4-Sided Dice (d4) Roller & Probability Calculator

Choose how many four-sided dice to roll and a target total. Get the exact probability of that sum, plus the mean, standard deviation, and cumulative (at least / at most) odds.

Quick Facts

Method
Exact counting via generating-function convolution
Each d4 face (1-4) has probability 1/4. Totals for N dice range from N to 4N.

Your Results

Calculated
P(sum = target)
-
Exact probability of that total
Ways / total outcomes
-
Combinations giving the target
P(sum ≥ target)
-
At least the target
P(sum ≤ target)
-
At most the target

Ready

Set the number of dice and a target total, then calculate.

Understanding the four-sided die (d4)

A four-sided die, written d4, is the pyramid-shaped (tetrahedral) die used in tabletop role-playing games such as Dungeons & Dragons. Its four faces are numbered 1, 2, 3 and 4, and on a fair die each face is equally likely, so a single roll has probability 1/4 = 0.25 (25%) for any given number. This calculator handles both a single d4 and pools of several d4 rolled together, giving the exact probability of any total.

The single-die facts

  • Probability of each face: P(1) = P(2) = P(3) = P(4) = 1/4 = 25%.
  • Mean (expected value): (1 + 2 + 3 + 4) / 4 = 10/4 = 2.5.
  • Variance: for a fair die with s faces the variance is (s² − 1)/12; for s = 4 that is (16 − 1)/12 = 15/12 = 1.25.
  • Standard deviation: √1.25 ≈ 1.118.

Rolling several d4 at once (NdN sums)

When you roll N four-sided dice and add them up, the total ranges from N (all ones) to 4N (all fours). The dice are independent, so there are exactly 4N equally likely ordered outcomes. The probability of a given sum S is simply (number of ordered ways to make S) ÷ 4N. This calculator counts those ways exactly using the generating function (x + x² + x³ + x⁴)N — no simulation or approximation is involved.

Because sums are built from many small dice, the distribution is triangular for two dice and bell-shaped for three or more (a consequence of the central limit theorem). For N dice the mean total is 2.5N and the standard deviation is √(1.25N).

Reference distribution for 2d4

Two four-sided dice have 4² = 16 equally likely outcomes and totals from 2 to 8:

  • Sum 2: 1 way — 1/16 = 6.25%
  • Sum 3: 2 ways — 2/16 = 12.5%
  • Sum 4: 3 ways — 3/16 = 18.75%
  • Sum 5: 4 ways — 4/16 = 25% (the most likely total)
  • Sum 6: 3 ways — 3/16 = 18.75%
  • Sum 7: 2 ways — 2/16 = 12.5%
  • Sum 8: 1 way — 1/16 = 6.25%

The counts (1, 2, 3, 4, 3, 2, 1) add to 16, and the distribution is symmetric around the mean of 5.

Frequently Asked Questions

What are the odds of rolling a specific number on one d4?
On a single fair four-sided die every face — 1, 2, 3 or 4 — is equally likely, so the probability of any one of them is 1/4 = 0.25, or 25%. The odds are 1 in 4 (equivalently 3-to-1 against).
What is the most likely total when rolling multiple d4?
The most likely total is the one closest to the mean, 2.5 × N. For 2d4 the peak is 5 (probability 25%); for 3d4 the peak totals are 7 and 8 (each with the highest count). Extreme totals — all ones or all fours — are always the rarest, each occurring in just one of 4^N outcomes.
How does this differ from rolling a six-sided die?
The counting method is identical, but a d4 has faces 1–4 (mean 2.5, variance 1.25) while a d6 has faces 1–6 (mean 3.5, variance ≈ 2.917). A d4 therefore has a narrower spread, and its single-die distribution runs 1–4 instead of 1–6.
Are these exact probabilities or a simulation?
They are exact. The calculator counts every combination that produces each total by expanding the polynomial (x + x² + x³ + x⁴)^N, then divides by 4^N. There is no random sampling, so the numbers do not change between runs.

Related calculators