If you have ever played tabletop role-playing games like Dungeons and Dragons or any game that uses dice, you probably know about the twenty-sided die, also called the d twenty. This die is special because it has twenty faces, each with a number from one to twenty. Understanding the probability of rolling certain numbers on this die can help you make better decisions during your games or simply satisfy your curiosity about chance and randomness.
20-Sided Dice Probability Calculator
In this article, we will cover everything you need to know about the probability of a twenty-sided die. We will explain basic probability ideas, how to calculate the chance of specific rolls, how to calculate the chance of rolling numbers in ranges, what odds mean, and how these ideas apply in real games.
What Is a Twenty-Sided Die?
A twenty-sided die is a solid shape with twenty flat faces. Each face shows a different number from one to twenty. When you roll the die, it can land on any of these faces. Assuming the die is fair, which means that it is not weighted or biased, each face has the same chance of appearing.
This type of die is very popular in role-playing games for deciding the results of actions. For example, you might roll the die to see if your character succeeds in hitting an enemy or solving a puzzle.
Basic Ideas of Probability for a Twenty-Sided Die
What Is Probability?
Probability is a way to measure how likely something is to happen. It is usually expressed as a number between zero and one, where zero means the event cannot happen and one means the event will always happen. In everyday language, you can think of probability as a percentage. For example, zero point five probability means fifty percent chance.
Probability of Rolling a Specific Number on a Twenty-Sided Die
Because the die has twenty sides, and each side is equally likely, the chance of rolling any one particular number is one divided by twenty. In other words, each number from one to twenty has a five percent chance of being rolled.
For example, the chance of rolling a seven on a single roll is five percent.
Probability of Rolling Certain Numbers or Ranges
Often, you want to know more than the chance of a single number. You may want to know the chance of rolling a number higher than some number, or lower than some number, or between two numbers.
Probability of Rolling a Number Equal To or Greater Than a Certain Value
Suppose you want to know the chance of rolling a number equal to or greater than fifteen. The numbers that count are fifteen, sixteen, seventeen, eighteen, nineteen, and twenty. There are six such numbers.
Because each number is equally likely and there are twenty numbers total, the chance is six divided by twenty, or thirty percent.
Probability of Rolling a Number Equal To or Less Than a Certain Value
If you want the chance of rolling five or less, the numbers that count are one, two, three, four, and five. There are five numbers here.
So the chance is five divided by twenty, which equals twenty-five percent.
Probability of Rolling a Number Within a Range
What if you want to roll a number between eight and twelve inclusive? The numbers are eight, nine, ten, eleven, and twelve. Five numbers in total.
Again, the chance is five divided by twenty, which is twenty-five percent.
Odds Versus Probability
It is important to understand that probability and odds are related but different ways of describing chance.
- Probability is the chance that an event will happen out of all possible outcomes.
- Odds compare the chance that the event will happen to the chance it will not happen.
For example, if the probability of rolling a number greater than or equal to fifteen is thirty percent, then the chance it will not happen is seventy percent. The odds in favor are thirty to seventy, which simplifies to three to seven.
How to Calculate Probability for Multiple Rolls
Sometimes, you might want to know the chance of rolling a certain number or range at least once over many rolls.
For example, what is the chance of rolling a twenty (the highest number) at least once in three rolls?
To calculate this, it is easier to find the chance of the opposite event — that is, never rolling a twenty in three rolls — and then subtract this from one hundred percent.
- The chance of not rolling a twenty in one roll is ninety-five percent.
- For three rolls, the chance of not rolling a twenty at all is ninety-five percent times ninety-five percent times ninety-five percent.
- Multiplying ninety-five percent by itself three times gives approximately eighty-five point seven percent.
- Therefore, the chance of rolling at least one twenty in three rolls is one hundred percent minus eighty-five point seven percent, which equals fourteen point three percent.
Common Probability Questions About a Twenty-Sided Die
What Is the Chance of Rolling a Natural Twenty?
In games like Dungeons and Dragons, rolling a natural twenty, which means getting a twenty on a d twenty, is often a critical success.
Since each number has an equal chance, the chance of rolling a natural twenty on one roll is one out of twenty, or five percent.
What Is the Chance of Rolling Below Ten?
Numbers below ten are one through nine, which are nine numbers.
So the chance is nine divided by twenty, or forty-five percent.
What Is the Chance of Rolling Between Five and Fifteen?
Numbers between five and fifteen inclusive are five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, and fifteen. That is eleven numbers.
So the chance is eleven divided by twenty, or fifty-five percent.
Using Probability to Improve Your Game Strategy
Understanding the probabilities of a twenty-sided die can help you make better decisions during gameplay.
For example, if you know that the chance of rolling fifteen or higher is only thirty percent, you might want to improve your character’s skills or equipment to increase your chances of success.
You can also calculate the chance of succeeding if you have to roll a certain number or higher multiple times, helping you plan your actions.
Summary Table of Common Probabilities for a d20
Here is a summary of common chances for various rolls on a twenty-sided die:
| Roll Condition | Number of Possible Outcomes | Probability Percentage |
|---|---|---|
| Rolling a specific number | 1 | 5 |
| Rolling fifteen or higher | 6 | 30 |
| Rolling five or lower | 5 | 25 |
| Rolling between eight and twelve | 5 | 25 |
| Rolling below ten | 9 | 45 |
| Rolling between five and fifteen | 11 | 55 |
Real-World Applications of d20 Probability
While the twenty-sided die is mostly used for gaming, the concepts of equal probability and chance apply to many real-world situations:
- Random selection in lotteries or raffles
- Decision making in uncertain conditions
- Teaching probability and statistics
Advanced Concepts: Probability Distributions and Expected Value
Probability Distribution
The probability distribution of a twenty-sided die is uniform because every number has the same chance.
Expected Value
The expected value is the average number you expect to get if you roll the die many times. To calculate this, you add up all the numbers on the die and divide by the total number of sides.
For a d twenty, the expected value is the sum of all numbers from one to twenty divided by twenty.
The sum of numbers from one to twenty is two hundred and ten.
Dividing two hundred and ten by twenty gives ten point five.
So on average, you expect to roll ten point five.
Conclusion
The twenty-sided die is a fascinating object with simple yet rich probability properties. Each face has an equal chance of five percent to appear, and you can calculate the chance of rolling numbers within certain ranges by counting how many numbers fit the criteria and dividing by twenty.
Knowing these probabilities can enhance your gameplay, help you understand odds, and improve your decision-making when rolling dice.