Dice notation explained (NdS+M)
How to read tabletop dice notation like 2d6+3, d20 and 4d8-1, what the count, sides and modifier mean, and the odds each roll produces.
Reading the NdS+M pattern
Dice notation is a compact recipe with three parts. The number before the d is the count, how many dice you roll. The number after the d is the sides, the faces on each die. An optional trailing sign and number is the modifier, a flat amount added to or subtracted from the sum. So 2d6+3 means roll two six-sided dice and add three. When the count is left off, as in d20, it is understood to be one, so d20 is simply 1d20.
Why one die and many dice behave differently
A single die is flat, every face is equally likely, so a lone d20 gives each number from 1 to 20 the same chance. Add more dice and the shape changes. Two or more dice summed cluster toward the middle because there are many ways to reach an average total and only one way to hit the extremes. That is why 2d6 peaks at 7 and only rarely gives 2 or 12, and why swapping 1d12 for 2d6 makes results feel steadier even though both span a similar range.
What the modifier really adds
The modifier is applied once to the final sum, not to every die, which is a common point of confusion. In 3d6+2 you roll three dice, add them together, then add a single 2. This shifts the whole range of possible totals up or down without changing the spread. A negative modifier such as 4d8-1 slides the range the other way. Because the shift is uniform, the modifier is often used to represent a fixed bonus like a skill or penalty on top of a random roll.
Common rolls and where they appear
A handful of rolls show up constantly at the table. An attack or check is often 1d20 plus a modifier. Damage is frequently a small pool like 2d6 or 3d6. Character ability scores are classically 4d6, sometimes dropping the lowest die, though this tool sums all four. Percentile checks use d100 for a 1 in 100 read. Knowing these patterns lets you type the notation straight in rather than reaching for physical dice.