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History of Games of Chance
The Chinese I Ching or “Book of Chnage” shows that
• deterministic thinking is not at all hereditary, but learned;
• random experiments were used as a means to make predictions and
decisions from ancient times.
Although, random experiments and games of chance are not really identical,
they are taken as being equivalent here.
The random experiment of the I Ching is probably deliberately selected as
a drawing experiment with natural yarrow stalks in order to create a direct
connection between man and nature. However, this kind of experiments
were soon abandoned by producing special tools, as for example dice, which
allowed to perform the experiment everywhere under same conditions.
Game of dice in ancient India
Already in ancient times random experiments were also used in India for the
purpose of pastime and entertainment.
Game of dice and chariot race are the two amusement to which
the vedic Indian1 abandon themselves with passion.
Originally, the nuts of the Vibhidaka tree were used in India as dice which had
five sides. Later the Vibhidaka nuts were replaced by artificially produced
dice (pasakas =prosm with four sides) made of wood. ivory or even gold. The
sides were numbered from 1 to 4 and probably three such dice were thrown.
Astragals in Greece and Rome
In Greece games of chance were originally played by means of astragals.
Later they were made from clay, glass, ivory, etc. Astragals are small bones
of sheep or goat in the ankle that articulate with the leg bones to form the
ankle joint. The bones have four sides which were not numbered because
each side was different.
The astragal game was also popular in Egypt (ca. 3000 B.C.) and in many
other cultures of the Middle East. It is noteworthy that the probabilites for
the different outcomes of an astragal game are not the same.
1
2000-1000 B.C.
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Game of dice in Greece and Rome
Besides the astragal game there was also the game with artificially made
more or less regular cubes, which were called “kybos” in Greece and “alea”
in Rome. “Alea” was also used with the general meaning of random, risk or
hazard. Thus, Caesar shall have shouted when he crossed the river Rubicon
“Iacta alea est” or “anerriphtho kybos” = “the die has been cast!”)
The oldest finding of a die made of clay in northern Iraq (Tepe Gawra) stem
from the third millennium B.C. Already soon there was the nowadays used
configuration with the sum 7 of opposite sides. Generally three dice were
thrown.
The four sides of an astragal, hind leg of a sheep.
Dice made of bronze and bone from Rome 1st to 3rd century AD.
Mathematical treatment of throwing three dice
The problem to be investigated was to determine the number of different
possibilities to obtain certain outcomes. The first written investigations stem
from the 13th century AD:
• A fictitious auto-biography of Ovid.
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• In Dante’s (1265-1321) Divina Comedia.
There are, however, no indication at that time that probabilities were already
considered or that the idea of of equiprobable outcomes were considered or
that any attempt was made to quantify randomness.
Besides entertainment, dive served served as oracles and it was believed that
the fate or the gods would dispose the dice in each individual case. The
question how the results of large sequences would look like arose no interest
at all.