Polyhedral Madness

Beyond the fact that polyhedral dice are just cool, the objective of the Adventure System was for the die range from d4 to d12 and the use of Advantage and Disadvantage to “feel” right.  Meaning, does a character’s advancement from a d4 to a d8 feel meaningful? Does a character with Strength d8 seem stronger than a character with a Strength d4? Is having Advantage or Disadvantage clearly impacting outcomes?

Well, the answer to all of those questions is a definitive yes. For most players the “feel” of it is all that matters. From a game design perspective, however, and for those who like math or statistics, here is how it all works out in the Adventure System.

Reading the Table
The Polyhedral Probabilities table provides the percent chance of getting the specified result or greater (across the top) with a certain die type (down the side) and how Disadvantage (Dis) and Advantage (Adv) affects that chance. The table provides the chances of getting a Critical Failure (a natural 1) or getting a success against the standard Difficulty Numbers (DN) of 4, 8, and 12. It accounts for a die needing to max to achieve the result. For example, a character with d6 skill has a 50.0% chance of getting a 4 or more on a standard check. If they have Advantage, their chance improves to 75.0%. If they have Disadvantage, it drops to 25.0%. If that character attempted a heroic task (DN 12) their chance of success drop to 2.9%.

Polyhedral Probabilities
Critical Failure Success Success Success
Die Dis 1 Adv Dis 4 Adv Dis 8 Adv Dis 12 Adv
d4 43.8% 25.0% 6.3% 6.3% 25.0% 43.8% 1.6% 6.3% 10.9% 0.4% 1.6% 2.7%
d6 30.6% 16.7% 2.8% 25.0% 50.0% 75.0% 2.3% 13.9% 25.5% 0.5% 2.9% 5.1%
d8 23.4% 12.5% 1.6% 39.1% 62.5% 84.6% 1.6% 12.5% 23.4% 1.0% 8.1% 14.6%
d10 19.0% 10.0% 1.0% 49.0% 70.0% 91.0% 9.0% 30.0% 51.0% 0.9% 9.0% 17.1%
d12 16.0% 8.3% 0.7% 56.3% 75.0% 93.8% 17.4% 41.7% 66.0% 0.7% 8.3% 16.0%

What Does It All Mean
The dice mechanics and probabilities deliver on the “feel” we intended as the objective of the Adventure System. Here are some examples of exactly how that happens behind the scenes.

Untrained Skill Use: Unless the referee rules otherwise, a character is able to use a skill untrained at a d2 (rules lingo for a d4 with Disadvantage) in the Adventure System. Envision if you will, a dwarf (Notice d2; untrained), a rogue (Notice d4), and an elf (Notice d8) are travelling through goblin infested woods. The referee calls for a Notice check (DN 4) to spot a band of goblins. The dwarf only has a 6.3% chance of noticing the goblins while the rogue has 25% chance and the elf a 62.5% chance. This demonstrates how even the short five die step range from d4 to d12 results in markedly different chances of success that will stand out in play or “feel” right. Overtime, the elf is going seem far more perceptive than the dwarf.

Critical Failure: Another key element of skill use, particularly untrained, is the chance of a Critical Failure that prevents the use of Fate to alter the result. Often, an untrained character is simply hoping not to get a Critical Failure. Presuming they need a 4 to succeed, as long as they don’t get a 1, they have an 83.3% chance of getting a 2 or move on their Fate die if they choose to spend Fate and have a Spirit d6. Unfortunately, an untrained character has a 43.8% chance of getting a Critical Failure. Compared to our dwarf, the rogue has a 25% chance of a critical failure, but the elf’s chance is half that at 12.5%.

Opposed Checks: Since the DN of an opposed check is the opponent’s passive skill, the probabilities provide a powerful differentiating tool. Let us imagine, after a successful adventure, the drunken rogue decides to challenge the dwarf to an arm wrestling contest. The rogue is Strength d4 and the dwarf Strength d8. The rogue’s DN is a 5 (passive of the dwarf’s d8) and the dwarf’s DN is a 3 (passive of the rogue’s d4). Well, neither of these numbers is on the chart, but the rogue only has a 25% chance of getting a 5 or greater, while the dwarf has a 75% chance of getting a 3 or greater. Only if they both succeed on their check does the higher result number win. In this case, the rogue will average a 2.5 while the dwarf will average a 4.5. The dwarf has a significantly better chance on average of winning the contest. The referee might also decide that the character must accumulate three successes to win making the contest more interesting and – maybe – a bit more back and forth. In this case, it would more likely result in the dwarf dominating the contest.

Character Improvement: All of these numbers also feed into the noticeable change in probabilities as a character improves. Let us say our dwarf has a falling out with his companions and decides to take to thieving requiring him to improve his Notice. Right from the start, moving from 43.8% chance of a Critical Failure and a 6.3% chance of success (against a 4) when untrained to a 25% chance of a Critical Failure and success at Notice d4 is going to very clear progress. That continues when the dwarf advances to Notice d6 and the chance of a Critical Failure drops to a 16.7% and the chance of success jumps up to 50%.

Here is the Math
Calculating the probability on a single die works like this:

Probability = number of desired outcomes ÷ number of possible outcomes

For example, when the dwarf needed a 3 or better in the arm wrestling contest, there are six desired outcomes (3, 4, 5, 6, 7, 8) out of 8: 6 ÷ 8 = .75 or 75%

Calculating Advantage or Disadvantage essentially uses the same method by combining the two dice. The easy part is determining the number of possible outcomes by multiplying the number of sides on each die. For example, there are 4 x 4 or 16 possible outcomes when rolling a d4 with Advantage or Disadvantage. Unfortunately, calculating the number of desired outcomes is a bit more work. For example, the chance of a Critical Failure on a d4 with Disadvantage has 7 potential desired outcomes (1-1, 1-2, 1-3, 1-4, 2-1, 3-1, 4-1): 7 ÷ 16 = .438 or 43.8%

If a max die is required to succeed, we need to combine the two independent die rolls together:

Probability = probability of outcome one × probability of outcome two

For example, to succeed against DN 8 with a d6, first you need to roll a 6 and then you need to roll at least a 2 on the second die: (1 ÷ 6 = .167) x (5 ÷ 6 = .833) = .139 or 13.9%

Ultimately, I am just glad this all shakes out and feels right in the Adventure System. My muse and the limited part of my brain that can do math don’t always play well together…