jdn dice and averages

7/28/2019 JDN Dice and Averages

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Die Rolls and AveragesFor Pencils and Paper Role-Playing Games

Copyright ©2013 J.D. Neal ● All Rights Reserved.

This tutorial is mainly comcerned with ordinary dicemarked in evenly incremented integers, such as a six-sided or ten-sided die; other die types may give

different results and should be analyzed on their own.Some of the following numbers were rounded for convenience; if you need precision, do the mathyourself.

The probability of a number occurring as a result of a die roll is mainly a concern for success-or-failuresituations, or for random picks from a table withunique entries. When creating quantities (such as howmuch damage a hit does in combat), averages are abigger concern, in the long run.

In some circumstances the probability can havesome affect on play: consider using a Quasi-d4 (seebelow) versus a regular d4. If the referee createscreatures who can take 2 points of damage, then aone-hit kill is certainly easier with a quasi-d4 than witha common d4. But such things tend to be both flukesand the product of unimaginative referees. Variedgame play means that such things are not asimportant.

Likewise, if there is a significant difference in thenumber range created, then the die rolls can besignificantly different, even if they have the sameaverage. For example, 1d8 (1-8), 1d6+1 (2-7), andd4+2 (3-6), and d2+3 (4-5) all have an average of 4.5,but the smaller the die used the narrower the number range.

Calculating Averages

The average (mean) roll of a standard die will bethe total of the end points divided by 2. For example, ad6 (marked 1 to 6) will average 3.5 (6 + 1 = 7; 7 / 2 =3.5) while a d8 will average 4.5 (1 + 8 = 9; 9 / 2 = 4.5)and a d10 is 5.5 (1 + 10 = 11; 11 / 2 = 5.5).

Individual rolls will vary from 1 to 8, but whenrolling a d8 one expects a final result around 45 for tenrolls. The final total could be anything but 45 (sincethis is a random roll), but in the long run 45 will beabout the expected normal average for 10 rolls of ad8.

Note how a roll of d6+1 (a six-sided die plus 1) has

the exact same average as a d8: 4.5 (2 + 7 = 9; 9 / 2 =4/.5). A roll of ten d6+1s will not create the same highand low numbers as a d8 (being limited to 2 to 7) butwill give the same general average as a d8 over 10rolls – barring random flukes.

If dice are marked in common fashion, than theaverage of multiple rolls is also the total of the endpoints divided by 2. Thus 3d6 (for 3 to 18) has an

average of 21 divided by 2, which is 10.5, meaningrolls tend to average 10 or 11.

Subtracting a number from a die roll might create a

different average than the end points divided by 2,depending on how the subtraction is handled.Consider this:

Die Roll Result Total Avg1d6 1 2 3 4 5 6 21 3.51d6-1 limited to 1 1 1 2 3 4 5 16 2.671d6-1 not limited 0 1 2 3 4 5 15 2.5

If 1d6-1 is limited to 1, the average is not the sameas 1 + 5 = 6 / 2 = 3, since 1 occurs 33% of the time.

Other methods of Calculating Averages

Not all dice are standard dice: some might bemarked in unusual ways or interpreted in odd ways.The average may or may not be the sum of the

end points divided by 2. (Averaging dice are indeeddesigned to create a specific average that often doesequal the end points divided by 2; but not all die rollsor dice will do that.)

Another way of calculating averages is to total thepossible numbers and divide by the number of sideson the die, or multiply the frequency of a number occurring times itself to get a weighted value andsumming them up.

Consider a d6 roll:

1d6 Frequency Frequency WeightedNumber as Fraction as Decimal Value

1 1/6 16.7% 0.22 1/6 16.7% 0.33 1/6 16.7% 0.54 1/6 16.7% 0.7

5 1/6 16.7% 0.8

6 1/6 16.7% 1.0

21 Total: 3.5

21 / 6 = 3.5

1+6 = 7 : 7/ 2 = 3.5

Below is an “averaging die” (a d6 marked toexclude 1 and 6): a 2 or 5 will occur more often thanother numbers; and a 1 or 6 will never occur; but theaverage is the same as a normal six-sided die.

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Averaging d6 WeightedNumber Frequency Value

2 16.7% 0.32 16.7% 0.33 16.7% 0.54 16.7% 0.75 16.7% 0.85 16.7% 0.8

21 Total: 3.521 / 6 = 3.5

2+5 = 7 : 7/ 2 = 3.5

Consider the average of a d4 and a quasi-d4 (a d6marked as shown):

d4 Weighted Quasi-d4 Weighted# Freq. Value # Freq. Value1 25.0% 0.3 1 16.7% 0.22 25.0% 0.5 2 16.7% 0.33 25.0% 0.8 3 16.7% 0.54 25.0% 1.0 4 16.7% 0.7

2 16.7% 0.33 16.7% 0.5

Avg: 2.5 Average: 2.5

Following are tables showing averages for a d5, ad6-1 limited to 1, and rolling a d6 and counting 6 as 3.Note how the die roll of 1d6-1 does not create theaverage of 1+5 (which would be 3): oddly marked diceor certain die roll interpretations can affect averagesand probabilities.

d5 Weighted d6 (count 6 as 3) Weighted 1d6-1 (limited to minimum of 1)

Number Frequency Value Number Frequency Value Number Frequency Value

1 20.0% 0.2 1 16.7% 0.2 1 16.7% 0.22 20.0% 0.4 2 16.7% 0.3 1 16.7% 0.23 20.0% 0.6 3 16.7% 0.5 2 16.7% 0.34 20.0% 0.8 4 16.7% 0.7 3 16.7% 0.55 20.0% 1.0 5 16.7% 0.8 4 16.7% 0.7

3 16.7% 0.5 5 16.7% 0.8Average: 3 Average: 3 Average: 2.7

Median Of a Die Roll

The mathematical median of a typical die roll is the“middle number”. If the total of the end points is odd,then there are two mid points, the one before and after the mean. If it is even, then the median is also theaverage. Consider the following die rolls, their averageand median:

Die Roll Average Median Number(s)d3 2 2d4 2.5 2, 3d6 3.5 3, 4d8 4.5 4, 52d4 5 5d10 5.5 5, 6d12 6.5 6, 72d6 7 7d16 8.5 8, 92d8 9 9d18 9.5 9, 10

d20 10.5 10, 11

This is a consideration in game design for whenthe designer wants to know what the middlenumber(s) are so they can design a mechanic, suchas a to-hit roll or saving throw. For example: supposethe referee wants to design a mechanic (such asleaping a chasm) based on a d20 die roll. The

average is 10.5, but they cannot start the concept at10.5 because said number will never actually occur onany single die roll. Instead, they have to start with theidea that the number 1 to 10 represent 1/2 of thepossible number created by the die roll; and 11 to 20the other half of the number space. A 50/50 chance of success means the character must roll 11 or higher to

succeed.Suppose they consider using 2d6 as the die roll to

use for the mechanic. The average is also the median:7. There is no 50% mark: 2 to 7 or 7 to 12 both occupy58% of the number space available.

Best of Two d6s Analysis

Some people prefer to simplify their game by usinga d6 for damage rolls. Some of them prefer to thefollowing scheme (they aren't interested in creating avariable scale; they want it simple and easy, notcomplex):

Unarmed the lowest of two dice

Standard 1-handed melee one die (d6)

2-handed melee the best of two dice

At first glance the best of two d6s is an interestingand novel approach: indeed, the author adored itwhen he found it. In reality, it isn't much more different

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than rolling 1d6+1 (one d6 and add 1). If anything, it isa little inferior to 1d6+1 (which gives the sameaverage as a d8 die roll). Yes, it does result in a 1-6number range versus 2-7, but what is the practicalresult? After all, with the best of two d6s you have toroll and compare two dice, versus rolling one andadding 1 to the result.

Below is an analysis of the two die rolls. A simple

grid is used to determine the combinations that thebest of 2d6 will generate. The related numbers werethen counted, and a weighted average andpercentage chance of each occuring was computed.The author both manually rolled two dice and set upformulas in a spreadsheet to compare the equivalentof 100 rolls for practical purposes. Warning: a singletest will not necessarily return useful data: sometimesthey gave a different weighted average; sometimesthey were much the same (as is expected for randomrolls, which being random are not predictable).

The 1 added to the die for 1d6+1 kicks the overallaverage up higher than picking the best of two dice.

Consider how a 6 occurs 30.5% of the time on thebest of 2d6: a 6 or 7 will occur 32% of the time with1d6+1. A 5 or 6 occurs about 55% of the time with thebest of 2d6: while 5, 6, or 7 occurs about 50% of thetime with 1d6+1. A 1 occurs occasionally with the bestof 2d6 (1 in 36 rolls) while it never occurs with 1d6+1.

Table #1: Best of 2d6 Analysis gridd6 #2

d6 #1 1 2 3 4 5 61 1 2 3 4 5 62 2 2 3 4 5 63 3 3 3 4 5 64 4 4 4 4 5 65 5 5 5 5 5 66 6 6 6 6 6 6

Table #2: Best of 2d6 Statistics# Occs. Weighted Average % Chance1 1 .03 2.78%2 3 .17 8.33%3 5 .42 13.89%4 7 .78 19.44%5 9 1.25 25.00%

6 11 1.83 30.56%

TOTALS: 36 4.47 100.00%

Table #3: d6+1 Statisticsd6+1 Occs. Weighted Average % Chance

2 1 .33 16.67%3 1 .50 16.67%4 1 .67 16.67%5 1 .83 16.67%6 1 1.00 16.67%

7 1 1.17 16.67%

TOTALS: 6 4.50 16.67%

Table #4: 100 Random Testsd6+1 (A+1) Best of 2d6 (A or B) d6 A d6 B

7 6 6 43 4 2 42 2 1 24 5 3 56 5 5 44 3 3 13 2 2 15 6 4 64 3 3 2

3 2 2 14 3 3 14 4 3 43 2 2 22 3 1 36 6 5 64 3 3 25 4 4 35 6 4 67 6 6 43 2 2 24 4 3 47 6 6 1

4 4 3 47 6 6 34 3 3 24 3 3 17 6 6 43 5 2 54 6 3 65 4 4 13 4 2 44 4 3 42 4 1 45 6 4 67 6 6 5

7 6 6 26 5 5 46 5 5 12 1 1 14 3 3 13 3 2 34 5 3 56 5 5 14 3 3 2

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7 6 6 43 2 2 25 6 4 63 6 2 62 6 1 62 2 1 24 4 3 42 5 1 5

6 5 5 43 5 2 57 6 6 16 5 5 37 6 6 32 1 1 12 2 1 22 4 1 46 5 5 14 3 3 16 5 5 46 5 5 13 2 2 2

2 1 1 14 3 3 32 6 1 65 4 4 47 6 6 34 3 3 13 3 2 34 3 3 13 3 2 37 6 6 37 6 6 52 5 1 54 3 3 13 3 2 36 5 5 33 5 2 54 3 3 37 6 6 33 6 2 62 6 1 62 5 1 52 1 1 15 4 4 27 6 6 17 6 6 35 4 4 4

2 4 1 46 5 5 14 3 3 32 3 1 33 5 2 56 5 5 24 3 3 36 5 5 1

427 4204.27 4.2

Lowest of Two d6s Analysis

The lowest (lesser) of two d6s (from the lead-in of the above discussion) creates almost the sameaverage as a d4 roll. The number range is extendedpast 4 to 5 or 6 - but there is only a 3/36 (1/12) chanceof a 5 occurring and 1/36 chance of a 6 occuring. Notehow the lesser of 2d6 INCREASES the chance of a 1or 2 occuring (55% versus 50% on a d4); and reducesthe chance of a 3 or 4 (let alone higher number)occuring.

d6 #2

d6 #1 1 2 3 4 5 6

1 1 1 1 1 1 1

2 1 2 2 2 2 2

3 1 2 3 3 3 3

4 1 2 3 4 4 4

5 1 2 3 4 5 5

6 1 2 3 4 5 6

# Freq Weighted Average % Chance

1 11 .31 30.56%2 9 .50 25.00%

3 7 .58 19.44%

4 5 .56 13.89%

5 3 .42 8.33%

6 1 .17 2.78%

Totals: 36 2.53 100.00%

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VARIOUS AVERAGESd4 Weighted Quasi-d4 Weighted# Freq. Value # Freq. Value1 25.0% 0.3 1 16.7% 0.22 25.0% 0.5 2 16.7% 0.33 25.0% 0.8 3 16.7% 0.54 25.0% 1.0 4 16.7% 0.7

2 16.7% 0.3

3 16.7% 0.5Avg: 2.5 Average: 2.5

d5 Weighted Quasi d5 (d6 :count 6 as 3) Weighted 1d6-1 (limited to minimum of 1)

Number Frequency Value Number Frequency Value Number Frequency Value

1 20.0% 0.2 1 16.7% 0.2 1 16.7% 0.22 20.0% 0.4 2 16.7% 0.3 1 16.7% 0.23 20.0% 0.6 3 16.7% 0.5 2 16.7% 0.34 20.0% 0.8 4 16.7% 0.7 3 16.7% 0.55 20.0% 1.0 5 16.7% 0.8 4 16.7% 0.7

3 16.7% 0.5 5 16.7% 0.8Average: 3 Average: 3 Average: 2.7

1d6 Frequency Frequency Weighted Averaging d6 WeightedNumber as Fraction as Decimal Value Number Frequency Value

1 1/6 16.7% 0.2 2 16.7% 0.32 1/6 16.7% 0.3 2 16.7% 0.33 1/6 16.7% 0.5 3 16.7% 0.54 1/6 16.7% 0.7 4 16.7% 0.7

5 1/6 16.7% 0.8 5 16.7% 0.8

6 1/6 16.7% 1.0 5 16.7% 0.8

21 Total: 3.5 21 Total: 3.5

21 / 6 = 3.5 21 / 6 = 3.5

1+6 = 7 : 7/ 2 = 3.5 2+5 = 7 : 7/ 2 = 3.5

Using a d6 to recreate other dice tends to fail past a d6 due to the narrowed number ranges. This does, though,illustrate how different number combinations can create the same average:

Quasi-d8 Quasi-d8 Quasi-d10 Quasi-d10

Weighted Weighted Weighted Weighted

Mark ed Freq Average Mark . Freq Average Mark . Freq Average Mark . Freq Average

1 16.7% 0.2 1 16.7% 0.2 1 16.7% 0.2 1 16.7% 0.2

2 16.7% 0.3 2 16.7% 0.3 3 16.7% 0.5 2 16.7% 0.3

3 16.7% 0.5 4 16.7% 0.7 4 16.7% 0.7 5 16.7% 0.8

6 16.7% 1.0 5 16.7% 0.8 7 16.7% 1.2 6 16.7% 1.0

7 16.7% 1.2 7 16.7% 1.2 8 16.7% 1.3 9 16.7% 1.5

8 16.7% 1.3 8 16.7% 1.3 10 16.7% 1.7 10 16.7% 1.7

Sum: 27 4.5 27 4.5 33 5.5 33 5.5

/6 = 4.5 4.5 5.5 5.5

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