“ the ratio
of the number of favourable cases to the number of all the cases “ or

**P(E) =**

__Number of outcomes favourable to E__**Number of all possible outcomes of the experiment**

— PROBABILITY
is a measure of the degree to which an occurrence is certain [or uncertain].

**Concept of Probability**

v The concept of Probability originated in the
16

^{th}century when an Italian physician and mathematician J.Cardan wrote the first book on the subject , The Book on Games of Chance. Since its inception, the study of probability has attracted the attention of great mathematicians such as James Bernoulli (1654-1705) ,
v Probability
theory is a very fascinating subject which can be studied at various
mathematical levels. Probability is the foundation of statistical theory and
applications.

v Several
mathematicians like Pascal, James Bernoulli, De-Moivre, Bayes applied the
theory of permutations and combinations to quantify or calculate probability.
Today the probability theory has become one of the fundamental technique in the
development of Statistics.

v The
term “

*probability*” in Statistics refers to the chances of occurrence of an event among a large number of possibilities.**GAMBLER’S FALLACY**

— The

**, also known as the**__gambler's fallacy__**is the mistaken belief that if something happens more frequently than normal during some period, then it will happen less frequently in the future (presumably as a means of**__Monte Carlo fallacy__,*balancing*nature).**: A coin is tossed 5 times, and all the results comes as ‘ HEADS ’. So during the 6**__For eg__^{th}attempt , the person may feel that**and hence falls prey to the Fallacy by predicting a Tail ,not thinking that**__A TAIL IS DUE____the PROBABILITY OF OBTAINING EITHER HEADS OR TAILS REMAINS SAME__
— The use of the term

**originates from the most famous example of this phenomenon, which occurred in a Monte Carlo Casino on 18/8/1913.Then,a ball fell in black***Monte Carlo fallacy***15 TIMES . Because of such an occuring, gambler’s started betting on RED.But they fell prey to the Fallacy as**__RED TURNED-UP after 26 attempts.__**APPLICATION OF PROBABILITY**

— Probability
theory is applied in day to day life in risk assessments and in trade on
financial markets

— Another
Significant application of probability theory in everyday life is reliability.
Many consumer products, such as automobiles and consumer electronics use
reliability theory in product design to reduce the probability of failure.

**TERMINOLOGIES**

**Random Experiment:**

— If
an experiment or trial is repeated under the same conditions for any number of times
and it is possible to count the total number of outcomes is called as “

*Random Experiment*”.**Sample Space:**

— The
set of all possible outcomes of a random experiment is known as “Sample Space”
and denoted by set S. [this is similar to Universal set in Set Theory] The
outcomes of the random experiment are called sample points or outcomes.

**Exhaustive Events:**

— The
total number of all possible elementary outcomes in a random experiment is
known as ‘

*exhaustive events*’. In other words, a set is said to be exhaustive, when no other possibilities exists.**Favorable Events:**

— The
elementary outcomes which entail or favor the happening of an event is known as
‘

*favorable events*’ i.e., the outcomes which help in the occurrence of that event.**Equally likely or Equi-probable Events:**

— Outcomes
are said to be ‘

*equally likely*’ if there is no reason to expect one outcome to occur in preference to another. i.e., among all exhaustive outcomes, each of them has equal chance of occurrence.**Complementary Events:**

— Let
E denote occurrence of event. The complement of E denotes the non occurrence of
event E. Complement of E is denoted by ‘Ä’’.

**Independent Events:**

— Two
or more events are said to be ‘independent’, in a series of a trials if the
outcome of one event is does not affect the outcome of the other event or vise
versa.

— In
other words, several events are said to be ‘dependents’ if the occurrence of an
event is affected by the occurrence of any number of remaining events, in a
series of trials.

**Mutually Exclusive Events:**

— Events
are said to be ‘

*mutually exclusive*’ if the occurrence of an event totally prevents occurrence of all other events in a trial. In other words, two events A and B cannot occur simultaneously.**THREE TYPES OF PROBABILITY**

**1.**

**Theoretical probability**:

For
theoretical reasons, we assume that all

*n*possible outcomes of a particular experiment are equally likely, and we assign a probability of to each possible outcome.**2.**

**Relative frequency**

Relative Frequency is based on
observation or actual measurements.

**3.**

**Personal or subjective probability:**

These
are values (between 0 and 1 or 0 and 100%) assigned by individuals based on how
likely they think events are to occur.

**MEASUREMENT OF PROBABILITY**

There
are three approaches to construct a measure of probability of occurrence of an
event. They are:

—

**Classical or Mathematical Approach:**
In this approach we assume that an
experiment or trial results in any one of many possible outcomes, each outcome
being Equi-probable or equally-likely.

**Definition:**If a trial results in ‘n’ exhaustive, mutually exclusive, equally likely and independent outcomes, and if ‘m’ of them are favorable for the happening of the event E, then the probability ‘P’ of occurrence of the event ‘E’ is given by-

—

**Empirical or Statistical Approach:**
This
approach is also called the ‘frequency’ approach to probability. Here the
probability is obtained by actually performing the experiment large number of
times. As the number of trials n increases, we get more accurate result.

**Definition:**Consider a random experiment which is repeated large number of times under essentially homogeneous and identical conditions.

—

**Axiomatic Approach:**

This approach was
proposed by Russian Mathematician A.N.Kolmogorov in1933.

‘Axioms’ are statements
which are reasonably true and are accepted as such, without seeking any proof.

**Definition:**Let S be the sample space associated with a random experiment. Let A be any event in S. then P(A) is the probability of occurrence of A if the following axioms are satisfied.

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