In Dungeons and Dragons, all characters have backstories. The iconic d20 also has a heroic backstory as a wonder of probability and randomness. It is the workhorse of combat encounters, uncanny dodges, and conniving persuasion checks.
I fell in love with the randomness of the d20 back in the 1980s. The d20 was not so easy and ubiquitous, and I learned to generate random numbers from 1 to 20 on my Dad's HP41c calculator by following an article in Dragon Magazine. The article "The Electric Eye—Computer Technology and Terminology—Dice from a Calculator" was published in Dragon Magazine #45 in January 1981. Thanks to the miracle of the internet, the article has survived and is an amusing read 44 years later.
Despite technological advances, the programmable calculator never replaced the tangible joy of rolling dice. Players who use virtual dice are often converted to the cult of physical dice. There's no substitute for rolling dice by hand—especially the extraordinary 20-sided die, the icosahedron.
Players tie themselves into knots, searching for meaning in this elusive twenty-sided dice. Let's explore some of the properties of the icosahedron that make it so fascinating.
The Natural 20 and the Natural 1 - Statistical Significance?
The d20 has a uniform distribution of 0.05, which is a fancy way of saying that every single number has an equal 5% chance of occurring. Statistically, 0.05 is a value we often use to indicate significance. When testing a hypothesis about something, 5% is a number that gained traction in the 1920s. In his 1925 book Statistical Methods for Research Workers, Ronald Fisher came up with the idea that something that occurs that only has a 5% chance of happening is statistically significant and unlikely to be pure chance. Hence, when your warrior roles that natural 20, he is, in all likelihood, a true heroic persona. Conversely, when you roll a natural one on that intelligence check, he is also dumb as a rock. It is this all too human statistical fallacy that makes D&D so much. We can ascribe a story to the randomness of a dice mimicking the same statistics that drive much out of real-world analysis.
Cooking the Books: Rolling with Advantage and Disadvantage
The dungeon master's job is to tell stories from seemingly random occurrences, with maybe a little bit of nudging the numbers. The dungeon master can let the player roll the d20 twice and take the higher number. Say they want to make it harder for the player to succeed.
The DM can use the probability scores to tweak the chances for success and failure on a roll. Pick the DC and then multiply the roll percentage for every roll.
DC Table with Disadvantage
If the DM wants to build obstacles for players, they will disadvantage them by having them roll twice and take the lesser role. The failure table shows the raw values of failure probability. If you know little about the players' bonuses, add those to the DC level to make it more challenging.
Difficulty | DC Level | Formula | Success Probability |
Very Easy | 5 | 0.8 x 0.8 = 0.64 | 64% |
Easy | 10 | 0.55 x 0.55 = 0.3025 | 30% |
Moderate | 15 | 0.3 x 0.3 = 0.09 | 9% |
Hard | 20 | 0.05 x 0.05 = 0.0025 | 0.25% |
DC Table for Success with Advantage
Sometimes, the DM is generous and will allow the players to roll twice and take the higher of the two rolls. Consider this table if you are gracious and want to optimize for success.
Difficulty | DC Level | Formula | Success Probability |
Very Easy | 5 | 1 - (0.2 x 0.2) = 0.96 | 96% |
Easy | 10 | 1 - (0.55 x 0.55) = 0.7975 | 80% |
Moderate | 15 | 1 - (0.7 x 0.7) = 0.51 | 51% |
Hard | 20 | 1 - (0.95.x 0.95) = 0.0975 | 10% |
The Gamblers Ruin and Dice Jail
The DM's joy is messing with their players' heads. Knowing the probabilities and using them to manipulate how the players respond emotionally to randomness and variance leads to the practice of "Dice Jail." Any bad outcome is undoubtedly the fault of the dice, not probability. Therefore, bad dice go to jail.
Put the players on an endless treadmill to outwit basic mathematics and watch as hilarity ensues. As the dice observe the principles of mathematics, one by one, they will all end up in dice jail until the player is out of dice, and that's when the DM lowers the boom with a high-stakes roll, and the player must use an unlucky dice to great drama.
Conclusion
Dungeons and Dragons is a joyous storytelling, improvisation, and probability mashup. The d20 has become a symbol of adventure, drama, and the human need to find meaning. Even though patterns in mathematical probability are illusions, the meaning that players find in each other telling the story is very real, and that is how the d20 transforms mathematics into magic.
Comments