Understanding custom dice rolls
Tabletop games, probability homework, and game design all use the shorthand NdM+K: roll N dice, each with M sides numbered 1 through M, sum them, then add a flat modifier K. This calculator takes any N, M, and K and returns the exact statistics of the total — its lowest and highest possible values, its average, its spread, and how many distinct ways the dice can land. "3d6+2" means three six-sided dice plus 2; "1d20" means a single twenty-sided die; "4d4-1" means four four-sided dice minus 1.
The formulas
A single fair die of M sides is a discrete uniform variable on {1, 2, …, M}. Because the N dice are independent, means and variances add. Let S be the total of NdM+K:
- Minimum: Smin = N + K (every die shows 1).
- Maximum: Smax = N×M + K (every die shows M).
- Mean (expected total): E[S] = N×(M+1)/2 + K. One die averages (M+1)/2, so N of them average N×(M+1)/2, and the modifier just shifts the center by K.
- Variance: Var[S] = N×(M²−1)/12. A single die has variance (M²−1)/12; adding a constant K does not change variance.
- Standard deviation: σ = √(N×(M²−1)/12).
- Distinct ordered outcomes: MN (each die has M faces, chosen independently). Note that many of these map to the same total, which is why middle totals are far more likely than extremes.
Common reference points
- 1d6: min 1, max 6, mean 3.5, σ ≈ 1.708. The classic single cube.
- 2d6: min 2, max 12, mean 7, σ ≈ 2.415. The most common roll in board games; 7 is the single most likely total.
- 3d6: min 3, max 18, mean 10.5, σ ≈ 2.958. The traditional ability-score roll in many role-playing games.
- 1d20: min 1, max 20, mean 10.5, σ ≈ 5.766. Flat and swingy — every value is equally likely.
- 4d6: min 4, max 24, mean 14, σ ≈ 3.416.
Why the distribution matters
The single die (1dM) is flat: every face is equally likely. But adding dice together produces a bell-shaped distribution by the central limit theorem — the more dice, the tighter and more symmetric the hump around the mean. That is why 3d6 (rolls cluster near 10–11) feels very different from 1d18 (any value 1–18 equally likely) even though both have the same average of 10.5. The mean tells you the center; the standard deviation tells you how far a typical roll strays from it.
Frequently Asked Questions
What does NdM+K notation mean?
What is the average roll of NdM+K?
Why is 7 the most likely result on 2d6?
Does adding dice make the result more or less predictable?
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 |
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| Category | % of Net Income | Examples |
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| Needs | 50% | Rent, groceries, utilities, insurance |
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| 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 |