Statistics How To

Dice Rolling Probability: Statistics and Dice

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

dice rolling probability

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][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!

Like the explanation? Check out our Practically Cheating Statistics Handbook for hundreds more solved problems.

Check out our statistics YouTube channel for hundreds of probability and statistics help videos!


If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Comments are now closed for this post. Need help or want to post a correction? Please post a comment on our Facebook page and I'll do my best to help!
Dice Rolling Probability: Statistics and Dice was last modified: October 12th, 2017 by Stephanie