Two Dice Probability: Calculate Sums and Outcomes

This two-dice probability calculator finds the chance of a specified result when two fair, six-sided dice are rolled. Enter the number of favorable outcomes and the total possible outcomes to get the probability as a percentage; the defaults show the chance of rolling a sum of 7.

Formula

Probability is the number of favorable outcomes divided by the total number of equally likely outcomes. To express it as a percentage, multiply that fraction by 100:

P = F / T; percentage = 100 × F / T

• F is the count of outcomes that meet the condition, with no unit.
• T is the count of all possible outcomes, with no unit.
• P is the probability, a fraction from 0 to 1; the percentage is the same probability on a 0%–100% scale.

For two fair six-sided dice, each die has six faces and the dice rolls are independent. There are 6 × 6 = 36 equally likely ordered outcomes, so T is 36. The calculator’s default F of 6 represents the six ways to make a sum of 7.

Worked Example

Using the calculator defaults, find the chance of a sum of 7:

1. Total outcomes: 6 × 6 = 36.
2. Favorable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), so F = 6.
3. Probability: 6 / 36 = 1 / 6 = 0.1667 when rounded to four decimal places.
4. Percentage: 100 × 6 / 36 = 16.67% when rounded to two decimal places.

The calculator therefore returns 16.67% and a probability fraction of 0.1667. The exact fraction is 1/6; the decimal and percentage are rounded representations.

How Two-Dice Probability Works

For two distinguishable dice, count each result as an ordered pair: the first number is the first die and the second number is the other. For example, (2,5) and (5,2) are separate outcomes, even though both have a sum of 7. This is why there are 36 equally likely pairs rather than just 21 visually distinct combinations if order is ignored.

To calculate a probability, first define the event precisely, then list or count the pairs that satisfy it. Divide that count by 36. For a standard sum, the possible totals run from 2 through 12, but they do not all have the same probability:

• A sum of 2 has 1 outcome: (1,1), so its probability is 1/36, or about 2.78%.
• A sum of 6 has 5 outcomes, so its probability is 5/36, or about 13.89%.
• A sum of 7 has 6 outcomes, so its probability is 6/36, or about 16.67%.
• A sum of 12 has 1 outcome: (6,6), so its probability is 1/36, or about 2.78%.

Seven is the most likely total because it can be formed by more pairs than any other sum. This does not make it certain, or guarantee that it will appear within a small number of rolls.

Using the Calculator for Other Events

The calculator works for any event as long as you enter matching counts: F must be the number of favorable ordered outcomes, and T must be the total number of possible ordered outcomes. For two fair six-sided dice, T is usually 36. For example, the event “at least one die shows a 6” has 11 favorable pairs: six pairs with the first die showing 6, six with the second showing 6, minus the one pair (6,6) counted twice. Its probability is 11/36, or about 30.56%.

Other examples include doubles, an even sum, or a sum greater than 9. Count the pairs meeting the condition rather than counting only the possible totals. If a question concerns a single die, a different number of dice, or dice with more or fewer faces, update the total and favorable counts to match that situation.

Common Counting Mistakes and Limits

• Counting totals instead of outcomes: There are 11 possible sums, from 2 to 12, but they are not equally likely. Dividing one by 11 does not give the chance of a particular sum.
• Ignoring order: If you list only unordered pairs, account for the fact that (2,5) and (5,2) are distinct rolls. Doubles have only one ordering; most other pairs have two.
• Using the wrong total: Thirty-six assumes two independent, fair, six-sided dice. A different number of dice or faces changes the sample space.
• Rounding too early: Keep the exact fraction while calculating, then round the final decimal or percentage. A rounded percentage may not convert back to an exact count.

The calculation assumes each face is equally likely and that one die’s result does not affect the other. It describes the theoretical chance for a roll, not a prediction of the next roll. If dice are biased or the rolls are dependent, the 36 pairs may not be equally likely, and a simple favorable-count ratio is not sufficient.

Frequently Asked Questions

How many outcomes are there when rolling two dice?
There are 36 ordered outcomes when each of two six-sided dice can show 1 through 6: 6 × 6 = 36.

What is the probability of rolling a 7 with two dice?
Six of the 36 ordered outcomes sum to 7, so the probability is 6/36 = 1/6, or about 16.67%.

Are all sums from 2 to 12 equally likely?
No. A sum of 7 has six possible ordered pairs, while sums of 2 and 12 each have only one. The number of combinations varies by total.

What is the chance of rolling doubles?
There are six doubles: (1,1) through (6,6). The chance is 6/36 = 1/6, or about 16.67%.

Does rolling the same result repeatedly change the next roll’s probability?
Not for independent rolls of fair dice. Each new roll has the same probabilities, regardless of earlier results.

Why count (2,5) and (5,2) separately?
The dice are treated as distinguishable outcomes, so one die showing 2 while the other shows 5 is a different ordered result from the reverse. Both happen to have the same sum.

Related

• Predictive Value: Calculate PPV and NPV
• Calculate Percentile Rank: Formula and Examples
• Mann-Whitney U Test: Formula and Calculation

in f | Favorable outcomes | - | 6 | 0
in t | Total outcomes | - | 36 | 1
out p | Probability | % | 100 * f / t | 2
out r | Probability as a fraction | - | f / t | 4