MA2216/ST2131 Probability
Notes 1
A Brief Historical Account.
Three hundred and fifty years ago, it is said, correspondences between
two famous French mathematicians,
Blaise Pascal
and
Pierre de Fermat
,
gave birth to a mathematical theory of probability.
The story goes like this: Pascal was approached by a French nobleman,
Antoine Gombaud, Chevalier de M´
er´
e, who was interested in gambling.
Indeed, Chevalier de M´
er´
e gambled frequently and made a fortune. He
bet on a roll of a die that at least one 6 would appear during a total of four
rolls. From his own experience, Chevalier de M´
er´
e knew that he was more
successful than not with this game of chance.
However, if he bet that he
would get a total of 12, or a double 6, on twentyfour rolls of two dice, he
realized that his old approach to the game resulted in more money. He asked
Pascal why his new approach was not as profitable.
This problem and others posed by de M´
er´
e led to an exchange of letters
between Pascal and Fermat. This took place in 1654. Out of such correspon
dences, some fundamental principles of probability theory were formulated
for the first time.
The first book on probability was said to be written by a Dutch scientist
Christian Huygens
in 1657.
Understandably, the socalled probability
theory developed during that period dealt with primarily problems associated
with gambling (i.e., games of chance), and hence, became popular rapidly.
Subsequently,
Jakob Bernoulli
(16541705) and
Abraham de Moivre
(16671754) made some major contributions.
In 1812, Pierre de Laplace
(17491827) introduced a host of new ideas and mathematical techniques in
one of his books. He also applied probabilistic ideas to many scientific and
practical problems.
Since then, many have done significant works; among
the most important are Chebyshev, Markov, von Mises, Kolmogorov, Paul
L´
evy, etc.
1
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An axiomatic approach that forms the basis for the modern theory was
presented in 1933 by Kolmogorov, a Russian mathematician, in his celebrated
monograph “
Foundations of Probability Theory
.” Nowadays, the the
ory of probability has already established itself as an independent branch of
mathematical systems. Applications can be found in many practical fields,
such as engineering, statistical mechanics (physics), genetics, insurance, and
financial mathematics.
What to be covered?
About 15% on
combinatorial probability
and 85% on
calculus of prob
ability
.
Course Contents
1.
Basic Concepts of Probability
Sample space & events; Some basic counting schemes (e.g. permutation,
combinations, etc.); Axioms of probability; Some simple properties of
probabilities; Union of events; Conditional probability; Independence of
events.
2.
Discrete Random Variables
Random variables; Distribution functions; Discrete density functions;
Definition of the expectation (of a function of a random variable);
The Bernoulli & binomial random variables; The Poisson random vari
able; Poisson approximation to the binomial distribution; The geometric
& negative binomial random variable; The hypergeometric random vari
able;
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 Fall '08
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 Probability theory, Blaise Pascal

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