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Probability theory
Probability theory
is the branch of
mathematics
concerned with analysis of
random
phenomena.
[1]
The central objects of probability theory are
random variables
,
stochastic
processes
, and
events
: mathematical abstractions of
nondeterministic
events or measured
quantities that may either be single occurrences or evolve over time in an apparently random
fashion. Although an individual coin toss or the roll of a
die
is a random event, if repeated
many times the sequence of random events will exhibit certain statistical patterns, which can
be studied and predicted. Two representative mathematical results describing such patterns
are the
law of large numbers
and the
central limit theorem
.
As a mathematical foundation for
statistics
, probability theory is essential to many human
activities that involve quantitative analysis of large sets of data. Methods of probability theory
also apply to descriptions of complex systems given only partial knowledge of their state, as
in
statistical mechanics
. A great discovery of twentieth century
physics
was the probabilistic
nature of physical phenomena at atomic scales, described in
quantum mechanics
.
History
The mathematical theory of
probability
has its roots in attempts to analyze
games of chance
by
Gerolamo Cardano
in the sixteenth century, and by
Pierre de Fermat
and
Blaise Pascal
in
the seventeenth century (for example the "
problem of points
").
Christiaan Huygens
published
a book on the subject in 1657.
[2]
Initially, probability theory mainly considered
discrete
events, and its methods were mainly
combinatorial
. Eventually,
analytical
considerations compelled the incorporation of
continuous
variables into the theory.
This culminated in modern probability theory, the foundations of which were laid by
Andrey
Nikolaevich Kolmogorov
. Kolmogorov combined the notion of
sample space
, introduced by
Richard von Mises
, and
measure theory
and presented his
axiom system
for probability
theory in 1933. Fairly quickly this became the undisputed
axiomatic basis
for modern
probability theory.
[3]
[
edit
] Treatment
Most introductions to probability theory treat discrete probability distributions and continuous
probability distributions separately. The more mathematically advanced measure theory based
treatment of probability covers both the discrete, the continuous, any mix of these two and
more.
[
edit
] Discrete probability distributions
Main article:
Discrete probability distribution
1
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View Full DocumentDiscrete probability theory
deals with events that occur in
countable
sample spaces.
Examples: Throwing
dice
, experiments with
decks of cards
, and
random walk
.
Classical definition:
Initially the probability of an event to occur was defined as number of
cases favorable for the event, over the number of total outcomes possible in an equiprobable
sample space.
For example, if the event is "occurrence of an even number when a
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