Descriptive statistics and probabilityExample: Tossing a coin, throwing a die, selecting a card from a pack of playing cards, etc.

**Basic concepts of
Probability**

**Recall**

If an
experiment or trial can be repeated under the same conditions, any number of
times and it is possible to count the total number of outcomes, but individual
result ie., individual outcome is not predictable, then the experiment is known
as random experiment.

Example:
Tossing a coin, throwing a die, selecting a card from a pack of playing cards,
etc.

The result of a random experiment
will be called an outcome.

Any particular performance of a
random experiment is called a trial and outcome or combinations of outcomes are
termed as events.

The total
number of possible outcomes of a random experiment is known as the exhaustive
events.

The number of cases favourable to
an event in a trial is the number of outcomes which entail the happening of the
event.

Events
are said to be mutually exclusive if the happening of any one of them precludes
the happening of all the others, ie., if no two or more of them can happen
simultaneously in the same trial. Symbolically the event *A* and B are mutually exclusive if *A* âˆ© *B* = âˆ… .

Events
(two or more) of an experiment are said to be equally likely, if any one of
them cannot be expected to occur in preference to the others.

If a
random experiment or trial results in â€˜nâ€™ exhaustive, mutually exclusive and
equally likely outcomes (or cases), out of which m are favourable to the
occurrence of an event E, then the probability â€˜pâ€™ of occurrence (or happening)
of E, usually denoted by P(E), is given by

**Properties**

(i) 0â‰¤*P*(*E*)â‰¤1

(ii) Sum
of all the probability equal to 1.

(iii) If *P*(*E*)=0
then E is an impossible event.

**For
example : **A coin is tossed. Find the probability of getting a head

**Solution:
**The total
possible outcomes of an experiment {H,T}

Therefore
*n* = 2

The favourable
outcome for getting a head {H}. Therefore *m*
=1. Thus the required probability is

The modern approach to
probability is purely axiomatic and it is based on the set theory concepts. In
order to study, the theory of probability with an axiomatic approach it is
necessary to define certain basic concepts. They are

**(i) Sample Space: **Each
possible outcome of an experiment that can be repeated** **under similar or identical conditions is called a sample point and
the collection of sample points is called the sample space, denoted by S.

**(ii) Event: **Any subset of a sample space is
called an event.

**(iii) Mutually Exclusive events: **Two events** ***A*** **and** ***B***
**are said to be mutually** **exclusive
events if *A* âˆ© *B* = Ï• i.e., if
*A* and *B* are disjoint sets.

Example:
Consider S = { 1,2,3,4,5}

Let A = the set of odd numbers = {1,3,5}

and B = the set of even numbers = {2,4}

Then Aâˆ©B =
Ï•

Therefore
the events *A* and *B* are mutually exclusive

Statement meaning in terms of Set
theory approach

Let *E* be an experiment. Let *S* be a sample space associated with *E*. With every event in *S* we associate a real number denoted by *P*(*A*)
called the probability of the event *A *satisfying
the following axioms.

Axiom 1
: P(A)â‰¥0

Axiom 2
: P(S)=1

Axiom 3 : If
*A*_{1} , *A*_{2} ,......, *A _{n}* be a sequence of

*P *(* A*_{1}* UA*_{2}*U*â€¦*UA _{n} *)

**Theorem 1:**

Then *P*( âˆ… ) = 0
i.e., probability of an impossible event is zero.

**Theorem 2:**

Let *S* be the sample space and *A*
be an event in *S* then P( ) â‰¡ 1â€“ *P*(*A*)

**Theorem 3: Addition Theorem**

If *A* and *B* are any two
events then *P* ( *A* âˆª *B* ) = *P*( *A* ) + *P*( *B* ) âˆ’ *P*( *A* âˆ© *B*)

(i) If
the two events *A* and *B* are mutually exclusive then Aâˆ©*B* = âˆ…

âˆ´*P*(*A*âˆ©*B*)
= 0

â‡’*P*(*AUB*) =* P*(*A*) +* P*(*B*)

(ii) The addition theorem may be
extended to any three events *A,B,C*
and we have

*P *(* A*âˆª* B *âˆª* C*)* *=* P *(* A*)* *+* P *(* B*)+* P *(* C*)* *âˆ’* P *(* A*âˆ©* B *)* *âˆ’* P*(* A *âˆ©* C*)* *âˆ’* P *(* B *âˆ©* C*) + *P *(* A*âˆ©* B*âˆ©*C*)

It is believed that the students
might be familiar with the above concepts and our present syllabus continues
from the following.

Tags : Probability , 11th Business Mathematics and Statistics(EMS) : Chapter 8 : Descriptive statistics and probability

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11th Business Mathematics and Statistics(EMS) : Chapter 8 : Descriptive statistics and probability : Basic concepts of Probability | Probability

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