Fair Dice

Dice are usually cubes of consistent material. Symmetry suggests that a consistent cube has an equivalent likelihood of landing on every one of its six faces after a vigorous roll, so it is said to be fair.

Is there any other Fair Polyhedral?

Firstly, polyhedral means a three-dimensional object composed of a finite number of 2-dimensional figure faces, essentially a boundary between the interior and exterior of a solid. 

Similarly, the four different regular solids are:

• A tetrahedron may be a 3-dimensional form or an object that has 4 triangular faces.

• An octahedron may be a polyhedron with 8 faces, 12 edges, and half-a-dozen vertices.

• The dodecahedron is any polyhedron with 12 flat faces (platonic).

• Icosahedron may be a polyhedron with 20 faces. 

Are these Fair Solids?

To answer this question, we need to understand what is meant by fair. 

This means that any face will be remodeled into any other face by a rotation, a mirror image, or a combined rotation and reflection, which turns the polyhedron into itself. 

There are other polyhedra that are fair, but not fair by symmetry.

Fair Dice Meaning

Fair dice mean that each of the faces has the same probability of landing facing up. A standard 6-sided die, for example, can be considered “fair” when each of the faces has the same probability of ⅙. 

“FAIR” meaning in mathematics: A probability experiment may be considered “FAIR” when all the outcomes are equivalent or when the expected value of some random variable is 0.

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 6-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.

Factors Affecting whether the Rolled Die is Fair

There are many factors which make things get more complicated when you start considering shapes for dice, other than a cube.

There are 3 main factors that influence whether a die rolled is fair:

1. It is the geometric shape of dice.

2. It is the physics of the roll.

3. It is the real-world environment, like the surface you are rolling on.

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 consist of special qualities. Also referred to as a casino die, a professional die 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 that the weight is similar on each side 

Spherical Dice

A die can be of spherical shape with the addition of an internal cavity in the shape of a dual polyhedron. The weight of the die will settle in one of the points of the internal cavity and a small internal weight will settle with one point of the cavity held downwards by the weight.

Physical Characteristics of a Fair-Sided Die

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

• A regular white and black playing die 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 contain an even number of sides.

Note: The type of rolling surface and how used or weary the die is impact the outcome of the roll.

Single Fair Die

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

Two Fair Dice

If the two dice ('singular' 2 - fair die) are fair and independent, each has 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.

36 Outcomes with 2 Dice

When you roll 2 dice, there can now be 36 different ways with which the dice can come up. You can conclude that the figure arrived at by multiplying the number of ways the first dice can come up is 6, the number of ways the second dice can come up is also 6. So, 6X6 is 36.

How to Calculate Probability?

Step 1 - Determine a single event with a single outcome.

Step 2 - Identify the total number of outcomes that can occur.

Step 3 - Divide the number of events by the number of possible outcomes.

One Die Rolled

Learning to calculate dice probability is the opportunity of getting a finite number with one dice (1, 2, 3, 4, 5, and 6). The basic rule of probability is to calculate it by taking the number of possible outcomes in differentiation to the outcome that you are expecting. So, for a die, there are 6 faces/sides, and for any roll, there are 6 equivalent likelihoods. There is only one outcome you are expecting, no matter which one number you choose.

Formula:

Probability = number of desired outcomes  \[\div\] number of possible outcomes

For odds of rolling a specific number on a dice,

Probability =  \[3 \div 6 = 0.5\]

Probabilities in Percentage

Probabilities are given as numbers whose values vary from 0 to 1. Any event having probability 0 is an impossible event and the event having probability as 1 is a sure event and has 100% probability.

So the chance of rolling a 6 on a single die would be exactly 50%.

Two or More Dice Rolled

When 2 dice are rolled, the probabilities are still simple to work out. To know the possible outcomes of getting 6s when 2 dice are rolled, you are calculating “independent possibilities”. This is because the result of one die does not depend on the result of the other die at all. This essentially leaves you with 2 separate one-in-six opportunities.

The rule for independent probabilities is when you multiply the particular possibilities together to get the result.

Formula:

Probability of Both Dice = Probability of Outcome One \[\times\] Probability of Outcome Two

This is easiest when you work in fractions; for rolling matching numbers from two dice, you have two ⅙ chances.

Probability = \[\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\]

In numerical results, complete the final division. 

\[\frac{1}{36}\] = 1 36 = 0.0278

Probability in percentage is 2.78%.

For the probability of getting 2 specific different numbers on 2 dice,

For example, if you are looking for a 4 and a 5, it does not matter which die you roll with 4 or with which you roll 5. In that case, it would be best to think about it as we were thinking in the previous section. Out of a total of 36 possible results, we will be interested in only two outcomes. 

Probabilities = number of desired outcomes number of possible outcomes = 2/36 = 0.0556.

In percentage, 0.0556 X 100 = 5.56 % (this is twice as likely as rolling two 6s).

Solved Examples

Example 1. When a fair die is rolled one time, find 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

= ⅓, that is, 33333, which is a probability

= 33.3%.

Example 2. 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 0.25.

It is 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 shapes known as "trapezohedron" can be stretched out to any even number of faces that also acts as a fair dice.