Probability and Statistics > Probability > Dice rolling probability: Statistics and Dice

## Dice rolling probability: Statistics and Dice Overview

It’s very common to find questions about dice rolling in **probability and statistics**. You might be asked the probability of rolling a five and a seven, or rolling a twelve, or rolling a double-six. While you *could* technically use a formula or two (like a combinations formula), you really have to understand each number that goes into the formula. And that’s not always simple. By far the easiest (visual) way to solve these types of problems (ones that involve finding the probability of rolling a certain combination or set of numbers) is by writing out a **sample space**.

## Dice Rolling Probability: Sample Space

A sample space is just the set of all possible probabilities. In simple terms, **you have to figure out every possibility for what might happen.** With dice rolling, your sample space is going to be every possible dice roll.

For example, in order to know what the odds are of rolling a 4 or a 7 from a set of two dice, we would first need to find out what all the possible combinations. We could roll a double one [1][1], or a one and a two [1][2]. In fact, there are 36 possible combinations.

## Dice Rolling Probability: Steps

**Step 1:** Write out your sample space. For two dice, the **36** different possibilities are:

[1][1], [1][2], [1][3], [1][4], [1][5], [1][6],

[2][1], [2][2], [2][3], [2][4], [2][5], [2][6],

[3][1], [3][2], [3][3], [3][4], [3][5], [3][6],

[4][1], [4][2], [4][3], [4][4], [4][5], [4][6],

[5][1], [5][2], [5][3], [5][4], [5][5], [5][6],

[6][1], [6][2], [6][3], [6][4], [6][5], [6][6]

**Step 2:** Look at your sample space and find how many add up to 4 or 7 (because we’re looking for the probability of rolling one of those numbers).

[1][1], [1][2], **[1][3],** [1][4], [1][5], **[1][6],**

[2][1],** [2][2],** [2][3], [2][4],**[2][5]**, [2][6],

**[3][1],** [3][2], [3][3], **[3][4],** [3][5], [3][6],

[4][1], [4][2], **[4][3],** [4][4], [4][5], [4][6],

[5][1], **[5][2],** [5][3], [5][4], [5][5], [5][6],**
[6][1],** [6][2], [6][3], [6][4], [6][5], [6][6].

There are **9** possible combinations.

**Step 3:** Take the answer from step 2, and divide it by the size of your total sample space from step 1:

9 / 36 = **.25**

You’re done!

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