4d5

4d5 is a dice notation used in tabletop gaming to denote rolling four five-sided dice and summing the results. Each die is fair and shows the integers 1 through 5, with equal probability for each face. The four results are added to produce a single total.

Statistical properties of 4d5: The sum of four independent uniform(1–5) variables has a mean of 12 and a variance of 8, corresponding to a standard deviation of about 2.83. The possible totals range from 4 to 20. The distribution is symmetric around 12, since replacing each roll x with 6−x maps totals s to 24−s with equal probability.

Probability and calculation: The probability of obtaining a total s is the coefficient of x^s in (x+x^2+x^3+x^4+x^5)^4, equivalently out of 5^4 = 625 equally likely outcomes. The distribution increases toward the center and decreases toward the extremes, with 12 as the most probable total.

Uses and notes: 4d5 is used when a game calls for the sum of four five-sided dice, such as certain damage or skill checks in some systems. It is less common than 4d6 but appears in specific rule sets and homebrew content. In cases where dice labeling differs (for example if faces are labeled 0–4 on a d5), the mean and probabilities change accordingly, so the notation would reflect the intended dice labeling.