Die Rolls and Averages Averages For Pencils and Paper Role-Playing Games Copyright ©2013 J.D. Neal ● All Rights Reserved. This tutorial is mainly comcerned with ordinary dice marked in evenly incremented integers, such as a sixsided sided or ten-si ten-sided ded die; other other die types may give give different results and should be analyzed on their own. Some Some of the follow following ing number numbers s were were rounde rounded d for conv conven enie ienc nce; e; if you you need need prec precis isio ion, n, do the the math math yourself. The probability of a number occurring as a result of a die roll is mainly a concern for success-or-failure situat situation ions, s, or for rando random m picks picks from from a table table with with unique entries. When creating quantities (such as how much damage a hit does in combat), averages are a bigger concern, in the long run. In some circumstances the probability can have some affect on play: consider using a Quasi-d4 (see below) below) versus versus a regula regularr d4. If the referee referee create creates s creatures who can take 2 points of damage, then a one-hit kill is certainly easier with a quasi-d4 than with a common d4. But such things tend to be both flukes and the produc productt of unimag unimagina inativ tive e refere referees. es. Varied aried game game play lay mean means s that hat such uch thing hings s are are not as important. Likewise, if there is a significant difference in the numb number er rang range e creat created, ed, then then the the die die roll rolls s can can be signif significa icantl ntly y differ different, ent, even even if they they have have the same same average. average. For example, example, 1d8 (1-8), (1-8), 1d6+1 (2-7), and d4+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.
average of 21 divided by 2, which is 10.5, meaning rolls 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, depe depend ndin ing g on how how the the subt subtra ract ctio ion n is hand handle led. d. Consider this: Die Roll 1d6 1d6-1 limited to 1 1d6-1 not limited
1 1 0
Result 2 3 4 5 1 2 3 4 1 2 3 4
Total Av Avg 6 21 3. 3 .5 5 16 2.67 5 15 2.5
If 1d6-1 is limited to 1, the average is not the same as 1 + 5 = 6 / 2 = 3, 3 , since 1 occurs 33% of the time. Other methods of Calculating Averages Not all dice dice are standar standard d dice: dice: some some might might be marked in unusual ways or interpreted in odd ways. The average may or may not be the sum of the end points divided by 2. (Averagi (Averaging ng dice are indeed designed to create a specific average that often does equal the end points divided by 2; but not all die rolls or dice will do that.) Another way of calculating averages is to total the possible numbers and divide by the number of sides on the die, or multi multiply ply the freque frequency ncy of a number number occurr occurring ing times times itsel itselff to get a weight weighted ed value value and summing them up. Consider a d6 roll:
Calculating Averages The average average (mean) (mean) roll of a standa standard rd die die will will be the total of the end points divided by 2. For example, a d6 (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). Indivi Individua duall rolls rolls will will vary vary from from 1 to 8, but when rolling a d8 one expects a final result around 45 for ten rolls. The final total could be anything but 45 (since this is a random roll), but in the long run 45 will be about the expected expected normal average for 10 rolls of a d8. 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 high and low numbers as a d8 (being limited to 2 to 7) but will give give the same general average average as a d8 over 10 rolls – barring random flukes. If dice are marked in common fashion, than the average of multiple rolls is also the total of the end points divided by 2. Thus 3d6 (for 3 to 18) has an
1d6 Frequency Number as Fraction 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 21
Frequency as Decimal 16.7% 16.7% 16.7% 16.7% 16.7% 16.7%
Weighted Value 0.2 0.3 0.5 0.7 0.8 1.0
Total:
3.5
21 / 6 =
3.5
1+6 = 7 : 7/ 2 =
3.5
Belo Below w is an “ave “avera ragi ging ng die” die” (a d6 mark marked ed to exclude exclude 1 and 6): a 2 or 5 will occur more often than other numbers; and a 1 or 6 will never occur; but the average is the same as a normal six-sided die.
1
Averaging d6 Number 2 2 3 4 5 5 21
Frequency 16.7% 16.7% 16.7% 16.7% 16.7% 16.7% Total: 21 / 6 = 2+5 = 7 : 7/ 2 =
Weighted Value 0.3 0.3 0.5 0.7 0.8 0.8 3.5 3.5 3.5
d4 # 1 2 3 4
Number 1 2 3 4 5
Weighted Frequency 20.0% 20.0% 20.0% 20.0% 20.0%
Value 0.2 0.4 0.6 0.8 1.0
Average:
3
Average 2 2.5 3.5 4.5 5 5.5 6.5 7 8.5 9 9.5 10.5
2.5
d6 (count 6 as 3) Number 1 2 3 4 5 3
Weighted 1d6-1 (limited to minimum of 1)
Frequency 16.7% 16.7% 16.7% 16.7% 16.7% 16.7% Average:
Value 0.2 0.3 0.5 0.7 0.8 0.5 3
Number 1 1 2 3 4 5
Frequency 16.7% 16.7% 16.7% 16.7% 16.7% 16.7% Average:
Value 0.2 0.2 0.3 0.5 0.7 0.8 2.7
average is 10.5, but they cannot start the concept at 10.5 because said number will never actually occur on any single die roll. Instead, they have to start with the idea that the number 1 to 10 represent 1/2 of the possible possible number created created by the die roll; roll; and 11 to 20 the 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 occupy 58% of the number space available.
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 the average. Consider the following die rolls, their average and median: Die Roll d3 d4 d6 d8 2d4 d10 d12 2d6 d16 2d8 d18 d20
Avg:
Quasi-d4 Weighted # Freq. Value 1 16.7% 0.2 2 16.7% 0.3 3 16.7% 0.5 4 16.7% 0.7 2 16.7% 0.3 3 16.7% 0.5 Average: 2.5
Following are tables showing averages for a d5, a d6-1 limited to 1, and rolling a d6 and counting 6 as 3. Note how the die roll of 1d6-1 does not create the average of 1+5 (which would be 3): oddly marked dice or certain certain die roll interpreta interpretations tions can affect affect averages averages and probabilities.
Consider the average of a d4 and a quasi-d4 (a d6 marked as shown):
d5
Freq. 25.0% 25.0% 25.0% 25.0%
Weighted Value 0.3 0.5 0.8 1.0
Median Number(s) 2 2, 3 3, 4 4, 5 5 5, 6 6, 7 7 8, 9 9 9, 10 10, 11
Best of Two d6s Analysis Some people prefer to simplify their game by using a d6 for for dama damage ge roll rolls. s. Some Some of them them pref prefer er to the the following scheme (they aren't interested in creating a vari variabl able e scal scale; e; they they want want it simp simple le and and easy easy,, not not complex):
This is a consideration in game design for when the the desi design gner er want ants to know know what hat the the midd middlle number(s) are so they can design a mechanic, such as a to-hit roll or saving throw. For example: suppose the the refe refere ree e want wants s to desig design n a mech mechan anic ic (suc (such h as leap leapin ing g a chas chasm) m) base based d on a d20 d20 die die roll roll.. The The
Unarmed
the lowest of two dice
Standar dard 11-hande nded me melee
one one di die (d (d6)
2-handed melee
the best of two dice
At first glance the best of two d6s is an interesting and novel novel approa approach: ch: indeed indeed,, the author author adored adored it when he found it. In reality, reality, it isn't much more different 2
than rolling 1d6+1 (one d6 and add 1). If anything, it is a litt little le infer inferio iorr to 1d6+ 1d6+1 1 (whi (which ch give gives s the the same same average as a d8 die roll). Yes, it does result in a 1-6 number range versus 2-7, but what is the practical result? result? After After all, with the best of two d6s you have to roll roll and compar compare e two dice, versus versus rolli rolling ng one and adding 1 to the result. Below is an analysis of the two die rolls. A simple grid is used to determine determine the combinati combinations ons that the best of 2d6 will generate. generate. The related numbers were then then count ounted ed,, and and a weigh eightted aver averag age e and and percentage percentage chance of each occuring was computed. computed. The author both manually rolled two dice and set up formulas in a spreadsheet to compare the equivalent of 100 rolls for practical practical purposes. Warning: Warning: a single single test will not necessarily return useful data: sometimes they gave a different weighted average; sometimes they were much the same (as is expected for random rolls, which being random are not predictable). The 1 added to the die for 1d6+1 kicks the overall average up higher than picking the best of two dice. Consider how a 6 occurs 30.5% of the time on the best of 2d6: a 6 or 7 will occur 32% of the time with 1d6+1. A 5 or 6 occurs about 55% of the time with the best of 2d6: while 5, 6, or 7 occurs about 50% of the time with 1d6+1. A 1 occurs occasionally with the best of 2d6 (1 in 36 rolls) while it never occurs with 1d6+1. Table #1: Best of 2d6 Analysis grid d6 #2 d6 #1 1 2 3 4 1 1 2 3 4 2 2 2 3 4 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 Table #2: Best of 2d6 Statistics # Occs. Weighted Average 1 1 .03 2 3 .17 3 5 .42 4 7 .78 5 9 1.25 6 11 1.83 TOTALS:
36
4.47
5 5 5 5 5 5 6
Table #3: d6+1 Statistics d6+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
Table #4: 100 Random Tests d6+1 d6+1 (A+1) (A+1) Best Best of 2d6 (A or B) d6 A 7 6 6 3 4 2 2 2 1 4 5 3 6 5 5 4 3 3 3 2 2 5 6 4 4 3 3 3 2 2 4 3 3 4 4 3 3 2 2 2 3 1 6 6 5 4 3 3 5 4 4 5 6 4 7 6 6 3 2 2 4 4 3 7 6 6 4 4 3 7 6 6 4 3 3 4 3 3 7 6 6 3 5 2 4 6 3 5 4 4 3 4 2 4 4 3 2 4 1 5 6 4 7 6 6 7 6 6 6 5 5 6 5 5 2 1 1 4 3 3 3 3 2 4 5 3 6 5 5 4 3 3
6 6 6 6 6 6 6
% Chance 2.78% 8.33% 13.89% 19.44% 25.00% 30.56% 100.00%
3
16.67%
d6 B 4 4 2 5 4 1 1 6 2 1 1 4 2 3 6 2 3 6 4 2 4 1 4 3 2 1 4 5 6 1 4 4 4 6 5 2 4 1 1 1 3 5 1 2
7 3 5 3 2 2 4 2 6 3 7 6 7 2 2 2 6 4 6 6 3 2 4 2 5 7 4 3 4 3 7 7 2 4 3 6 3 4 7 3 2 2 2 5 7 7 5
6 2 6 6 6 2 4 5 5 5 6 5 6 1 2 4 5 3 5 5 2 1 3 6 4 6 3 3 3 3 6 6 5 3 3 5 5 3 6 6 6 5 1 4 6 6 4
6 2 4 2 1 1 3 1 5 2 6 5 6 1 1 1 5 3 5 5 2 1 3 1 4 6 3 2 3 2 6 6 1 3 2 5 2 3 6 2 1 1 1 4 6 6 4
4 2 6 6 6 2 4 5 4 5 1 3 3 1 2 4 1 1 4 1 2 1 3 6 4 3 1 3 1 3 3 5 5 1 3 3 5 3 3 6 6 5 1 2 1 3 4
2 6 4 2 3 6 4 6 427 4.27
4 5 3 3 5 5 3 5 420 4.2
1 5 3 1 2 5 3 5
4 1 3 3 5 2 3 1
Lowest of Two d6s Analysis The lowest (lesser) of two d6s (from the lead-in of the the abov above e disc discus ussi sion on)) crea creattes almo almost st the the same same average as a d4 roll. The number range is extended past 4 to 5 or 6 - but there is only a 3/36 (1/12) chance of a 5 occurring and 1/36 chance of a 6 occuring. Note how the lesser of 2d6 INCREASES the chance of a 1 or 2 occuring (55% versus 50% on a d4); and reduces the the chanc chance e of a 3 or 4 (let (let alon alone e high higher er numb number) er) occuring. d6 #2 d6 #1 1 1 1 2 1
4
2 1 2
3 1 2
4 1 2
5 1 2
6 1 2
2 2 2 2
3 3 3 3
3 4 4 4
3 4 5 5
3 4 5 6
3 4 5 6
1 1 1 1
# 1 2 3
Freq 11 9 7
Weighted Average .31 .50 .58
% Chance 30.56% 25.00% 19.44%
4 5 6 Totals:
5 3 1 36
.56 .42 .17 2.53
13.89% 8.33% 2.78% 100.00%
d4 # 1 2 3 4
VARIOUS AVERAGES Weighted Quasi-d4 Value # Freq. 0.3 1 16.7% 0.5 2 16.7% 0.8 3 16.7% 1.0 4 16.7% 2 16.7% 3 16.7% 2.5 Average:
Freq. 25.0% 25.0% 25.0% 25.0%
Avg: d5
Weighted
Number Frequency 1 20.0% 2 20.0% 3 20.0% 4 20.0% 5 20.0% Average: 1d6 Frequency Number as Fraction 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6
Quasi d5 (d6 :count 6 as 3)
Value 0.2 0.4 0.6 0.8 1.0
Number 1 2 3 4 5 3
Frequency as Decimal 16.7% 16.7% 16.7% 16.7% 16.7% 16.7%
21
Weighted 1d6-1 (limited to minimum of 1)
Frequency 16.7% 16.7% 16.7% 16.7% 16.7% 16.7% Average:
3
Weighted Value 0.2 0.3 0.5 0.7 0.3 0.5 2.5
Value 0.2 0.3 0.5 0.7 0.8 0.5 3
Weighted Averaging d6 Value Number 0.2 2 0.3 2 0.5 3 0.7 4 0.8 5 1.0 5
Number 1 1 2 3 4 5
Frequency 16.7% 16.7% 16.7% 16.7% 16.7% 16.7% Average:
Value 0.2 0.2 0.3 0.5 0.7 0.8 2.7
Frequency 16.7% 16.7% 16.7% 16.7% 16.7% 16.7%
Weighted Value 0.3 0.3 0.5 0.7 0.8 0.8
Total:
3.5
Total:
3.5
21
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
Weighted Marked Freq Average Mark. 1 2 3 6
Freq
Quasi-d10 Weighted Average Mark.
Freq
Quasi-d10
Weighted Weighted Average Mark. Freq Average
16.7% 16.7% 16.7% 16.7%
0.2 0.3 0.5 1.0
1 2 4 5
16.7% 16.7% 16.7% 16.7%
0.2 0.3 0.7 0.8
1 3 4 7
16.7% 16.7% 16.7% 16.7%
0.2 0.5 0.7 1.2
1 2 5 6
16.7% 16.7% 16.7% 16.7%
0.2 0.3 0.8 1.0
7 16.7% 8 16.7% Sum: 27 27 /6 = 4.5
1.2 1.3 4.5
7 8 27 4.5
16.7% 16.7%
1.2 1.3 4.5
8 10 33 5.5
16.7% 16.7%
1.3 1.7 5.5
9 10 33 5.5
16.7% 16.7%
1.5 1.7 5.5
5
