Probability theorem has been used to help understand everything from simple to very
complicated systems. It also helps us realize the uncertainty of the universe. Albert Einstein
always said, " God does not play dice". But today we know that this is not true- God does
play dice. An event can have many outcomes.
Let us now examine the basic elements of the probability theorem. It is made up of postulates or axioms. A statement, also known as an axiom, which is taken to be true without proof. Postulates are the basic structure from which lemmas and theorems are derived.
Definition of Probability
Probability is the branch of mathematics which studies the possible outcomes of given events together with their relative likelihoods and distributions. In common usage, the word ``probability'' is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a Percentage between 0 and 100%. The analysis of events governed by probability is called statistics.
There are several competing interpretations of the actual "meaning'' of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution.
A properly normalized function which assigns a probability "density'' to each possible outcome within some interval is called a probability function, and its cumulative value (integral for a continuous distribution or sum for a discrete distribution) is called a Distribution Function.
Probabilities are defined to obey certain assumptions, called the probability axioms.
Given an event E in a sample space S or probability space, which is either finite with N elements or countably infinite with N = ¥
elements, then one can write
A Sample Space contain the Union (È ) of all possible events Ei .
In terms of the axiomatic approach, the probability space is left unspecified. Only its general properties, stipulated by the following definition, are used.
Let E and F denote subsets of S. Further, let F' = not-F be the complement of F , so that
F È F' = S.
Then the set E can be written as
E = E Ç S = E Ç (F È F' ) = ( E Ç F ) È (E Ç F').
Where Ç denotes the intersection.
A probability space (W , F , P ) consists of:
The Analytic Approach To Probability Space
The probability space used is simply the interval (0,1) with the uniform probability distribution:
W =
(0,1),P
([a, b]) = b - a for 0 < a < b < 1.A quantity P(Ei), called the probability of eventEi , is defined such that
1. 0 £ P( Ei ) < 1
2. P(S) = 1
3. Additivity:
P(E1 È
E2) = P(E1) + P(E2) where
E 1 and E2
are mutually exclusive.
4. Countable additivity:
for
n = 1, 2, ..., N where
E1,E2, ... are mutually exclusive (i.e.,
).
is the empty set.
Uncertainty, Quantum Mechanics
and Relativity 
In 1926, a German scientist, Werner Heisenberg explained the following: the more accurately one try to measure the position of a particle, the less accurately one can measure its speed. Heisenberg's Uncertainty Principal is a fundamental property of our universe. A new theory emerged in the 1920s that took uncertainty, probability and chance into account. That was quantum mechanics. According to the theory of quantum mechanics, particles no longer had separate, well-defined positions and velocities that could be observed. Instead, they had a quantum state, which was a combination of position and velocity. This theory does not predict a single definite result for an observation, but a number of different possible outcomes and how likely each of these is. In other words, it predicts what the probability is for a certain outcome. The result of quantum mechanics was to introduce an unavoidable element of unpredictability or randomness into science. It takes into account the fact that an event can have many outcomes. Quantum mechanics has become an exceptionally successful theory and underlies nearly all of modern science and technology
Einstein's general theory of relativity, also known as the classical theory, does not take into
account the uncertainty principal. Because of the weak gravitational fields normally
experienced, this does not lead to any discrepancies. Einstein had a difficult time accepting
the fact that the universe was governed by chance. In a sense, classical general relativity, by
not taking into account of chance and probability, predicts its own downfall.
This paper aims to explain the foundations of the probability theorem, starting first from
the axioms or postulates. Then it goes on to examine its implications on some of the most
powerful theories and principals used to explain the universe. It is truly fascinating to learn
that the probability theorem, with its simply defined axioms, is the foundation of many
theories in physics, cosmology, biology and chemistry.
References
Doob, J. L. ``The Development of Rigor in Mathematical Probability (1900-1950).'' Amer. Math. Monthly 103, 586-595, 1996.
Hawking, Stephan, W. " A Brief History of Time - From Big Bang to Black Holes." Bantom Books, 1988.
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 26-28, 1984.
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