Die

Ordinary dice

The most current dice are small Cube S from 1 to 2 cm on side (16 mm being the standard), thus having 6 numbered faces from 1 to 6, generally using reasons for points. Traditionally, the sum of the numbers located on two opposite faces is equal to 7 (what is the case since at least antiquity); consequently, the numbered faces 1,2 and 3 are touched in a top of the die. Two choices are thus possible: to place these faces in the direction of the needles of a watch or in the other direction around this top.

The dice are thrown in order to provide random random numbers, generally for the games of chance, and are thus an example of Générateur of random random numbers. However, as the numbers are usually illustrated using holes, some faces are seen withdrawing more material than others, which causes a light statistical skew. This skew can be reduced, as in the case of the Asian dice where the numbered face 1 has a hole largely larger than the others, or in the case of dice used in the casinos where marks are practiced on surface.

From the practical point of view, the dice are thrown, simple or in group, with the hand or using a container dedicated to this use, on a plane surface. The face taken into account for each die is that which is located on the top when it stops.

Alternatives

Noncubic dice

Certain dice have the form of a Polyèdre other than the cube. Formerly little employed in the play, they became more popular since the years 1950 particularly with the introduction of the Wargame S, roleplays, card decks to collect and some board games. These dice are generally out of plastic and their faces rather have numbers than reasons of points.

If it is about an innovation at modern times, it seems that certain old cultures used some (in particular, two icosahedral dice dating from the ancient Rome are exposed to the British Museum of London).

The Platonic solid are used in a current way for the dice with 4,6,8,12 and 20 faces. Other forms can be found for dice with 2,3,5,7,10,16,24,30,34,50 or 100 faces, but separately the die with 10 faces, they are used little.

A great number of different distributions of Probabilité S can be obtained using these Des. For example, two dice with 10 faces can be used to produce a number ranging between 1 and 100 (one of the dice giving the figure of tens, the other that of the units, pulling “00” correspondent with 100 or 0 following practiced play) in order to obtain a linear distribution of Pourcentage S. By adding the results with several dice, it is possible to approach a normal distribution ; by eliminating pullings more (or less) raised, to modify these distributions, etc using these techniques, the plays can approach with sufficient realism the probabilities of the events which they simulate.

The equiprobability of these dice (i.e. equal probability to obtain any of its faces) is prone to controversy; the dice with 6 faces used in the casinos have the legal requirement be equiprobable. The manufactoring processes used for the other types of dice do not have any obligation of this kind.

Spherical dice exist too; their function is identical to that of the dice with 6 faces, but they have a cavity interns octahedral in which a weight moves and causes their stop in a direction among six. They require however a plane surface and horizontal to function correctly.

The forms most usually used, apart from the cubic dice with 6 faces, are:

the Tetrahedron, with 4 faces. These dice not rolling almost, they comprise three numbers on each face, each one registered along an edge, arranged in such way that located on the edge of the bottom of the three visible faces is the same one; this number is that taken into account during a throw.
the Octahedral , with 8 faces. Each face is triangular. The sum of the opposite faces is generally equal to 9.
the pentagonal Trapézoèdre, with 10 faces. The only die running which is not a Platonic solid. It is generally used per pair to generate the numbers from 0 to 100, one appearing tens (00, 10,20… up to 90), the other the units (from 0 to 9). The position of faces 00 and 0 accounts for 0 or 100 according to the play.
the Dodecahedron, with 12 faces. Each face is a regular pentagon.
the Icosahedral , with 20 faces.

Among the rarer forms:

1 face: Sphere where there is registered the figures from 1 to 6.
2 faces: Cylinder. It acts neither more nor less than one coin having one 1 on a face and one 2 on the other. When such a pulling is necessary, pulling with Pile or face is traditionally employed.
3 faces: truncated triangular prism and rounded, it is often replaced by a die with 6 faces of which the result is divided by 2, rounded with the higher entirety.
5 faces: pentagonal Prism.
7 faces: heptagonal Prism.
12 faces: rhombic Dodecahedron.
14 faces: Bipyramide hexagonal.
16 faces: Bipyramide octagonal.
24 faces: Tétrakihexaèdre.
30 faces: Triacontahedral rhombic, or Die with 30 faces.
37 faces, to replace a caster of casino.
50 faces: Bipyramide icosakaipentagonale .
100 faces: Zocchièdre.

Classification

The majority of the faces of the dice are numbered by an uninterrupted succession of integers, begin with one (or zero), expressed by holes or figures. Exceptions exist however:

Die lapping machine or Videau, used amongst other things with the Backgammon, increasing numbers 2,4,8,16,32 and 64 and symbolizing the current multiplicative coefficient of the initial setting. This die is not thrown and is simply used to note the stake.
Dice for the play of Poker of ace where the figures of the charts to be played are represented: ace, king, lady, servant, ten and nine.
specific Die to play Mah-jong.
coloured Dice, each face carrying a different color.
Dice comprising of the drawings on the faces, used for example to determine certain occurrences of Sets of figurines or the positions in an erotic play.
In the play of plate Formula Die, the dice represent speeds of the car: their starting figures go increasing, so that one needs much chance to double a car in the 3rd speed with the die the 2nd speed.
the dice with 10 official faces of the play Vampire: the Masquerade comprise a Ankh instead of the 1, to point out the " life éternelle" vampires.

Probabilities

For a simple throw of only one die with 6 faces, which one considers balanced, the Probabilité of obtaining any value 1 to 6 is exactly of 1/6. Pulling thus follows a discrete uniform Loi. The pulling of N dice follows a Loi multinomiale whose probabilities p ₁, p ₂,…, p ₆ are all equal to 1/6, if the die is not pipe.

If two dice are thrown and that one adds the numbers obtained on the two higher faces, pullings are not any more distributed in a uniform way but follow a triangular distribution:

The most probable pulling is then 7.

With three dice or more, the distribution approaches a normal distribution with the addition each die (consequence of the Théorème of the central limit). The exact probability distribution $F_i$ for a $i$ number of dice can be calculated by Convolution repeated probability distribution of a simple die with itself:

F_i (m) = \ sum_n {F_1 (N) F_ {i-1} (m - N)} \,

To determine if a die is pipe

A die is known as “pipe” if the law is not uniform any more. When it is intentional, one arranges oneself so that a result more frequently left, or on the contrary less frequently, the others faces having the same probability of appearance between them. If it is about a nonintentional defect, each face will have a clean probability.

If one throws the die several times of continuation, one will not obtain a strict alternation of values. For example, if one twice draws a die from continuation, one has 6 chances out of 36, that is to say 16,66… % of chances, to obtain the same result twice (each doubled bloom is likely 1/36 to appear, and there are 6 doubled blooms); in a case on six, one obtains the same throw twice. The frequency observed for each event will be seen to approach the theoretical frequency on a great number of launching, for example 100.

If one makes N throws, to know if die is balanced (i.e. if there are indeed 1/6 of chances to have each figure), it is necessary to use a Test of the χ ² of adequacy to five degrees of freedom (since there are six results but that their probabilities are complementary). The minimal number of throws is of 30 (5 divided by the theoretical frequency, 1/6 = 0,166…, cf Test of the χ ² > Test conditions ). If one calls O_i the number of throws giving the figure I , one has the table of results according to:

with ∑_I O_i = N

The χ ² is

\ chi^2 =

\ sum_ {i=1} ^6 \ frac {(O_i - N \ times 1/6) ^2} {N \ times 1/6}

\ frac {6} {N} \ cdot \ sum_ {i1} ^6 (O_i - n/6) ^2

The probability p that the die is balanced is given according to the tabulées values of the χ ².

For example, if the χ ² is lower or equal to 0,55, the die has 99% of chances to be balanced (1% of chances to be pipe); if the χ ² is equal to or higher than 15,09, the die has 1% of chances to be balanced (99% of chances to be pipe).

History

The dice probably draw their origin from the bones of the ankles (specifically the astragale) of animals the such ox. It is not possible to precisely determine the appearance of the dice and their distinction of the ossicles, the ancient writers pretense to confuse the two plays. It is certain on the other hand that they date from prehistoric times . Their presence in old tombs of the valley of Indus seems to point towards an Asian origin. The play of die is mentioned in the Rig-Veda and the Indian Atharvaveda .

The sets of dice were popular in Rome, particularly during the records days of the Roman Empire, although they were prohibited, except during the Saturnales. Horace described for example what it presented like a typical young man of the time, which wasted its time with the dice rather than to overcome its horse. To play of the money to the dice was the subject of several specific laws; one of them ruled that no lawsuit could be required by a person who authorized the bets in her house, even if it had been attacked or if one had cheated against him. The professional players were however current and certain their dice pipes were preserved.

Tacite reported that the Germanic tribes adored the dice particularly and were ready to bring into play their own freedom after having lost all the remainder. Several centuries later, the dice became the pastime of the knight S and of the schools and the Guilde S of dice existed.

In India, the dice were used in particular to play Chaturanga, one of the ancestors of the Jeu of failures. Chaturanga would have been played with dice with 8 marked faces 2,3,4 and 5, each one indicating one of the types of parts of the play like front being played this turn. One found besides in France of the sets of failures close to Chaturanga, dating from the Romance time and being also played with dice, where the king presented the attributes of Charlemagne.

In many Asian countries, the dice are since always a popular pastime.

Expressions

“the dice are thrown”, translation of the Latin risk jacta is (literally: the fate is thrown by it), sentence pronounced by Jules César after it had crossed the Rubicon.

This sentence means that an irreversible action was made, and that the future is between the hands of the Hasard.

“a toss of the dice”, represents the chance.

Here an example: this operation was exploited a toss of the dice . This sentence means that important part of the aforesaid the operation was achieved by the chance, by the chance.

Another example is the famous sentence of Stephan Mallarmé: " A toss of the dice never will not abolish the hasard."

“God does not play dice” of Albert Einstein during his confrontation with the Quantum physics.

Meaning by there its feeling (and this for what it will pass the remainder of its life) of a prédictible Universe.

what he will also explain by saying that if one is not able to apprehend the whole of the Universe, it is quite simply that one does not have yet the totality of the laws which govern this Universe; but that once they are had, it becomes possible then, in theory, without taking into account an infinite hypothetical time of calculation, to determine the characteristics passed, present and to come from any element composing the Universe.

See too

Internal bonds

Jet of die (roleplay)
Set of dice

External bonds

'' Dice '' ( Wolfram MathWorld , analyzes probabilities with the dice)
'' Fair Dice '' (study of the various polyhedrons leading to balanced dice)
the AttaKuBe board game puts at the catches two camps dice with the completely original figures.
the play Star Trek: The Next Generation Collectible Dice Game proposes '' very original dice '' for each vessel which clashes there.
Idem for the play Dragon Dice .

References

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