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Rolling a Die

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Rolling a Die is an important concept in Mathematics and its concepts are highly used in solving various problems of Probability. A die is a solid structure that is a cube with six faces and each of its faces is marked with a number from 1 to 6. And rolling a die there is equal probability that it lands on any of its faces, so each face has an equal probability of appearing. Now there are six faces so the probability of any one face is 1/6. The concept of Rolling a die is used in various games like Snake and Ladders, Ludo, Monopoly, various casino games, and others.

In this article we will learn about, What Rolling a Die means, the Probability of rolling a fair die, and others in detail.

What Does Rolling a Die Mean?

Rolling a die means throwing an unbiased die such that it lands on any of its faces. The die is in the shape of a cube and all its faces are marked from 1 to 6 and the probability of each of its faces is equal and is equal to 1/6.

Every time a die is rolled there are equal probability of landing on any of its face and the six possible outcome of rolling a die is {1, 2, 3, 4, 5, 6}. The image added below shows various faces of a die.

Rolling a Die

Probability of Rolling a Fair Dice

Rolling a die we get any number from 1 to 6. The outcome of any number in case of rolling a single die is 1/6. This is calculated using the formula,

Probability = (Favorable Value)/(Total Values)

If we throw 2 dies we the possible outcome 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)}

These all possible outcome have equal probability. Suppose if we throw two dice and the outcome is (1, 5) then the probability of occurring (1, 5) is 1/36.

Probability of a Number on Dice

We see that there are six faces in a die and rolling a dice we get any one number on the top and the probability of appearing each number on the top is equal. If we throw a die then the probability of any number that comes on the top is 1/6 where we have one favorable and six total event.

For example, if we roll a die then the probability of occurring 6 is,

Favourable Outcome = 6

Total Outcome = {1, 2, 3, 4, 5, 6}

Probability of occurring 6 = 1/6

Possible Outcomes when Two Dice are Rolled

A standard die has six sides numbering 1, 2, 3, 4, 5, and 6. If the die is fair, then each of these outcomes is equally likely event. Since there are six possible outcomes. The probability of getting any side of the die is 1/6. The probability of obtaining a 1 is 1/6, the probability of obtaining a 2 is 1/6, and so on.

The number of total possible outcomes is equal to the total numbers of the first die (6) multiplied by the total numbers of the second die (6), which is 36. So, the total possible outcomes when two dice are thrown together is 36.  

The equally likely outcomes of rolling two dice are shown below:

(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)

Example: Suppose if we throw two die then what is the probability that the sum of two numbers is 10.

Favourable Outcome where the sum is 10 = (5, 5), (4, 6) and (6, 4)

Total Outcome = 36

Probability that the sum is 10 = 3/36 = 1/12

Rolling a Die Examples

Example 1: If we through a die than what is the probability that we get an even number.

Solution:

Total Outcome = {1, 2, 3, 4, 5, 6}

Favourable Outcome = {2, 3, 5}

Probability of Geting a Prime Number = 3/6 = 1/2

Example 2: If we through two dies than what is the probability that we get an prime numbers on both dies.

Solution:

Total Outcome = 36

Favourable Outcome = {(2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (5, 2), (5, 5)}

Probability of Geting a Prime Number = 7/36

Read More,

Practice Questions on Rolling a Die

Q1. If you roll a die twice, what is the probability of getting a 6 on the second roll given that the first roll was a 6?

Q2. If you roll two die, what is the probability of getting a sum of 10 or more given that at least one of the dice shows a 6?

Q3. Kabir rolls a 6-faced die. What is the probability that he gets a Prime Number?

Q4. If you roll two die, what is the probability of getting a sum of 8 or more?

FAQs on Rolling a Die

1. What is the Meaning of Rolling a Die?

Rolling a Die means that we throw a die with six faces where probability of getting each number is equal.

2. What is the Rolling of a Die an Example of?

Rolling a die is an example Discrete Random Variable. We roll a die when we play Ludo, Snakes and Ladder and others.

3. What are the Probabilities of Rolling a Die?

The probability of occurrence of any one face when we throw a die is 1/6. Where all the possible outcomes are {1, 2, 3, 4, 5, 6}.

4. How many Faces in a Fair Die?

A fair die is a cube and it has six faces.

5. What is the Probability of Rolling a Die?

The probability of occurrence of any number on a die is 1/6.

6. What is the Probability of Rolling 4 on a Die?

The probability of rolling 4 on a die is 1/6.

7. What is the Probability of Rolling an Odd Number in a Die?

Theere are 3 even numbers in a die and probability of rolling an odd number = 3/6 = 1/2.

8. What is the Probability of Rolling an Even Number in a Die?

Theere are 3 odd numbers in a die and probability of rolling an odd number = 3/6 = 1/2.



Last Updated : 17 Nov, 2023
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