Transcript PROBABILITY THEORY

PROBABILITY THEORY:
a probable chronology
Mathematical Analysis for
Computer Science
Lecture 1
Definition?
• American Heritage Dictionary defines the Probability
Theory as the branch of Mathematics that studies the
likelihood of occurrence of random events in order to
predict the behavior of defined systems.
• So, Probability Theory, being a branch of Mathematics, is
an exact, deductive science that studies uncertain
quantities related to random events.
• A strange marriage of mathematical certainty and
uncertainty of randomness?
Origin of Probability Theory
• People started to use the principles of probability many
years ago.
• The elements of probability were applied for census of
population in the ancient countries such as China, India
and Egypt.
• The same methods were used for estimation of the
overall strength of enemy army.
• Theory of probabilities as a real science came into
existence in the middle of 17th century in France.
French Society in the 1650’s
• The time of royals and
musketry, beautiful ladies and
noble cavaliers!
• Gambling was popular and
fashionable, not restricted by
law
• As the games became more
complicated and the stakes
became larger there was a
need for mathematical
methods for computing
chances.
Enter the Mathematician
• In 1654 French courtiers Antoine Gombaud, Chevalier
de Méré (1607-1648) consulted Blaise Pascal (16231662) in Paris about an apparent contradiction
concerning a popular dice game.
• He posed some questions (known now as de Méré's
problems ):
– How many throws of two dice are required for a number of
double six appear events will be more than a half of total
throws?
– problème des partis (problem of points): How to share the
wagered money between two gamblers if the game
interrupted untimely?
Principles of Probability Theory
formulated
• Pascal began to correspond with his
friend Pierre de Fermat (16011665) about these problems.
• The correspondence between
Pascal and Fermat is the origin of
the mathematical study of
probability.
• Although a few special problems on
games of chance had been solved
by some Italian mathematicians in
the 15th and 16th centuries, no
general theory was developed
before this famous correspondence.
Classical Probability
• The method they developed is now called the
classical approach to computing probabilities.
• The method:
Suppose a game has n equally likely outcomes, of which m
outcomes correspond to winning. Then the probability of
winning is m / n.
• The classical method requires a game to be broken down into
equally likely outcomes.
– It is not always possible to do this.
– It is not always clear when possibilities are equally likely.
Experience
• Another method, known as the frequency method had
also been used for some time.
• This method consists of repeating a game a large
number of times under the same conditions. The
probability of winning is then approximately equal to the
proportion of wins in the repeats.
• This method was used by Pascal and Fermat to verify
results obtained by the classical method.
First Book
• The Dutch scientist Christian Huygens (1629-1695), a
teacher of Leibniz (1646-1716), learned of this
correspondence and shortly thereafter (in 1657)
published the first book on probability; entitled De
Ratiociniis in Ludo Aleae.
• It was a treatise on problems associated with gambling.
• Huygens tract remained the only text on probability for 50
years.
Early Generalizations
• Jacob Bernoulli (1654-1705) proved that the frequency
method and the classical method are consistent with one
another in his book Ars Conjectandi in 1713.
• Abraham De Moivre (1667-1754) provided many tools
to make the classical method more useful, including the
multiplication rule, in his book The Doctrine of Chances
in 1718.
• The book was popular, eventually going through three
editions.
From Games to Science
• Throughout the 18th century, the application of
probability moved from games of chance to scientific
problems:
– Mathematical theory of life insurance -life tables.
– Biological problems -what is the probability of being born
female or male?
Applied Probability
• Pierre-Simon Laplace (1749-1827) presented a
mathematical theory of probability with an emphasis on
scientific applications in his 1812 book Théorie Analytique
des Probabilités.
– Before Laplace, probability theory was solely concerned
with developing a mathematical analysis of games of
chance.
– Laplace applied probabilistic ideas to many scientific and
practical problems.
• The theory of errors, actuarial mathematics, and
statistical mechanics are examples of some of the
important applications of probability theory developed in
the l9th century.
Probability Applied
• Development of probability theory has been stimulated
by the variety of its applications.
• Conversely, each advance in the theory has enlarged the
scope of its influence.
• Mathematical statistics is one important branch of
applied probability; other applications occur in such
widely different fields as genetics, psychology,
economics, and engineering.
• Many workers have contributed to the theory since
Laplace's time; among the most important are
Chebyshev, Markov, von Mises, and Kolmogorov.
Stagnation and Frustration
• After the publication of Laplace’s book, the mathematical
development of probability stagnated for many years.
• By 1850, many mathematicians found the classical
method to be unrealistic for general use and were
attempting to redefine probability in terms of the
frequency method.
• These attempts were never fully accepted and the
stagnation continued.
Axiomatic Development
• One of the difficulties is to
– arrive at a definition of probability that is precise enough for use in
mathematics
– comprehensive enough to be applicable to a wide range of
phenomena.
• The search for a widely acceptable definition took nearly three
centuries and was marked by much controversy.
• The matter was finally resolved in the 20th century by treating
probability theory on an axiomatic basis.
• In 1933 a monograph Grundbegriffe der
Wahrscheinlichkeitsrechnun (Foundations of the Calculus of
Probabilities) by Russian mathematician A. Kolmogorov
outlined an axiomatic approach that forms the basis for the
modern theory.
• He built up probability theory from fundamental axioms in a way
comparable with Euclid's treatment of geometry.
Let’s play a
Game of Chance