D20 Dice Roller

What this d20 roller does

The d20, or twenty sided die, is the die that decides nearly everything in Dungeons and Dragons and most other tabletop roleplaying games. This roller reproduces it in the browser: press roll and the page returns a whole number from 1 to 20, with the option to add a modifier or roll with advantage and disadvantage, the two special modes that fifth edition D&D introduced.

Each mode shows what happened on every individual die, which matters because advantage and disadvantage both roll two dice and the display proves which die was kept. The mode picker also computes the exact probability of succeeding at a difficulty check, so the roller doubles as a probability reference at the table.

Nothing leaves the browser. The faces come from the random number function built into the page, and no result is uploaded, logged or shared. There is no install, no account, and no notation to learn beyond the three modes.

The geometry of the icosahedron

A standard d20 is a regular icosahedron, one of the five Platonic solids. It has 20 identical triangular faces, 30 edges and 12 vertices, and five triangles meet at every vertex, a shape catalogued in detail by Wolfram MathWorld. Because every face is congruent and the solid is symmetric, each face has an equal chance of landing face down, which is what makes the die fair, a symmetry the standard references on dice probability derive from the Platonic solid geometry.

The numbering is not arbitrary. On a properly manufactured die, opposite faces sum to 21: the 1 sits across from the 20, the 2 across from the 19, and so on. This arrangement spreads the high and low numbers evenly around the solid, which reduces any bias from small imperfections in the plastic or in how the die is thrown.

Probability of a single d20 roll

Because all 20 faces are equally likely, the math is flat. Any specific number has a 1 / 20 = 5 percent chance. The average of all faces is (1 + 2 + ... + 20) / 20 = 210 / 20 = 10.5. Half the time you roll 10 or lower, and half the time 11 or higher.

Success in D&D is usually defined as rolling at least a target number called the difficulty class, or DC. If a check has DC 15 and the roll has no modifier, the chance of meeting or beating it is the count of faces from 15 to 20 divided by 20, which is 6 / 20 = 30 percent.

Add a modifier and the count changes by the size of the modifier. The general formula for the chance that a roll of X plus a modifier M meets or beats a DC is P(X + M >= DC) = (21 - (DC - M)) / 20. For DC 15 with a +3 modifier, DC - M = 12, so the chance is (21 - 12) / 20 = 9 / 20 = 45 percent.

Worked example: DC 15 with a +3 modifier

Take a skill check with DC 15 and a character whose relevant modifier is +3. The raw die needs to reach 15 - 3 = 12 for the total to hit 15. Faces 12 through 20 are successes, which is 20 - 12 + 1 = 9 faces out of 20.

The probability of success without any special mode is 9 / 20 = 0.45, or 45 percent.

Now roll with advantage. The character rolls two d20s and keeps the higher. The probability that at least one of two independent rolls meets the target is 1 minus the probability that both miss. Each individual roll misses with probability 1 - 0.45 = 0.55, so both missing is 0.55 x 0.55 = 0.3025. The success probability is therefore 1 - 0.3025 = 0.6975, or 69.75 percent.

With disadvantage, the character keeps the lower of two rolls. Both dice must succeed, so the probability is 0.45 x 0.45 = 0.2025, or 20.25 percent.

The gap is wide. Moving from disadvantage to advantage swings this particular check by 69.75 - 20.25 = 49.5 percentage points, which is why the two conditions carry so much weight in play.

How advantage and disadvantage reshape the odds

The general advantage formula is P(advantage) = 1 - (1 - p) squared, where p is the normal single roll chance of success. Disadvantage is the mirror: P(disadvantage) = p squared. The square is what makes both effects nonlinear. The further the normal chance sits from 50 percent, the more advantage or disadvantage moves it.

At p = 0.5, advantage produces 1 - 0.25 = 0.75 and disadvantage produces 0.25, a symmetric 25 point swing either way. At p = 0.9, advantage gives 1 - 0.01 = 0.99, a 9 point gain. At p = 0.1, advantage gives 1 - 0.81 = 0.19, nearly double. The same square drives disadvantage: at p = 0.9 it produces 0.81, a small loss, and at p = 0.1 it produces 0.01, a near total collapse.

Advantage and disadvantage also reshape the average roll. A single d20 averages 10.5. With advantage, the higher of two d20s averages 13.825, a gain of 3.325. With disadvantage, the lower of two averages 7.175, a loss of 3.325. In fifth edition terms, that swing is worth more than most ability score modifiers.

Critical hits and critical failures

Fifth edition defines two special results on the d20. A natural 20, the die showing 20 before any modifier, is a critical hit on an attack roll: the attack hits regardless of the target's armor class and damage dice are rolled twice. A natural 1 is a critical failure: the attack misses no matter how large the modifier.

The probability of each is 1 / 20 = 5 percent on a single die. Advantage and disadvantage change both in the same squared way that they change success chances. With advantage, at least one of two dice shows a 20 with probability 1 - (19/20) squared = 1 - 0.9025 = 0.0975, or 9.75 percent, nearly double. With disadvantage, both dice must show 20, which has probability (1/20) squared = 0.0025, or 0.25 percent.

The same logic applies to natural 1s in reverse. Advantage makes a critical failure nearly vanish, from 5 percent to 0.25 percent, because both dice would have to land on 1. Disadvantage nearly doubles it to 9.75 percent.

One detail matters for the math: a natural 20 on the die is not the same as a total of 20. A roll of 17 with a +3 modifier has a total of 20 but is not a critical hit. Only the face value of the die triggers the critical, which is why the two events have different probabilities.

Worked example: comparing a +4 modifier to advantage

Players often ask whether a flat bonus or advantage helps more. The answer depends on the baseline chance, and the comparison is easy to run with the formulas above.

Take a DC 14 check with no modifier. The raw die needs a 14 or higher, which is 7 faces out of 20, so p = 7 / 20 = 0.35.

Option one, a +4 modifier: the die now needs a 10 or higher, which is 11 faces, so the chance is 11 / 20 = 0.55, or 55 percent. That is a 20 point gain over the baseline 35 percent.

Option two, advantage with the original +0 modifier: 1 - (1 - 0.35) squared = 1 - 0.4225 = 0.5775, or 57.75 percent. That is a 22.75 point gain, slightly better than the +4 modifier.

Now shift the baseline. Against DC 18 with no modifier, p = 3 / 20 = 0.15. A +4 modifier moves the need to 14 or higher, which is 7 faces, so 35 percent. Advantage alone gives 1 - 0.85 squared = 1 - 0.7225 = 0.2775, or 27.75 percent. The flat modifier wins at this end because 4 full points on the die is worth more than the squared bonus when p is small.

Rule of thumb: against middle difficulty targets the two options are close, and against very hard targets a flat bonus beats advantage, while advantage protects better against very easy tasks going wrong.

Where the d20 shows up at the table

Fifth edition D&D, published by Wizards of the Coast, uses the d20 for four kinds of rolls: attack rolls, saving throws, ability checks, and death saving throws. Each uses the same mechanic, roll the die, add a modifier, compare to a DC or armor class. Three of the four benefit from proficiency bonuses and ability modifiers, while death saving throws use the raw die with no modifier at all, needing a 10 or higher to succeed.

Outside of D&D, the d20 system spawned a family of games that share the same roll mechanic, and many indie systems use a d20 with different modifiers or success tiers. The d20 also appears in board games as a random event die or a turn counter, where its flat 5 percent per face gives a uniform spread that a d6 cannot match.

Physical dice wear unevenly and can develop bias, and casino grade precision is not the norm for plastic gaming dice. A digital roller sidesteps that concern because the random number generator treats every face identically by construction. The tradeoff is that a physical die has table presence, so many groups use both: physical dice for feel, a roller for statistics or for games on a screen.

Whatever the setting, the underlying math stays the same. One die, twenty faces, five percent each, and a formula that converts modifiers and modes into exact probabilities.