Chapter 13
PROBABILITY
13.1 Overview
13.1.1 Conditional Probability
If E and F are two events associated with the same sample space of a random
experiment, then the conditional probability of the event E under the condition that the
event F has occurred, written as P (E | F), is given by
P(E ∩ F)
P(E | F) = , P(F) ≠ 0
P(F)
13.1.2 Properties of Conditional Probability
Let E and F be events associated with the sample space S of an experiment. Then:
(i) P (S | F) = P (F | F) = 1
(ii) P [(A ∪ B) | F] = P (A | F) + P (B | F) – P [(A ∩ B | F)],
where A and B are any two events associated with S.
(iii) P (E′ | F) = 1 – P (E | F)
13.1.3 Multiplication Theorem on Probability
Let E and F be two events associated with a sample space of an experiment. Then
P (E ∩ F) = P (E) P (F | E), P (E) ≠ 0
= P (F) P (E | F), P (F) ≠ 0
If E, F and G are three events associated with a sample space, then
P (E ∩ F ∩ G) = P (E) P (F | E) P (G | E ∩ F)
, PROBABILITY 259
13.1.4 Independent Events
Let E and F be two events associated with a sample space S. If the probability of
occurrence of one of them is not affected by the occurrence of the other, then we say
that the two events are independent. Thus, two events E and F will be independent, if
(a) P (F | E) = P (F), provided P (E) ≠ 0
(b) P (E | F) = P (E), provided P (F) ≠ 0
Using the multiplication theorem on probability, we have
(c) P (E ∩ F) = P (E) P (F)
Three events A, B and C are said to be mutually independent if all the following
conditions hold:
P (A ∩ B) = P (A) P (B)
P (A ∩ C) = P (A) P (C)
P (B ∩ C) = P (B) P (C)
and P (A ∩ B ∩ C) = P (A) P (B) P (C)
13.1.5 Partition of a Sample Space
A set of events E1, E2,...., En is said to represent a partition of a sample space S if
(a) Ei ∩ Ej = φ, i ≠ j; i, j = 1, 2, 3,......, n
(b) Ei ∪ E2∪ ... ∪ En = S, and
(c) Each Ei ≠ φ, i. e, P (Ei) > 0 for all i = 1, 2, ..., n
13.1.6 Theorem of Total Probability
Let {E1, E, ..., En} be a partition of the sample space S. Let A be any event associated
with S, then
n
P (A) = ∑ P(E
j=1
j )P(A | E j )
, 260 MATHEMATICS
13.1.7 Bayes’ Theorem
If E1, E2,..., En are mutually exclusive and exhaustive events associated with a sample
space, and A is any event of non zero probability, then
P(E i )P(A | Ei )
P(E i | A) = n
∑ P(E )P(A | E )
i =1
i i
13.1.8 Random Variable and its Probability Distribution
A random variable is a real valued function whose domain is the sample space of a
random experiment.
The probability distribution of a random variable X is the system of numbers
X : x1 x2 ... xn
P (X) : p1 p2 ... pn
n
where pi > 0, i =1, 2,..., n, ∑p
i =1
i = 1.
13.1.9 Mean and Variance of a Random Variable
Let X be a random variable assuming values x 1, x 2,...., x n with probabilities
n
p1, p2, ..., pn, respectively such that pi ≥ 0, ∑p i =1
i = 1 . Mean of X, denoted by µ [or
expected value of X denoted by E (X)] is defined as
n
μ = E (X) = ∑ xi pi
i =1
and variance, denoted by σ2, is defined as
n n
2 = ∑ ( xi – ) 2 pi =
i =1
∑x
i =1
2
i pi – 2
PROBABILITY
13.1 Overview
13.1.1 Conditional Probability
If E and F are two events associated with the same sample space of a random
experiment, then the conditional probability of the event E under the condition that the
event F has occurred, written as P (E | F), is given by
P(E ∩ F)
P(E | F) = , P(F) ≠ 0
P(F)
13.1.2 Properties of Conditional Probability
Let E and F be events associated with the sample space S of an experiment. Then:
(i) P (S | F) = P (F | F) = 1
(ii) P [(A ∪ B) | F] = P (A | F) + P (B | F) – P [(A ∩ B | F)],
where A and B are any two events associated with S.
(iii) P (E′ | F) = 1 – P (E | F)
13.1.3 Multiplication Theorem on Probability
Let E and F be two events associated with a sample space of an experiment. Then
P (E ∩ F) = P (E) P (F | E), P (E) ≠ 0
= P (F) P (E | F), P (F) ≠ 0
If E, F and G are three events associated with a sample space, then
P (E ∩ F ∩ G) = P (E) P (F | E) P (G | E ∩ F)
, PROBABILITY 259
13.1.4 Independent Events
Let E and F be two events associated with a sample space S. If the probability of
occurrence of one of them is not affected by the occurrence of the other, then we say
that the two events are independent. Thus, two events E and F will be independent, if
(a) P (F | E) = P (F), provided P (E) ≠ 0
(b) P (E | F) = P (E), provided P (F) ≠ 0
Using the multiplication theorem on probability, we have
(c) P (E ∩ F) = P (E) P (F)
Three events A, B and C are said to be mutually independent if all the following
conditions hold:
P (A ∩ B) = P (A) P (B)
P (A ∩ C) = P (A) P (C)
P (B ∩ C) = P (B) P (C)
and P (A ∩ B ∩ C) = P (A) P (B) P (C)
13.1.5 Partition of a Sample Space
A set of events E1, E2,...., En is said to represent a partition of a sample space S if
(a) Ei ∩ Ej = φ, i ≠ j; i, j = 1, 2, 3,......, n
(b) Ei ∪ E2∪ ... ∪ En = S, and
(c) Each Ei ≠ φ, i. e, P (Ei) > 0 for all i = 1, 2, ..., n
13.1.6 Theorem of Total Probability
Let {E1, E, ..., En} be a partition of the sample space S. Let A be any event associated
with S, then
n
P (A) = ∑ P(E
j=1
j )P(A | E j )
, 260 MATHEMATICS
13.1.7 Bayes’ Theorem
If E1, E2,..., En are mutually exclusive and exhaustive events associated with a sample
space, and A is any event of non zero probability, then
P(E i )P(A | Ei )
P(E i | A) = n
∑ P(E )P(A | E )
i =1
i i
13.1.8 Random Variable and its Probability Distribution
A random variable is a real valued function whose domain is the sample space of a
random experiment.
The probability distribution of a random variable X is the system of numbers
X : x1 x2 ... xn
P (X) : p1 p2 ... pn
n
where pi > 0, i =1, 2,..., n, ∑p
i =1
i = 1.
13.1.9 Mean and Variance of a Random Variable
Let X be a random variable assuming values x 1, x 2,...., x n with probabilities
n
p1, p2, ..., pn, respectively such that pi ≥ 0, ∑p i =1
i = 1 . Mean of X, denoted by µ [or
expected value of X denoted by E (X)] is defined as
n
μ = E (X) = ∑ xi pi
i =1
and variance, denoted by σ2, is defined as
n n
2 = ∑ ( xi – ) 2 pi =
i =1
∑x
i =1
2
i pi – 2
