# Fair Dice

## Fair Dice Meaning

A fair 6-sided die is typically numbered from 1 through 6 (standard), meaning that a single roll will give you one of any of those numbers: 1, 2, 3, 4, 5, or 6. This fair six-sided die consists of a 1/6 chance of a 2 and a 1/6 chance of a 5, and because they’re mutually exclusive, you just add them together to obtain a 1/3 chance of obtaining either a 2 or 5.

Some other fair dice roll would have some other possibility. For a fair 7 sided die (1 through 7), it’s 1/7 for each.

### Use of Fair Dice

A die (plural "dice") is any solid object that has markings on each face that can be used to form a random number. a fair dice roll is quite useful when playing games of chance!

### Professional Dice

For very fair dice there are casino-grade dice that consists of special qualities. Also referred to as a casino dice, a professional dice has the following characteristics:

• Partially transparent that we can even see any bubbles or weights inside.

• Closest possible to exact cubes.

• Sharp corners and edges.

• Dots filled in, so the weight is similar on each side.

### Physical Characteristics of a Fair Sided Dice

The ideal world of mathematics is quite in contrast to reality. That being said the Physical characteristics of a fair dice sides include:

• A regular white and black playing dice has holes for the number dots that are drilled and then filled with paint—a paint which is lighter than the substance of the die, so the one side is ever-so-slightly heavier than the 6th side.

• All fair dice contains an even number of sides.

Note: The type of rolling surface and how used or weary the dice impacts the outcome of the roll.

### Single Fair Die

In Mathematics, a single fair die or one fair dice is what we mean that there is an equally likely chance of landing on any face.  So long as there is an equal possibility of landing on any face then they are Fair Dice.

### Two Fair Dice

If the two dice ('singular' 2 -fair die) are fair and independent, each having an equally likely possibility, (a,b). Usually, when the two dice are fair and independent, the chances of any event occurring are the number of elements in the event divided by 36.

### Solved Examples

Example:

When a fair die is rolled one time what is the probability of getting any one of the following outcomes:

{1, 2, 3, 4, 5, 6}

Solution:

N(S) = 6

Probability of getting 2 = 1/6

Probability of getting 5 = 1/6

Probability of getting 2 or 5 is 1/6 + 1/6 = 2/6

= 1/3 and that’s.33333, which is a probability of.

= 33.3%.

Example:

Find out the probability of either rolling a 5 or a 6 on a pair of fair dice?

Solution:

There are basically 4 ways to make a 5, and 5 ways to make a 6.

Thus, the possibility of throwing either a 5 or a 6 on any given fair dice roll is 9/36, or 1 in 4 or say,

To make 5 you would require (2, 3) or (3, 2) or (1, 4) or (4, 1)

To make 6 you need (3, 3) or (2, 4) or (4, 2), (1, 5) or (5, 1)

Therefore, there are a total of 9 ways to make 5 or 6.

However, in total there are 36 possible outcomes,

Thus, we get 9/36 = 1/4 or say 0.25.

Not really a good bet then.

### Fun Facts

• A coin is also a fair die that has a 50-50 probability for two alternatives.

• All the 5 Platonic solids are fair dice.

• There are a total of 30 families of fair dice.

• A 20-sided Roman die is 1,800 years old.

• A class of shape known as "trapezohedron" can be stretched out to any even number of faces that also acts as a fair dice.

Q1. When is a Die Called Fair?

Answer: A die is termed as being fair by symmetry "if and only if its symmetry category acts transitively on its faces". This states that you could place each individual face of the die on top and the entire geometric shape of the die remains symmetrical and unaltered.

Q2. What Makes Dice Fair?

Answer: At its simplest, a fair die states that each of the faces has a similar probability of landing facing up. A standard fair six-sided die, for example, can be regarded as "fair" if each of the faces consists of a probability of 1/6. But there are a number of factors that come into play, and things get more complicated when you begin considering fair dice shapes other than a cube.

Q3. What are the Factors that Make a Dice Fair?

Answer: There are 3 main factors that impact whether a dice roll is fair. First, of all, it is the geometric shape of the dice. Secondly, it is the physics of the dice roll. Third, is a real-world environment, such as the surface you are rolling on. However, the most crucial factor is whether a die is "fair by symmetry".

Moreover, there are specific ways in which one die can be "more fair" than another. This becomes clear when you take into account how many "symmetries" a die has, or, simply to put, how many ways you are able to rotate the die so that its shape remains unaltered although different parts of the die end up in different positions. A six-sided cube die, for example, has 24 different symmetries. A 4-sided die also known as a tetrahedral die, in the shape of a pyramid, consists of only 12 different symmetries.