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Take Your Chances

The Statistics and Probability of Dice

Copyright 1999 by Gary Renaud and Lowell R. Matthews

The rolling of dice lies at the core of all but a very few role-playing games—even computer RPG's, where those dice rolls are simulated by some kind of pseudo-random number generator. Truly fair dice do indeed generate random numbers, and every roll is independent of the one before—which can lead to the well-known gamblers' fallacy. Decks of cards have memory; dice rolls do not. For those reasons, we thought a closer look at the mathematics behind the statistics and probability of dice was worth discussion.

Linear Probability Distributions

For the sake of discussion, we will assert that all our dice are fair. To be truly fair, a die must have a homogeneous density and faces with equal areas and equal distances from its center of mass. The five classic Pythagorean solids, tetrahedron, cube, octahedron, dodecahedron, and icosahedron, all meet the facial requirements and produce dice of four, six, eight, twelve, and twenty sides (abbreviated d20). Dice of ten and thirty sides are also popular, and we have heard of dice with seven, 34, and 100 sides. Higher ranges of values, still linear, can be constructed from two or more dice read as different orders of magnitude, e.g., a d10 read as 0–9 plus another appropriate die read as (1–N times; 10) can make dice of 40, 60, 80, 100, 120, or 200 sides. With a little math, we could make many more linear combinations.

A single die produces a linear probability distribution, i.e., every result has an equal probability of being observed on a particular throw. For a six-sided die (d6), each value (1–6) has a 1/6 or 16.7% chance of occurring. If we want to create a linear distribution, the procedure is easy. We take the difference between the desired low and high values, then roll a die of that size (or larger, throwing away out-of-bounds numbers), then add the offset back in to reach the desired result. For example, let's say we want to generate a linear random integer between 0 and 10. The problem here is that our set of results has eleven members, which does not match any convenient die. One easy solution is to use a d12, counting 12 as 0 and ignoring 11.

Triangular Probability Distributions

Two dice rolled together and summed produce a triangular probability distribution. In the case of 2d6, there are 62 = 36 possible results. If we were to read the dice in order, we would produce a linear distribution, but when we add the two values together, combinations towards the middle of the value range occur more frequently than those at the extremes, distributed symmetrically about the mean. If we use two dice of different sizes, the distribution takes the shape of a trapezoid; i.e., the top of the triangle is flattened. For example, let's look at 2d6 and d8+d4. Both have a mean or average value of 7 and a range of 2–12. So, they're the same, right? No, as shown in Table I below (probabilities were rounded to the nearest percent).

Table I: 2d6 vs. d8+d4

 

Occurrences

Probabilities

Result

2d6

d8+d4

2d6

d8+d4

2 1 1 3% 3%
3 2 2 6% 6%
4 3 3 8% 9%
5 4 4 11% 13%
6 5 4 14% 13%
7 6 4 17% 13%
8 5 4 14% 13%
9 4 4 11% 13%
10 3 3 8% 9%
11 2 2 6% 6%
12 1 1 3% 3%

Binary combinations, usually of two equal dice, are frequent in many RPG's, so we have included a wide selection in the tables below. The column labels are R for result or total, N for number of occurrences, and P for probability of each occurrence.

Table II: 2d4

R

N

P

2

1

6%

3

2

13%

4

3

19%

5

4

25%

6

3

19%

7

2

13%

8

1

6%


Table III: 2d8

R

N

P

2

1

2%

3

2

3%

4

3

5%

5

4

6%

6

5

8%

7

6

9%

8

7

11%

9

8

13%

10

7

11%

11

6

9%

12

5

8%

13

4

6%

14

3

5%

15

2

3%

16

1

2%


Table IV: 2d10

R

N

P

2

1

1%

3

2

2%

4

3

3%

5

4

4%

6

5

5%

7

6

6%

8

7

7%

9

8

8%

10

9

9%

11

10

10%

12

9

9%

13

8

8%

14

7

7%

15

6

6%

16

5

5%

17

4

4%

18

3

3%

19

2

2%

20

1

1%


Table V: 2d12

R

N

P

2

1

1%

3

2

1%

4

3

2%

5

4

3%

6

5

4%

7

6

4%

8

7

5%

9

8

6%

10

9

7%

11

10

7%

12

11

8%

13

12

9%

14

11

8%

15

10

7%

16

9

7%

17

8

6%

18

7

5%

19

6

4%

20

5

4%

21

4

3%

22

3

2%

23

2

1%

24

1

1%


Probability Distributions Approaching Normal

Three or more dice rolled together and summed produce a probability distribution that begins to approach the bell-curve shape of the true statistical normal distribution. The more dice we roll, the closer we get to the true normal distribution, but since we are dealing with discrete numbers (i.e. integers), we can never truly reach it. The number of possible results is equal to the product of the value of each die, so in the case of 3d6, there are 63 = 216 possible results with the distribution shown in Table VI and Figure 1.

Table VI: 3d6

R

N

P

3

1

0.5%

4

3

1.4%

5

6

2.8%

6

10

4.6%

7

15

6.9%

8

21

9.7%

9

25

11.6%

10

27

12.5%

11

27

12.5%

12

25

11.6%

13

21

9.7%

14

15

6.9%

15

10

4.6%

16

6

2.8%

17

3

1.4%

18

1

0.5%



Figure 1: 3d6 Chart

If we want to create a particular (approximately) normal distribution, then the two important parameters to consider are the mean and range. The low and high values of the range are the sums of the lows and highs on each die, respectively, and the mean is halfway between the low and the high. In the case of 3d6, the range is 3–18 and the mean is 10.5. Selecting the right mean and range is usually all that matters for most game work. The fewer dice we use, the flatter our distributions. That is, the outlying numbers will be more likely to occur than for a true normal distribution, while the central numbers will be less likely. To balance this, the "tails" of the distributions are shorter than a true normal distribution, in which they reach to infinity. In the tables below are the results for 3d4 and 4d4.

Table VII: 3d4

R

N

P

3

1

2%

4

3

5%

5

6

9%

6

10

16%

7

12

19%

8

12

19%

9

10

16%

10

6

9%

11

3

5%

12

1

2%


Table VIII: 4d4

R

N

P

4

1

0.4%

5

4

1.6%

6

10

3.9%

7

20

7.8%

8

31

12.1%

9

40

15.6%

10

44

17.2%

11

40

15.6%

12

31

12.1%

13

20

7.8%

14

10

3.9%

15

4

1.6%

16

1

0.4%


The Open-Ended Roll

For the ever-popular open-ended roll used so frequently in Rolemaster and its siblings, the distribution is "stepped," essentially linear over the 90 middle numbers (06–95), which are each 1% likely to occur (90% total chance). In the first set of open-ended numbers (high or low), each value is about 0.05% likely (about 4.75% total chance on each side). The next set of open-ended numbers (obtained from an 01–05 roll followed by a 96–100 roll on the low end, or by two 96–100 rolls in a row on the other) are about 0.24% likely cumulatively, with an individual percentage chance that is vanishingly small. We recommend that you forget trying anything that requires three or more open-ended rolls. Gary and Lowell have each seen about four in 15 years of RM play! One was a spell failure by one of Gary's characters; another was an attack against one of Lowell's characters (it scored a maximum result in spite of a huge defensive bonus and Deflections and Invisibility spells)!

The Divided and Multiplied Rolls

With the little-known "divided" roll, we can generate probability distributions which differ from normal in that they are asymmetric about the mean. For example, let's take a d8 and divide it by a d4, then round down or truncate the result. This technique will produce an integer value ranging from 0 to 8. The results, shown in Table VII and Figure 2, are interesting and can be quite useful.

Table VII: d8/d4

R

N

P

0

6

19%

1

10

31%

2

7

22%

3

3

9%

4

2

6%

5

1

3%

6

1

3%

7

1

3%

8

1

3%



Figure 2: d8/d4 Chart

Note that all of the results are clustered towards the low end of the value range, yet there is a small chance of getting a comparatively large number. The range is 0–8, yet the mean is just shade over 2, while the median is 1.5 (it falls right between the values 1 and 2). Note also that the number of possible results is 8 times; 4 = 32, the same as if the d8 and d4 were added.

So why is this useful? Well, let's consider the distribution of levels in a typical population. We should find a group of very low-level individuals (mostly children), a relatively large group of people with journeyman-grade skills (e.g., experienced farmers, journeymen craftsmen, and trained soldiers), and a very few people with higher skill levels (e.g., master craftsmen and elite soldiers). If we use larger dice (say, d100/d20), we will get a much wider range, yet the median value will stay low.

Interesting, no? Gary uses a similar scheme to generate random NPC levels. He modifies it based on class, city size, and other factors, but the core is a divided roll.

At first glance, it would appear that multiplying dice would give a similar result to dividing, but this is in fact not the case as shown in Figure 3. Multiplying dice results in a distribution which shows peaks for those numbers with a relatively large number of divisors, numbers which when multiplied will produce a particular result. This technique would not appear to be as useful as its counterpart.

 

Figure 3: d8 times;d4 Chart

Editor's Note

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