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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 doublesix. 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 (6sided) dice roll probability table
The following table shows the probabilities for rolling a certain number with a twodice 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 6sided 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. 
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Comments? Need to post a correction? Please Contact Us. A branch of mathematics that deals with the happening of a random event is termed probability. It is used in Maths to predict how likely events are to happen. The probability of any event can only be between 0 and 1 and it can also be written in the form of a percentage. The probability of event A is generally written as P(A). Here P represents the possibility and A represents the event. It states how likely an event is about to happen. The probability of an event can exist only between 0 and 1 where 0 indicates that event is not going to happen i.e. Impossibility and 1 indicates that it is going to happen for sure i.e. Certainty
If we are not sure about the outcome of an event, we take help of the probabilities of certain outcomes—how likely they occur. For a proper understanding of probability we take an example as tossing a coin: There will be two possible outcomes—heads or tails. The probability of getting heads is half. You might already know that the probability is half/half or 50% as the event is an equally likely event and is complementary so the possibility of getting heads or tails is 50%. Formula of Probability
Probability of an event, P(A) = Favorable outcomes / Total number of outcomes
Some Terms of Probability Theory Experiment: An operation or trial done to produce an outcome is called an experiment.
 Sample Space: An experiment together constitutes a sample space for all the possible outcomes. For example, the sample space of tossing a coin is head and tail.
 Favourable Outcome: An event that has produced the required result is called a favourable outcome. For example, If we roll two dice at the same time then the possible or favourable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).
 Trial: A trial means doing a random experiment.
 Random Experiment: A random experiment is an experiment that has a welldefined set of outcomes. For example, when we toss a coin, we would get ahead or tail but we are not sure about the outcome that which one will appear.
 Event: An event is the outcome of a random experiment.
 Equally Likely Events: Equally likely events are rare events that have the same chances or probability of occurring. Here The outcome of one event is independent of the other. For instance, when we toss a coin, there are equal chances of getting a head or a tail.
 Exhaustive Events: An exhaustive event is when the set of all outcomes of an experiment is equal to the sample space.
 Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events. For example, the climate can be either cold or hot. We cannot experience the same weather again and again.
 Complementary Events: The Possibility of only two outcomes which is an event will occur or not. Like a person will eat or not eat the food, buying a bike or not buying a bike, etc. are examples of complementary events.
Addition rule: Union of two events, say A and B, then P(A or B) = P(A) + P(B) – P(A∩B) P(A ∪ B) = P(A) + P(B) – P(A∩B)
Complementary rule: If there are two possible events of an experiment so the probability of one event will be the Complement of another event. For example – if A and B are two possible events, then P(B) = 1 – P(A) or P(A’) = 1 – P(A). P(A) + P(A′) = 1.
Conditional rule: When the probability of an event is given and the second is required for which first is given, then P(B, given A) = P(A and B), P(A, given B). It can be vice versa P(B∣A) = P(A∩B)/P(A)
Multiplication rule: Intersection of two other events i.e. events A and B need to occur simultaneously. Then P(A and B) = P(A)⋅P(B). P(A∩B) = P(A)⋅P(B∣A)
Find the probability of getting a number less than 5 in a single dice throw.Solution : When the dice is rolled then there will be 6 outcomes. Total number of favorable outcome { set of outcome } = {1, 2, 3, 4, 5, 6 } = 6 Now as per the question Probability of getting a number less than 5 in a single throw is 4 Numbers less than 5 are { 1,2,3,4} therefore favorable outcome will be = 4 P(A) = Favorable outcomes / Total number of outcomes = 4/6 = 2/3
Hence the probability of getting a number less than 5 in a single throw of a die is 2/3
Similar QuestionsQuestion 1: Find the probability of getting a number less than 4 in a single throw of a die. Solution: When the dice is rolled then there will be 6 outcomes Total number of favorable outcome { set of outcome } = {1 ,2 ,3 ,4 , 5, 6 } = 6 Now as per the question probability of getting a number less than 4 in a single throw is 3 Numbers less than 4 are { 1,2,3} Therefore favorable outcome will be = 3 P(A) = Favorable outcomes / Total number of outcomes = 3/6 = 1/2 Hence the probability of getting a number less than 4 in a single throw of a die is 1/2
Question 2: Find the probability of getting a number more than 4 in a single throw of a die. Solution: When the dice is rolled then there will be 6 outcomes. Total number of favorable outcome { set of outcome } = {1 ,2 ,3 ,4 , 5, 6 } = 6 Now as per the question probability of getting a number more than 4 in a single throw is 2 Numbers more than 4 are { 5,6} Therefore favorable outcome will be = 2 P(A) = Favorable outcomes / Total number of outcomes
= 2/6 = 1/3 Hence the probability of getting a number more than 4 in a single throw of a die is 1/3
Question 3: Find the probability of getting a number 5 in a single throw of a die. Solution: When the dice is rolled then there will be 6 outcomes. Total number of favorable outcome { set of outcome } = {1 ,2 ,3 ,4 , 5, 6 } = 6 Now as per the question probability of getting a number 5 in a single throw is 1 Therefore favorable outcome will be = 1 P(A) = Favorable outcomes / Total number of outcomes = 1/6 Hence the probability of getting a number 5 in a single throw of a die is 1/6
Question 4: What is the chance of rolling a 3 two times in a row? Solution: P(A) = Favorable outcomes / Total number of outcomes Probability of getting 3 = 1/6. Rolling dice is an independent event, it is not dependent on how many times it’s been rolled. Probability of getting 3 two times in a row = probability of getting 3 first time × probability of getting 3 second time. Probability of getting 3 two times in a row = (1/6) × (1/6) = 1/36. Hence, the probability of getting 3 two times in a row 2.77 %.
