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Author Topic: Probability Theory and You  (Read 6111 times)

VisionStorm

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« Reply #90 on: May 15, 2020, 10:20:07 am »
Quote from: Cloyer Bulse;1130040
The distinction between intelligence and wisdom is very real and appropriately represented in AD&D.


You basically snipped my actual points to address the opening statement you took issue with without refuting a single actual argument. Then you made a bunch of unsupported claims about religion making us wise and worked backwards from a set of pseudo-religions D&D assumptions to basically "jump through hoops trying justify the existence of both as separate stats by going into nuance and falling back on their "books smarts" vs "common sense" distinctions". Just like I mentioned in my post that people normally did during this type of discussion.

Steven Mitchell

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« Reply #91 on: May 15, 2020, 11:07:20 am »
Vision Storm, you seem to be confusing "Doesn't do what I think it should do" with "Doesn't do what the designers intended or said they would do".  It makes your arguments difficult to follow.

VisionStorm

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« Reply #92 on: May 15, 2020, 12:48:59 pm »
Quote from: Steven Mitchell;1130055
Vision Storm, you seem to be confusing "Doesn't do what I think it should do" with "Doesn't do what the designers intended or said they would do".  It makes your arguments difficult to follow.


I would need specific examples to understand exactly what you mean. Though I do tend to speculate on the designers intent in some instances, or to look at or judge rules beyond just a specific system's implementation, but more in term of possibilities or whether or not a rule actually serves a practical purpose in practice (beyond theory or the designers stated intent).

Though, I am aware that what I consider ideal or optimal and what the designers might have wanted might be different. But there are cases where what the designers might intend for a rule or ability doesn't really pan out in actual play or most situations, which is part of some of the criticisms I sometimes have.

jhkim

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« Reply #93 on: May 15, 2020, 04:29:37 pm »
Quote from: jhkim
In terms of system design, you can get a less swingy system by any of (1) reducing the size of the die like using 1d10 instead of 1d20, (2) replacing the die with more smaller dice for a tighter bell curve, like 3d6 instead of 1d20; or (3) increasing the range of the stats. It's not like the only choices are 65% or 99%.
Quote from: VisionStorm;1130049
But this type of issue only happens when making plain ability score contests without Proficiencies. There's no need to adjust the entire task resolution mechanic to account just for this type of specialized scenario. I'm also not sure that Conan needs a 99% success rate in a Strength contest in the middle of combat, where lots of variables are in play. Specially when we consider that chimps are technically stronger than humans in real life, which means that just because a creature is small that doesn't mean that they're automatically weak.
It's a matter of taste whether one likes a more swingy system or not, but for me, it's vastly *more* of an issue with skills than it is with raw attributes.

To my intuition, skill should make a huge difference in terms of chance of success. I mentioned earlier that there should be a huge difference between a high school dropout, an undergraduate math major, and a top PhD mathematician in terms of solving a math problem. An undergrad can solve math problems where the dropout can't even understand the question. For another angle, consider driving -- what are the chances for someone new to driving, an average licensed person, and a racecar driver? Someone with a license should have a 99.99% chance to drive on the highway to work in the morning without incident, but that should be quite difficult for someone new to driving (under 50%, say).

To take a D&D5 example, consider a desert nomad (barbarian with outlander background) and a pirate (rogue with sailor background). They are each trying to sail a ship to the island as quickly as possible. The rogue has proficiency bonus with vehicles (water) - but that only makes a difference of +10%. Their chances aren't very different.

Pat

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« Reply #94 on: May 15, 2020, 08:12:19 pm »
Quote from: jhkim;1130088

To my intuition, skill should make a huge difference in terms of chance of success. I mentioned earlier that there should be a huge difference between a high school dropout, an undergraduate math major, and a top PhD mathematician in terms of solving a math problem. An undergrad can solve math problems where the dropout can't even understand the question. For another angle, consider driving -- what are the chances for someone new to driving, an average licensed person, and a racecar driver? Someone with a license should have a 99.99% chance to drive on the highway to work in the morning without incident, but that should be quite difficult for someone new to driving (under 50%, say).

I don't agree. PhD mathematicians aren't always the best at basic math, whereas an undergrad math major is probably someone who both has recently used those skills, and has a knack for it. Also, academic disciplines are highly specialized, and people are only truly experts in a narrow area. The PhD might be great at things in their specialty, mediocre at some of the basics, and left floundering when it comes to something outside their expertise. Conversely, the math major might be better at routine math, but have no relevant background when it comes to more advanced topics.

In a lot of ways, skill is more about what you can do, than increasing your odds. Similarly, driving to work everyday is less about technical skill than it is about safe habits. Experience helps, because it increases situational awareness. But being exceptionally skilled at high speed chases might actually increase your chance of accident, because someone with that training is attuned to taking risks. True, that skill might help with diagnosing and compensating for problems with the vehicle, or assessing dangers like the potential for black ice, or just not panicking. But it's primarily about being exceptionally capable in extreme circumstances, not being better at routine things.

mightybrain

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« Reply #95 on: May 15, 2020, 08:28:29 pm »
Quote from: jhkim;1130088
To take a D&D5 example, consider a desert nomad (barbarian with outlander background) and a pirate (rogue with sailor background). They are each trying to sail a ship to the island as quickly as possible. The rogue has proficiency bonus with vehicles (water) - but that only makes a difference of +10%. Their chances aren't very different.

As far as I can tell, a ship can't move at all in 5e unless you have the minimum number of skilled crew. The fact that skilled crew are specified separately to unskilled crew implies that anyone without water vehicle proficiency can't act as crew. And without enough crew the ship can't move at all.

A better example might be driving a wagon or chariot.

Note, when they published Xanathar's guide they addressed a problem with tool proficiencies in general. For example, if you were driving a chariot and the DM called for a check, it would almost certainly be an animal handling roll, and if you already had proficiency with that, your vehicle proficiency would effectively add nothing. To patch this up they suggest that if a skill and tool proficiency both apply then you can roll with advantage.

mightybrain

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« Reply #96 on: May 15, 2020, 09:07:18 pm »
Quote from: jhkim;1130088
To my intuition, skill should make a huge difference in terms of chance of success. I mentioned earlier that there should be a huge difference between a high school dropout, an undergraduate math major, and a top PhD mathematician in terms of solving a math problem. An undergrad can solve math problems where the dropout can't even understand the question.

This could well be the case if the problem had a DC so high that the unskilled player could not reach it even with a 20, but the high level character with expertise could (expertise can go up to +12 at high level).

VisionStorm

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« Reply #97 on: May 15, 2020, 09:09:44 pm »
Quote from: jhkim;1130088
It's a matter of taste whether one likes a more swingy system or not, but for me, it's vastly *more* of an issue with skills than it is with raw attributes.

To my intuition, skill should make a huge difference in terms of chance of success. I mentioned earlier that there should be a huge difference between a high school dropout, an undergraduate math major, and a top PhD mathematician in terms of solving a math problem. An undergrad can solve math problems where the dropout can't even understand the question. For another angle, consider driving -- what are the chances for someone new to driving, an average licensed person, and a racecar driver? Someone with a license should have a 99.99% chance to drive on the highway to work in the morning without incident, but that should be quite difficult for someone new to driving (under 50%, say).

To take a D&D5 example, consider a desert nomad (barbarian with outlander background) and a pirate (rogue with sailor background). They are each trying to sail a ship to the island as quickly as possible. The rogue has proficiency bonus with vehicles (water) - but that only makes a difference of +10%. Their chances aren't very different.


Personally, I'm willing to accept a certain degree of swinginess because it's a game and that allows more room to maneuver without characters with high ability completely overwhelming lower ability characters, particularly in combat, which I believe was the thinking behind the range of modifiers allowed in 5e. Though, I suppose maybe 5e takes it a little too far, so I may have to end up reassessing my own system's ability range at some point, since it uses a similar range to 5e.

But some of these are outlier scenarios. The academic stuff in particular is hard to appropriately handle in terms of the game rules, since a lot of times it deals with expert knowledge (sometimes layers of it) that you either have or you don't. Or it may deal with research that could take years to complete, on top of requiring very specific expert knowledge, which might entail a lot of very low probability ability checks leading up to the big discoveries in terms of the game rules, depending on how you handle them mechanically. I would handle specific knowledge as a separate requirement you would need to purchase as a separate ability (similar to a Feat) in addition to using a more general skill level for rolls. If you don't have the specific knowledge for a task (such as speaking a specific language) you may simply be unable to attempt it, or may suffer extreme penalties if allowed.

I'm also not sure about the driving stuff because driving is a very low failure rate type of task, unless you're performing stunts or driving at excessively high speeds. I never even crashed till like a month after getting my license, when my car slipped making a turn at an intersection and did a 180 on me, and I didn't know you were supposed to (cautiously) accelerate when a car starts to spin to attempt to regain traction on the tires to stop it from spinning (which is counterintuitive). If driving untrained had a 50% or less success rate I should've crashed a lot more.

GeekyBugle

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« Reply #98 on: May 16, 2020, 03:11:33 am »
Quote from: jhkim;1129985
mightybrain is correct that linear vs bell-curve doesn't directly address this. The issue is *die roll variance* vs *stat range*. 3d6 has less variance than 1d20, but so does 1d10.

1d20 has standard deviation of 5.77
3d6 has standard deviation of 2.96
1d10 has standard deviation of 2.87
1d6 has standard deviation of 1.71

It's a question of the standard deviation of the roll versus the stat difference between master and weakling (i.e. Strength mod between Conan and a kobold, for example).

In terms of system design, you can get a less swingy system by any of (1) reducing the size of the die like using 1d10 instead of 1d20, (2) replacing the die with more smaller dice for a tighter bell curve, like 3d6 instead of 1d20; or (3) increasing the range of the stats. It's not like the only choices are 65% or 99%.


1 Die doesn't have a bell curve, each time you roll you have equal chance to get any number on the die.

With 3d6 most of the rolls will end in the middle, the more to the extremes you go the less probable the result is.

It's a matter of how many ways you have to get number X

In 1d20 you have 1/20 always.

In 3d6 it depends on the number 3 & 18 you have 1 way to get so 1/18

4 & 17 you have 2 different ways so 2/18 for any of those two numbers.

And so on.

Until you reach the middle which has the most combinations possible, meaning it will get rolled more often than any other number.

This means the system has less swing and is more stable.

A difference of +-1 in modifiers has a huge weight in comparing 1d20 to 3d6.

And I will not continue arguing about rolls because I have read all of you do it before and you just don't get it, some of you even argue that a single die has a bell curve.
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nDervish

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« Reply #99 on: May 16, 2020, 09:09:00 am »
Quote from: nDervish;1130037
As jhkim pointed out, the modifier ranges you used are about a third less than the standard deviations of each roll (+/-2 vs. SD 2.96 for 3d6, +/-4 vs. SD 5.77 for 1d20), so it's no surprise that the results would be tilted in your favor, but still pretty random.  If you want the results to go 99% "the right way", then you need to use larger modifiers, so that the results will be shaped primarily by the modifiers, rather than by the randomness of the roll.

If you use 3d6 + 20 against random opponents rolling 3d6 with modifiers from -2 to +2 you'll win about 100% of contests
If you use d20 + 20 against random opponents rolling d20 with modifiers from -4 to +4 you'll win about 100% of contests

Quote from: mightybrain;1130038
Ah, the old 50 strength ploy. Even the mighty Tarrasque only has a +10!


Ehm, no, I'm not talking about "50 strength".  I'm not talking about D&D (or any particular system) at all.  I'm talking about the raw numbers and how they compare:  If you want to ensure that a more-capable opponent is (nearly-)guaranteed to defeat a less-capable opponent, then you use modifiers on the roll which are large enough that the modifier will be more significant than the likely variance in the raw dice results.  How you get the modifiers which are large enough to overcome the variance in the raw die result is a question of how the dice mechanic is designed, which was supposed to be the original point of this thread, before it got bogged down in D&D Strength modifiers.

But, as I mentioned in my first reply in this thread, there are other (not-D&D) systems out there which do use modifiers large enough to make the raw die result all but irrelevant.  For example, there's Ars Magica, which can see you rolling 1d10+20 against someone rolling 1d10+5 - unless you're in a "stress" situation (in which case those are exploding 1d10 rolls), there's no need to even roll at all because it's not possible for the lower-skilled person to roll high enough to beat the higher-skilled.  Whether you consider that a bug or a feature is a matter of personal taste (I've gone back and forth on it myself), but it is an option for how to design your dice mechanic, even if it's not the option that D&D5e chose to take.

mightybrain

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« Reply #100 on: May 16, 2020, 10:02:06 am »
Quote from: GeekyBugle;1130139
In 3d6 it depends on the number 3 & 18 you have 1 way to get so [strike]1/18[/strike] 1/216

4 & 17 you have [strike]2[/strike] 3 different ways so [strike]2/18[/strike] 3/216 for any of those two numbers.

And so on.


Fixed that for you.

GeekyBugle

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« Reply #101 on: May 16, 2020, 10:32:55 am »
“During times of universal deceit, telling the truth becomes a revolutionary act.”

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mightybrain

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« Reply #102 on: May 16, 2020, 03:07:01 pm »
Here's a chart of a simulation of contests between characters with different modifiers:

[table=class: grid]
  [tr]
    [td]mod[/td]
    [td]-4(-2)[/td]
    [td]-2(-1)[/td]
    [td]0(0)[/td]
    [td]+2(+1)[/td]
    [td]+4(+2)[/td]
  [/tr]
  [tr]
    [td]-4(-2)[/td]
    [td]47.5% (45.5%)[/td]
    [td]57.0% (54.7%)[/td]
    [td]65.9% (63.6%)[/td]
    [td]73.8% (72.3%)[/td]
    [td]80.5% (79.1%)[/td]
  [/tr]
  [tr]
    [td]-2(-1)[/td]
    [td]38.4% (36.5%)[/td]
    [td]47.5% (45.6%)[/td]
    [td]57.0% (54.9%)[/td]
    [td]65.9% (63.9%)[/td]
    [td]73.8% (72.0%)[/td]
  [/tr]
  [tr]
    [td]0(0)[/td]
    [td]29.9% (27.9%)[/td]
    [td]38.2% (36.3%)[/td]
    [td]47.3% (45.2%)[/td]
    [td]57.3% (54.5%)[/td]
    [td]66.0% (63.8%)[/td]
  [/tr]
  [tr]
    [td]+2(+1)[/td]
    [td]22.6% (20.6%)[/td]
    [td]29.7% (27.6%)[/td]
    [td]38.2% (36.1%)[/td]
    [td]47.6% (45.1%)[/td]
    [td]57.6% (54.3%)[/td]
  [/tr]
  [tr]
    [td]+4(+2)[/td]
    [td]16.5% (14.4%)[/td]
    [td]22.8% (20.5%)[/td]
    [td]29.9% (27.9%)[/td]
    [td]38.2% (36.4%)[/td]
    [td]47.5% (45.1%)[/td]
  [/tr]
[/table]

The value in each cell is the percentage of times that the character with the modifier in the column beat the character with the modifier in the row using the d20 + modifier method. The values in brackets represent the same contest using 3d6 + (modifier). Each test was run 100,000 times.

You can see that the figures are fairly close but with +2% to +3% for the d20 method. This is explained by the fact that the 3d6 method results in more draws. Here is the chart for the percentage of draws.

[table=class: grid]
  [tr]
    [td]mod[/td]
    [td]-4(-2)[/td]
    [td]-2(-1)[/td]
    [td]0(0)[/td]
    [td]+2(+1)[/td]
    [td]+4(+2)[/td]
  [/tr]
  [tr]
    [td]-4(-2)[/td]
    [td]5.0% (9.2%)[/td]
    [td]4.5% (9.3%)[/td]
    [td]4.0% (8.3%)[/td]
    [td]3.5% (7.3%)[/td]
    [td]3.1% (6.1%)[/td]
  [/tr]
  [tr]
    [td]-2(-1)[/td]
    [td]4.4% (9.0%)[/td]
    [td]5.0% (9.3%)[/td]
    [td]4.5% (9.1%)[/td]
    [td]4.0% (8.3%)[/td]
    [td]3.6% (7.4%)[/td]
  [/tr]
  [tr]
    [td]0(0)[/td]
    [td]4.1% (8.4%)[/td]
    [td]4.5% (9.1%)[/td]
    [td]5.0% (9.3%)[/td]
    [td]4.6% (8.9%)[/td]
    [td]4.0% (8.4%)[/td]
  [/tr]
  [tr]
    [td]+2(+1)[/td]
    [td]3.6% (7.4%)[/td]
    [td]4.0% (8.3%)[/td]
    [td]4.6% (9.1%)[/td]
    [td]5.1% (9.3%)[/td]
    [td]4.5% (8.9%)[/td]
  [/tr]
  [tr]
    [td]+4(+2)[/td]
    [td]3.0% (6.3%)[/td]
    [td]3.4% (7.4%)[/td]
    [td]4.0% (8.3%)[/td]
    [td]4.5% (9.1%)[/td]
    [td]5.0% (9.3%)[/td]
  [/tr]
[/table]

If you adjust for the draws, the probabilities for d20+mod and 3d6+1/2mod are pretty much the same. (At least within the range of values we're interested in.)

jhkim

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« Reply #103 on: May 16, 2020, 07:17:32 pm »
Quote from: jhkim
The issue is *die roll variance* vs *stat range*. 3d6 has less variance than 1d20, but so does 1d10.

1d20 has standard deviation of 5.77
3d6 has standard deviation of 2.96
1d10 has standard deviation of 2.87
1d6 has standard deviation of 1.71

It's a question of the standard deviation of the roll versus the stat difference between master and weakling (i.e. Strength mod between Conan and a kobold, for example).

In terms of system design, you can get a less swingy system by any of (1) reducing the size of the die like using 1d10 instead of 1d20, (2) replacing the die with more smaller dice for a tighter bell curve, like 3d6 instead of 1d20; or (3) increasing the range of the stats. It's not like the only choices are 65% or 99%.
Quote from: GeekyBugle;1130139
This means the system has less swing and is more stable.

A difference of +-1 in modifiers has a huge weight in comparing 1d20 to 3d6.

And I will not continue arguing about rolls because I have read all of you do it before and you just don't get it, some of you even argue that a single die has a bell curve.
I agree with you that 3d6 has a bell curve while 1d20 has a flat distribution. But the point is that 1d20 and 3d6 aren't the only choices. 1d10 also has less variance than 1d20. So let's look at real systems.

In D&D5, let's consider a Contest of Strength between a kobold (Strength mod -1) and Conan (Strength mod +5). So the kobold rolls 1d20 and subtracts 1, compared to Conan's 1d20 plus 5. There is a 73.75% chance that Conan will win, a 3.5% chance of a tie, and a 22.75% chance that the kobold will win. (Note that there was a previous claim that the kobold only had a 9% chance of winning, but I don't think that's right.)

Let's look at Interlock instead, which is the system for the Cyberpunk RPG (from R Talsorian). Here Strength goes from 2 to 10, with average 5. In a contest, each character rolls their stat plus 1d10. So the equivalent to a kobold would be Strength 4, while Conan is Strength 10. So here, there is a 90% chance that Conan will win, a 4% chance of a tie, and a 6% chance that the kobold will win.

That's a system with much less swing in it, despite not having a bell curve.

A bell curve 3d6 also has less swing than 1d20. Let's take GURPS. It's a little unclear what Conan's strength would be in GURPS -- there isn't a strict maximum and disagreement about what is a standard human max, and I don't have the GURPS Conan books. This thread rated him as ST 17, though. If we take GURPS as Conan with 17 ST compared to ST 9, then a quick contest of Strength will result that Conan has an 94.6% chance to win, a 1.56% chance of tie, and the kobold has an 3.8% chance to win.

Quote from: mightybrain;1130202
If you adjust for the draws, the probabilities for d20+mod and 3d6+1/2mod are pretty much the same. (At least within the range of values we're interested in.)
I haven't checked all of them, but those numbers seem reasonable. I mentioned that the standard deviation of 3d6 is about half that of 1d20. I don't know where you're getting those mods from, though. I think it's more concrete to use real RPG systems.

GeekyBugle

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« Reply #104 on: May 16, 2020, 10:41:14 pm »
Quote from: jhkim;1130241
I agree with you that 3d6 has a bell curve while 1d20 has a flat distribution. But the point is that 1d20 and 3d6 aren't the only choices. 1d10 also has less variance than 1d20. So let's look at real systems.

In D&D5, let's consider a Contest of Strength between a kobold (Strength mod -1) and Conan (Strength mod +5). So the kobold rolls 1d20 and subtracts 1, compared to Conan's 1d20 plus 5. There is a 73.75% chance that Conan will win, a 3.5% chance of a tie, and a 22.75% chance that the kobold will win. (Note that there was a previous claim that the kobold only had a 9% chance of winning, but I don't think that's right.)

Let's look at Interlock instead, which is the system for the Cyberpunk RPG (from R Talsorian). Here Strength goes from 2 to 10, with average 5. In a contest, each character rolls their stat plus 1d10. So the equivalent to a kobold would be Strength 4, while Conan is Strength 10. So here, there is a 90% chance that Conan will win, a 4% chance of a tie, and a 6% chance that the kobold will win.

That's a system with much less swing in it, despite not having a bell curve.

A bell curve 3d6 also has less swing than 1d20. Let's take GURPS. It's a little unclear what Conan's strength would be in GURPS -- there isn't a strict maximum and disagreement about what is a standard human max, and I don't have the GURPS Conan books. This thread rated him as ST 17, though. If we take GURPS as Conan with 17 ST compared to ST 9, then a quick contest of Strength will result that Conan has an 94.6% chance to win, a 1.56% chance of tie, and the kobold has an 3.8% chance to win.


I haven't checked all of them, but those numbers seem reasonable. I mentioned that the standard deviation of 3d6 is about half that of 1d20. I don't know where you're getting those mods from, though. I think it's more concrete to use real RPG systems.

Now, take Interlock and switch 1d10 for 2d6.

And in a contest between Conan and a Kobold (how did the both of them end in the same world?) the result is automatic, Conan wins.
“During times of universal deceit, telling the truth becomes a revolutionary act.”

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https://www.youtube.com/channel/UCjC7-w5KDKNiD-k0tVo1DPw?view_as=subscriber