine for this, as would 4DF or a D&D style D20.  Dice pools where you might vary the number of dice being rolled add a round trip to the conversation.  This also includes D&D's advantage/disadvantage mechanics if it's not always obvious when it applies.
« Last Edit: May 09, 2020, 06:21:00 pm by Nobby-W »
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S'mon

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« Reply #7 on: May 09, 2020, 06:50:48 pm »
I love D6 system, where you roll your skill dice vs a target number and you get more dice the more skilled you are. It avoids all the problems the D20 system has, since 2d6+ is always a bell curve, which range grows slowly, but always a bell curve.
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mightybrain

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« Reply #8 on: May 09, 2020, 08:07:05 pm »
Although the d20 result probability itself is flat, it is modified and compared against targets that are bell curves so the final effect is the same.

Cave Bear

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« Reply #9 on: May 09, 2020, 09:18:36 pm »
Quote from: HappyDaze;1129383
I'm not a fan of card draw resolutions, especially when the deck is not reshuffled between draws. This is because previous draws will influence the outcome of later draws--if I did great last turn with good draws, then this turn I'm not as likely to get draws as good since those cards are no longer in the deck. Likewise the reverse is getting the "you suck" hand and knowing that your chance of doing better next turn is higher. In both cases, I'd prefer that each outcome was independent of the last, so rolling dice has always been my preference.



What you're describing is a feature, not a flaw. It models the rise and fall of fortune in a way that's less prone to streaks than dice-based systems.

Trond

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« Reply #10 on: May 09, 2020, 09:28:17 pm »
My friends and I used to figure out the probabilities of all sorts of systems :)
I DID have a few players who liked it simple though, and BRP (percentage system) was always popular.

Having said that, I'm not that picky, as long as I don't notice holes or inconsistencies in the logic and probabilities of the rules.

Theory of Games

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« Reply #11 on: May 09, 2020, 10:15:07 pm »
D100. Seems more "even" in a silly subjective way.

D20 aquired great Fame being as "swingy" as possible, but it can drive PC parties to TPKs like no other solid. It can "feel treacherous" at times.

Steven Mitchell

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« Reply #12 on: May 09, 2020, 10:26:11 pm »
My aesthetic preference for the math alone is a system that uses 2d10 + (small) mods versus target numbers.  It's a relatively flat curve, and the percentages are easily understood once you show someone a chart of the outcomes. It's a compromise between something like 3d6 and d20.  

For simple handling, I think a system using only a d12 would be rather nice, as you can do some fun things with 12 being evenly divisible by 1, 2, 3, 4, and 6.  

That's in theory.  In practice, I find that I don't want to play the kinds of games that would best fit such somewhat narrow ranges.  That is, I don't necessarily need "zero to hero", but I do want some substantial power level differences in the math itself, not just add-on abilities.  The same way that modifiers over -3 or +3 in 3d6 start to skew things (unless balanced by competing modifiers, which is its own issue), with 2d10, -4 to +4 is the reasonable limit, with maybe -5 and +5 for extreme edge cases.  I think, for example, that I'd like Hero System better if it were built from the ground up using 2d10, but I no longer have enough interest in the game to put in the work to test the theory.  Likewise, the proper sort of game that would use a d12 is either very light or the complexity is embedded in units--so far not my preference.

I'm happy enough with any moderately simple system in an existing game, d20 + mods, percentage rolls, etc. Some of the clever system have more appeal than others, but I seldom play with a group that would appreciate them.  For the dice roll itself to invoke decision points is more OOC game than I usually want with casual players, since they can't internalize the mechanics well enough for it stop interfering with their actions in character.

Libramarian

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« Reply #13 on: May 10, 2020, 02:04:19 am »
I like linear for binary pass/fail rolls. The effect of modifiers is more intuitive, and I don't understand what people mean when they say the d20 is "swingy", unless they're narrating attacks that miss by 8 as "worse" misses than attacks that miss by 1. Just stop doing that.

I like curves for rolls with more than two possible outcomes, because then you can have different probabilities for different outcomes, but still assign each outcome the same number of possible results, for easier memorization.

E.g. 2d6 --> 2: Hostile, 3-5: Unfriendly, 6-8: Uncertain, 9-11: Friendly, 12: Helpful

is easier to memorize (3, 6, 9, 12) than

1d20 --> 1: Hostile, 2-5: Unfriendly, 6-15: Uncertain, 16-19: Friendly, 20: Helpful

Mishihari

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« Reply #14 on: May 10, 2020, 02:19:05 am »
For dice choice, it's all about how much granularity is important.  (Of course) d100 gives 1% granularity, d20 5%, and d6 16%.  While I have some sentimentality for the d20, I've come to the conclusion that a d6 gives sufficient granularity for my purposes.  For a d20, I'm not much concerned about a mod smaller than +3, which is equivalent to a +1 on a d6, so why bother with all of those extra numbers?  d20 or d100 does better in systems where you add dozens of tiny mods to get a final result, but I don't care for those anyway.

For a simple system d6+skill+mods vs target, roll over, is great.  If simplicity is not my primary goal, I prefer d6-d6 with both dice exploding.  The latter has some nice properties, such as a fumble below a negative threshold always being possible but becoming very rare as one becomes skilled, a small chance to do almost anything, and added excitement when a die explodes.

I also really like systems where margin of success or failure matters.  Frex, on the system I'm working on now, margin of success on an attack is damage done, which speeds things up a bit by eliminating the damage die roll.
« Last Edit: May 10, 2020, 02:23:08 am by Mishihari »