3d4

3d4 is a common dice notation used in tabletop role-playing games to denote rolling three four-sided dice and summing their results. In this notation, XdY means roll X dice, each with Y sides, and add the results together. For 3d4, three independent four-sided dice are rolled and their values are added to produce a total.

A four-sided die, or d4, is a polyhedral die with four faces numbered 1 to 4. When performing 3d4, the three dice are rolled independently and the numbers shown are summed to yield a total in the range from 3 to 12. The distribution of possible totals is not uniform; some sums occur more frequently than others.

Statistics for 3d4: the expected value is 7.5, since each d4 has a mean of 2.5. The variance of a single d4 is 1.25, so the variance of the sum of three independent d4s is 3.75, with a standard deviation of about 1.94. The probability distribution across totals is symmetric around 7.5, with the following exact probabilities: 3 (1/64), 4 (3/64), 5 (6/64), 6 (10/64), 7 (12/64), 8 (12/64), 9 (10/64), 10 (6/64), 11 (3/64), 12 (1/64).

In use, 3d4 provides simple, predictable random values suitable for light damage, healing, or minor effects in various game systems. It can also be used in probability exercises to illustrate the summing of independent discrete uniform variables, and modifiers may be added to the total as dictated by game rules. See also dice notation and discrete probability.