The differences between using 1d20 and using 2d6 are a frequent topic of discussion. This post takes a closer look at the actual math to clarify what these differences really are. Three types of dice systems are considered: systems in which players and referees roll against each other, systems in which players roll against a target number, and systems in which player rolls can have three possible results (success, partial or mixed success, and failure).
Opposed Rolls
In opposed-roll systems in which the magnitude of the difference between opposing rolls is not considered, the only possible outcomes are player wins, referee wins, or tie. The type and number of dice are of little consequence in such systems as the player and the referee will always each have an equal chance of winning. The only difference is the likelihood of a tie: 5% for opposed 1d20 rolls versus 11.27% for opposed 2d6 rolls.
More common are opposed-roll systems where the magnitude of the difference between opposing rolls does have meaning. In such systems, the mixed result is commonly expanded from a pure tie to rolls that are “close.” In addition, a very large difference between rolls may be interpreted as an extreme success or failure. So we have 5 possible results: Extreme success or failure, regular success or failure, and close (tie.) For these systems, using 1d20 allows greater flexibility in determining what counts as close and what counts as extreme.
For opposed 1d20 rolls:
if close is a difference of: | the chance of close rolls is: |
2 or less | 23.5% |
3 or less | 32% |
4 or less | 40% |
5 or less | 47.5% |
if extreme success or failure is a difference of: | the chance of extreme success or failure is: |
13 or greater | 7% |
14 or greater | 5.25% |
15 or greater | 3.75% |
For opposed 2d6 rolls:
if close is a difference of: | the chance of close rolls is: |
1 or less | 32.87% |
2 or less | 52.17% |
if extreme success or failure is a difference of: | the chance of extreme success or failure is: |
6 or more | 5.4% |
7 or more | 2.7% |
Clearly with opposed 1d20 rolls one has more options for defining close and extreme results, allowing more flexibility to tailor the game to suit the table.
Some opposed-roll systems may also count rolling the maximum or minimum possible to indicate extreme success or failure. The chance of the player rolling a 1 or a 20 on 2d6 and the referee not rolling the same number is 4.75%. The chance of a player rolling a 2 or a 12 on 2d6 and the referee not rolling the same number is 2.7%. So for these systems, 1d20 also provides more frequent extreme results than 2d6.
Rolls Against a Target Number
1d20
target number: | chance of rolling target number or higher: |
13 | 40% |
14 | 35% |
15 | 30% |
2d6
target number: | chance of rolling target number or higher: |
8 | 41.67% |
9 | 27.78% |
10 | 16.67% |
11 | 8.34% |
For target-number systems, 1d20 once again provides greater granularity for setting different target numbers. It also makes it quite easy to know the probablity of hitting a target number without having a table–it simply varies by 5% at each step. With 2d6, the steps are not equal as the target number increases, making judging the odds of hitting the number non-intuitive. And once again 1d20 provides more extreme results in games where the maximum and minimum rolls have special meaning: the chance of rolling a 1 or a 20 on 1d20 is 5%; the chance of rolling a 2 or 12 on 2d6 is 2.78%.
Three-Way Player Rolls
1d20
range | chance of rolling in range |
1–8 | 40% |
9–16 | 40% |
17–20 | 20% |
1d20 with alternative ranges
range | chance of rolling in range |
1–6 | 30% |
7–14 | 60% |
15–20 | 30% |
2d6
range | chance of rolling in range |
2–6 | 41.67% |
7–9 | 41.67% |
10–12 | 16.67% |
2d6 with alternative ranges
range | chance of rolling in range |
2–5 | 27.78% |
6–8 | 44.44% |
9–12 | 27.78% |
Using 1d20, one can easily adjust the probabilities of success, mixed success, and failure in 5% increments to suit their table or the specific situation at hand. With 2d6, the options are more limited and small adjustments to the ranges make bigger changes to the probabilities.
At this point one might ask, is there anything 2d6 can do that 1d20 can’t do? When we look at the probabilities of various results, the bell curve of 2d6 results doesn’t actually make much of a difference; 1d20 can be made to approximate the same probabilities by adjusting target numbers or result ranges. The only things you can do with 2d6 that you can't do with 1d20 are make the odds of rolling the maximum or minimum less than 5%, and make it difficult to guess the odds of a particular result without a table.
In conclusion, the type of dice you use is far less important than an awareness of the probabilities of various results and of your ability to adjust those probabilities to achieve the kind of game you want.
Big thanks to anydice.com for helping with the math!
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