A gambler bets half the money in his pocket on the toss of a coin,
winning an equal amount for a head and losing his money if the
result is a tail. After 2n plays he has won exactly n times. Has he
more money than he started with?
You have two bags, four red balls and four white balls. You must
put all the balls in the bags although you are allowed to have one
bag empty. How should you distribute the balls between the two bags
so as to make the probability of choosing a red ball as small as
possible and what will the probability be in that case?
To win on a scratch card you have to uncover three numbers that add
up to more than fifteen. What is the probability of winning a
Published June 2008,February 2011.
In the time before the mathematical idea of randomness was discovered, people thought that everything that happened was part of the will of supernatural beings, the gods, who looked down upon human affairs and decided to 'tip the balance' one way or another to influence events. Hence, sacrifices were made and rituals performed to discover the 'will of the gods' or to try to influence human
affairs. This idea still prevails, and many people all over the world use lucky charms, engage in superstitious practices, use horoscopes, and still have some kind of belief that there are such ways of influencing their lives. The gods may be dead, but 'Lady Luck' still survives.
The astragalus is a small bone, about an inch cube, found in the heel of hoofed mammals. Astragali have six sides but are not symmetrical, so there is no way of knowing which way they will eventually come to rest. For many ancient civilizations, astragali were used by priests to discover the opinions of their gods. It was customary in divination rites to roll, or cast, five astragali. Typically,
each possible configuration was associated with the name of a god and carried with it the sought-after advice.
The astragali have been used from classical times for gambling, and similar stones are still in use today for games like 'fivestones' or 'jacks'.
Gradually, over thousands of years, astragali were replace by dice [see note 1 below], and pottery dice have been found in Egyptian tombs. The earliest die known was made from pottery and excavated in Northern Iraq dating from about 3,000 BCE. It has dots arranged as in (Die A).
Die (B), from about 1400 BCE found in a tomb in Egypt, shows consecutive numbers opposite each other.
Dice with other markings like the names or portraits of gods have been found, probably used for special games or rituals, and others where some numbers are repeated, or 'loaded', for special purposes or possibly for cheating (Die C).
Once the Greeks had worked out the geometry of the polyhedra, dice of other shapes began to be constructed. However, whether cube or polyhedral, the shapes were not entirely regular and were therefore biased.
Over time, gamblers would get used to using the same dice, and have an intuitive idea of how they would fall, but given another set of dice, the odds would be different. Later, as the manufacture of dice became more exact, some ideas of the possible combinations of number began to emerge.
There were many other forms of rituals hoping to overcome the randomness of nature and man's condition. A few of these which became of particular mathematical interest are geomancy, the nine square grid or magic square, and temple designs, the ancestors of board games.
Geomancy means divination of or by the earth , and is a system of 16 mathematically related arrangements of stones, beans or other available small objects used to make decisions, answer questions, or foretell the future. The stones are cast upon the ground and the pattern formed is interpreted. The symbols
represent a series of binary 'opposites' like good and evil, male or female, sadness and happiness, etc. Combinations of these opposites can be used to represent odd and even numbers.
The Nine square grid is said to come from an ancient system for the division of land, probably from feudal India. In China the nine-square configuration was supposed to be an ideal arrangement, with eight farmers' fields surrounding a central well. The grid of nine squares, or a circle divided into nine sections by straight lines often appears as a central form in Tibetan sacred diagrams. In
Scotland, the pattern was used at Beltane (the eve of May) where eight squares were cut out from the turf, and a bonfire lit on the central square.
In this way, from practical beginnings in different cultures, the nine-square grid acquired mystic importance and symbolised divine order, and the representation of control by the gods.
Magic Squares are directly related to the Sacred Grid, supposedly being the numerical mystery which underlies their physical form. The simplest magic square is the square of nine, ascribed to Saturn, where each row and column adds up to 15; the total of the rows and the columns is 45, and the diagonals 30. The 4x4 square with row and column numbers 34 is
assigned to Jupiter, the 5x5 with row or column numbers 65 to Mars, and so on for the Sun, Venus, Mercury and the 9x9 square with row or column numbers 369, to the Moon [an NRICH article on Magic Squares can be found here at nrich.maths.org/1337].
As with other devices, these magic squares are all said to have correspondences to different numbers, various deities, days of the week, natural objects, different qualities, and so on. In the Hindu Temple Yantra [see note 2 below] you can see the nine squares, the 'sacred space', or source of energy, in the centre.
Board Games are clearly linked with divination, astrology and sacred geometry, and the designs of the boards can show their sacred or occult origins. The popular game of 'snakes and ladders' is controlled by the throw of dice, and the ladders and snakes originally referring to good and bad fortune, now refer to good and bad 'luck' in the progress of the
game. In some cases the designs of the boards are the same as the plans of temples and holy cities with a 'sacred space' in the centre.
In ancient times, few people could understand even the simplest arithmetic and geometry, and the confusion of mathematics with magic has a long history.
People who had knowledge of the regular movements of the heavens were able to predict the position of planets, and the particular the times when astronomical events appeared in certain sections of the sky. In ancient civilisations these were highly skilled technicians, called 'priests', and their activities were partly scientific, and partly religious. In Europe, after the arrival of
Christianity, the religious aspect of these practices was condemned as superstition. Because numbers were used in these processes, anyone who used numbers was regarded with considerable suspicion. In this way genuine mathematicians were looked upon with suspicion by the ignorant, and the titles of Astrologer, Mathematician and Conjurer were virtually synonymous.
An early Bishop of the Church, St. Augustine of Hippo (354-430 CE) once said:
"The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell."
Augustine was arguing that belief in astrology denies the freedom of the will.
Roger Bacon (1214 - 1292), often called England's first Scientist, had a reputation as a 'great necromancer' because of his ingenious experiments and John Dee (1527 - 1609) probably one of the foremost mathematicians in Europe of his time, gained a reputation as a 'Conjuror' while he was at Oxford because he was respnsible for developing a simple mechanical device by which an actor appeared to
fly, and people claimed he was in league with the devil. [see note 3 below]
Following the foundation of the Oxford chairs in mathematics and astronomy in 1619, some parents kept their sons away from the university in fear of them becoming contaminated by the 'Black Art'.
As the predictive power of astronomy and other practical uses of mathematics became apparent, mathematicians were able to dispel the idea that many events were not controlled by the goddess Fortuna, but could be explained in a rational way.
Since dice were used in gambling, in religious ceremonies and for divination, it is believed that those who used the dice had a good intuitive idea of the likely frequency of various number combinations. The first printed document showing the possibilities with three dice was the Latin poem De Vetula , which shows all the combinations for the fall of
three dice, and is believed to have been written in the early 13th century. The idea of using binomial coefficients to calculate the possibilities appears in the poem, but is not taken up until much later [see note 4 below].
Since the Christian Church was against gaming, and there was much superstition about divination, it is not surprising that a theory of probability did not begin to appear until the 16th century. Cardano, writing with considerable personal knowledge of gambling, recognised that if the die was honest, each face would have an equal chance of appearing. His
manuscript, Liber De Ludo Aleae , was written about 1526 but only found after his death, and not published until 1663. He gave tables of the results for one, two and three dice, but these are not all correct. However, Cardano is credited with recognising that the abstraction of the 'honest die' is the key to a theory of probability based on mathematical
By the mid 16th century the theory of probability became established on a rigorous basis with the work of Pascal and Fermat. However, as we have seen, the idea of the application of 'Pascal's Triangle' had been suggested as early as the 13th century but forgotten for some 200 years. The triangle itself was known and published before, by Stifel (Arithmetica Integra 1543) Tartaglia (Trattato
1556) Stevin (Arithmetic 1625) Pierre Herigone (Cours Mathematique 1634), and we also know it was known to the Chinese and the Arabs by the mid 13th century, but Pascal was the first to apply it to probability.
Use the notes tab at the top of this article or click here .
Jenkins, G.W. & Slack J.L. (1979) Classroom Experiments with Dice . St Albans. Tarquin Publications
Woods, G. Symmetry Dice . St Albans. Tarquin Publications
Benson, S. (2005) Ways to think about Mathematics:Activities and Investigations for Grade 6 . California. Corwin Press. This is a useful book with many examples of activities. There are sections about probability and binomial coefficients.
Here is a shop for all kinds of dice : large; small; all colours; with numbers; with spots; blank; arithmetic symbols; money symbols; polyhedral; round (yes round!); loaded; and for cheating! http://www.dice.co.uk/index.htm