You Roll 2 Fair Six-Sided Dice Calculate The Following Probabilities

When you roll two fair six-sided dice, there are 36 possible outcomes (since each die has 6 faces and the dice are independent). This calculator helps you determine probabilities for various scenarios involving two dice.

Introduction

Rolling two six-sided dice is a fundamental probability problem that appears in many games and statistical applications. Understanding the probabilities involved can help you make better decisions in games like craps, backgammon, or even when analyzing random events.

This guide will explain how to calculate probabilities when rolling two fair six-sided dice, including probabilities for sums, specific numbers, and other common scenarios.

Basic Probabilities

The basic probabilities when rolling two dice are:

Probability of any specific number (e.g., rolling a 3 on one die): 1/6 or approximately 16.67%
Probability of any specific combination (e.g., rolling a 2 on the first die and a 5 on the second die): 1/36 or approximately 2.78%
Probability of any specific sum (e.g., rolling a sum of 7): varies (see next section)

Probability of a specific number: P(number) = 1/6 ≈ 0.1667 or 16.67%

Probability of a specific combination: P(combination) = 1/36 ≈ 0.0278 or 2.78%

Probability of Sums

The probability of rolling a specific sum when two dice are rolled can be calculated using the following formula:

Probability of sum S: P(S) = Number of ways to get sum S / Total number of outcomes

Total number of outcomes = 6 × 6 = 36

Here are the probabilities for each possible sum:

Sum	Number of Ways	Probability
2	1	1/36 ≈ 2.78%
3	2	2/36 ≈ 5.56%
4	3	3/36 ≈ 8.33%
5	4	4/36 ≈ 11.11%
6	5	5/36 ≈ 13.89%
7	6	6/36 ≈ 16.67%
8	5	5/36 ≈ 13.89%
9	4	4/36 ≈ 11.11%
10	3	3/36 ≈ 8.33%
11	2	2/36 ≈ 5.56%
12	1	1/36 ≈ 2.78%

For example, the probability of rolling a sum of 7 is 6/36 or approximately 16.67%. This is the most likely sum when rolling two dice.

Probability of Specific Numbers

You can also calculate the probability of rolling specific numbers on each die or combinations of numbers.

Probability of specific numbers: P(A and B) = (Number of ways to get A on first die × Number of ways to get B on second die) / Total number of outcomes

Since each die is independent, P(A and B) = P(A) × P(B) = (1/6) × (1/6) = 1/36

For example, the probability of rolling a 3 on the first die and a 5 on the second die is 1/36 or approximately 2.78%.

Comparison Table

Here's a comparison table showing the probabilities for different scenarios:

Scenario	Probability	Example
Rolling a specific number on one die	1/6 ≈ 16.67%	Probability of rolling a 4 on one die
Rolling a specific sum	Varies (2/36 to 6/36)	Probability of rolling a sum of 7
Rolling specific numbers on both dice	1/36 ≈ 2.78%	Probability of rolling a 2 on the first die and a 6 on the second die
Rolling at least one specific number	Varies	Probability of rolling at least one 5

FAQ

What is the probability of rolling a sum of 7 with two dice?: The probability of rolling a sum of 7 is 6/36 or approximately 16.67%. This is because there are 6 ways to roll a sum of 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).
What is the probability of rolling doubles (both dice show the same number)?: The probability of rolling doubles is 6/36 or approximately 16.67%. There are 6 possible doubles (2-2, 3-3, 4-4, 5-5, 6-6).
What is the probability of rolling a sum of 2 or 12?: The probability of rolling a sum of 2 or 12 is 2/36 or approximately 5.56%. There is only 1 way to roll a sum of 2 (1+1) and 1 way to roll a sum of 12 (6+6).
What is the probability of rolling at least one 5?: The probability of rolling at least one 5 is 11/36 or approximately 30.56%. This can be calculated by finding the probability of not rolling any 5s (25/36) and subtracting it from 1.