User:LRichardson/Essays/2d6 to 1d20 Conversion
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Contents
Abstract[edit]
This essay discusses the relative probabilities of a 2d6 vs a 1d20 roll of dice. As large numbers of 2d6 rolls can be very time consuming, three methods of converting a 1d20 roll into a 2d6 result are discussed, along with their relative merits and flaws.
Purpose[edit]
In most Battletech games the cannon rules call for many instances of 2d6 being thrown. This allows for common 6 sided dice to be used to play and provides a probabilistic distribution that allows for unusual events, such as head shots, to occur relatively infrequently. Unfortunately, the use of multiple dice that need to be summed also can considerably slow a game down, especially if multiple rolls need to be made per each phase of a game. Some games use dice with more sides to allow a single die to be thrown with a reasonable number of possible outcomes. Dice can be found in numerous configurations with 100, 30, and 24 sided dice being only somewhat exotic. Typical polyhedral dice sets only have the D20 as the largest die available. As such, these rules seek to use the d20, as the largest common die to replace the throwing of 2d6 in order to simplify game play.
Issues[edit]
In throwing 2d6 there are 36 possible specific outcomes (permutations) with 11 general outcomes (combinations), ie the sum of the dice, being noted. This unfortunately does not easily correspond to the d20's twenty possible outcomes. As such, some sort of translation needs to be made to go between 2d6 and 1d20 rolls. In BattleTech 2d6 rolls are usually either a threshold roll or a list roll. A threshold roll is where there is a to-hit number that must be met or exceeded to be successful. A list roll is where there is a table of possible outcomes that the 2d6 roll is compared against.
In defining a threshold roll there is usually a base-to-hit number that is modified by various conditions to ultimately arrive at a final modified-to-hit number. Due to the binomial distribution of 2d6, this has the effect of making a modifier of one or two points to a base to hit of six to be much less significant than a base to hit of ten. List rolls also are affected by the binomial distribution of 2d6. Such list rolls allow that certain events, such as a basic center torso hit in BattleTech, are far more common than other events, such as a head hit. Any equivalence table from 1d20 to 2d6 ought to reflect the games original distribution to some degree.
Some observations about 2d6:
Table 1: 2d6 probability[edit]
| Die Roll (2D6) | # of chances | Probability | Cumulative |
|---|---|---|---|
| 2 | 1 | 0.028 | 0.028 |
| 3 | 2 | 0.056 | 0.083 |
| 4 | 3 | 0.083 | 0.167 |
| 5 | 4 | 0.111 | 0.278 |
| 6 | 5 | 0.139 | 0.417 |
| 7 | 6 | 0.167 | 0.583 |
| 8 | 5 | 0.139 | 0.722 |
| 9 | 4 | 0.111 | 0.833 |
| 10 | 3 | 0.056 | 0.917 |
| 11 | 2 | 0.056 | 0.972 |
| 12 | 1 | 0.028 | 1.000 |
This notes the individual probability of each outcome and calculates the cumulative probability. The cumulative P is the upper bound of a given result with the cumulative of the previous result being the lower bound. For a "direct" translation the cumulative probability can be multiplied by 20 and rounded to the nearest 1 to get the equivalent on the roll of 1d20. In the table below the 2d6 column has been spelled out as a result rather than a number to avoid confusion.
Table 2: Direct 2d20 Equivalence[edit]
| Die Roll (2D6) | Cumulative P | Equivalent 1d20 |
|---|---|---|
| "two" | 0.028 | 1 |
| "three" | 0.083 | 2 |
| "four" | 0.167 | 3 |
| "five" | 0.278 | 4-6 |
| "six" | 0.417 | 7-8 |
| "seven" | 0.583 | 9-12 |
| "eight" | 0.722 | 13-14 |
| "nine" | 0.833 | 15-17 |
| "ten" | 0.917 | 18 |
| "eleven" | 0.972 | 19 |
| "twelve" | 1.000 | 20 |
This correspondence, while correctly rounded, due to the particular nature of the relation between 20 and 36 is not satisfactory. Specifically, the odds of rolling a "five" or "nine" on the 2d6 side of the table are %33 greater than rolling a "six" or "eight". This does not keep with the bell shaped binomial distribution of 2d6 results.
So, a manually modified table is shown here that better keeps with the overall trend of 2d6:
Table 3: Modified 2d20 Equivalence[edit]
| Die Roll (2D6) | Cumulative P | Equivalent 1d20 |
|---|---|---|
| "two" | 0.028 | 1 |
| "three" | 0.083 | 2 |
| "four" | 0.167 | 3 |
| "five" | 0.278 | 4-5 |
| "six" | 0.417 | 6-8 |
| "seven" | 0.583 | 9-12 |
| "eight" | 0.722 | 13-15 |
| "nine" | 0.833 | 16-17 |
| "ten" | 0.917 | 18 |
| "eleven" | 0.972 | 19 |
| "twelve" | 1.000 | 20 |
This table too has an issue however. The odds of rolling a "twelve" or a "two" become 1/20 each, as opposed to 1/36. While this might be acceptable for threshold rolls, such as resolving hits, if this table was used to determine hit locations (for example) the odds of both head-shots and critical center torso hits nearly doubles. While some player groups might welcome this increased lethality, a great many will be less than enthused about their hard earned `Mechs being twice as likely to be taken out by "freak" head-shots.
This can be remedied by making the roll of 20 on the d20 a wildcard result. This is to say that the 20 is rerolled and if the result is 1-10, treat it as a "two"; if the result is a 11-20, treat the result as a "twelve". This has the end result of making head-shots and CT critical rolls slightly less common, something that many players might welcome. The rest of the hit possibilities are then reshuffled in the table and the result is as follows:
Table 3: Modified 2d20 Equivalence with Boxcar and Snake-Eye Consideration[edit]
| Die Roll (2D6) | Cumulative P | Equivalent 1d20 |
|---|---|---|
| "two" | 0.028 | 20* |
| "three" | 0.083 | 1 |
| "four" | 0.167 | 2-3 |
| "five" | 0.278 | 4-5 |
| "six" | 0.417 | 6-8 |
| "seven" | 0.583 | 9-11 |
| "eight" | 0.722 | 12-14 |
| "nine" | 0.833 | 15-16 |
| "ten" | 0.917 | 17-18 |
| "eleven" | 0.972 | 19 |
| "twelve" | 1.000 | 20* |
- On a roll of 20, re-roll: If result is 1-10, count as a "two", if result is 11-20, count as a "twelve".
Through the use of these table any roll that calls for a 2d6 can be rolled on a single D20 and compared.
For threshold rolls there remains the issue of to-hit modifiers. There are two obvious ways this could be handled. The first is to find the 2d6 to-hit number and then roll 1d20 and compare the result to the chart. This method keeps closer to the relative weights of multiple to-hit penalties. A less "true" method that is a little faster is to find the base-to-hit number normally to find the needed roll on the 1d20 then modify that d20 values by the modifiers that apply as published. In either event the rolling of events that require the sum of 2d6, such as hits, hit locations, cluster hit numbers and numbers of critical hits is much faster. This is especially true if multiple dice of varying colours are rolled simultaneously.
Further essays in this series will address specific application of these concepts to BattleTech along with concise translated tables suitable for direct game use.