- 6 Sided Dice probability (worked example for two dice).
- Two (6-sided) dice roll probability table
- Single die roll probability tables.

Watch the video for three examples:

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## Dice roll probability: 6 Sided Dice Example

It’s very common to find questions about dice rolling in **probability and statistics**. You might be asked the probability of rolling a variety of results for a 6 Sided Dice: five and a seven, a double twelve, or 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 Roll Probability for 6 Sided Dice: Sample Spaces

A sample space is just **the set of all possible results**. 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.

**Example question**: What is the probability of rolling a 4 or 7 for two 6 sided dice?

In order to know what the odds are of rolling a 4 or a 7 from a set of two dice, you first need to find out all the possible combinations. You 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 (i.e. all of the possible results). 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). The rolls that add up to 4 or 7 are in **bold**:

[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. What I mean by the “size of your sample space” is just all of the possible combinations you listed. In this case, Step 1 had 36 possibilities, so:

9 / 36 = **.25**

You’re done!

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## Two (6-sided) dice roll probability table

The following table shows the probabilities for rolling a certain number with a two-dice roll. If you want the probabilities of rolling a set of numbers (e.g. a 4 and 7, or 5 and 6), add the probabilities from the table together. For example, if you wanted to know the probability of rolling a 4, or a 7:

3/36 + 6/36 = 9/36.

Roll a… | Probability |
---|---|

2 | 1/36 (2.778%) |

3 | 2/36 (5.556%) |

4 | 3/36 (8.333%) |

5 | 4/36 (11.111%) |

6 | 5/36 (13.889%) |

7 | 6/36 (16.667%) |

8 | 5/36 (13.889%) |

9 | 4/36 (11.111%) |

10 | 3/36 (8.333%) |

11 | 2/36 (5.556%) |

12 | 1/36 (2.778%) |

Probability of rolling a certain number *or less* for two 6-sided dice.

Roll a… | Probability |
---|---|

2 | 1/36 (2.778%) |

3 | 3/36 (8.333%) |

4 | 6/36 (16.667%) |

5 | 10/36 (27.778%) |

6 | 15/36 (41.667%) |

7 | 21/36 (58.333%) |

8 | 26/36 (72.222%) |

9 | 30/36 (83.333%) |

10 | 33/36 (91.667%) |

11 | 35/36 (97.222%) |

12 | 36/36 (100%) |

## Dice Roll Probability Tables

**Contents:**

1. Probability of a certain number (e.g. roll a 5).

2. Probability of rolling a certain number *or less* (e.g. roll a 5 *or* less).

3. Probability of rolling *less than a certain number* (e.g. roll less than a 5).

4. Probability of rolling a certain number *or more* (e.g. roll a 5 *or *more).

5. Probability of rolling *more than a certain number* (e.g. roll more than a 5).

## Probability of a certain number with a Single Die.

Roll a… | Probability |
---|---|

1 | 1/6 (16.667%) |

2 | 1/6 (16.667%) |

3 | 1/6 (16.667%) |

4 | 1/6 (16.667%) |

5 | 1/6 (16.667%) |

6 | 1/6 (16.667%) |

## Probability of rolling a certain number *or less* with one die

.

Roll a…or less | Probability |
---|---|

1 | 1/6 (16.667%) |

2 | 2/6 (33.333%) |

3 | 3/6 (50.000%) |

4 | 4/6 (66.667%) |

5 | 5/6 (83.333%) |

6 | 6/6 (100%) |

## Probability of rolling *less than certain number* with one die

.

Roll less than a… | Probability |
---|---|

1 | 0/6 (0%) |

2 | 1/6 (16.667%) |

3 | 2/6 (33.33%) |

4 | 3/6 (50%) |

5 | 4/6 (66.667%) |

6 | 5/6 (83.33%) |

## Probability of rolling a certain number *or more*.

Roll a…or more | Probability |
---|---|

1 | 6/6(100%) |

2 | 5/6 (83.333%) |

3 | 4/6 (66.667%) |

4 | 3/6 (50%) |

5 | 2/6 (33.333%) |

6 | 1/6 (16.667%) |

## Probability of rolling *more than a certain number* (e.g. roll more than a 5).

Roll more than a… | Probability |
---|---|

1 | 5/6(83.33%) |

2 | 4/6 (66.67%) |

3 | 3/6 (50%) |

4 | 4/6 (66.667%) |

5 | 1/6 (66.67%) |

6 | 0/6 (0%) |

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## References

Dodge, Y. (2008). The Concise Encyclopedia of Statistics. Springer.

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.

Salkind, N. (2016). Statistics for People Who (Think They) Hate Statistics: Using Microsoft Excel 4th Edition.