3d6 Dice Roll Probabilities

[MagicPegasus]

Can someone post a more acurate 3d6 dice roll probability chart? The one below doesn't seem to be completely accurate. Also, if you post it, can you do it to at least 0.1 % accuracy?

 

Example:

 

3d6 Roll Percentage

3- 0.5

4- 3

5- 7

6- 12

7- 19

8- 28

9- 39

10- 50

11- 60

12- 70

13- 80

14- 87

15- 92

16- 96

17- 98

18- 100

 

There is a 1 in 216 chance of rolling a 3 on 3d6. That's 0.46 %. And I need to see the percentages to 0.1 % or 0.01 %. I'd like to see how these numbers are being rounded.)

[BNakagawa]

Re: 3d6 Dice Roll Probabilities

 

3 = .463% (1/216)

4 = 1.39% (3/216)

5 = 2.78% (6/216)

6 = 4.63% (10/216)

7 = 6.94% (15/216)

8 = 9.72% (21/216)

9 = 11.57% (25/216)

10 = 12.5% (27/216)

 

The other half are the same as the first, order reversed.

[Lord Liaden]

Re: 3d6 Dice Roll Probabilities

 

Should anyone be interested, this section of the "Free Stuff" part of the website has a spreadsheet which will calculate the percentage chance of rolling any number on any quantity of d6 up to 30, to two decimal places.

[Cmonkey]

Re: 3d6 Dice Roll Probabilities

 

Try this site.

[Killer Shrike]

Re: 3d6 Dice Roll ProbabilitiesFrom http://www.darkshire.net/~jhkim/rpg/systemdesign/dice-methods.html :

	1d20		3d6
	Pct	Odds	Pct	Odds
1	5%	19.0 : 1	-	-
2	10%	9 : 1	-	-
3	15%	5.7 : 1	0.5%	215 : 1
4	20%	4 : 1	1.9%	53 : 1
5	25%	3 : 1	4.6%	20.6 : 1
6	30%	2.3 : 1	9.3%	9.8 : 1
7	35%	1.8 : 1	16.2%	5.2 : 1
8	40%	1.5 : 1	25.9%	2.9 : 1
9	45%	1.2 : 1	37.5%	1.7 : 1
10	50%	1 : 1	50.0%	1 : 1
11	55%	1 : 1.2	62.5%	1 : 1.7
12	60%	1 : 1.5	74.1%	1 : 2.9
13	65%	1 : 1.8	83.8%	1 : 5.2
14	70%	1 : 2.3	90.7%	1 : 9.8
15	75%	1 : 3	95.4%	1 : 20.6
16	80%	1 : 4	98.1%	1 : 53
17	85%	1 : 5.7	99.5%	1 : 215
18	90%	1 : 9	100%	1 : inf
19	95%	1 : 19	-	-
20	100%	1 : inf	-	-

         Although it has the same average and a similar range, the 3d6 roll has a lower variance than the 1d20 roll. This can be expressed in the RMS (or Root-Mean-Square). The 3d6 roll has RMS=3.0, while 1d20 has RMS=5.8, nearly double.

        The important thing to note is the behavior. The 1d20 roll always has a difference of 5% for each +-1. However, the difference in odds changes dramatically. Around the center odds change by x1.2 or so, but at the edge they go to x2. In contrast, the 3d6 roll tends to roughly double the odds of success or failure with each +-1. The percentage change varies from large at the center (12%) to less than 1% at the edge.

        What does this mean for RPG play? Well, a linear roll is going to have a lot more rolls at fair odds. Half of the 1d20 rolls are between 3:1 and 1:3 odds. By comparison, a bell-curve roll will tend to have more "almost-sure-things" and "long-shot" rolls.

[MagicPegasus]

Re: 3d6 Dice Roll Probabilities

 

Thanks very much for everone who responded. I got my chart and recalculated the Hit Location Table with it. It turns out that the average STUN Multiplier from it is 2.86. So that table definately makes killing attacks more powerfull. And that's not even including called shots.

[Gary]

Re: 3d6 Dice Roll Probabilities

 

Thanks very much for everone who responded. I got my chart and recalculated the Hit Location Table with it. It turns out that the average STUN Multiplier from it is 2.86. So that table definately makes killing attacks more powerfull. And that's not even including called shots.

 

 

Hit locations are not more powerful in general than the normal stun multiple. It all depends on the defense of the target you're hitting. 25 or so defense is approximately the break even point for a 4d6 RKA. Above 25, the stun multiple is better, and below the HLC is better, although any defense level between 23-27 is essentially a wash.

 

The raw average of the HLC chart is higher, but the stun multiple has a much greater chance of getting a high stun multiple which will skew the net stun results.

[Trebuchet]

Re: 3d6 Dice Roll Probabilities

 

Chart showing percentage probabilities of rolling a given number or less on 3d6

 

Roll/Success %

3- 0.5 %

4- 1.9

5- 4.6

6- 9.3

7- 16.2

8- 25.9

9- 37.5

10- 50.0

11- 62.5

12- 74.1

13- 83.8

14- 90.7

15- 95.4

16- 98.1

17- 99.5

18- 100.0

[Foxx!]

Re: 3d6 Dice Roll Probabilities

 

Heroes!

 

Is this chart in one of the 5th edition books? I think I saw it as a side bar somewhere, but I cannot find it now.

 

Cheers

[prestidigitator]

Re: 3d6 Dice Roll Probabilities

 

From http://www.darkshire.net/~jhkim/rpg/systemdesign/dice-methods.html

...

That was a cool site, Killer Shrike. From it, I found this link to UeberDice, a neat little Java application that does dice rolling nicely, although it has some annoying bugs for its more complex features and is not very customizable. The cool part about this application is that you can do recursion and (in a way) assign values to a variable. For example, here is how I discovered I could roll normal damage dice:

switch(, val != val, 0,
  {switch(val-1, <0, 0,
     recurse+switch(1d6, ==1, [1, 0], ==6, [6, 2], [val, 1]))}
)

If you want to try this, you should:

1. Go into the, "Options->Options..." dialog and set the, "Maximum recursion depth," to the maximum dice in any roll (I just set it to 20), and
   
2. Replace in my expression above with the actual number of damage dice to roll.

The value it outputs will be in the form "[, ]", where is the sum of the dice and is the Normal Body Count.

 

Unfortunately you cannot get much more complex than this without the program behaving in an unexpected fashion, so I cannot figure out how to, for example, roll both an attack and the above damage in one go. So I figured I could just open one window for attack rolls and one for damage.

 

Here's a similar expression for Killing Damage:

switch(, val != val, 0,
  sum((val / 3)d6) + switch(val % 3, ==0, 0, ==1, 1, 1d3)
) * switch(1d6, ==1, [1, 1], [val-1, 1])

where is the number of Damage Classes. The output is again in the form, "[, ]".

[prestidigitator]

Re: 3d6 Dice Roll Probabilities

 

Heroes!

 

Is this chart in one of the 5th edition books? I think I saw it as a side bar somewhere, but I cannot find it now.

I think it is actually in a side bar for either the Bestiary or Grimoire. Can't remember for sure.

[Foxx!]

Re: 3d6 Dice Roll Probabilities

 

I think it is actually in a side bar for either the Bestiary or Grimoire. Can't remember for sure.

prestidigitator!

 

Thank you. It's on p.34 of the Bestiary. It's a bell curve that shows the probability of each number.

 

Cheers

[AngleWyrm]

Re: 3d6 Dice Roll Probabilities

 

There's a handy math formula for calculating any number of dice with any number of sides. And they don't have to be all the same kind either. It goes like this:

 

Represent the sides of a die as powers of x.

1d2--a coin toss-- looks like:

(x^1 + x^2)

1d4 looks like:

(x^1 + x^2 + x^3 + x^4)

1d6 looks like so:

(x^1 + x^2 + x^3 + x^4 + x^5 + x^6)

 

Ok we got a pattern going. To roll several die, multiply them together. So rolling 3d6 is:

 

(x^1 + x^2 + x^3 + x^4 + x^5 + x^6) *

(x^1 + x^2 + x^3 + x^4 + x^5 + x^6) *

(x^1 + x^2 + x^3 + x^4 + x^5 + x^6)

 

In this case, since all the dice are the same kind, I could have written it as (x^1 + x^2 + x^3 + x^4 + x^5 + x^6)^3.

Whatever. Not really important.

 

The best part is when you actually multiply them together. When you do, each term lists the chances of getting that total. At first it looks rediculously messy--but it's way cool. Don't be scared--they're just numbers:

 

1x^3 + 3x^4 + 6x^5 + 10x^6 + 15x^7 + 21x^8 + 25x^9 + 27x^10 + 27x^11 + 25x^12 + 21x^13 + 15x^14 + 10x^15 + 6x^16 + 3x^17 + x^18

 

So the first term says there is 1 way to get a 3. The next term says there are 3 ways to get a 4. The third term says there are 6 ways(chances) to get a sum of 5, and so on. For probability, divide the chances for a given roll by the total chances (add em all up).

 

Incredible, huh?

[Frenchman]

Re: 3d6 Dice Roll Probabilities

 

The best part is when you actually multiply them together. When you do, each term lists the chances of getting that total. At first it looks rediculously messy--but it's way cool. Don't be scared--they're just numbers:

 

1x^3 + 3x^4 + 6x^5 + 10x^6 + 15x^7 + 21x^8 + 25x^9 + 27x^10 + 27x^11 + 25x^12 + 21x^13 + 15x^14 + 10x^15 + 6x^16 + 3x^17 + x^18

 

Incredible, huh?

 

I remember when I figured this out for myself back in junior high when we were first introduced to variables and exponents - I was flabbergasted when I saw that the x-multiple and exponent corresponded to the possible rolls, and I've been using that simple (if tedious for lots of dice) formula to make random charts and tables ever since, usually without access to any calculators or computers.

[Pariah]

Re: 3d6 Dice Roll Probabilities

 

From our gaming group's wiki: Roll the Bones

 

Roll The Bones

A Study in Probability

by Pariah

 

One thing that makes Hero System different that most other RPGs is the dice. Champions players don't habitually carry around a bunch of dice that make you think of Kepler trying to align the planets within Platonian solids. We use six-sided dice (abbreviated 'd6'), the kind you used as a kid to play Monopoly, Risk, or Yahtzee. We use d6 and only d6. We're pretty exclusive that way. We only use the fancy dice to indicate on the map where the bad guys are—if we use them at all.

 

So, where D&D might call on a player to roll a d20 (read as 'a twenty-sided die', a geometric shape called an icosahedron), Hero System calls for the player to roll 3d6 (read as 'three six-sided dice'). Why would anyone prefer rolling three dice when you could only roll one? The answer is simple. It's called "The Bell Curve."

 

Let's say that in order for your character to perform a certain skill, you have to roll ten or less (abbreviated '10-') on the dice. With either a single d20 or with 3d6, the probability of doing so is 50%. So far, so good. Now, let's say that with experience, you character's roll changes so that now you have to roll eleven or less (11-). The probability of doing so on a d20 is 55%, which is not bad. But on 3d6, the probability jumps to 62.5%. The net effect that the mid-range is opened up substantially with 3d6, while the extremes remain highly improbable. (One effect of this is that a person rolling a d20 is ten times as likely to roll a critical failure and get his/her character killed at an inopportune moment.)

 

The table below, which I constructed based on my own study and knowledge of probabilities back in the mid-90's, shows how likely each outcome is when 3d6 are rolled. There are 216 possibilities that can arise when rolling 3d6: each of the three dice can achieve six different results, which leads to 6^3 (6x6x6=216) different total outcomes. Only one of these outcomes produces a result of 3: a '1' on all three dice (abbreviated, for our purposes, as '111'). There are three different ways a 4 can result: each of the three dice could produce a 2, while the others show 1, i.e., 112, 121, and 211. There are six different ways that a 5 can be produced: 113, 131, 311, 122, 212, and 221. And so on.

 

The table below is organized as follows. The first column shows each possible value from rolling 3d6, 3-18. The second column shows how many different ways (out of 216) that value can be obtained. The third column shows the probability or that value coming up [P(x)]. And the final column—the really useful one—shows the cumulative probability of all values equal to or less than the current value turning up [P(x-)]. So, as can be seen from the table, the chance of rolling 8- is about one in four, while the probability of rolling 14- is about nine in ten.

 

 

Value Occurrences P(x)% P(x-)%

=============================

3..............1...............0.46.......0.46

4..............3...............1.38.......1.85

5..............6...............2.78.......4.63

6............10...............4.63.......9.26

7............15...............6.94.......16.2

8............21...............9.72.......25.9

9............25.............11.57.......37.5

10...........27.............12.50......50.0

11...........27.............12.50.......62.5

12...........25.............11.57.......74.1

13...........21...............9.72.......83.8

14...........15...............6.94.......90.7

15...........10...............4.63.......95.4

16.............6...............2.78.......98.1

17.............3...............1.38.......99.5

18.............1...............0.46........100

 

All hail the Hero System!

 

--

Verbose, but I hope informative as well. And yes, if anyone was wondering, I AM a math & science geek.

[Sean Waters]

Re: 3d6 Dice Roll Probabilities

 

Here's what I use. It is great.

 

http://www.fnordistan.com/smallroller.html

[Maur]

Re: 3d6 Dice Roll Probabilities

 

I use a Perl module (http://search.cpan.org/~jad/Games-Dice-Probability-0.02/lib/Games/Dice/Probability.pm) to calculate odds. It can even handle things like xd6-yd6. So, if you wanted to calculate the odds of beating someone in an opposed check you could do something like:

 

(Your skill - 3d6) - (Opponents skill - 3d6). The results that give you a positive value are the ones where you win

 

ex: I have a skill of 14- and my opponent has an 11-. What is the probability that I will roll and get more of a success than he will?

 

Formula: (14 - 3d6) - (11 - 3d6)

results in:

-12	1	2.14E-05
-11	6	0.000128601
-10	21	0.000450103
-9	56	0.001200274
-8	126	0.002700617
-7	252	0.005401235
-6	456	0.009773663
-5	756	0.016203704
-4	1161	0.024884259
-3	1666	0.035708162
-2	2247	0.048161008
-1	2856	0.061213992
0	3431	0.073538237
1	3906	0.083719136
2	4221	0.090470679
3	4332	0.092849794
4	4221	0.090470679
5	3906	0.083719136
6	3431	0.073538237
7	2856	0.061213992
8	2247	0.048161008
9	1666	0.035708162
10	1161	0.024884259
11	756	0.016203704
12	456	0.009773663
13	252	0.005401235
14	126	0.002700617
15	56	0.001200274
16	21	0.000450103
17	6	0.000128601
18	1	2.14E-05

 

With a total probability of getting more than 0 (tie between opponents) is 72.0615%

[Sean Waters]

Re: 3d6 Dice Roll Probabilities

 

I use a Perl module (http://search.cpan.org/~jad/Games-Dice-Probability-0.02/lib/Games/Dice/Probability.pm) to calculate odds. It can even handle things like xd6-yd6. So, if you wanted to calculate the odds of beating someone in an opposed check you could do something like:

 

 

 

Cool. What's PERL?

[Maur]

Re: 3d6 Dice Roll Probabilities

 

Practical Extraction and Reporting Language

 

A very powerful scripting language.

 

Perl.org <-- main perl site

activestate.com <-- good place to get a version of it for Windows

 

It is installed with almost every form of *NIX.

[Sean Waters]

Re: 3d6 Dice Roll Probabilities

 

Practical Extraction and Reporting Language

 

A very powerful scripting language.

 

Perl.org <-- main perl site

activestate.com <-- good place to get a version of it for Windows

 

It is installed with almost every form of *NIX.

 

Cool. Now all I need to do is learn all about comupters

[Comic]

Re: 3d6 Dice Roll Probabilities

 

I love Perl.

 

It's like that extra language the guy who invented Esperanto invented after Esperanto, because he was bored.

 

Certainly, far more useful overall than Klingon, though not nearly as likely to have funny stories told about it.

 

Now, Forth. There was a language.