Understanding the Dice Roller
This tool rolls virtual dice using standard tabletop notation and reports both a random result and the exact statistics of that roll. Dice notation is written as NdS+M, where N is the number of dice, S is the number of sides on each die, and M is an optional flat modifier added to (or subtracted from) the sum. For example, 3d6+2 means "roll three six-sided dice, add them together, then add 2."
The formulas
For a roll of N dice with S sides each plus a modifier M, three key values are exact and do not require simulation:
- Minimum total: every die shows 1, so the lowest possible sum is N + M.
- Maximum total: every die shows its highest face S, so the peak is N × S + M.
- Average (expected value): a single fair S-sided die averages (S+1)/2, so N dice plus a modifier average N × (S+1)/2 + M.
The random result itself is generated by rolling each die independently and uniformly — every face from 1 to S is equally likely — then summing and applying the modifier. Because the dice are independent, the totals cluster around the average and form a bell-shaped distribution as N grows (a consequence of the Central Limit Theorem).
Common reference points
- 1d6 (one standard die): range 1–6, average 3.5.
- 2d6 (two dice, as in many board games): range 2–12, average 7 — 7 is the single most likely sum.
- 3d6 (classic ability-score roll): range 3–18, average 10.5.
- 1d20 (a twenty-sided die): range 1–20, average 10.5.
- 1d100 (percentile die): range 1–100, average 50.5.
Why the distribution matters
Rolling one die gives a flat (uniform) distribution: every outcome is equally likely. Rolling several dice and summing does not. With 2d6, a total of 7 can happen six ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) but a total of 2 only one way (1+1), so 7 is six times more likely than 2. Adding more dice narrows the spread relative to the average, which is why 3d6 produces middling scores far more often than extreme ones.
Frequently Asked Questions
What does 3d6+2 mean?
What is the average roll of a single die?
Are these rolls truly random?
Why is 7 the most common result on two dice?
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 |