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?
What is the most likely total when rolling multiple d4?
How does this differ from rolling a six-sided die?
Are these exact probabilities or a simulation?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |