A very simple dice question... I think...

[bobert]

I can build a galaxy in my head with a working history across several planets... but I cannot find a formula that explains whether or not rolling one 12 sided die yields the same odds of rolling a value of 12 as rolling two 6 sided dice. I would assume the odds are slightly worse on rolling two 6 sided dice as in actuality that would be rolling one 6 sided die followed by another 6 sided die. Is this correct? I hope this hasn't come across as convoluded. My grammar is not the best, but at least I try... which is more than can be said for most now'days .
Thank you for the responses.

[M. Burrell]

I believe you're correct when you say that your chances are slightly less for rolling '12' on 2d6. If I recall correctly, there is a bell-curve of probability that makes rolling middling results more likely on 2d6 and more 'extreme' (1,2,11,12) less so.
12 on a 1d12 is, however, genuinely more random as there is always a one-in-twelve chance.

[Vordark]

This is the frequency distribution of 2d6:

2 = 2.8%
3 = 5.6%
4 = 8.3%
5 = 11.1%
6 = 13.9%
7 = 16.7%
8 = 13.9%
9 = 11.1%
10 = 8.3%
11 = 5.6%
12 = 2.8%

On a 1d12, the odds of rolling each result is roughly 8.3%.

Hope this helps.

[MacLeod]

While we are on the subject... does someone have the probabilities for 2D20...? =D I've been fiddling around with the idea of using a bell curve (something I don't normally do) with a reeaally wide curve.

[Wordman]

To see why the two are different, it is usually easier to "see" if you actually build the grid of results for the 2d6 like so:

Code Select Expand

             die 1
           1  2  3  4  5  6
         +-----------------
       1 | 2  3  4  5  6  7
       2 | 3  4  5  6  7  8
die 2  3 | 4  5  6  7  8  9
       4 | 5  6  7  8  9 10
       5 | 6  7  8  9 10 11
       6 | 7  8  9 10 11 12   


This shows all 36 possible results for a 2d6. You can easily see that the dice combine to make a result of 7 much more often (six of the possible results, or 6/36 = 16.7%) than they do to make a result of 12 (one result, or 1/36 = 2.8%).

In contrast, a similar chart for a d12 looks sort of ridiculous, because there is only one die involved:

Code Select Expand

                    die 1
           1  2  3  4  5  6  7  8  9 10 11 12
         +-----------------------------------
           1  2  3  4  5  6  7  8  9 10 11 12


Each result on a d12 is equally likely. That is there are only 12 possible results, and each on only appears once (or 1/12 = 8.3%).

You can figure out the 2d20 similarly to the 2d20 grid, though there are more efficient ways. Or, you can just use the probability calculator that has been posted about quite a bit lately.

[Jasper Flick]

Wordman beat me to it, with an excellent explanation!

Thanks man, now I don't need to toot my own horn.

[MacLeod]

Sweet Jesus. =) I've never bothered looking at the dice/math threads because I like linear for the most part... but I am thinking about checking out some new avenues. Thanks for the explanation and link.

[Vordark]

Thanks man, now I don't need to toot my own horn.

I think I have to pay more attention to people's sigs.  I've been answering a lot of these questions lately (here and elsewhere), but with this tool it's more of a "teach them to fish" sort of thing.  Thanks!

[Wordman]

Thanks man, now I don't need to toot my own horn.

You can count on me to praise your calculator both here and elsewhere. It is an excellent tool. As are the docs that go with it.

I've been fiddling around with the idea of using a bell curve (something I don't normally do) with a reeaally wide curve.

You can do a lot with dice of the form xdy. They give symmetric distributions, and can be tinkered with to find the behavior you want. Generally, as you increase x, rolls will get more and more likely to hit an average result. That is, the curve gets "pointier", with a lot of results in the middle and very few on the edges. Increasing y is a bit harder to explain the impact of. Basically, it gives the system more a "finer grain", with more possible "steps" between the results. This usually doesn't actually matter than much by itself. When it does matter is when in mixes with other rules, particularly things like bonuses. For example, consider two systems where one rolls a 4d4 and the other rolls a 4d20. Now imagine that you award a +1 to the roll. That +1 obviously has much more of an effect on the outcome of the 4d4 roll than the 4d20 roll, because the 4d20 results have a finer "resolution". Another way to look at increasing y is to think of two people making a roll and comparing the result; as you increase y, the chance of a tie goes down.

You might compare, for example, a 5d6 vs. a 3d10. These have similar ranges (5-30 vs. 3-30), slightly different averages, and much different probabilities.

[MacLeod]

Thanks for the expanded explanation. =)
I'm exploring other dice methods because as of right now, I am trying to reduce the impact of someone who has a terrible die roll bonus (+6) versus someone with a great die roll bonus (+16). Such extremes aren't precisely normal but I want to make sure the game stays playable after bonuses/penalties come into play.

Here... maybe you can help me with this problem even further? I've no head for probabilities and what-not. x_x
Basic Dice Mechanic
1D20 is rolled, attacker adds his attack stat to the result, defender subtracts his defense stat from that. The final value is compared against a never changing table (like Talislanta) with the following breakdown; 1 ~ 4=Critical Fault 5 ~ 8=Miss 9 ~ 16=Hit 17~=Critical Hit.
As you can see, its an attacker's game. However, I don't want stats to be the end all be all of the game... I want them to matter but I still want some tension to exist. As I said before, bonuses range from +6 (rarely occurs), to +10 (common occurrence) all the way up to +16 (also rare). So I'm thinking that maybe 2D20 will work. Rewards high values but also gives plenty of opportunity for experts to be beaten. At least, that is what I am hoping.

Whaddya think? And, thanks in advance for any kernels of wisdom. =D

[dindenver]

Matt,
  The trick is scaling the bonuses to the dice used. let's say your game has an old school mechanic. And the following feeds into each roll:
1) Dice
2) Skill
3) Stat
4) Modifiers

So, if the ranges are as follows:
1) Dice 1d20
2) Skill +0 to +10
3) Stat -2 to +5
4) Modifiers: -2 to +2

What you are saying is, Luck is the biggest factor in any task. Skill counts for 40% more than natural ability and Natural abilit is almost 100% more important than positioning and environmental factors.

So, in your example, if you want a +6 attacker to have a shot with a +16 defender a d20 should work fine, assuming that is all your bonuses added up already.

  I hope that helps.

[MacLeod]

I think the problem may be that the game has maneuvers that differ from character to character... some have defense bonuses and others have attack bonuses. They are usually simply +2. This escalates the bonuses, so it can be a +6 vs. +18 (in a very rare, rare circumstance). Not so good.
I haven't actually had any playtesting so perhaps I should make a go at that before I think myself to death. =)

[Vulpinoid]

Thanks for the expanded explanation. =)
I'm exploring other dice methods because as of right now, I am trying to reduce the impact of someone who has a terrible die roll bonus (+6) versus someone with a great die roll bonus (+16). Such extremes aren't precisely normal but I want to make sure the game stays playable after bonuses/penalties come into play.

Why are you doing this?

How do you think this will make the game more playable, rather than just less risky?

It might be good for the story to have one character who simply can't be overcome by someone else. This might open the opportunity for a good "Teamwork" game mechanism.

Consider what the game is about before really tweaking the mechanisms.

You might be tainting a good thing...

Just some ideas...

V

[MacLeod]

I agree, for sure. I haven't even play tested it yet, I'm just getting wrapped up in thinking about stuff. I built the game so that all characters will have one or more weaknesses, several average stats and then one, MAYBE TWO, strengths. The game is 100% focused on gameplay and balance so I'm obsessing... I'll play test the thing before I go about making any major dice mechanic changes. =)