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Author Topic: Probability Theory and You  (Read 10252 times)

Kuroth

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« Reply #120 on: May 30, 2020, 04:30:20 AM »
Matthew Conroy has an interesting document (perhaps fun for the maturing math mind) over on his page that he has updated over the years, which has a number of dice questions and such.  Pretty good for your Summer math.

https://www.madandmoonly.com/doctormatt/mathematics/dice1.pdf
« Last Edit: May 30, 2020, 04:34:03 AM by Kuroth »
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Lunamancer

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« Reply #121 on: June 13, 2020, 10:46:00 PM »
Quote from: Jamfke;1129371
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!)?


Personally, I don't find the belaboring of differences in dicing mechanisms in RPGs to be all that fruitful. Virtually all analysis suffers from a fatal flaw. Confusing x for f(x). What that means is, yeah, I could use 3d6 rather than d20 and claim it's because of the bell curve. But the fact is, when I'm just trying to figure out if someone hits or not, I really don't care what the chances of rolling exactly a 6, or exactly a 7, or exactly an 8. What matters is the probability that it's a hit. And that's going to end up being a percentage, regardless of whether I'm using d100, d20, 3d6, 9d12 keep the best 3, drawing cards, or playing rock paper scissors.

And so I prefer to just use a percentile system. Skip the pretense and keep it simple.

d20 is a close second and more appropriate in cases where doing math on the dice is common, kind of like D&D's to hit roll. It's not just a question of how good the attacker is and their associated probability of success. That probability varies on a sliding scale according to the opponent's defensive capabilities. It's just easier to add a +5 adjustment than a +25% adjustment.

And then my third choice is anything that just follows directly from what it is that's happening. If something seems 1 in 3 likely, I'm probably just going to roll a d6 and see if it comes up 1 or 2. I'm not going to try to shoehorn it into a core mechanic involving a target number. Nor am I going to round it to 33% just so I can use percentile.

One thing I have heard a player express was, "I want to use all the damn dice I paid for." So I do prefer to mix it up.
That's my two cents anyway. Carry on, crawler.

Tu ne cede malis sed contra audentior ito.

S'mon

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« Reply #122 on: June 14, 2020, 01:21:21 AM »
Quote from: Lunamancer;1134036
One thing I have heard a player express was, "I want to use all the damn dice I paid for."

Yeah, some of my players are iffy on playing D6 System for just this reason!! :-O

RPGPundit

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« Reply #123 on: June 20, 2020, 06:02:41 AM »
Yeah, I don't mind D6 systems, but if you have all the crazy dice, you want to use them. I was originally totally wrong on the subject of DCC, when they announced you'd need non-standard dice to play, thinking that this would kill the popularity of the game. I was an idiot about it; of course the weirdo dice being part of DCC just became an excuse for players to buy them!
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Tom Kalbfus
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« Reply #124 on: June 21, 2020, 10:05:08 PM »
Classic Traveller is just such a game, translating d6 only rolls into percentage probabilities is a pain in the you know what. Also if I have all the polyhedral dice, I tend to want to use them in building encounters tables, I don't like to figure out how to make those tables using only d6s. When I use 1 d6, it is a short encounter table with only 6 encounters, if I use 2d6s, then the probability of an encounter is higher towards the middle of the list and lower towards the ends, then there is d66, which is a square table with 2 resultant d6 rolls, one along the X axis the other along the y axis, which is the equivalent of rolling a 36 sided die or a d36. Having all the polyhedral dice means I can use any number on my encounter tables with variable probabilities so long as the total of probabilities don't exceed 100%, typically I use either percentile Dice of d20s.

GameDaddy
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« Reply #125 on: June 22, 2020, 12:57:16 AM »
Quote from: Lunamancer;1134036
And so I prefer to just use a percentile system. Skip the pretense and keep it simple.

d20 is a close second and more appropriate in cases where doing math on the dice is common, kind of like D&D's to hit roll. It's not just a question of how good the attacker is and their associated probability of success. That probability varies on a sliding scale according to the opponent's defensive capabilities. It's just easier to add a +5 adjustment than a +25% adjustment.

If players want to use all the dice, I just run D&D.

Percentile system is of course Basic Roleplaying / Runequest. The purpose of other dice is to skew the probability curve slightly.  d20 for example, is much more action oriented using a critical hit or critical fail mechanic, Literally a 1 in 20 chance of having a dramatic success or failure (Will usually happen about once a four hour session. D100 not so much crits and critical failures 01 and 100 respectively, happen about once a month if a group is playing weekly sessions. Using dice pools (like 3d6 for example) introduces bell curves, meaning that most things will be median, and the probability critical hits and failures drops dramatically to where you may only see one every six months or so, or even longer with more than three dice in the pool (1 in 216 chance of rolling three 6's for example... 1 in 1296 chance of rolling four sixes).
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Tom Kalbfus
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« Reply #126 on: June 22, 2020, 01:23:13 PM »
Anyone ever use an Excel Spreadsheet for dice?
I have, and when I do, I have no reason to stick with d6s.
For example here is what I write for the various die rolls:
1d4 =RANDBETWEEN(1,4)
1d6 =RANDBETWEEN(1,6)
2d6 =RANDBETWEEN(1,6)+RANDBETWEEN(1,6)
1d8 =RANDBETWEEN(1,8)
1d10 =RANDBETWEEN(1,10)
1d12 =RANDBETWEEN(1,12)
1d20 =RANDBETWEEN(1,20)
d100 =RANDBETWEEN(0,99)

Lunamancer

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« Reply #127 on: June 22, 2020, 11:56:19 PM »
Quote from: GameDaddy;1135576
If players want to use all the dice, I just run D&D.

Percentile system is of course Basic Roleplaying / Runequest. The purpose of other dice is to skew the probability curve slightly.  d20 for example, is much more action oriented using a critical hit or critical fail mechanic, Literally a 1 in 20 chance of having a dramatic success or failure (Will usually happen about once a four hour session. D100 not so much crits and critical failures 01 and 100 respectively, happen about once a month if a group is playing weekly sessions. Using dice pools (like 3d6 for example) introduces bell curves, meaning that most things will be median, and the probability critical hits and failures drops dramatically to where you may only see one every six months or so, or even longer with more than three dice in the pool (1 in 216 chance of rolling three 6's for example... 1 in 1296 chance of rolling four sixes).

This is what I was getting at when I said people who analyze probabilities in RPG mechanics often confuse x for f(x). What you say is true for x.  It's usually not true for f(x). In the examples you name, f(x) is always one of four possible outcomes--hit, miss, crit, and fumble. In each example, the crits and fumbles are the extremes and rarer. The hit/miss results are more moderate and far more likely. So the probability distribution for f(x) is as good a bell curve as one can hope for with only 4 data points. Notice even the mechanics that have linear distributions for x produce bell curve distributions for f(x).

I would also point out that it is not necessarily the case that multiple dice even make for rarer extremes in f(x)--there's no reason to assume, for instance, that only 1 and 100 are the crits and fumbles on the d100. Two of the percentile RPGs I play use one-tenth the base probability for success as the probability for critical (e.g. if your percentile rating is 60,  you crit on 1-6, succeed on 7-60). The other percentile RPG I play generally has it at one-fifth. The 3E/d20 system had kind of a neat way of doing crits. It varied by weapon, but I think most often, if you rolled a 19-20, it was a potential crit, calling for a second successful hit roll to confirm the crit. This essentially made the odds of crit equal to 1/10 the odds of hitting at all. Same as the first two d100 systems I play.

So my position is that it's f(x) that really matters while x is largely irrelevant. And since the probability distribution of x does not necessarily correlate to the probability distribution of f(x) (and in fact does not in the vast majority of RPG mechanics I've seen), I consider probability analysis of game mechanics which invariably focus on x to be distractions and not very insightful.

Also, because I focus on f(x) rather than x, I have increased appreciation for the most standard plain Jane mechanics. Like D&D's roll d20 to hit, then roll funny shaped die for damage. Say a fighter needs a 13 to hit and does a d8 damage. When I look at the distribution of f(x,y) (x being the hit roll variable, y being the damage roll variable), I see there's a 60% chance of doing no damage, 5% chance of doing 1 damage, 5% chance of doing 2 damage, and so on. Much like the 4-data point "bell curve", this also creates a crude curve if you can forgive the limitations of the low resolution data set we're using. And that curve is more like a Pareto distribution--in this example, 83% of all damage the fighter deals will come from the best 25% of the fighter's attacks. Pretty close to 80/20. This satisfies my sense of naturalism far better than chasing bell curves, and strikes me as far more sophisticated as it keeps the game deceptively simple.
That's my two cents anyway. Carry on, crawler.

Tu ne cede malis sed contra audentior ito.

amacris

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« Reply #128 on: June 23, 2020, 12:33:39 AM »
Quote from: jhkim;1129689
I would ask you to think for a bit and picture this in your head. One of the strongest men in the world - an Olympic weight-lifter or the equivalent - has to arm-wrestle a little runt with below-average strength. He loses the first match, and he says "Can we do 2 out of 3? That would be more fair."  He then goes on to win.

That is a result utterly divorced from reality. If I saw it in a movie, I would think it was intentional parody or something. There is simply no way it would happen.

The reason why arm wrestling contests end up in 3-2 victories is because the contestants are very closely matched due to ranking. The same is true of most other tournaments -- whether physical or mental. The way to get tense games is by carefully ranking who is testing against who. The top champion will almost always beat someone in the middle of the rankings. But someone in the middle of the rankings is still really good, and will almost always beat a beginner. That's true in chess, arm-wrestling, and many other contests.


In designing my logarithmic superhero game Ascendant, I looked really closely at real-world variance in outcomes - for instance, one of the things I simulated in the game was bowling by professional bowlers, and another was random chess matches of people with different chess scores. The research for Ascendant says "jhkim is 100% right". The difference between peak performance and sub-par performance is far more profound than a typical D20 game would have us believe.

amacris

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« Reply #129 on: June 23, 2020, 12:38:36 AM »
Quote from: Cloyer Bulse;1131642
Human intelligence comes at a severe cost: the existential crisis. Animals know nothing of the future, they know nothing of themselves, therefore they are inured to the evils of the world. But humans can be crippled by despair and self-doubt. If a people are afflicted by widespread despair, then they are subject to extinction. Religion seems to counteract this fundamental failure of the human mind. ALL human cultures have religion, therefore it seems that religion must perform a critical function.

Empiricism is a purely objective analysis of reality. However, it has a basic flaw: we are subjects, not objects; we live in the bivalent world of experience. Empirical thought can tell us the physical composition of a chair, such as its density and molecular composition, but it cannot tell us whether that chair is comfortable or uncomfortable. The mythological stories of religion tell us about the world we live in and they tell us how to navigate that world without our minds coming apart. They are first and foremost stories about us and our experiences with other humans. The Sky and Earth of Sumerian mythology are not the sky and Earth of science, rather they are archetypal personalities, categories of apprehension that we use to understand the world that we must navigate in order to survive.

Irreligious people are thus more vulnerable to crippling depression and anxiety. Intelligence in itself is not a shield, and in fact renders one more vulnerable. In our modern society, people rely more heavily on psychiatric medication to remain relatively functional sans the practices of magic and religion, but even so, depression and anxiety have continued to increase. The current mass hysteria, a stress test for our civilization and triggered so easily, shows how close we are to the total mental breakdown of our entire society.

One must remember however that superstition, a core element of pagan religion, is a double edged sword. On the one hand it provides resistance to anxiety, but on the other hand failure to perform the required rituals increases anxiety. Monotheism is an advancement, as it uses the conscious will, not superstition, as a shield (as in "the power of positive thinking") -- pay attention to the litanies and chants used in Catholicism, and sit in on a conservative Catholic service. Using the Rosary is just as effective at combating anxiety as any modern medication, if not more so.

Here is an example of how it works: When Trobriand islanders fish in a lagoon, they rely only on their science and technology to perform their task. But when they go out into the open ocean, a potentially chaotic environment which they cannot control and which provokes anxiety, they use magic and religion. This calms their anxieties and allows them to effect their science and technology without the diminishment of its efficacy. Thus, in this context, one can see that the use of magic is actually logical because it shows an understanding of how the human brain functions, not empirical understanding (i.e. not by using intelligence -- none of them have PhD's in psychology), but nevertheless that knowledge is embodied through the practice of their ancient religion (i.e. by using wisdom).

Belief is purely an act of will. Belief can be invoked by the rote practice of ritual, as in paganism, or it can be invoked by a conscious act of will, as in monotheism. Either way, it is a vital tool in the abnegation of the demons that haunt us.


Dude. I would like to subscribe to your newsletter. Seriously. This is good, deep stuff!

amacris

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« Reply #130 on: June 23, 2020, 12:48:56 AM »
Quote from: Libramarian;1129857
I'm not sure on what world the average man can bench press 345lb, but it's not Earth! A typical man can't bench press 185lb without strength training.

Larratt in that video mentions it's his first time bench pressing in many years. A 255lb bench with years of detraining is very impressive. He would have no trouble benching 315+ with focused training. He's an 18 Strength in D&D-land.

Bjornsson's enormous strength is well beyond an 18, but his size is only possible with huge dosages of anabolic steroids, and probably exogenous insulin as well. There no humans in D&D-land as big as him. He's a good model for a 19 Str Ogre, I guess.

Edit: I see where you made the mistake - the 30 x Strength rule applies to lifting things off the ground using your entire body, i.e. a deadlift, not a bench press. Men typically can deadlift twice as much as they bench press. Bjornsson recently deadlifted 1104lb.

Since D&D stats are rolled on a bell curve, I have always interpreted an 18 to represent someone who is in the top 1 out of 216 people. That's about three standard deviations above the mean.
18 Strength is the strongest kid in your high school class.
18 Dexterity is the best soccer player in your high school class.
18 Intelligence is the valedictorian of your high school class.
18 Charisma is the most popular girl in your high school.
18 Wisdom is ... I'm not gonna touch that one

For this reason I tend to think that gamers underestimate how many 18s a heroic character could have. There are plenty of people we all are familiar with who have multiple 18s, and it's not unrealistic at all, especially because in real life high stats tend to correlate with each other.

In any case, people like Bjornsson (STR) or Albert Einstein (INT) or Kim Kardashian (CHA) are more like five standard deviations beyond the mean and can't really be represented with 3-18. AD&D got closer for Strength when it added 18(00) strength. 18(00) strength would be in 21,600, or about IQ 180.

Zalman

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« Reply #131 on: June 23, 2020, 10:33:59 AM »
Quote from: Libramarian;1129857
Bjornsson's ... There no humans in D&D-land as big as him.

There aren't really any other humans in Real Life Land as big as him either. :D
Old School? Back in my day we just called it "School."

jhkim

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« Reply #132 on: June 23, 2020, 09:17:12 PM »
Quote from: jhkim
I would ask you to think for a bit and picture this in your head. One of the strongest men in the world - an Olympic weight-lifter or the equivalent - has to arm-wrestle a little runt with below-average strength. He loses the first match, and he says "Can we do 2 out of 3? That would be more fair." He then goes on to win.

That is a result utterly divorced from reality. If I saw it in a movie, I would think it was intentional parody or something. There is simply no way it would happen.

The reason why arm wrestling contests end up in 3-2 victories is because the contestants are very closely matched due to ranking. The same is true of most other tournaments -- whether physical or mental. The way to get tense games is by carefully ranking who is testing against who. The top champion will almost always beat someone in the middle of the rankings. But someone in the middle of the rankings is still really good, and will almost always beat a beginner. That's true in chess, arm-wrestling, and many other contests.
Quote from: amacris;1135770
In designing my logarithmic superhero game Ascendant, I looked really closely at real-world variance in outcomes - for instance, one of the things I simulated in the game was bowling by professional bowlers, and another was random chess matches of people with different chess scores. The research for Ascendant says "jhkim is 100% right". The difference between peak performance and sub-par performance is far more profound than a typical D20 game would have us believe.
Thanks amacris. That's what just Kickstartered in March, right?

https://www.kickstarter.com/projects/autarch/ascendant/description

From a glance at the resolution chart, it looks like the probabilities shift a lot -- so +7 vs +0 is an automatic win. But on the other hand, it's a very compressed scale where each step is x2. Any comments on how that revelation affected how you designed your chart?