    # What is the probability of getting sum of 8 with two dice? Contents:

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Probability: Dice Rolling 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 , or a one and a two . 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:

, , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

, , , , , .

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:

, , , , , ,
, , , ,, ,
, , , , , ,
, , , , , ,
, , , , , ,
,
, , , , .

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!

## 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%)

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

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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.

---------------------------------------------------------------------------  Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Probability means Possibility. 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.

The higher or lesser the probability of an event, the more likely it is that the event will occur or not respectively. For example – An unbiased coin is tossed once. So the total number of outcomes can be 2 only i.e. either “heads” or “tails”. The probability of both outcomes is equal i.e. 50% or 1/2.

So, the probability of an event is Favorable outcomes/Total number of outcomes. It is denoted with the parenthesis i.e. P(Event).

P(Event) = N(Favorable Outcomes) / N (Total Outcomes)

Note: If the probability of occurring of an event A is 1/3 then the probability of not occurring of event A is 1-P(A) i.e. 1- (1/3) = 2/3

What is Sample Space?

All the possible outcomes of an event are called Sample spaces.

Examples-

• A six-faced dice is rolled once. So, total outcomes can be 6 and
Sample space will be [1, 2, 3, 4, 5, 6]
• An unbiased coin is tossed, So, total outcomes can be 2 and
Sample space will be [Head, Tail]
• If two dice are rolled together then total outcomes will be 36 and
Sample space will be  [(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) ]

### Types of Events

Independent Events: If two events (A and B) are independent then their probability will be

P(A and B) = P (A ∩ B) = P(A).P(B) i.e. P(A) * P(B)

Example: If two coins are flipped, then the chance of both being tails is 1/2 * 1/2 = 1/4

Mutually exclusive events:

• If event A and event B can’t occur simultaneously, then they are called mutually exclusive events.
• If two events are mutually exclusive, then the probability of both occurring is denoted as P (A ∩ B) and
P (A and B) = P (A ∩ B) = 0
• If two events are mutually exclusive, then the probability of either occurring is denoted as P (A ∪ B)
P (A or B) = P (A ∪ B)
= P (A) + P (B) − P (A ∩ B)
= P (A) + P (B) − 0
= P (A) + P (B)

Example: The chance of rolling a 2 or 3 on a six-faced die is P (2 or 3) = P (2) + P (3) = 1/6 + 1/6 = 1/3

Not Mutually exclusive events: If the events are not mutually exclusive then

P (A or B) = P (A ∪ B) = P (A) + P (B) − P (A and B)

What is Conditional Probability?

For the probability of some event A, the occurrence of some other event B is given. It is written as P (A ∣ B)

P (A ∣ B) = P (A ∩ B) / P (B)

Example- In a bag of 3 black balls and 2 yellow balls (5 balls in total), the probability of taking a black ball is 3/5, and to take a second ball, the probability of it being either a black ball or a yellow ball depends on the previously taken out ball. Since, if a black ball was taken, then the probability of picking a black ball again would be 1/4, since only 2 black and 2 yellow balls would have been remaining, if a yellow ball was taken previously, the probability of taking a black ball will be 3/4.

Solution:

When two dice are rolled together then total outcomes are 36 and
Sample space is [ (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) ]

So, pairs with sum 8 are (2, 6) (3, 5) (4, 4) (5, 3) (6, 2) i.e. total 5 pairs

So, in 5 ways we can roll a 8 with two dice.

### Similar Questions

Question 1: How many ways can you roll a 7 with two dice?

Solution:

When two dice are rolled together then total outcomes are 36 and
Sample space is [ (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) ]

So, pairs with sum 7 are (1,6) (2, 5) (3,4) (4, 3) (5, 2) (6,1) i.e. total 6 pairs

So, in 6 ways we can roll a 7 with two dice.

Question 2: How many ways can you roll a 6 with two dice?

Solution:

When two dice are rolled together then total outcomes are 36 and
Sample space is [ (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) ]

So, pairs with sum 6 are (1,5) (2, 4) (3,3) (4, 2) (5, 1) i.e. total 5 pairs

So, in 5 ways we can roll a 6 with two dice.

Question 3: How many ways can you roll a 5 with two dice?

Solution:

When two dice are rolled together then total outcomes are 36 and
Sample space is [ (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) ]

So, pairs with sum 5 are (1,4) (2, 3) (3,2) (4, 1) i.e. total 4 pairs

So, in 4 ways we can roll a 5 with two dice. 