Probability Concepts and Random Variable

Sathyabama Institute of Science and Technology
SMTA1402 - Probability and Statistics

Unit-1 Probability Concepts and Random Variable

Random Experiment

An experiment whose outcome or result can be predicted with
certainty is called a Deterministic experiment.
Although all possible outcomes of an experiment may be known in
advance the outcome of a particular performance of the experiment cannot
be predicted owing to a number of unknown causes. Such an experiment is
called a Random experiment.
(e.g.) Whenever a fair dice is thrown, it is known that any of the 6 possible
outcomes will occur, but it cannot be predicted what exactly the outcome
will be.

Sample Space

The set of all possible outcomes which are assumed equally likely.

Event

A sub-set of S consisting of possible outcomes.

Mathematical definition of Probability

Let S be the sample space and A be an event associated with a random
experiment. Let n(S) and n(A) be the number of elements of S and A. then
the probability of event A occurring is denoted as P(A), is denoted by
n( A)
P( A) 
n( S )
Note: 1. It is obvious that 0  P(A)  1.
2. If A is an impossible event, P(A) = 0.
3. If A is a certain event , P(A) = 1.

A set of events is said to be mutually exclusive if the occurrence of any one
them excludes the occurrence of the others. That is, set of the events does
not occur simultaneously,
P(A1  A2  A3 ….. An,….. ) = 0 A set of events is said to be mutually
exclusive if the occurrence of any one them excludes the occurrence of the
others. That is, set of the events does not occur simultaneously,
P(A1  A2  A3 ….. An,….. ) = 0

Axiomatic definition of Probability

Let S be the sample space and A be an event associated with a random
experiment. Then the probability of the event A, P(A) is defined as a real
number satisfying the following axioms.
1. 0  P(A)  1
2. P(S) = 1
3. If A and B are mutually exclusive events, P(A  B) = P(A) + P(B)
and
4. If A1, A2 A3,….., An,….. are mutually exclusive events,
P(A1  A2  A3 ….. An,….. ) = P(A1) + P(A2) + P(A3) + ….. +
P(An)…..



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Important Theorems

Theorem 1: Probability of impossible event is zero.

Proof: Let S be sample space (certain events) and  be the impossible
event.
Certain events and impossible events are mutually exclusive.
P(S  ) = P(S) + P() (Axiom 3)
S=S
P(S) = P(S) + P()
P() = 0, hence the result.

Theorem 2: If A is the complementary event of A, P( A )  1  P( A)  1 .

Proof: Let A be the occurrence of the event
A be the non-occurrence of the event .
Occurrence and non-occurrence of the event are mutually exclusive.
P( A  A )  P( A)  P( A )
A A  S  P( A  A )  P( S )  1
 1  P( A)  P( A )
P( A )  1  P( A)  1 .

Theorem 3: (Addition theorem)
If A and B are any 2 events,
P(A  B) = P(A) + P(B)  P(A  B)  P(A) + P(B).

Proof: We know, A  AB  AB and B  A B  AB
 P( A)  P( AB )  P( AB) and P( B)  P( A B)  P( AB) (Axiom 3)
P( A)  P( B)  P( AB )  P( AB)  P( A B)  P( AB)
 P( A  B)  P( A  B)
P(A  B) = P(A) + P(B)  P(A  B)  P(A) + P(B).

Note: The theorem can be extended to any 3 events, A,B and C
P(A  B  C) = P(A) + P(B) +P(C)  P(A  B)  P(B  C)  P(C  A) +
P(A  B  C)


Theorem 4: If B  A, P(B)  P(A).

Proof: A and AB are mutually exclusive events such that B  AB  A
 P( B  AB )  P( A)
P( B)  P( AB )  P( A) (Axiom 3)
P ( B )  P ( A)

Conditional Probability

The conditional probability of an event B, assuming that the event A has
happened, is denoted by P(B/A) and defined as
P( A  B)
P( B / A)  , provided P(A)  0
P( A)




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Product theorem of probability

Rewriting the definition of conditional probability, We get
P( A  B)  P( A) P( A / B)
The product theorem can be extended to 3 events, A, B and C as follows:
P( A  B  C )  P( A) P( B / A) P(C / A  B)
Note: 1. If A  B, P(B/A) = 1, since A  B = A.
P( B)
2. If B  A, P(B/A)  P(B), since A  B = B, and  P( B),
P( A)
As P(A)  P(S) = 1.
3. If A and B are mutually exclusive events, P(B/A) = 0, since P(A
 B) = 0.
4. If P(A) > P(B), P(A/B) > P(B/A).
5. If A1  A2, P(A1/B)  P(A2/B).

Independent Events

A set of events is said to be independent if the occurrence of any one
of them does not depend on the occurrence or non-occurrence of the others.
If the two events A and B are independent, the product theorem takes
the form P(A  B) = P(A)  P(B), Conversely, if P(A  B) = P(A)  P(B),
the events are said to be independent (pair wise independent).
The product theorem can be extended to any number of independent
events, If A1 A2 A3 ….. An are n independent events, then
P(A1  A2  A3 ….. An) = P(A1)  P(A2 ) P(A3 )….. P(An)

Theorem 4:

If the events A and B are independent, the events A and B are also
independent.

Proof:
The events A  B and A  B are mutually exclusive such that (A  B) 
( A  B) = B
 P(A  B) + P( A  B) = P(B)
P( A  B) = P(B)  P(A  B)
= P(B)  P(A) P(B) (A and B are
independent)
= P(B) [1  P(A)]
= P( A ) P(B).
Theorem 5:
If the events A and B are independent, the events A and B are also
independent.

Proof:
P( A  B ) = P A  B  = 1  P(A  B)
= 1  [ P(A) + P(B)  P(A  B)] (Addition theorem)
= [1  P(A)]  P(B) [1  P(A)]
= P( A )P( B ).




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