(5,5), and (6,6).
Independent events are those which do not affect each other, for example rolling two dice are
independent events, as the rolling of the first die does not effect the outcome of the rolling of the
second die. If events are described as mutually exclusive it means that if one happens, then it prevents
the other from happening. So tossing a coin is a mutually exclusive event as it can result in a head or a
tail but not both. The sum of the probabilities of mutually exclusive events is always equal to one. For
example, if one has a bag containing three marbles, each of a different colour, the probability of selecting
each colour would be 1/3.
1/3 + 1/3 + 1/3 = 1
To find out the probability of two or more mutually exclusive events occurring, their individual
probabilities are added together. So, in the above example, the probability of selecting either a blue
marble or a red marble is 1/3 + 1/3 = 2/3 The probability of two independent events both occurring is
smaller than the probability of one such event occurring. For example, the probability of throwing a
three when rolling a die is 1/6, but the probability of throwing two threes when rolling two dice is 1/36.
Probability theory was developed by the French mathematicians Blaise Pascal and Pierre de
Fermat in the 17th century, initially in response to a request to calculate the odds of being dealt various
hands at cards. Today probability plays a major part in the mathematics of atomic theory and finds
application in insurance and statistical studies.
Pascal, Blaise (1623-1662)
Blaise Pascal was a French philosopher and mathematician. He contributed to the development
of hydraulics, calculus, and the mathematical theory of probability. Mathematics Pascal's work in
mathematics widened general understanding of conic sections, introduced an algebraic notational
system that rivalled that of Descartes and made use of the arithmetical triangle (called Pascal's triangle)
in the study of probabilities. Together with Fermat, Pascal studied two specific problems of probability:
the first concerned the probability that a player will obtain a certain face of a dice in a given number of
throws; and the second was to determine the portion of the stakes returnable to each player of several
if a game is interrupted. Pascal used the arithmetical triangle to derive combinational analysis. Pascal's
triangle is a triangular array of numbers in which each number is the sum of the pair of numbers above it.
In general the nth (n = 0, 1, 2, ...) row of the triangle gives the binomial coefficients nCr, with r = 0, 1, ...,
n. In 1657-59, Pascal also perfected his `theory of indivisibles´ - the forerunner of integral calculus -,
which enabled him to study problems involving infinitesimals, such as the calculations of areas and
volumes.
Calculating machine
Between 1642 and 1645, Pascal constructed a machine to carry out the processes of addition
and subtraction, and then organised the manufacture and sale of these first calculating machines. At
least seven of these `computers´ still exist. One was presented to Queen Christina of Sweden in 1652.
Pascal was born in Clermont- Ferrand. In Paris in his teens he met mathematicians Descartes and Fermat.
From 1654 Pascal was closely involved with the Jansenist monastery of Port Royal. He defended a
prominent Jansenist, Antoine Arnauld (1612-1694), against the Jesuits in his Lettres
provinciales/Provincial Letters 1656. His Pensées 1670 was part of an unfinished defence of the
Christian religion. His last project was to design a public transport system for Paris, which was
inaugurated in 1662. Pascal's pioneering work on fluid pressure laid the foundations for both hydraulics
and meteorology. In his honour, the SI unit for pressure is called the pascal. It is equal to one Newton
per square metre. (©Helicon Publishing Ltd, printed from the Hutchinson Educational Encyclopedia,
2000)