How 6-sided dice probability works
A standard six-sided die (a d6) is a fair cube with faces numbered 1 through 6. On a single roll, each face is equally likely, so the probability of any one specific number is 1/6 ≈ 16.67%. Dice problems get interesting when you roll several dice and ask about the sum of the faces. This calculator answers exactly that: given N dice and a target total, it counts how many of the possible rolls hit that total and divides by the number of rolls that can occur.
The core formula
Every ordered outcome of N dice is equally likely, and there are 6N of them (6 faces on the first die × 6 on the second × … × 6 on the Nth). So:
P(sum = t) = (number of ordered rolls that total t) ÷ 6N
For a single die the answer is trivial — one way each for the totals 1 through 6, so 1/6 apiece. For two dice there are 62 = 36 ordered outcomes. The calculator finds the numerator by building the full distribution of sums with a repeated convolution: start with one die, then combine it with the next die face by face until all N dice are accounted for. That gives an exact integer count for every reachable sum, so the probabilities are exact fractions, not simulations or estimates.
Why the middle sums are most likely
With two dice the possible totals run from 2 (rolling 1 and 1) to 12 (rolling 6 and 6). There is only one way to make 2 and only one way to make 12, but six different ways to make 7 — (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). That is why 7 is the most common roll in games like craps: P(7) = 6/36 = 1/6 ≈ 16.67%. As you add more dice, the distribution of sums bunches even more tightly around the middle and takes on the familiar bell shape (a consequence of the Central Limit Theorem).
Two-dice reference table
These are the exact probabilities for the sum of two standard dice — the single most common dice-probability question:
| Sum | Ways | Probability | Percent |
|---|---|---|---|
| 2 | 1 | 1/36 | 2.78% |
| 3 | 2 | 1/18 | 5.56% |
| 4 | 3 | 1/12 | 8.33% |
| 5 | 4 | 1/9 | 11.11% |
| 6 | 5 | 5/36 | 13.89% |
| 7 | 6 | 1/6 | 16.67% |
| 8 | 5 | 5/36 | 13.89% |
| 9 | 4 | 1/9 | 11.11% |
| 10 | 3 | 1/12 | 8.33% |
| 11 | 2 | 1/18 | 5.56% |
| 12 | 1 | 1/36 | 2.78% |
The 11 possible sums add up to 36/36 = 1, and the table is symmetric around 7 — a sum of 6 is exactly as likely as a sum of 8, and 2 is as likely as 12.
Frequently Asked Questions
What is the probability of rolling a 7 with two dice?
What are the odds of rolling doubles?
Does this calculator work for more than two dice?
What is the difference between "exactly", "at least", and "at most"?
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 |