2.
THEORY OF
PROBABILITY
2.1BASIC
TERMS
The basic terms used in defining
probability are:
- Trial and
event
- Exhaustive
events
- Favourable
events or cases
- Mutually
exclusive events
- Equally
likely events
- Independent
events
2.1.1
Trial and event:
If we take an
experiment in which the experiment is repeated under essentially same
conditions, the outcome wont be unique but may result in any of the possible
outcomes. The experiment is known as a trial and the outcomes are known as
events or cases.
2.1.2
Exhaustive events:
The all possible outcomes of any
experiment is known as the exhaustive events.
2.1.3
Favourable events or cases:
The number of
events favourable to an event in an experiment is the number of outcomes which
entail the happening of the event.
2.1.4
Mutually exclusive events:
The happening of
any one of them precludes the happening of all the other events that is if no
two or more of them can happen simultaneously in the same trial.
2.1.5
Equally likely events:
Outcomes of an
experiment is said to be equally likely
if taking into consideration of all the relevant evidences tjere is no reason
to expect one in preference to the others.
2.1.6
Independent events:
Several events
are said to be independent if the happening of an event is not affected by the
supplementary knowledge concerning the occurrence of any number of the
remaining events.
2.2
MATHEMATICAL OR CLASSICAL PROBABILITY:
If an experiment
results in n exhaustive, mutually exclusive and equally likely cases and m of
them are favourable to the happening of an event E, the probability p of
happening of E is given by
p = P(E) =
Favourable number of cases/ exhaustive number of cases
2.3 SETS AND ELEMENTS OF SETS:
A set is defined as the collection or
aggregate of all possible objects. The objects comprising a set are called
elements or points of the set.
2.3.1
Operation on
sets:
The union of two given sets A and B,
denoted by
, is defined as a set consisting of all possible points which
belong to either A or B or both. Thus symbolically,
The intersection of two sets A and B,
denoted by 
, is defined as a set consisting of all those elements which
belong to both A and B, thus symbolically,
If A and B have no common point, that is
, then the sets A and B are said to be disjoint, mutually
exclusive or non overlapping.
The relative difference of a set A
from another set B, denoted by A B is
defined as a set consisting of those elements of A which do not belong to B.
symbolically,
The complement or negative of any set A,
denoted by 
is a set containing all elements of the universal set S, that
are not elements of A, that is
2.3.2
Algebra of sets:
Some important properties concerning
operations on sets. If A, B and C are the subsets of a universal set S, then
the following laws hold:
2.3.2.1
Commutative law:
2.3.2.2
Associative law:
2.3.2.3
Distributive law:
2.3.2.4
Complementary law:
2.3.2.5
Difference law:
2.3.2.6 De-
Morgans law distribution law:
2.3.2.7
Involution law:
2.3.2.8
Idempotency law:
2.4 AXIOMATIC APPROACH TO PROBABILITY:
The axiomatic
approach to probability which closely relates the theory of probability with
the modern metric theory of functions and also set theory was proposed by A.N.
Kolmogrov. The axiomatic definition of probability includes both the classical
and the statistical definition as particular cases and overcomes the
deficiencies of each of them. On this basis, it is possible to construct a
logically perfect structure of the modern theory of probability and at the same
time to satisfy the enchanced requirements of modern natural science. The
axiomatic development of mathematical theory of probability relies entirely
upon the logic of deduction.
2.5
RANDOM EXPERIMENT:
The theory of
probability provides mathematical models for real world phenomenon involving
games of chance such as the tossing of coins and dice. Any probabilistic
situation is called as a random experiment.
2.6 PROBABILITY MATHEMATICAL NOTION:
Suppose in a
large number of trials the sample space S contains N sample points. The event A
is defined by a description which is satisfied by NA of the
occurrences. The frequency interpretation of the probability P(A) of the event
A, tells us that P(A) = NA/ N.
A purely
mathematical definition of probability cannot give us the actual value of P(A)
and this must be considered as a function defined on all events. With this in
view, a mathematical definition of probability is enunciated as follows:
Given a sample
description space, probability is a function which assigns a non negative
real number to every event A, denoted by P(A) and is called the probability of
the event A.
2.6.1
Probability
function:
P(A) is the probability function defined
on a σ field B of events if the following properties hold:
- For each A
Є B, P(A) is defined, is real and P(A) ≥0
- P(S) =1
- If {An}
is any finite or infinite sequence of disjoint events in B, then
The above three axioms are termed as the
axiom of positiveness, certainty and union.
2.6.2
Law of addition
of probabilities:
Statement:
If
A and B are any two events and are not disjoint then
Proof:
We have
Since A and 
are disjoint,
Since
are disjoint
ΰ
2.6.3
Extension of
general law of addition of probabilities:
For n events A1, A2,
..
An we have
2.7 MULTIPLICATION LAW OF PROBABILITY AND CONDITIONAL
PROBABILITY:
For two events A and B
Where 
represents the conditional probability of occurrence of B
when the event A has already happened and 
is the conditional probability of happening of A, given that
B has already happened.
2.8 EXTENSION OF MULTIPLICATION LAW OF PROBABILITY:
For n events A1, A2,
..
An we have
Where 
represents the conditional probability of the event Ai given that the events
have already happened.
2.8.1
Independent
events:
an event B is
said to be independent of event A, if the conditional probability of B given A
that is is equal to the unconditional probability of B that is
if 
Since 
And since when B is independent of A, we must have 
or it follows that A is also independent of B. hence the
events A and B are independent if and only if
2.8.2
Pair wise
independent events:
A set of events
A1, A2,
.. An are said to be pair wise
independent if
2.9 BAYES THEOREM:
If E1,
E1,
. En are mutually disjoint events with P(Ei)
≠0, (i= 1, 2,
.,n) then for any arbitrary event A which is a subset of 
such that P(A) >0 we have
, i =1, 2,
n
Proof:
Since
, we have
Since 
(i=1,2
..n) are mutually disjoint events, we have by addition
theorem of probability
By compounded theorem of probability.
Also we have
2.10
GEOMETRIC
PROBABILITY:
If we are
interested in finding the probability that a point selected at random in a
given region will lie in a specified part of it, the classical definition of
probability is modified and extended to what is called geometrical probability
or probability in continuum. In this case, the general expression for
probability p is given by
P =Measure of
specified part of the region/ measure of the whole region
When measure
refers to the length, area or volume of the region if we are dealing with one,
two or three dimensional space respectively.