What is the probability of getting a sum of 5 or 8?

When rolling two dice, distinguish between them in some way: a first one and second one, a left and a right, a red and a green, etc. Let (a,b) denote a possible outcome of rolling the two die, with a the number on the top of the first die and b the number on the top of the second die. Note that each of a and b can be any of the integers from 1 through 6. Here is a listing of all the joint possibilities for (a,b):

With the sample space now identified, formal probability theory requires that we identify the possible events. These are always subsets of the sample space, and must form a sigma-algebra. In an example such as this, where the sample space is finite because it has only 36 different outcomes, it is perhaps easiest to simply declare ALL subsets of the sample space to be possible events. That will be a sigma-algebra and avoids what might otherwise be an annoying technical difficulty. We make that declaration with this example of two dice.

With the above declaration, the outcomes where the sum of the two dice is equal to 5 form an event. If we call this event E, we have

E={(1,4),(2,3),(3,2),(4,1)}.

Consider next the probability of E, P(E). Here we need more information. If the two dice are fair and independent , each possibility (a,b) is equally likely. Because there are 36 possibilities in all, and the sum of their probabilities must equal 1, each singleton event {(a,b)} is assigned probability equal to 1/36. Because E is composed of 4 such distinct singleton events, P(E)=4/36= 1/9.

In general, when the two dice are fair and independent, the probability of any event is the number of elements in the event divided by 36.

What if the dice aren't fair, or aren't independent of each other? Then each outcome {(a,b)} is assigned a probability (a number in [0,1]) whose sum over all 36 outcomes is equal to 1. These probabilities aren't all equal, and must be estimated by experiment or inferred from other hypotheses about how the dice are related and and how likely each number is on each of the dice. Then the probability of an event such as E is the sum of the probabilities of the singleton events {(a,b)} that make up E.

Probability is a numerical description of how likely an event is to occur. The probability of an event is in the range from 0 to 1 where 0 represents the impossibility of the event and 1 represents certainty over the thing. When the probability is higher, then there are more chances to occur the event. 

Terms used in Probability

The terms used in probability are experiment, random experiment, sample space, outcome, and event. Let’s take a look at the  definitions of these terms in brief,

• Experiment: An operation that produces some outcomes.

Example When we throw a die, there will be 6 numbers from which anyone can be up. So, the operation of rolling a die may be said to have 6 outcomes.

• Random Experiment: An operation in which all possible outcomes are known but the exact outcome is not predictable.

Example When we throw a die there can be 6 outcomes but we cannot say the exact number which will show up.

• Sample Space: All possible outcomes of an operation.

Example When we throw a die there can be six possible outcomes that is from {1,2,3,4,5,6} and represented by S.

• Outcome: Any possible result out of the Sample Space S.

Example When we throw a die, we might get 6.

• Event: Subset of a sample space that has to occur when an outcome belongs to an event and is represented by E.

Example When we roll a die there are six sample spaces {1, 2, 3, 4, 5, 6}. Let’s E occurs when “number is divisible by 2” then E ={2, 4, 6}. If the outcome is {2} which is a subset of E so it is considered an event that occurs otherwise event does not occur. Let’s look at the formula for an event occurring,

Probability of an event occur = Number of outcomes / Sample Space

What is the probability of getting a sum of 5 or 6 when a pair of dice is rolled?

Solution: 

Sample Space of one dice = 6

Sample Space of 2 dice = 6 × 6 = 36

Number of outcomes for sum of 5 = 4 {(1, 4), (2, 3), (3, 2), (4, 1)}

Number of outcomes for sum of 6 = 5 {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

Total Outcomes = 4 + 5 = 9

Probability of getting a sum of 5 or 6 = 9/36 = 1/4.

Sample Problems

Question 1: Probability of getting at least (minimum) one head while tossing two coins simultaneously.

Solution: 

Sample Space of one coin = 2

Sample Space of 2 coins = 2 × 2= 4

Number of outcomes for at least one head = 3 {(H, T),(T, H),(H, H)}

Probability of getting at least one head = 3/4.

Question 2: Probability of getting a sum of even number while rolling two dice.

Solution: 

Sample Space of one dice = 6

Sample Space of 2 dice = 6 × 6 = 36

Number of outcomes to get a sum of multiple of 4 = 9 ((1, 3),(2, 2),(2, 6),(3, 1),(3, 5),(4, 4),(5, 3),(6, 2),(6, 6))

What is the probability of rolling 5 or 8 on two dice?

Two (6-sided) dice roll probability table.

What is the probability of getting a sum of 5?

The probability of getting a sum of 5 on rolling two dice is 1/11.

What is the probability of getting a sum of 5 or a sum of 6?

What is the probability of getting a sum of 5 or 6 when a pair of dice is rolled? Probability of getting a sum of 5 or 6 = 9/36 = 1/4.

What is the probability of getting a sum of 8?

Probability of getting a sum of 8 is 5/36.