introduction to probablity, bays theorem ,

1. 1
Module 1
Probability
1. Introduction
In our daily life we come across many processes whose nature cannot be predicted in advance.
Such processes are referred to as random processes. The only way to derive information about
random processes is to conduct experiments. Each such experiment results in an outcome
which cannot be predicted beforehand. In fact even if the experiment is repeated under
identical conditions, due to presence of factors which are beyond control, outcomes of the
experiment may vary from trial to trial. However we may know in advance that each outcome
of the experiment will result in one of the several given possibilities. For example, in the cast of
a die under a fixed environment the outcome (number of dots on the upper face of the die)
cannot be predicted in advance and it varies from trial to trial. However we know in advance
that the outcome has to be among one of the numbers 1, 2, … , 6. Probability theory deals with
the modeling and study of random processes. The field of Statistics is closely related to
probability theory and it deals with drawing inferences from the data pertaining to random
processes.
Definition 1.1
(i) A random experiment is an experiment in which:
(a) the set of all possible outcomes of the experiment is known in advance;
(b) the outcome of a particular performance (trial) of the experiment cannot be
predicted in advance;
(c) the experiment can be repeated under identical conditions.
(ii) The collection of all possible outcomes of a random experiment is called the sample
space. A sample space will usually be denoted by ߗ. ▄
Example 1.1
(i) In the random experiment of casting a die one may take the sample space as
ߗ = ሼ1, 2, 3, 4, 5, 6ሽ, where ݅ ∈ ߗ indicates that the experiment results in ݅ሺ݅ = 1, … ,6ሻ
dots on the upper face of die.
(ii) In the random experiment of simultaneously flipping a coin and casting a die one may
take the sample space as
ߗ = ሼ‫,ܪ‬ ܶሽ × ሼ1, 2, … , 6ሽ = ൛ሺ‫,ݎ‬ ݅ሻ: ‫ ݎ‬ ∈ ሼ‫,ܪ‬ ܶሽ, ݅ ∈ ሼ1, 2, … , 6ሽൟ,

2. 2
where ሺ‫,ܪ‬ ݅ሻ൫ሺܶ, ݅ሻ൯ indicates that the flip of the coin resulted in head (tail) on the
upper face and the cast of the die resulted in ݅ሺ݅ = 1, 2, … , 6ሻ dots on the upper face.
(iii) Consider an experiment where a coin is tossed repeatedly until a head is observed. In
this case the sample space may be taken as ߗ = ሼ1, 2, … ሽ (or ߗ =
ሼT, TH, TTH, … ሽ),where ݅ ∈ ߗ (or TT ⋯ TH ∈ ߗ with ሺ݅ − 1ሻ Ts and one H) indicates
that the experiment terminates on the ݅-th trial with first ݅ − 1 trials resulting in tails on
the upper face and the ݅-th trial resulting in the head on the upper face.
(iv) In the random experiment of measuring lifetimes (in hours) of a particular brand of
batteries manufactured by a company one may take ߗ = ሾ0,70,000ሿ,where we have
assumed that no battery lasts for more than 70,000 hours. ▄
Definition 1.2
(i) Let ߗ be the sample space of a random experiment and let ‫ܧ‬ ⊆ ߗ. If the outcome of the
random experiment is a member of the set ‫ܧ‬ we say that the event ‫ܧ‬ has occurred.
(ii) Two events ‫ܧ‬ଵand ‫ܧ‬ଶare said to be mutually exclusive if they cannot occur simultaneously,
i.e., if ‫ܧ‬ଵ ∩ ‫ܧ‬ଶ = ߶, the empty set. ▄
In a random experiment some events may be more likely to occur than the others. For
example, in the cast of a fair die (a die that is not biased towards any particular outcome),
the occurrence of an odd number of dots on the upper face is more likely than the
occurrence of 2 or 4 dots on the upper face. Thus it may be desirable to quantify the
likelihoods of occurrences of various events. Probability of an event is a numerical measure
of chance with which that event occurs. To assign probabilities to various events associated
with a random experiment one may assign a real number ܲሺ‫ܧ‬ሻ ∈ ሾ0,1ሿ to each event ‫ܧ‬ with
the interpretation that there is a ൫100 × ܲሺ‫ܧ‬ሻ൯% chance that the event ‫ܧ‬ will occur and a
ቀ100 × ൫1 − ܲሺ‫ܧ‬ሻ൯ቁ % chance that the event ‫ܧ‬ will not occur. For example if the
probability of an event is 0.25 it would mean that there is a 25% chance that the event will
occur and that there is a 75% chance that the event will not occur. Note that, for any such
assignment of possibilities to be meaningful, one must have ܲሺߗሻ = 1. Now we will discuss
two methods of assigning probabilities.
I. Classical Method
This method of assigning probabilities is used for random experiments which result in a
finite number of equally likely outcomes. Let ߗ = ሼ߱ଵ, … , ߱௡ሽ be a finite sample space with
݊ ሺ∈ ℕሻ possible outcomes; here ℕ denotes the set of natural numbers. For ⊆ ߗ , let |‫|ܧ‬
denote the number of elements in ‫.ܧ‬ An outcome ߱ ∈ ߗ is said to be favorable to an event

3. 3
‫ܧ‬ if ߱ ∈ ‫.ܧ‬ In the classical method of assigning probabilities, the probability of an event ‫ܧ‬ is
given by
ܲሺ‫ܧ‬ሻ =
number of outocmes favorable to E
total number of outcomes
=
|‫|ܧ‬
|ߗ|
=
|‫|ܧ‬
݊
.
Note that probabilities assigned through classical method satisfy the following properties of
intuitive appeal:
(i) For any event ‫,ܧ‬ ܲሺ‫ܧ‬ሻ ≥ 0;
(ii) For mutually exclusive events ‫ܧ‬ଵ, ‫ܧ‬ଶ, … , ‫ܧ‬௡ ሺ i.e. , ‫ܧ‬௜ ∩ ‫ܧ‬௝ = ߶ , whenever ݅, ݆ ∈
ሼ1, … , ݊ሽ, ݅ ≠ ݆ሻ
ܲ ൭ራ ‫ܧ‬௜
௡
௜ୀଵ
൱ =
|⋃ E୧
୬
୧ୀଵ |
n
=
∑ |E୧|୬
୧ୀଵ
n
= ෍
|E୧|
n
୬
୧ୀଵ
= ෍ ܲሺ‫ܧ‬௜ሻ;
௡
୧ୀଵ
(iii) ܲሺߗሻ =
|ఆ|
|ఆ|
= 1 .
Example 1.2
Suppose that in a classroom we have 25 students (with registration numbers1, 2, … , 25) born in
the same year having 365 days. Suppose that we want to find the probability of the event ‫ܧ‬
that they all are born on different days of the year. Here an outcome consists of a sequence of
25 birthdays. Suppose that all such sequences are equally likely. Then
|ߗ| = 365ଶହ
, |E| = 365 × 364 × ⋯ × 341 =ଷ଺ହ
ܲଶହ and ܲሺ‫ܧ‬ሻ =
|ா|
|ఆ|
=
ଷ଺ହುమఱ
ଷ଺ହమఱ ∙
The classical method of assigning probabilities has a limited applicability as it can be used only
for random experiments which result in a finite number of equally likely outcomes. ▄
II. Relative Frequency Method
Suppose that we have independent repetitions of a random experiment (here independent
repetitions means that the outcome of one trial is not affected by the outcome of another trial)
under identical conditions. Let ݂ேሺ‫ܧ‬ሻ denote the number of times an event ‫ܧ‬ occurs (also
called the frequency of event ‫ܧ‬ in ܰ trials) in the first ܰ trials and let ‫ݎ‬ேሺ‫ܧ‬ሻ = ݂ேሺ‫ܧ‬ሻ/ܰ denote
the corresponding relative frequency. Using advanced probabilistic arguments (e.g., using Weak
Law of Large Numbers to be discussed in Module 7) it can be shown that, under mild
conditions, the relative frequencies stabilize (in certain sense) as ܰ gets large (i.e., for any
event ‫,ܧ‬ lim
ே→ஶ
r୒ሺEሻ exists in certain sense). In the relative frequency method of assigning
probabilities the probability of an event ‫ܧ‬ is given by

4. 4
ܲሺ‫ܧ‬ሻ = lim
ே→ஶ
‫ݎ‬ேሺ‫ܧ‬ሻ ൌ lim
ே→ஶ
݂ேሺ‫ܧ‬ሻ
ܰ
∙
Figure 1.1. Plot of relative frequencies (‫ݎ‬ேሺ‫ܧ‬ሻ) of number of heads against number of trials (N)
in the random experiment of tossing a fair coin (with probability of head in each trial as 0.5).
In practice, to assign probability to an event ‫,ܧ‬ the experiment is repeated a large (but fixed)
number of times (say ܰ times) and the approximation ܲሺ‫ܧ‬ሻ ൎ ‫ݎ‬ேሺ‫ܧ‬ሻ is used for assigning
probability to event ‫.ܧ‬ Note that probabilities assigned through relative frequency method also
satisfy the following properties of intuitive appeal:
(i) for any event ‫,ܧ‬ ܲሺ‫ܧ‬ሻ ൒ 0;
(ii) for mutually exclusive events ‫ܧ‬ଵ, ‫ܧ‬ଶ, … , ‫ܧ‬௡
ܲ ൭ራ‫ܧ‬௜
௡
௜ୀଵ
൱ ൌ ෍ ܲሺ‫ܧ‬௜ሻ
௡
௜ୀଵ
;
(iii) ܲሺߗሻ ൌ 1.
Although the relative frequency method seems to have more applicability than the classical
method it too has limitations. A major problem with the relative frequency method is that it is

5. 5
imprecise as it is based on an approximation൫ܲሺ‫ܧ‬ሻ ≈ ‫ݎ‬ேሺ‫ܧ‬ሻ൯. Another difficulty with relative
frequency method is that it assumes that the experiment can be repeated a large number of
times. This may not be always possible due to budgetary and other constraints (e.g., in
predicting the success of a new space technology it may not be possible to repeat the
experiment a large number of times due to high costs involved).
The following definitions will be useful in future discussions.
Definition 1.3
(i) A set ‫ܧ‬ is said to be finite if either ‫ܧ‬ = ߶ (the empty set) or if there exists a one-one and
onto function ݂: ሼ1,2, … , ݊ሽ → ‫ ܧ‬ሺor ݂: ‫ܧ‬ → ሼ1,2, … , ݊ሽሻ for some natural number ݊;
(ii) A set is said to be infinite if it is not finite;
(iii) A set ‫ܧ‬ is said to be countable if either ‫ܧ‬ = ߶ or if there is an onto function ݂: ℕ → ‫,ܧ‬
where ℕ denotes the set of natural numbers;
(iv) A set is said to be countably infinite if it is countable and infinite;
(v) A set is said to be uncountable if it is not countable;
(vi) A set ‫ܧ‬ is said to be continuum if there is a one-one and onto function ݂: ℝ →
‫ ܧ‬ሺor ݂: ‫ܧ‬ → ℝ ሻ, where ℝ denotes the set of real numbers. ▄
The following proposition, whose proof(s) can be found in any standard textbook on set theory,
provides some of the properties of finite, countable and uncountable sets.
Proposition 1.1
(i) Any finite set is countable;
(ii) If ‫ܣ‬ is a countable and ‫ܤ‬ ⊆ ‫ܣ‬ then ‫ܤ‬ is countable;
(iii) Any uncountable set is an infinite set;
(iv) If ‫ܣ‬ is an infinite set and ‫ܣ‬ ⊆ ‫ܤ‬ then ‫ܤ‬ is infinite;
(v) If ‫ܣ‬ is an uncountable set and ‫ܣ‬ ⊆ ‫ܤ‬ then ‫ܤ‬ is uncountable;
(vi) If ‫ܧ‬ is a finite set and ‫ܨ‬ is a set such that there exists a one-one and onto function
݂: ‫ܧ‬ → ‫ ܨ‬ሺor ݂: ‫ܨ‬ → ‫ܧ‬ሻ then ‫ܨ‬ is finite;
(vii) If ‫ܧ‬ is a countably infinite (continuum) set and ‫ܨ‬ is a set such that there exists a one-one
and onto function ݂: ‫ܧ‬ → ‫ ܨ‬ሺor ݂: ‫ܨ‬ → ‫ܧ‬ሻ then ‫ܨ‬ is countably infinite (continuum);
(viii) A set ‫ܧ‬ is countable if and only if either ‫ܧ‬ = ߶ or there exists a one-one and onto map
݂: ‫ܧ‬ → ℕ଴, for some ℕ଴ ⊆ ℕ;
(ix) A set ‫ܧ‬ is countable if, and only if, either ‫ܧ‬ is finite or there exists a one-one map
݂: ℕ → ‫;ܧ‬
(x) A set ‫ܧ‬ is countable if, and only if, either ‫ܧ‬ = ߶ or there exists a one-one map ݂: ‫ܧ‬ →
ℕ;

6. 6
(xi) A non empty countable set ‫ܧ‬ can be either written as ‫ܧ‬ = ሼ߱ଵ, ߱ଶ, … ߱௡ሽ, for some
݊ ∈ ℕ, or as ‫ܧ‬ = ሼ߱ଵ, ߱ଶ, … ሽ;
(xii) Unit interval ሺ0,1ሻ is uncountable. Hence any interval ሺܽ, ܾሻ, where −∞ < ܽ < ܾ < ∞,
is uncountable;
(xiii) ℕ × ℕ is countable;
(xiv) Let ߉ be a countable set and let ሼ‫ܣ‬ఈ: ߙ ∈ ߉ሽ be a (countable) collection of countable
sets. Then ⋃ఈ∈௸‫ܣ‬ఈ is countable. In other words, countable union of countable sets is
countable;
(xv) Any continuum set is uncountable. ▄
Example 1.3
(i) Define ݂: ℕ → ℕ by ݂ሺ݊ሻ = ݊, ݊ ∈ ℕ. Clearly ݂: ℕ → ℕ is one-one and onto. Thus ℕ is
countable. Also it can be easily seen (using the contradiction method) that ℕ is infinite.
Thus ℕ is countably infinite.
(ii) Let ℤ denote the set of integers. Define ݂: ℕ → ℤ by
݂ሺ݊ሻ = ൞
݊ − 1
2
, if ݊ is odd
−
݊
2
, if ݊ is even
Clearly ݂: ℕ → ℤ is one-one and onto. Therefore, using (i) above and Proportion 1.1 (vii),
ℤ is countably infinite. Now on using Proportion 1.1 (ii) it follows that any subset of ℤ is
countable.
(iii) Using the fact that ℕ is countably infinite and Proposition 1.1 (xiv) it is straight forward
to show that ℚ (the set of rational numbers) is countably infinite.
(iv) Define ݂: ℝ → ℝ and ݃: ℝ → ሺ0, 1ሻ by ݂ሺ‫ݔ‬ሻ = ‫,ݔ‬ ‫ݔ‬ ∈ ℝ, and ݃ ሺ‫ݔ‬ሻ =
ଵ
ଵା௘ೣ , ‫ݔ‬ ∈ ℝ. Then
݂: ℝ → ℝ and ݃: ℝ → ሺ0, 1ሻ are one-one and onto functions. It follows that ℝand (0, 1)
are continuum (using Proposition 1.1 (vii)). Further, for − ∞ < ܽ < ܾ < ∞ , let
ℎሺ‫ݔ‬ሻ = ሺܾ − ܽሻ‫ݔ‬ + ܽ, ‫ݔ‬ ∈ ሺ0, 1ሻ. Clearly ℎ: ሺ0,1ሻ → ሺܽ, ܾሻ is one-one and onto. Again
using proposition 1.1 (vii) it follows that any interval ሺܽ, ܾሻ is continuum. ▄
It is clear that it may not be possible to assign probabilities in a way that applies to every
situation. In the modern approach to probability theory one does not bother about how
probabilities are assigned. Assignment of probabilities to various subsets of the sample space ߗ
that is consistent with intuitively appealing properties (i)-(iii) of classical (or relative frequency)
method is done through probability modeling. In advanced courses on probability theory it is
shown that in many situations (especially when the sample space ߗ is continuum) it is not

7. 7
possible to assign probabilities to all subsets of ߗ such that properties (i)-(iii) of classical (or
relative frequency) method are satisfied. Therefore probabilities are assigned to only certain
types of subsets of ߗ.
In the following section we will discuss the modern approach to probability theory where we
will not be concerned with how probabilities are assigned to suitably chosen subsets of ߗ.
Rather we will define the concept of probability for certain types of subsets ߗ using a set of
axioms that are consistent with properties (i)-(iii) of classical (or relative frequency) method.
We will also study various properties of probability measures.
2. Axiomatic Approach to Probability and Properties of Probability Measure
We begin this section with the following definitions.
Definition 2.1
(i) A set whose elements are themselves set is called a class of sets. A class of sets will be
usually denoted by script letters ࣛ, ℬ, ࣝ, …. For example ࣛ = ൛ሼ1ሽ, ሼ1, 3ሽ, ሼ2, 5, 6ሽൟ;
(ii) Let ࣝ be a class of sets. A function ߤ: ࣝ → ℝ is called a set function. In other words, a
real-valued function whose domain is a class of sets is called a set function. ▄
As stated above, in many situations, it may not be possible to assign probabilities to all subsets
of the sample space ߗ such that properties (i)-(iii) of classical (or relative frequency) method
are satisfied. Therefore one begins with assigning probabilities to members of an appropriately
chosen class ࣝ of subsets of ߗ (e.g., if ߗ = ℝ, then ࣝ may be class of all open intervals in ℝ; if ߗ
is a countable set, then ࣝ may be class of all singletons ሼ߱ሽ, ߱ ∈ ߗ). We call the members of ࣝ
as basic sets. Starting from the basic sets in ࣝ assignment of probabilities is extended, in an
intuitively justified manner, to as many subsets of ߗ as possible keeping in mind that properties
(i)-(iii) of classical (or relative frequency) method are not violated. Let us denote by ℱ the class
of sets for which the probability assignments can be finally done. We call the class ℱ as event
space and elements of ℱare called events. It will be reasonable to assume that ℱ satisfies the
following properties: (i) ߗ ∈ ℱ, (ii) ‫ܣ‬ ∈ ℱ ⟹ ‫ܣ‬஼
= ߗ − ‫ܣ‬ ∈ ℱ ,and (iii)‫ܣ‬௜ ∈ ℱ, ݅ = 1,2, … ⇒
⋃ ‫ܣ‬௜ ∈ ℱஶ
௜ୀଵ . This leads to introduction of the following definition.
Definition 2.2
A sigma-field (ߪ-field) of subsets of ߗ is a class ℱ of subsets of ߗ satisfying the following
properties:
(i) ߗ ∈ ℱ;
(ii) ‫ܣ‬ ∈ ℱ ⇒ ‫ܣ‬௖
= ߗ − ‫ܣ‬ ∈ ℱ (closed under complements);

8. 8
(iii) ‫ܣ‬௜ ∈ ℱ, ݅ = 1, 2, … ⇒ ⋃ ‫ܣ‬௜ ∈ ℱஶ
௜ୀଵ (closed under countably infinite unions). ▄
Remark 2.1
(i) We expect the event space to be a ߪ-field;
(ii) Suppose that ℱ is a ߪ-field of subsets of ߗ. Then,
(a) ߶ ∈ ℱ ሺsince ߶ = ߗ௖ሻ
(b) ‫ܧ‬ଵ, ‫ܧ‬ଶ, … ∈ ℱ ⇒ ⋂ ‫ܧ‬௜ ∈ ℱஶ
௜ୀଵ ሺsince ⋂ ‫ܧ‬௜
ஶ
௜ୀଵ = ሺ⋃ ‫ܧ‬௜
௖ஶ
௜ୀଵ ሻ௖ሻ;
(c) ‫,ܧ‬ ‫ܨ‬ ∈ ℱ ⇒ ‫ܧ‬ − ‫ܨ‬ = ‫ܧ‬ ∩ ‫ܨ‬௖
∈ ℱ and ‫ ܧ‬Δ‫ܨ‬ ≝ ሺ‫ܧ‬ − ‫ܨ‬ሻ ∪ ሺ‫ܨ‬ − ‫ܧ‬ሻ ∈ ℱ;
(d) ‫ܧ‬ଵ, ‫ܧ‬ଶ, … , ‫ܧ‬௡ ∈ ℱ, for some ݊ ∈ ℕ, ⇒ ⋃ ‫ܧ‬௜ ∈ ℱ௡
௜ୀଵ and ⋂ ‫ܧ‬௜ ∈ ℱ௡
௜ୀଵ (take
‫ܧ‬௡ାଵ = ‫ܧ‬௡ାଶ = ⋯ = ߶so that ⋃ ‫ܧ‬௜
௡
௜ୀଵ = ⋃ ‫ܧ‬௜
∞
௜ୀଵ or ‫ܧ‬௡ାଵ = ‫ܧ‬௡ାଶ = ⋯ = ߗ so
that ⋂ ‫ܧ‬௜
௡
௜ୀଵ = ⋂ ‫ܧ‬௜
∞
௜ୀଵ );
(e) although the power set of ߗ൫࣪ሺߗሻ൯ is a ߪ-field of subsets of ߗ,in general, a ߪ-
field may not contain all subsets of ߗ. ▄
Example 2.1
(i) ℱ = ሼ߶, ߗሽ is a sigma field, called the trivial sigma-field;
(ii) Suppose that ‫ܣ‬ ⊆ ߗ. Then ℱ = ሼ‫,ܣ‬ ‫ܣ‬௖
, ߶, ߗሽ is a ߪ-field of subsets of ߗ. It is the
smallest sigma-field containing the set ‫;ܣ‬
(iii) Arbitrary intersection of ߪ-fields is a ߪ-field (see Problem 3 (i));
(iv) Let ࣝ be a class of subsets of ߗ and let ሼ‫ܨ‬ఈ ∶ ߙ ∈ ߉ሽ be the collection of all ߪ-fields
that contain ࣝ. Then
ℱ = ሩ ℱఈ
ఈ∈௸
is a ߪ-field and it is the smallest ߪ-field that contains class ࣝ (called the ߪ-field
generated by ࣝ and is denoted by ߪሺࣝሻ) (see Problem 3 (iii));
(v) Let ߗ = ℝ and let ࣤ be the class of all open intervals in ℝ. Then ℬଵ = ߪሺࣤሻ is called
the Borel ߪ-field on ℝ. The Borel ߪ-field in ℝ௞
(denoted by ℬ௞ ) is the ߪ-field
generated by class of all open rectangles in ℝ௞
. A set ‫ܤ‬ ∈ ℬ௞ is called a Borel set in
ℝ௞
; here ℝ௞
= ሼሺ‫ݔ‬ଵ, … , ‫ݔ‬௞ሻ: −∞ < ‫ݔ‬௜ < ∞, ݅ = 1, … , ݇ሽ denotes the ݇-dimensional
Euclidean space;
(vi) ℬଵ contains all singletons and hence all countable subsets of ℝ ቀሼܽሽ = ⋂ ቀܽ −ஶ
௡ୀଵ
ଵ
௡
, ܽ +
ଵ
௡
ቁቁ ∙ ▄
Let ࣝ be an appropriately chosen class of basic subsets of ߗ for which the probabilities can be
assigned to begin with (e.g., if ߗ = ℝ then ࣝ may be class of all open intervals in ℝ; if ߗ is a
countable set then ࣝ may be class of all singletons ሼ߱ሽ, ߱ ∈ ߗ). It turns out (a topic for an
advanced course in probability theory) that, for an appropriately chosen class ࣝ of basic sets,

9. 9
the assignment of probabilities that is consistent with properties (i)-(iii) of classical (or relative
frequency) method can be extended in an unique manner from ࣝ to ߪሺࣝሻ, the smallest ߪ-field
containing the class ࣝ. Therefore, generally the domain ℱ of a probability measure is taken to
be ߪሺࣝሻ, the ߪ-field generated by the class ࣝ of basic subsets of ߗ. We have stated before that
we will not care about how assignment of probabilities to various members of event space ℱ (a
ߪ-field of subsets of ߗ) is done. Rather we will be interested in properties of probability
measure defined on event space ℱ.
Let ߗ be a sample space associated with a random experiment and let ℱ be the event space (a
ߪ-field of subsets of ߗ). Recall that members of ℱ are called events. Now we provide a
mathematical definition of probability based on a set of axioms.
Definition 2.3
(i) Let ℱ be a ߪ-field of subsets of ߗ. A probability function (or a probability measure) is a
set function ܲ, defined on ℱ, satisfying the following three axioms:
(a) ܲሺ‫ܧ‬ሻ ≥ 0, ∀‫ܧ‬ ∈ ℱ; (Axiom 1: Non-negativity);
(b) If ‫ܧ‬ଵ, ‫ܧ‬ଶ, … is a countably infinite collection of mutually exclusive events ൫i. e., ‫ܧ‬௜ ∈
ℱ, ݅ = 1, 2, … , ‫ܧ‬௜ ∩ ‫ܧ‬௝ = ߶, ݅ ≠ ݆ ൯ then
ܲ ൭ራ ‫ܧ‬௜
∞
௜ୀଵ
൱ = ෍ ܲሺ‫ܧ‬௜ሻ
∞
ଵୀଵ
; ሺAxiom 2: Countably infinite additiveሻ
(c) ܲሺߗሻ = 1 (Axiom 3: Probability of the sample space is 1).
(ii) The triplet ሺߗ, ℱ, ܲሻ is called a probability space. ▄
Remark 2.2
(i) Note that if ‫ܧ‬ଵ, ‫ܧ‬ଶ, … is a countably infinite collection of sets in a ߪ-field ℱthen
⋃ ‫ܧ‬௜
ஶ
௜ୀଵ ∈ ℱ and, therefore, ܲሺ⋃ ‫ܧ‬௜
ஶ
௜ୀଵ ሻ is well defined;
(ii) In any probability space ሺߗ, ℱ, ܲሻ we have ܲሺߗሻ = 1 (or ܲሺ߶ሻ = 0; see Theorem 2.1 (i)
proved later) but if ܲሺ‫ܣ‬ሻ = 1 (or ܲሺ‫ܣ‬ሻ = 0), for some ‫ܣ‬ ∈ ℱ, then it does not mean that
‫ܣ‬ = ߗ ( or ‫ܣ‬ = ߶) (see Problem 14 (ii).
(iii) In general not all subsets of ߗ are events, i.e., not all subsets of ߗ are elements of ℱ.
(iv) When ߗ is countable it is possible to assign probabilities to all subsets of ߗ using Axiom
2 provided we can assign probabilities to singleton subsets ሼ‫ݔ‬ሽ of ߗ. To illustrate this let
ߗ = ሼ߱ଵ, ߱ଶ, … ሽ ሺor Ω = ሼ߱ଵ, … , ߱௡ሽ, for some n ∈ ℕሻ and let ܲሺሼ߱௜ሽሻ = ‫݌‬௜, ݅ =

10. 10
1, 2, … , so that 0 ≤ ‫݌‬௜ ≤ 1, ݅ = 1,2, … (see Theorem 2.1 (iii) below) and
∑ ‫݌‬௜ =ஶ
௜ୀଵ ∑ ܲሺሼ߱௜ሽሻஶ
௜ୀଵ = ܲሺ⋃ ሼ߱௜ሽஶ
௜ୀଵ ሻ = ܲሺߗሻ = 1. Then, for any ‫ܣ‬ ⊆ ߗ,
ܲሺ‫ܣ‬ሻ = ෍ ‫݌‬௜.
௜:ఠ೔ ∈஺
Thus in this case we may take ℱ = ܲሺߗሻ, the power set of ߗ. It is worth mentioning
here that if ߗ is countable and ࣝ = ൛ሼ߱ሽ ∶ ߱ ∈ ߗൟ (class of all singleton subsets of ߗ) is
the class of basic sets for which the assignment of the probabilities can be done, to
begin with, then ߪሺࣝሻ = ࣪ሺߗሻ (see Problem 5 (ii)).
(v) Due to some inconsistency problems, assignment of probabilities for all subsets of ߗ is
not possible when ߗ is continuum (e.g., if ߗ contains an interval). ▄
Theorem 2.1
Letሺߗ, ℱ, ܲሻbe a probability space. Then
(i) ܲሺ߶ሻ = 0;
(ii) ‫ܧ‬௜ ∈ ℱ, ݅ = 1, 2, … . ݊ , and ‫ܧ‬௜ ∩ ‫ܧ‬௝ = ߶, ݅ ≠ ݆ ⇒ ܲሺ⋃ ‫ܧ‬௜
௡
௜ୀଵ ሻ = ∑ ܲሺ‫ܧ‬௜ሻ௡
௜ୀଵ (finite
additivity);
(iii) ∀‫ܧ‬ ∈ ℱ, 0 ≤ ܲሺ‫ܧ‬ሻ ≤ 1 and ܲሺ‫ܧ‬௖ሻ = 1 − ܲሺ‫ܧ‬ሻ;
(iv) ‫ܧ‬ଵ, ‫ܧ‬ଶ ∈ ℱ and ‫ܧ‬ଵ ⊆ ‫ܧ‬ଶ ⇒ ܲሺ‫ܧ‬ଶ − ‫ܧ‬ଵሻ = ܲሺ‫ܧ‬ଶሻ − ܲሺ‫ܧ‬ଵሻ and ܲሺ‫ܧ‬ଵሻ ≤ ܲሺ‫ܧ‬ଶሻ
(monotonicity of probability measures);
(v) ‫ܧ‬ଵ, ‫ܧ‬ଶ ∈ ℱ ⇒ ܲሺ‫ܧ‬ଵ ∪ ‫ܧ‬ଶሻ = ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻ − ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ.
Proof.
(i) Let ‫ܧ‬ଵ = ߗ and ‫ܧ‬௜ = ߶, ݅ = 2, 3, …. Then ܲሺ‫ܧ‬ଵሻ = 1, (Axiom 3) ‫ܧ‬௜ ∈ ℱ, ݅ = 1, 2, … ,
‫ܧ‬ଵ = ⋃ ‫ܧ‬௜
ஶ
௜ୀଵ and ‫ܧ‬௜ ∩ ‫ܧ‬௝ = ߶, ݅ ≠ ݆. Therefore,
1 = ܲሺ‫ܧ‬ଵሻ = ܲ ൭ራ ‫ܧ‬௜
ஶ
௜ୀଵ
൱
= ෍ ܲሺ‫ܧ‬௜ሻ ሺusing Axiom 2ሻ
ஶ
௜ୀଵ
= 1 + ෍ ܲሺ߶ሻ
ஶ
௜ୀଶ
⇒ ෍ ܲሺ߶ሻ
ஶ
௜ୀଶ
= 0

11. 11
⇒ ܲሺ߶ሻ ൌ 0.
(ii) Let ‫ܧ‬௜ ൌ ߶, ݅ ൌ ݊ ൅ 1, ݊ ൅ 2, … . Then ‫ܧ‬௜ ∈ ࣠, ݅ ൌ 1, 2, … , ‫ܧ‬௜ ∩ ‫ܧ‬௝ ൌ ߶, ݅ ് ݆ and
ܲሺ‫ܧ‬௜ሻ ൌ 0, ݅ ൌ ݊ ൅ 1, ݊ ൅ 2, …. Therefore,
ܲ ൭ራ ‫ܧ‬௜
௡
ଵୀଵ
൱ ൌ ܲ ൭ራ ‫ܧ‬௜
ஶ
ଵୀଵ
൱
ൌ ෍ ܲሺ‫ܧ‬௜ሻ ሺusing Axiom 2ሻ
ஶ
௜ୀଵ
ൌ ෍ ܲሺ‫ܧ‬௜ሻ
௡
୧ୀଵ
.
(iii) Let ‫ܧ‬ ∈ ࣠. Then ߗ ൌ ‫ܧ‬ ∪ ‫ܧ‬௖
and ‫ܧ‬ ∩ ‫ܧ‬஼
ൌ ߶. Therefore
1 ൌ ܲሺߗሻ
ൌ ܲሺ‫ܧ‬ ∪ ‫ܧ‬௖ሻ
ൌ ܲሺ‫ܧ‬ሻ ൅ ܲሺ‫ܧ‬௖ሻ (using (ii))
⇒ ܲሺ‫ܧ‬ሻ ൑ 1 and ܲሺ‫ܧ‬௖ሻ ൌ 1 െ ܲሺ‫ܧ‬ሻ (since ܲሺ‫ܧ‬௖
ሻ ∈ ሾ0,1ሿ)
⇒ 0 ൑ ܲሺ‫ܧ‬ሻ ൑ 1 and ܲሺ‫ܧ‬௖ሻ ൌ 1 െ ܲሺ‫ܧ‬ሻ.
(iv) Let ‫ܧ‬ଵ, ‫ܧ‬ଶ ∈ ࣠ and let ‫ܧ‬ଵ ⊆ ‫ܧ‬ଶ . Then ‫ܧ‬ଶ െ ‫ܧ‬ଵ ∈ ࣠, ‫ܧ‬ଶ ൌ ‫ܧ‬ଵ ∪ ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ and ‫ܧ‬ଵ ∩
ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ ൌ ߶.
Figure 2.1
Therefore,

12. 12
ܲሺ‫ܧ‬ଶሻ = ܲ൫‫ܧ‬ଵ ∪ ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ൯
ൌ ܲሺ‫ܧ‬ଵሻ ൅ ܲሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ (using (ii))
⇒ ܲሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ ൌ ܲሺ‫ܧ‬ଶሻ െ ܲሺ‫ܧ‬ଵሻ.
As ܲሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ ൒ 0, it follows that ܲሺ‫ܧ‬ଵሻ ൑ ܲሺ‫ܧ‬ଶሻ.
(v) Let ‫ܧ‬ଵ, ‫ܧ‬ଶ ∈ ࣠. Then ‫ܧ‬ଶ െ ‫ܧ‬ଵ ∈ ࣠, ‫ܧ‬ଵ ∩ ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ ൌ ߶ and ‫ܧ‬ଵ ∪ ‫ܧ‬ଶ ൌ ‫ܧ‬ଵ ∪
ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ.
Figure 2.2
Therefore,
ܲሺ‫ܧ‬ଵ ∪ ‫ܧ‬ଶሻ ൌ ܲ൫‫ܧ‬ଵ ∪ ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ൯
ൌ ܲሺ‫ܧ‬ଵሻ ൅ ܲሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ (using (ii)) (2.1)
Also ሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ ∩ ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ ൌ ߶ and ‫ܧ‬ଶ ൌ ሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ ∪ ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ. Therefore,
Figure 2.3
ܲሺ‫ܧ‬ଶሻ ൌ ܲ൫ሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ ∪ ሺ‫ܧ‬ଶ െ ‫ܧ‬ଵሻ൯

13. 13
= ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ + ܲሺ‫ܧ‬ଶ − ‫ܧ‬ଵሻ (using (ii)
⇒ ܲሺ‫ܧ‬ଶ − ‫ܧ‬ଵሻ = ܲሺ‫ܧ‬ଶሻ − ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ ∙ (2.2)
Using (2.1) and (2.2), we get
ܲሺ‫ܧ‬ଵ ∪ ‫ܧ‬ଶሻ = ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻ − ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ. ▄
Theorem 2.2 (Inclusion-Exclusion Formula)
Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫ܧ‬ଵ, ‫ܧ‬ଶ, … , ‫ܧ‬௡ ∈ ℱ ሺ݊ ∈ ℕ, ݊ ≥ 2ሻ. Then
ܲ ൭ራ ‫ܧ‬௜
௡
௜ୀଵ
൱ = ෍ ܵ௞,௡
௡
௞ୀଵ
,
where ܵଵ,௡ = ∑ ܲሺ‫ܧ‬௜ሻ௡
௜ୀଵ and, for ݇ ∈ ሼ2, 3, … , ݊ሽ,
ܵ௞,௡ = ሺ−1ሻ௞ିଵ
෍ ܲ൫‫ܧ‬௜ଵ
∩ ‫ܧ‬௜ଶ
∩ ⋯ ∩ ‫ܧ‬௜௞
൯.
ଵஸ௜భழ⋯ழ௜ೖஸ௡
Proof. We will use the principle of mathematical induction. Using Theorem 2.1 (v), we have
ܲሺ‫ܧ‬ଵ ∪ ‫ܧ‬ଶሻ = ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻ − ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ
= ܵଵ,ଶ + ܵଶ,ଶ,
where ܵଵ,ଶ = ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻ and ܵଶ,ଶ = −ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ. Thus the result is true for ݊ = 2. Now
suppose that the result is true for ݊ ∈ ሼ2, 3, … , ݉ሽ for some positive integer ݉ ሺ≥ 2ሻ. Then
ܲ ൭ራ ‫ܧ‬௜
௠ାଵ
௜ୀଵ
൱ = ܲ ቌ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ ∪ ‫ܧ‬௠ାଵቍ
= ܲ ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ + ܲሺ‫ܧ‬௠ାଵሻ − ܲ ቌ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ ∩ ‫ܧ‬௠ାଵቍ ሺusing the result for ݊ = 2ሻ
= ܲ ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ + ܲሺ‫ܧ‬௠ାଵሻ − ܲ ൭ራሺ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵሻ
௠
௜ୀଵ
൱
= ෍ ܵ௜,௠
௠
௜ୀଵ
+ ܲሺ‫ܧ‬௠ାଵሻ − ܲ ൭ራሺ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵሻ
௠
௜ୀଵ
൱ ሺusing the result for ݊ = ݉ሻ ሺ2.3ሻ
Let ‫ܨ‬௜ = ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵ, ݅ = 1, … . ݉. Then

14. 14
ܲ ൭ራሺ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵሻ
௠
௜ୀଵ
൱ = ܲ ൭ራ ‫ܨ‬௜
௠
௜ୀଵ
൱
= ∑ ܶ௞,௠ ௠
௞ୀଵ ሺagain using the result for ݊ = ݉ሻ , ሺ2.4ሻ
where ܶଵ,௠ = ∑ ܲሺ‫ܨ‬௜ሻ௠
௜ୀଵ = ∑ ܲሺ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵሻ௠
௜ୀଵ and, for ݇ ∈ ሼ2, 3, ⋯ , ݉ሽ,
ܶ௞,௠ = ሺ−1ሻ௞ିଵ
෍ ܲ൫‫ܨ‬௜భ
∩ ‫ܨ‬௜మ
∩ ⋯ ∩ ‫ܨ‬௜ೖ
൯
ଵஸ௜భழ௜మழ⋯ழ௜ೖஸ௠
= ሺ−1ሻ௞ିଵ
෍ ܲ൫‫ܧ‬௜భ
∩ ‫ܧ‬௜మ
∩ ⋯ ∩ ‫ܧ‬௜ೖ
∩ ‫ܧ‬௠ାଵ൯
ଵஸ௜భழ௜మழ⋯ழ௜ೖஸ௠
.
Using (2.4) in (2.3), we get
ܲሺ⋃ ‫ܧ‬௜
௠ାଵ
௜ୀଵ ሻ = ቀܵଵ,௠ + ܲሺ‫ܧ‬௠ାଵሻቁ + ൫ܵଶ,௠ − ܶଵ,௠൯ + ⋯ + ൫ܵ௠,௠ − ܶ௠ିଵ,௠൯ − ܶ௠,௠ .
Note that ܵଵ,௠ + ܲሺ‫ܧ‬௠ାଵሻ = ܵଵ,௠ାଵ, ܵ௞,௠ − ܶ௞ିଵ,௠ = ܵ௞,௠ାଵ, ݇ = 2,3, … , ݉ , and ܶ௠,௠ =
−ܵ௠ାଵ,௠ାଵ. Therefore,
ܲ ൭ራ ‫ܧ‬௜
௠ାଵ
௜ୀଵ
൱ = ܵଵ,௠ାଵ + ෍ ܵ௞,௠ାଵ
௠ାଵ
௞ୀଶ
= ෍ ܵ௞,௠ାଵ
௠ାଵ
௞ୀଵ
. ▄
Remark 2.3
(i) Let ‫ܧ‬ଵ, ‫ܧ‬ଶ … ∈ ℱ. Then
ܲሺ‫ܧ‬ଵ ∪ ‫ܧ‬ଶ ∪ ‫ܧ‬ଷሻ
= ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻ + ܲሺ‫ܧ‬ଷሻᇣᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇥ
ௌభ,య
−൫ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ + ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଷሻ + ܲሺ‫ܧ‬ଶ ∩ ‫ܧ‬ଷሻ൯ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇧᇥ
ௌమ,య
+ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶ ∩ ‫ܧ‬ଷሻᇣᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇥ
ௌయ,య
= ‫݌‬ଵ,ଷ − ‫݌‬ଶ,ଷ + ‫݌‬ଷ,ଷ,
where ‫݌‬ଵ,ଷ = ܵଵ,ଷ, ‫݌‬ଶ,ଷ = −ܵଶ,ଷ and ‫݌‬ଷ,ଷ = ܵଷ,ଷ.
In general,
ܲሺ⋃ ‫ܧ‬௜
௡
௜ୀଵ ሻ = ‫݌‬ଵ,௡ − ‫݌‬ଶ,௡ + ‫݌‬ଷ,௡ ⋯ + ሺ−1ሻ௡ିଵ
‫݌‬௡,௡,
where

15. 15
‫݌‬௜,௡ = ൜
ܵ௜,௡, if ݅ is odd
−ܵ௜,௡, if ݅ is even
, ݅ = 1, 2, … ݊.
(ii) We have
1 ≥ ܲሺ‫ܧ‬ଵ ∪ ‫ܧ‬ଶሻ = ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻ − ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ
⇒ ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ ≥ ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻ − 1.
The above inequality is known as Bonferroni’s inequality. ▄
Theorem 2.3
Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫ܧ‬ଵ, ‫ܧ‬ଶ, … , ‫ܧ‬௡ ∈ ℱ ሺ݊ ∈ ℕ, ݊ ≥ 2 ሻ. Then, under
the notations of Theorem 2.2,
(i) (Boole’s Inequality) ܵଵ,௡ + ܵଶ,௡ ≤ ܲሺ⋃ ‫ܧ‬௜
௡
ଵୀଵ ሻ ≤ ܵଵ,௡;
(ii) (Bonferroni’s Inequality) ܲሺ⋂ ‫ܧ‬௜
௡
ଵୀଵ ሻ ≥ ܵଵ,௡ − ሺ݊ − 1ሻ.
Proof.
(i) We will use the principle of mathematical induction. We have
ܲሺ‫ܧ‬ଵ ∪ ‫ܧ‬ଶሻ = ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻᇣᇧᇧᇧᇤᇧᇧᇧᇥ
ௌభ,మ
−ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻᇣᇧᇧᇧᇤᇧᇧᇧᇥ
ௌమ,మ
= ܵଵ,ଶ + ܵଶ,ଶ
≤ ܵଵ,ଶ,
where ܵଵ,ଶ = ܲሺ‫ܧ‬ଵሻ + ܲሺ‫ܧ‬ଶሻ and ܵଶ,ଶ = −ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ ≤ 0.
Thus the result is true for ݊ = 2. Now suppose that the result is true for ݊ ∈
ሼ2, 3, … , ݉ሽ for some positive integer ݉ ሺ≥ 2ሻ, i.e., suppose that for arbitrary events
‫ܨ‬ଵ, … , ‫ܨ‬௠ ∈ ℱ
ܲ ቌራ ‫ܨ‬௜
௞
௜ୀଵ
ቍ ≤ ෍ ܲሺ‫ܨ‬௜ሻ
௞
௜ୀଵ
, ݇ = 2, 3, … , ݉ ሺ2.5ሻ
and
ܲ ቌራ ‫ܨ‬௜
௞
௜ୀଵ
ቍ ≥ ෍ ܲሺ‫ܨ‬௜ሻ
௞
௜ୀଵ
− ෍ ܲ൫‫ܨ‬௜ ∩ ‫ܨ‬௝൯
ଵஸ௜ழ௝ஸ௞
, ݇ = 2, 3, … , ݉. ሺ2.6ሻ
Then

16. 16
ܲ ൭ራ ‫ܧ‬௜
௠ାଵ
௜ୀଵ
൱ = ܲ ቌ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ ∪ ‫ܧ‬௠ାଵቍ
≤ ܲ ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ + ܲሺ‫ܧ‬௠ାଵሻ ሺusing ሺ2.5ሻ for ݇ = 2ሻ
≤ ෍ ܲሺ‫ܧ‬௜ሻ
௠
௜ୀଵ
+ ܲሺ‫ܧ‬௠ାଵሻ ሺusing ሺ2.5ሻ for k = mሻ
= ෍ ܲሺ‫ܧ‬௜ሻ
௠ାଵ
௜ୀଵ
= ܵଵ,௠ାଵ. ሺ2.7ሻ
Also,
ܲ ൭ራ ‫ܧ‬௜
௠ାଵ
௜ୀଵ
൱ = ܲ ቌ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ ∪ ‫ܧ‬௠ାଵቍ
= ܲ ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ + ܲሺ‫ܧ‬௠ାଵሻ − ܲ ቌ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ ∩ ‫ܧ‬௠ାଵቍ ሺusing Theorem 2.2ሻ
= ܲ ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ + ܲሺ‫ܧ‬௠ାଵሻ − ܲ ൭ራሺ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵሻ
௠
௜ୀଵ
൱. ሺ2.8ሻ
Using (2.5), for ݇ = ݉, we get
ܲ ൭ራሺ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵሻ
௠
௜ୀଵ
൱ ≤ ෍ ܲ
௠
௜ୀଵ
ሺ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵሻ, ሺ2.9ሻ
and using (2.6), for ݇ = ݉, we get
ܲ ൭ራ ‫ܧ‬௜
௠
௜ୀଵ
൱ ≥ ܵଵ,௠ + ܵଶ,௠. ሺ2.10ሻ
Now using (2.9) and (2.10) in (2.8), we get

17. 17
ܲ ൭ራ ‫ܧ‬௜
௠ାଵ
௜ୀଵ
൱ ≥ ܵଵ,௠ + ܵଶ,௠ + ܲሺ‫ܧ‬௠ାଵሻ − ෍ ܲሺ‫ܧ‬௜ ∩ ‫ܧ‬௠ାଵሻ
௠
௜ୀଵ
= ෍ ܲሺ‫ܧ‬௜ሻ
௠ାଵ
௜ୀଵ
− ෍ ܲ൫‫ܧ‬௜ ∩ ‫ܧ‬௝൯
ଵஸ௜ழ௝ஸ௠ାଵ
= ܵଵ,௠ାଵ + ܵଶ,௠ାଵ. (2.11)
Combining (2.7) and (2.11), we get
ܵଵ,௠ାଵ + ܵଶ,௠ାଵ ≤ ܲ ൭ ራ ‫ܧ‬௜
௠ାଵ
ଵୀଵ
൱ ≤ ܵଵ,௠ାଵ,
and the assertion follows by principle of mathematical induction.
(ii) We have
ܲ ൭ሩ ‫ܧ‬௜
௡
୧ୀଵ
൱ = 1 − ܲ ቌ൭ሩ ‫ܧ‬௜
௡
୧ୀଵ
൱
௖
ቍ
= 1 − ܲሺራ E୧
௖
୬
୧ୀଵ
ሻ
≥ 1 − ෍ ܲ
௡
ଵୀଵ
ሺ‫ܧ‬௜
௖
ሻ ሺusing Booleᇱ
sinequalityሻ
= 1 − ෍൫1 − ܲሺ‫ܧ‬௜ሻ൯
௡
௜ୀଵ
= ෍ ܲሺ‫ܧ‬௜ሻ − ሺ݊ − 1ሻ. ▄
௡
௜ୀଵ
Remark 2.4
Under the notation of Theorem 2.2 we can in fact prove the following inequalities:
෍ ܵ௝,௡
ଶ௞
௝ୀଵ
≤ ܲ ቌራ ‫ܧ‬௝
௡
௝ୀଵ
ቍ ≤ ෍ ܵ௝,௡
ଶ௞ିଵ
௝ୀଵ
, ݇ = 1,2, … , ቂ
݊
2
ቃ,

18. 18
where ቂ
௡
ଶ
ቃ denotes the largest integer not exceeding
௡
ଶ
. ▄
Corollary 2.1
Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫ܧ‬ଵ, ‫ܧ‬ଶ, … , ‫ܧ‬௡ ∈ ℱ be events. Then
(i) ܲሺ‫ܧ‬௜ሻ = 0, ݅ = 1, … , ݊ ⇔ ܲሺ⋃ ‫ܧ‬௜
௡
௜ୀଵ ሻ = 0;
(ii) ܲሺ‫ܧ‬௜ሻ = 1, ݅ = 1, … , ݊ ⇔ ܲሺ⋂ ‫ܧ‬௜
௡
௜ୀଵ ሻ = 1.
Proof.
(i) First suppose that ܲሺ‫ܧ‬௜ሻ = 0, ݅ = 1, … , ݊. Using Boole’s inequality, we get
0 ≤ ܲ ൭ራ ‫ܧ‬௜
௡
௜ୀଵ
൱ ≤ ෍ ܲሺ‫ܧ‬௜ሻ
௡
௜ୀଵ
= 0.
It follows that ܲሺ⋃ ‫ܧ‬௜
௡
௜ୀଵ ሻ = 0.
Conversely, suppose that ܲ൫⋃ ‫ܧ‬௝
௡
௝ୀଵ ൯ = 0 . Then ‫ܧ‬௜ ⊆ ⋃ ‫ܧ‬௝
௡
௝ୀଵ , ݅ = 1, … , ݊ , and
therefore,
0 ≤ ܲሺ‫ܧ‬௜ሻ ≤ ܲ ቌራ ‫ܧ‬௝
௡
௃ୀଵ
ቍ = 0, ݅ = 1, … , ݊,
i.e., ܲሺ‫ܧ‬௜ሻ = 0, ݅ = 1, … , ݊.
(ii) We have
ܲሺ‫ܧ‬௜ሻ = 1, ݅ = 1, … , ݊ ⇔ ܲሺ‫ܧ‬௜
௖
ሻ = 0, ݅ = 1, … , ݊
⇔ ܲ ൭ራ ‫ܧ‬௜
௖
௡
௜ୀଵ
൱ = 0 ሺusing ሺiሻሻ
⇔ ܲ ቌ൭ራ ‫ܧ‬௜
௖
௡
௜ୀଵ
൱
௖
ቍ = 1,
⇔ ܲ ൭ሩ ‫ܧ‬௜
௡
௜ୀଵ
൱ = 1. ▄
Definition 2.4
A countable collection ሼ‫ܧ‬௜: ݅ ∈ ߉ሽ of events is said to be exhaustive if ܲሺ⋃ ‫ܧ‬௜௜∈௸ ሻ = 1. ▄

19. 19
Example 2.2 (Equally Likely Probability Models)
Consider a probability space ሺߗ, ℱ, ܲሻ. Suppose that, for some positive integer ݇ ≥ 2,
ߗ = ⋃ ‫ܥ‬௜
௞
௜ୀଵ , where ‫ܥ‬ଵ, ‫ܥ‬ଶ, … , ‫ܥ‬௞ are mutually exclusive, exhaustive and equally likely events,
i.e., ‫ܥ‬௜ ∩ ‫ܥ‬௝ = ߶, if ݅ ≠ ݆, ܲ൫⋃ ‫ܥ‬௜
௞
௜ୀଵ ൯ = ∑ ܲ௞
௜ୀଵ ሺ‫ܥ‬௜ሻ = 1 and ܲሺ‫ܥ‬ଵሻ = ⋯ = ܲሺ‫ܥ‬௞ሻ =
ଵ
௞
.Further
suppose that an event ‫ܧ‬ ∈ ℱ can be written as
‫ܧ‬ = ‫ܥ‬௜ଵ
∪ ‫ܥ‬௜ଶ
∪ ⋯ ∪ ‫ܥ‬௜௥
,
where ሼ݅ଵ, … , ݅௥ሽ ⊆ ሼ1, … , ݇ሽ, ‫ܥ‬௜௝
∩ ‫ܥ‬௜௞
= ߶, ݆ ≠ ݇ and ‫ݎ‬ ∈ ሼ2, … , ݇ሽ. Then
ܲሺ‫ܧ‬ሻ = ෍ ܲ ቀ‫ܥ‬௜௝
ቁ
௥
௝ୀଵ
=
‫ݎ‬
݇
.
Note that here ݇ is the total number of ways in which the random experiment can terminate
(number of partition sets ‫ܥ‬ଵ, … , ‫ܥ‬௞ ), and ‫ݎ‬ is the number of ways that are favorable to ‫ܧ‬ ∈ ℱ.
Thus, for any ‫ܧ‬ ∈ ℱ,
ܲሺ‫ܧ‬ሻ =
number of cases favorable to ‫ܧ‬
total number of cases
=
‫ݎ‬
݇
,
which is the same as classical method of assigning probabilities. Here the assumption that
‫ܥ‬ଵ, … , ‫ܥ‬௞ are equally likely is a part of probability modeling. ▄
For a finite sample space ߗ, when we say that an experiment has been performed at random
we mean that various possible outcomes in ߗ are equally likely. For example when we say that
two numbers are chosen at random, without replacement, from the set ሼ1, 2, 3ሽ then
ߗ = ൛ሼ1, 2ሽ, ሼ1, 3ሽ, ሼ2, 3ሽൟand ܲሺሼ1, 2ሽሻ = ܲሺሼ1, 3ሽሻ = ܲሺሼ2, 3ሽሻ =
ଵ
ଷ
, where ሼ݅, ݆ሽ indicates that
the experiment terminates with chosen numbers as ݅ and ݆, ݅, ݆ ∈ ሼ1, 2, 3ሽ, ݅ ≠ ݆.
Example 2.3
Suppose that five cards are drawn at random and without replacement from a deck of 52
cards. Here the sample space ߗ comprises of all ቀ
52
5
ቁ combinations of 5 cards. Thus number of
favorable cases= ቀ
52
5
ቁ = ݇, say. Let ‫ܥ‬ଵ, … , ‫ܥ‬௞ be singleton subsets of ߗ.Then ߗ = ⋃ ‫ܥ‬௜
௞
௜ୀଵ and
ܲሺ‫ܥ‬ଵሻ = ⋯ = ܲሺ‫ܥ‬௞ሻ =
ଵ
௞
. Let ‫ܧ‬ଵ be the event that each card is spade. Then
Number of cases favorable to ‫ܧ‬ଵ = ቀ
13
5
ቁ.

20. 20
Therefore,
ܲሺ‫ܧ‬ଵሻ =
ቀ
13
5
ቁ
ቀ
52
5
ቁ
∙
Now let ‫ܧ‬ଶ be the event that at least one of the drawn cards is spade. Then ‫ܧ‬ଶ
௖
is the event that
none of the drawn cards is spade, andnumber of cases favorable to ‫ܧ‬ଶ
௖
= ቀ
39
5
ቁ ∙ Therefore,
ܲሺ‫ܧ‬ଶ
௖ሻ =
ቀ
39
5
ቁ
ቀ
52
5
ቁ
,
and ܲሺ‫ܧ‬ଶሻ = 1 − ܲሺ‫ܧ‬ଶ
௖ሻ = 1 −
ቀଷଽ
ହ
ቁ
ቀହଶ
ହ
ቁ
∙
Let ‫ܧ‬ଷ be the event that among the drawn cards three are kings and two are queens. Then
number of cases favorable to ‫ܧ‬ଷ = ቀ
4
3
ቁ ቀ
4
2
ቁ and, therefore,
ܲሺ‫ܧ‬ଷሻ =
ቀ
4
3
ቁ ቀ
4
2
ቁ
ቀ
52
5
ቁ
∙
Similarly, if ‫ܧ‬ସ is the event that among the drawn cards two are kings, two are queens and one
is jack, then
ܲሺ‫ܧ‬ସሻ =
ቀ
4
2
ቁ ቀ
4
2
ቁ ቀ
4
1
ቁ
ቀ
52
5
ቁ
. ▄
Example 2.4
Suppose that we have ݊ ሺ≥ 2ሻ letters and corresponding ݊ addressed envelopes. If these
letters are inserted at random in ݊ envelopes find the probability that no letter is inserted into
the correct envelope.
Solution. Let us label the letters as ‫ܮ‬ଵ, ‫ܮ‬ଶ, … , ‫ܮ‬௡ and respective envelopes as ‫ܣ‬ଵ, ‫ܣ‬ଶ, … , ‫ܣ‬௡. Let
‫ܧ‬௜ denote the event that letter ‫ܮ‬௜ is (correctly) inserted into envelope ‫ܣ‬௜, ݅ = 1, 2, … , ݊. We
need to find ܲሺ⋂ ‫ܧ‬௜
௖௡
௜ୀଵ ሻ. We have

21. 21
ܲ ൭ሩ ‫ܧ‬௜
௖
௡
௜ୀଵ
൱ = ܲ ቌ൭ራ ‫ܧ‬௜
௡
௜ୀଵ
൱
௖
ቍ = 1 − ܲ ൭ራ ‫ܧ‬௜
௡
௜ୀଵ
൱ = 1 − ෍ ܵ௞,௡,
௡
௞ୀଵ
where, for ݇ ∈ ሼ1, 2, … , ݊ሽ,
ܵ௞,௡ = ሺ−1ሻ௞ିଵ
෍ ܲ൫‫ܧ‬௜భ
∩ ‫ܧ‬௜మ
∩ ⋯ ∩ ‫ܧ‬௜ೖ
൯.
ଵஸ௜భழ௜మழ⋯ழ௜ೖஸ௡
Note that ݊ letters can be inserted into ݊ envelopes in ݊! ways. Also, for 1 ≤ ݅ଵ < ݅ଶ < ⋯ <
݅௞ ≤ ݊, ‫ܧ‬௜భ
∩ ‫ܧ‬௜మ
∩ ⋯ ∩ ‫ܧ‬௜ೖ
is the event that letters ‫ܮ‬௜భ
, ‫ܮ‬௜మ
, … , ‫ܮ‬௜ೖ
are inserted into correct
envelopes. Clearly number of cases favorable to this event is ሺ݊ − ݇ሻ!. Therefore, for
1 ≤ ݅ଵ < ݅ଶ < ⋯ < ݅௞ ≤ ݊,
ܲ൫‫ܧ‬௜భ
∩ ‫ܧ‬௜మ
∩ ⋯ ∩ ‫ܧ‬௜ೖ
൯ =
ሺ݊ − ݇ሻ!
݊!
⇒ ܵ௞,௡ = ሺ−1ሻ௞ିଵ
෍
ሺ݊ − ݇ሻ!
݊!
1≤݅1<݅2<⋯<݅݇≤݊
= ሺ−1ሻ௞ିଵ
ቀ
݊
݇
ቁ
ሺ݊ − ݇ሻ!
݊!
=
ሺ−1ሻ௞ିଵ
݇!
⇒ ܲ ൭ሩ ‫ܧ‬௜
௖
௡
௜ୀଵ
൱ =
1
2!
−
1
3!
+
1
4!
− ⋯ +
ሺ−1ሻ௡
݊!
. ▄
3. Conditional Probability and Independence of Events
Let ሺߗ, ℱ, ܲሻ be a given probability space. In many situations we may not be interested in the
whole space ߗ. Rather we may be interested in a subset ‫ܤ‬ ∈ ℱ of the sample space ߗ. This may
happen, for example, when we know apriori that the outcome of the experiment has to be an
element of ‫ܤ‬ ∈ ℱ.
Example 3.1
Consider a random experiment of shuffling a deck of 52 cards in such a way that all 52!
arrangements of cards (when looked from top to bottom) are equally likely.

22. 22
Here,
ߗ =all 52! permutations of cards,
and
ℱ = ࣪ሺΩሻ.
Now suppose that it is noticed that the bottom card is the king of heart. In the light of this
information, sample space ‫ܤ‬ comprises of 51! arrangements of 52 cards with bottom card as
king of heart.Define the event
‫ :ܭ‬top card is king.
For ‫ܧ‬ ∈ ℱ, define
ܲሺ‫ܧ‬ሻ = probability of event ‫ܧ‬ under sample space ߗ,
ܲ஻ሺ‫ܧ‬ሻ = probability of event ‫ܧ‬ under sample space ‫.ܤ‬
Clearly,
ܲ஻ሺ‫ܭ‬ሻ =
ଷ×ହ଴!
ହଵ!
.
Note that
ܲ஻ሺ‫ܭ‬ሻ =
3 × 50!
51!
=
ଷ×ହ଴!
ହଶ!
ହଵ!
ହଶ!
=
ܲሺ‫ܭ‬ ∩ ‫ܤ‬ሻ
ܲሺ‫ܤ‬ሻ
i. e. , ܲ஻ሺ‫ܭ‬ሻ =
ܲሺ‫ܭ‬ ∩ ‫ܤ‬ሻ
ܲሺ‫ܤ‬ሻ
. ሺ3.1ሻ
We call ܲ஻ሺ‫ܭ‬ሻ the conditional probability of event ‫ܭ‬ given that the experiment will result in an
outcome in ‫ܤ‬ (i.e., the experiment will result in an outcome ߱ ∈ ‫ܤ‬ ) and ܲሺ‫ܭ‬ሻ the
unconditional probability of event ‫.ܭ‬ ▄
Example 3.1 lays ground for introduction of the concept of conditional probability.
Let ሺߗ, ℱ, ܲሻ be a given probability space. Suppose that we know in advance that the outcome
of the experiment has to be an element of ‫ܤ‬ ∈ ℱ, where ܲሺ‫ܤ‬ሻ > 0. In such situations the
sample space is ‫ܤ‬ and natural contenders for the membership of the event space are

23. 23
ሼ‫ܣ‬ ∩ ‫ܤ‬ ∶ ‫ܣ‬ ∈ ࣠ሽ. This raises the question whether ࣠஻ ൌ ሼ‫ܣ‬ ∩ ‫ܤ‬ ∶ ‫ܣ‬ ∈ ࣠ሽ is an event space?
i.e., whether ࣠஻ ൌ ሼ‫ܣ‬ ∩ ‫ܤ‬ ∶ ‫ܣ‬ ∈ ࣠ሽ is a sigma-field of subsets of ‫?ܤ‬
Theorem 3.1
Let ࣠ be a ߪ-field of subsets ߗ and let ‫ܤ‬ ∈ ࣠. Define ࣠஻ ൌ ሼ‫ܣ‬ ∩ ‫ܤ‬ ∶ ‫ܣ‬ ∈ ࣠ሽ. Then ࣠஻ is a ߪ-
field of subsets of ‫ܤ‬and ࣠஻ ⊆ ࣠.
Proof. Since ‫ܤ‬ ∈ ࣠ and ࣠஻ ൌ ሼ‫ܣ‬ ∩ ‫ܤ‬ ∶ ‫ܣ‬ ∈ ࣠ሽ it is obvious that ࣠஻ ⊆ ࣠. We have ߗ ∈ ࣠ and
therefore
‫ܤ‬ ൌ ߗ ∩ ‫ܤ‬ ∈ ࣠஻. ሺ3.2ሻ
Also,
‫ܥ‬ ∈ ࣠஻ ⇒ C ൌ A ∩ ‫ܤ‬ for same ‫ܣ‬ ∈ ࣠
⇒ ‫ܥ‬௖
ൌ ‫ܤ‬ െ ‫ܥ‬ ൌ ሺߗ െ ‫ܣ‬ሻᇣᇧᇤᇧᇥ
∈࣠
∩ ‫ܤ‬ (since ‫ܣ‬ ∈ ࣠)
Figure 3.1
⇒ ‫ܥ‬஼
ൌ ‫ܤ‬ െ ‫ ܥ‬ ∈ ࣠஻, (3.3)
i.e., ࣠஻ is closed under complements with respect to ‫.ܤ‬
Now suppose that ‫ܥ‬௜ ∈ ࣠஻, ݅ ൌ 1,2, ….Then‫ܥ‬௜ ൌ ‫ܣ‬௜ ∩ ‫,ܤ‬ for some‫ܣ‬௜ ∈ ࣠, ݅ ൌ 1,2, …. Therefore,
ራ ‫ܥ‬௜
ஶ
௜ୀଵ
ൌ ൭ራ ‫ܣ‬௜
ஶ
௜ୀଵ
൱
ᇣᇧᇧᇤᇧᇧᇥ
∈࣠
∩ ‫ ܤ‬ሺsince ‫ܣ‬௜ ∈ ࣠, ݅ ൌ 1,2, … ሻ

24. 24
∈ ℱ஻, ሺ3.4ሻ
i.e., ℱ஻ is closed under countable unions.
Now (3.2), (3.3) and (3.4) imply that ℱ஻is a ߪ-field of subsets of ‫.ܤ‬ ▄
Equation (3.1) suggests considering the set function ܲ஻: ℱ஻ → ℝ defined by
ܲ஻ሺ‫ܥ‬ሻ =
ܲሺ‫ܥ‬ሻ
ܲሺ‫ܤ‬ሻ
, ‫ܥ‬ ∈ ℱ஻ = ሼ‫ܣ‬ ∩ ‫:ܤ‬ ‫ܣ‬ ∈ ℱሽ.
Note that, for ‫ܥ‬ ∈ ℱ஻, ܲሺ‫ܥ‬ሻ is well defined as ℱ஻ ⊆ ℱ.
Let us define another set function ܲሺ∙ |‫ܤ‬ሻ ∶ ℱ → ℝ by
Pሺ‫ܤ|ܣ‬ሻ ൌ ܲ஻ሺ‫ܣ‬ ∩ ‫ܤ‬ሻ =
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ
ܲሺ‫ܤ‬ሻ
, ‫ܣ‬ ∈ ℱ.
Theorem 3.2
Let ሺߗ, ℱ, ܲሻbe a probability space and let ‫ܤ‬ ∈ ℱ be such that ܲሺ‫ܤ‬ሻ > 0. Then ሺ‫,ܤ‬ ℱ஻, ܲ஻ ሻ
and ൫ߗ, ℱ, ܲሺ⋅ |‫ܤ‬ሻ൯ are probability spaces.
Proof. Clearly
ܲ஻ሺ‫ܥ‬ሻ ൌ
௉ሺ஼ሻ
௉ሺ஻ሻ
൒ 0, ∀ ‫ܥ‬ ∈ ℱ஻.
Let ‫ܥ‬௜ ∈ ℱ஻, ݅ = 1, 2, … be mutually exclusive.Then ‫ܥ‬௜ ∈ ℱ, ݅ = 1, 2, … (since ℱ஻ ⊆ ℱ), and
ܲ஻ ൭ራ ‫ܥ‬௜
ஶ
௜ୀଵ
൱ =
ܲሺ⋃ ‫ܥ‬௜
ஶ
௜ୀଵ ሻ
ܲሺ‫ܤ‬ሻ
=
∑ ܲሺ‫ܥ‬௜ሻஶ
௜ୀଵ
ܲሺ‫ܤ‬ሻ
= ෍
ܲሺ‫ܥ‬௜ሻ
ܲሺ‫ܤ‬ሻ
ஶ
௜ୀଵ
= ෍ ܲ஻
ஶ
௜ୀଵ
ሺ‫ܥ‬௜ሻ, ሺ3.5ሻ
i.e., ܲ஻ is countable additive on ℱ஻.

25. 25
Also
ܲ஻ሺ‫ܤ‬ሻ =
ܲሺ‫ܤ‬ሻ
ܲሺ‫ܤ‬ሻ
= 1 ∙
Thus ܲ஻ is a probability measure on ℱ஻.
Note that ܲሺ‫ܤ|ܣ‬ሻ ൒ 0, ∀ ‫ܣ‬ ∈ ℱ and
ܲሺߗ|Bሻ ൌ
ܲሺߗ ∩ Bሻ
ܲሺ‫ܤ‬ሻ
=
ܲሺ‫ܤ‬ሻ
ܲሺ‫ܤ‬ሻ
= 1 ∙
Let ‫ܧ‬௜ ∈ ℱ, ݅ = 1,2, … be mutually exclusive. Then ‫ܥ‬௜ = ‫ܧ‬௜ ∩ ‫ܤ‬ ∈ ℱ஻, ݅ = 1, 2, … are mutually
exclusive and
ܲ ൭ራ ‫ܧ‬௜|‫ܤ‬
ஶ
௜ୀଵ
൱ ൌ ܲ஻ ൭ራ ‫ܥ‬௜
ஶ
௜ୀଵ
൱ ൌ ෍ ܲ஻ሺ‫ܥ‬௜ሻ
ஶ
௜ୀଵ
ൌ ෍ ܲ஻
ஶ
௜ୀଵ
ሺ‫ܧ‬௜ ∩ ‫ܤ‬ሻ = ෍ ܲ
ஶ
௜ୀଵ
ሺ‫ܧ‬௜|‫ܤ‬ሻ. ሺusing ሺ3.5ሻሻ
It follows thatܲሺ∙ |‫ܤ‬ሻ is a probability measure on ℱ. ▄
Note that domains of ܲ஻ሺ∙ሻ and ܲሺ∙ |‫ܤ‬ሻ are ℱ஻ and ℱ respectively. Moreover,
ܲሺ‫ܤ|ܣ‬ሻ ൌ ܲ஻ሺ‫ܣ‬ ∩ ‫ܤ‬ሻ =
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ
ܲሺ‫ܤ‬ሻ
, ‫ܣ‬ ∈ ℱ.
Definition 3.1
Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫ܤ‬ ∈ ℱ be a fixed event such that ܲሺ‫ܤ‬ሻ > 0. Define
the set function ܲሺ∙ |‫ܤ‬ሻ: ℱ → ℝ by
ܲሺ‫ܤ|ܣ‬ሻ ൌ ܲ஻ሺ‫ܣ‬ ∩ ‫ܤ‬ሻ =
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ
ܲሺ‫ܤ‬ሻ
, ‫ܣ‬ ∈ ℱ.
We call ܲሺ‫ܤ|ܣ‬ሻ the conditional probability of event ‫ܣ‬ given that the outcome of the
experiment is in ‫ܤ‬ or simply the conditional probability of ‫ܣ‬ given ‫.ܤ‬ ▄
Example 3.2
Six cards are dealt at random (without replacement) from a deck of 52 cards. Find the
probability of getting all cards of heart in a hand (event A) given that there are at least 5 cards
of heart in the hand (event B).
Solution. We have,

26. 26
ܲሺ‫ܤ|ܣ‬ሻ =
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ
ܲሺ‫ܤ‬ሻ
.
Clearly,
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ = ܲሺ‫ܣ‬ሻ =
ቀଵଷ
଺ ቁ
ቀହଶ
଺
ቁ
,
and ܲሺ‫ܤ‬ሻ =
ቀଵଷ
ହ
ቁቀଷଽ
ଵ
ቁାቀଵଷ
଺ ቁ
ቀହଶ
଺
ቁ
∙
Therefore,
ܲሺ‫ܤ|ܣ‬ሻ =
ቀ
13
6
ቁ
ቀ
13
5
ቁ ቀ
39
1
ቁ + ቀ
13
6
ቁ
. ▄
Remark 3.1
For events ‫ܧ‬ଵ, … , ‫ܧ‬௡ ∈ ℱ ሺ݊ ≥ 2ሻ,
ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ = ܲሺ‫ܧ‬ଵሻ ܲሺ‫ܧ‬ଶ|‫ܧ‬ଵሻ, if ܲሺ‫ܧ‬ଵሻ > 0,
and
ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶ ∩ ‫ܧ‬ଷሻ = ܲ൫ሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ ∩ ‫ܧ‬ଷ൯
= ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻܲሺ‫ܧ‬ଷ|‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ
= ܲሺ‫ܧ‬ଵሻܲሺ‫ܧ‬ଶ|‫ܧ‬ଵሻ ܲሺ‫ܧ‬ଷ|‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ.
If ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ > 0 (which also guarantees that ܲሺ‫ܧ‬ଵሻ > 0, since ‫ܧ‬ଵ ∩ ‫ܧ‬ଶ ⊆ ‫ܧ‬ଵ).
Using principle of mathematical induction it can be shown that
ܲ ൭ሩ ‫ܧ‬௜
௡
௜ୀଵ
൱ = ܲሺ‫ܧ‬ଵሻܲሺ‫ܧ‬ଶ|‫ܧ‬ଵሻܲሺ‫ܧ‬ଷ|‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ ⋯ ܲሺ‫ܧ‬௡|‫ܧ‬ଵ ∩ ‫ܧ‬ଶ ∩ ⋯ ∩ ‫ܧ‬௡ିଵሻ,
provided ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶ ∩ ⋯ ∩ ‫ܧ‬௡ିଵሻ > 0 (which also guarantees that ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶ ∩ ⋯ ∩ ‫ܧ‬௜ሻ > 0,
݅ = 1, 2, ⋯ , ݊ − 1). ▄

27. 27
Example 3.3
An urn contains four red and six black balls. Two balls are drawn successively, at random and
without replacement, from the urn. Find the probability that the first draw resulted in a red ball
and the second draw resulted in a black ball.
Solution. Define the events
‫:ܣ‬ first draw results in a red ball;
‫:ܤ‬ second draw results in a black ball.
Then,
Required probability = ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ
= ܲሺ‫ܣ‬ሻܲሺ‫ܣ|ܤ‬ሻ
ൌ
4
10
×
6
9
=
12
45
. ▄
Let ሺߗ, ℱ, ܲሻ be a probability space. For a countable collection ሼ‫ܧ‬௜: ݅ ∈ ߉ሽ of mutually exclusive
and exhaustive events, the following theorem provides a relationship between marginal
probability ܲሺ‫ܧ‬ሻ of an event ‫ܧ‬ ∈ ℱ and joint probabilities ܲሺ‫ܧ‬ ∩ ‫ܧ‬௜ሻ of events ‫ܧ‬ and ‫ܧ‬௜, ݅ ∈ ߉.
Theorem 3.3 (Theorem of Total Probability)
Let ሺߗ, ℱ, ܲሻ be a probability space and let ሼ‫ܧ‬௜: ݅ ∈ ߉ሽ be a countable collection of mutually
exclusive and exhaustive events (i.e., ‫ܧ‬௜ ∩ ‫ܧ‬௝ = ߶, whenever ݅ ≠ ݆, and ܲሺ⋃ ‫ܧ‬௜௜∈௸ ሻ = 1) such
that ܲሺ‫ܧ‬௜ሻ > 0, ∀݅ ∈ ߉.Then, for any event ‫ܧ‬ ∈ ℱ,
ܲሺ‫ܧ‬ሻ = ෍ ܲሺ‫ܧ‬ ∩ ‫ܧ‬௜ሻ
௜∈௸
= ෍ ܲሺ‫ܧ|ܧ‬௜ሻ
௜∈௸
ܲሺ‫ܧ‬௜ሻ.
Proof. Let ‫ܨ‬ = ⋃ ‫ܧ‬௜௜∈௸ . Then ܲሺ‫ܨ‬ሻ = 1 and ܲሺ‫ܨ‬௖ሻ = 1 − ܲሺ‫ܨ‬ሻ = 0. Therefore,
ܲሺ‫ܧ‬ሻ = ܲሺ‫ܧ‬ ∩ ‫ܨ‬ሻ + ܲሺ‫ܧ‬ ∩ ‫ܨ‬௖
ሻ
= ܲሺ‫ܧ‬ ∩ ‫ܨ‬ሻ ሺ‫ܧ‬ ∩ ‫ܨ‬௖
⊆ ‫ܨ‬௖
⇒ 0 ≤ ܲሺ‫ܧ‬ ∩ ‫ܨ‬௖ሻ ≤ ܲሺ‫ܨ‬௖ሻ = 0ሻ
= ܲ ൭ራሺ‫ܧ‬ ∩ ‫ܧ‬௜ሻ
௜∈௸
൱ = ෍ ܲሺ‫ܧ‬ ∩ ‫ܧ‬௜ሻ
௜∈௸
ሺ‫ܧ‬௜‫ ݏ‬are disjoint
⇒ ‫ܧ‬௜ ∩ ‫ܧ‬s ሺ⊆ ‫ܧ‬௜ሻ are disjoint ሻ

28. 28
= ෍ ܲሺ‫ܧ|ܧ‬௜ሻ
௜∈௸
ܲሺ‫ܧ‬௜ሻ. ▄
Example 3.4
Urn ܷଵ contains 4 white and 6 black balls and urn ܷଶ contains 6 white and 4 black balls. A fair
die is cast and urn ܷଵ is selected if the upper face of die shows 5 or 6 dots. Otherwise urn ܷଶ is
selected. If a ball is drawn at random from the selected urn find the probability that the drawn
ball is white.
Solution. Define the events:
ܹ ∶ drawn ball is white;
‫ܧ‬ଵ ∶ urn ܷଵ is selected;
‫ܧ‬ଶ ∶ urn ܷଶis selected.
Then ሼ‫ܧ‬ଵ, ‫ܧ‬ଶሽ is a collection of mutually exclusive and exhaustive events. Therefore
ܲሺܹሻ = ܲሺ‫ܧ‬ଵሻ ܲሺܹ|‫ܧ‬ଵሻ ൅ ܲሺ‫ܧ‬ଶሻܲሺܹ|‫ܧ‬ଶሻ
ൌ
2
6
×
4
10
+
4
6
×
6
10
=
8
15
∙ ▄
The following theorem provides a method for finding the probability of occurrence of an event
in a past trial based on information on occurrences in future trials.
Theorem 3.4 (Bayes’ Theorem)
Let ሺߗ, ℱ, ܲሻ be a probability space and let ሼ‫ܧ‬௜: ݅ ∈ ߉ሽ be a countable collection of mutually
exclusive and exhaustive events with ܲሺ‫ܧ‬௜ሻ > 0, ݅ ∈ ߉. Then, for any event ‫ܧ‬ ∈ ℱ with
ܲሺ‫ܧ‬ሻ > 0, we have
ܲ൫‫ܧ‬௝|‫ܧ‬൯ ൌ
ܲ൫‫ܧ|ܧ‬௝൯ܲ൫‫ܧ‬௝൯
∑ ܲሺ‫ܧ|ܧ‬௜ሻܲሺ‫ܧ‬௜ሻ௜∈௸
, ݆ ∈ ߉ ∙
Proof. We have, for ݆ ∈ ߉,
ܲ൫‫ܧ‬௝|‫ܧ‬൯ ൌ
ܲ൫‫ܧ‬௝ ∩ ‫ܧ‬൯
ܲሺ‫ܧ‬ሻ

29. 29
=
ܲ൫‫ܧ|ܧ‬௝൯ܲ൫‫ܧ‬௝൯
ܲሺ‫ܧ‬ሻ
ൌ
ܲ൫‫ܧ|ܧ‬௝൯ܲ൫‫ܧ‬௝൯
∑ ܲሺ‫ܧ|ܧ‬௜ሻܲሺ‫ܧ‬௜ሻ௜∈௸
ሺusing Theorem of Total Probabilityሻ. ▄
Remark 3.2
(i) Suppose that the occurrence of any one of the mutually exclusive and exhaustive
events ‫ܧ‬௜, ݅ ∈ ߉, causes the occurrence of an event ‫.ܧ‬ Given that the event ‫ܧ‬ has
occurred, Bayes’ theorem provides the conditional probability that the event ‫ܧ‬ is
caused by occurrence of event ‫ܧ‬௝, ݆ ∈ ߉.
(ii) In Bayes’ theorem the probabilities ܲ൫‫ܧ‬௝൯, ݆ ∈ ߉, are referred to as prior probabilities
and the probabilities ܲ൫‫ܧ‬௝|‫ܧ‬൯, ݆ ∈ ߉, are referred to as posterior probabilities. ▄
To see an application of Bayes’ theorem let us revisit Example 3.4.
Example 3.5
Urn ܷଵcontains 4 white and 6 black balls and urn ܷଶ contains 6 white and 4 black balls. A fair
die is cast and urn ܷଵ is selected if the upper face of die shows five or six dots. Otherwise urn
ܷଶ is selected. A ball is drawn at random from the selected urn.
(i) Given that the drawn ball is white, find the conditional probability that it came from
urn ܷଵ;
(ii) Given that the drawn ball is white, find the conditional probability that it came from
urn ܷଶ.
Solution. Define the events:
ܹ ∶ drawn ball is white;
‫ܧ‬ଵ ∶ urn ܷଵ is selected
‫ܧ‬ଶ ∶ urn ܷଶ is selected
ൠ mutually & exhaustive events
(i) We have
ܲሺ‫ܧ‬ଵ|ܹሻ ൌ
ܲሺܹ|‫ܧ‬ଵሻ ܲሺ‫ܧ‬ଵሻ
ܲሺܹ|‫ܧ‬ଵሻܲሺ‫ܧ‬ଵሻ ൅ ܲሺܹ|‫ܧ‬ଶሻܲሺ‫ܧ‬ଶሻ
ൌ
ସ
ଵ଴
×
ଶ
଺
ସ
ଵ଴
×
ଶ
଺
൅
଺
ଵ଴
×
ସ
଺

30. 30
=
1
4
∙
(ii) Since ‫ܧ‬ଵ and ‫ܧ‬ଶ are mutually exclusive and ܲሺ‫ܧ‬ଵ ∪ ‫ܧ‬ଶ|ܹሻ ൌ ܲሺߗ|ܹሻ ൌ 1, we have
ܲሺ‫ܧ‬ଶ|ܹሻ ൌ 1 − ܲሺ‫ܧ‬ଵ|ܹሻ
ൌ
3
4
∙ ▄
In the above example
ܲሺ‫ܧ‬ଵ|ܹሻ ൌ
ଵ
ସ
<
ଵ
ଷ
ൌ ܲሺ‫ܧ‬ଵሻ,
and ܲሺ‫ܧ‬ଶ|ܹሻ ൌ
3
4
>
2
3
= ܲሺ‫ܧ‬ଶሻ,
i.e.,
(i) the probability of occurrence of event ‫ܧ‬ଵ decreases in the presence of the information
that the outcome will be an element of ܹ;
(ii) the probability of occurrence of event ‫ܧ‬ଶ increases in the presence of information that
the outcome will be an element of ܹ.
These phenomena are related to the concept of association defined in the sequel.
Note that
ܲሺ‫ܧ‬ଵ|ܹሻ < ܲሺ‫ܧ‬ଵሻ ⇔ ܲሺ‫ܧ‬ଵ ∩ ܹሻ < ܲሺ‫ܧ‬ଵሻܲሺܹሻ,
and
ܲሺ‫ܧ‬ଶ|ܹሻ > ܲሺ‫ܧ‬ଶሻ ⇔ ܲሺ‫ܧ‬ଶ ∩ ܹሻ > ܲሺ‫ܧ‬ଶሻܲሺܹሻ.
Definition 3.2
Letሺߗ, ℱ, ܲሻ be a probability space and let ‫ܣ‬ and ‫ܤ‬ be two events. Events ‫ܣ‬ and ‫ܤ‬ are said to
be
(i) negatively associated if ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ < ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻ;
(ii) positively associated if ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ > ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻ;
(iii) independent if ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ = ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻ. ▄
Remark 3.3

31. 31
(i) If ܲሺ‫ܤ‬ሻ = 0 then ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ = 0 = ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻ, ∀ ‫ܣ‬ ∈ ℱ, i.e., if ܲሺ‫ܤ‬ሻ = 0 then any
event ‫ܣ‬ ∈ ℱ and ‫ܤ‬ are independent;
(ii) If ܲሺ‫ܤ‬ሻ > 0 then ‫ܣ‬ and ‫ܤ‬ are independent If, and only if, ܲሺ‫ܤ|ܣ‬ሻ ൌ ܲሺ‫ܣ‬ሻ, i.e., if
ܲሺ‫ܤ‬ሻ > 0, then events ‫ܣ‬ and ‫ܤ‬ are independent if, and only if, the availability of the
information that event ‫ܤ‬ has occurred does not alter the probability of occurrence
of event ‫.ܣ‬ ▄
Now we define the concept of independence for arbitrary collection of events.
Definition 3.3
Let ሺߗ, ℱ, ܲሻ be a probability space. Let ߉ ⊆ ℝ be an index set and let ሼ‫ܧ‬ఈ: ߙ ∈ ߉ሽbe a
collection of events in ℱ.
(i) Events ሼ‫ܧ‬ఈ: ߙ ∈ ߉ሽ are said to be pair wise independent if any pair of events ‫ܧ‬ఈ and
‫ܧ‬ఉ, ߙ ≠ ߚ in the collection ൛‫ܧ‬௝: ݆ ∈ ߉ൟ are independent. i.e., if ܲ൫‫ܧ‬ఈ ∩ ‫ܧ‬ఉ൯ =
ܲሺ‫ܧ‬ఈሻܲ൫‫ܧ‬ఉ൯, whenever ߙ, ߚ ∈ ߉ and ߙ ≠ ߚ;
(ii) Let ߉ = ሼ1, 2, … , nሽ, for some ݊ ∈ ℕ, so that ሼ‫ܧ‬ఈ: ߙ ∈ ߉ሽ = ሼ‫ܧ‬ଵ, … , ‫ܧ‬௡ሽ is a finite
collection of events in ℱ. Events ‫ܧ‬ଵ, … , ‫ܧ‬௡ are said to be independent if, for any sub
collection ൛‫ܧ‬ఈଵ
, … , ‫ܧ‬ఈ௞
ൟ of ሼ‫ܧ‬ଵ, … , ‫ܧ‬௡ሽሺ݇ = 2,3, … , ݊ሻ
ܲ ቌሩ ‫ܧ‬ఈ௝
௞
௝ୀଵ
ቍ = ෑ ܲ
௞
௝ୀଵ
ቀ‫ܧ‬ఈ௝
ቁ. ሺ3.6ሻ
(iii) Let ߉ ⊆ ℝ be an arbitrary index set. Events ሼ‫ܧ‬ఈ: ߙ ∈ ߉ሽ are said to be independent if
any finite sub collection of events in ሼ‫ܧ‬ఈ: ߙ ∈ ߉ሽ forms a collection of independent
events. ▄
Remark 3.4
(i) To verify that ݊ events ‫ܧ‬ଵ, … , ‫ܧ‬௡ ∈ ℱ are independent one must verify 2௡
− ݊ −
1 ቀ= ∑ ቀ
݊
݆ቁ௡
௃ୀଶ ቁ conditions in (3.6). For example, to conclude that three events
‫ܧ‬ଵ, ‫ܧ‬ଶ and ‫ܧ‬ଷ are independent, the following 4 ሺ= 2ଷ
− 3 − 1ሻ conditions must be
verified:
ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶሻ = ܲሺ‫ܧ‬ଵሻܲሺ‫ܧ‬ଶሻ;
ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଷሻ = ܲሺ‫ܧ‬ଵሻܲሺ‫ܧ‬ଷሻ;

32. 32
ܲሺ‫ܧ‬ଶ ∩ ‫ܧ‬ଷሻ = ܲሺ‫ܧ‬ଶሻܲሺ‫ܧ‬ଷሻ;
ܲሺ‫ܧ‬ଵ ∩ ‫ܧ‬ଶ ∩ ‫ܧ‬ଷሻ = ܲሺ‫ܧ‬ଵሻܲሺ‫ܧ‬ଶሻܲሺ‫ܧ‬ଷሻ.
(ii) If events ‫ܧ‬ଵ, … , ‫ܧ‬௡ are independent then, for any permutation ሺߙଵ, … , ߙ௡ሻ of
ሺ1, … , ݊ሻ, the events ‫ܧ‬ఈଵ
, … , ‫ܧ‬ఈ௡
are also independent. Thus the notion of
independence is symmetric in the events involved.
(iv) Events in any sub collection of independent events are independent. In particular
independence of a collection of events implies their pair wise independence. ▄
The following example illustrates that, in general, pair wise independence of a collection of
events may not imply their independence.
Example 3.6
Let ߗ = ሼ1, 2, 3, 4ሽ and let ℱ = ࣪ሺߗሻ , the power set of ߗ . Consider the probability
space ሺߗ, ℱ, Pሻ, where ܲሺሼ݅ሽሻ =
ଵ
ସ
, ݅ = 1, 2, 3, 4 . Let ‫ܣ‬ = ሼ1, 4ሽ, ‫ܤ‬ = ሼ2, 4ሽ and ‫ܥ‬ = ሼ3, 4ሽ.
Then,
ܲሺ‫ܣ‬ሻ = ܲሺ‫ܤ‬ሻ = ܲሺ‫ܥ‬ሻ =
ଵ
ଶ
,
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ = ܲሺ‫ܣ‬ ∩ ‫ܥ‬ሻ = ܲሺ‫ܤ‬ ∩ ‫ܥ‬ሻ = ܲሺሼ4ሽሻ =
ଵ
ସ
,
and ܲሺ‫ܣ‬ ∩ ‫ܤ‬ ∩ ‫ܥ‬ሻ = ܲሺሼ4ሽሻ =
ଵ
ସ
∙
Clearly,
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ = ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻ; ܲሺ‫ܣ‬ ∩ ‫ܥ‬ሻ = ܲሺ‫ܣ‬ሻܲሺ‫ܥ‬ሻ, and ܲሺ‫ܤ‬ ∩ ‫ܥ‬ሻ = ܲሺ‫ܤ‬ሻܲሺ‫ܥ‬ሻ,
i.e., ‫,ܣ‬ ‫ܤ‬ and ‫ܥ‬ are pairwise independent.
However,
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ ∩ ‫ܥ‬ሻ =
ଵ
ସ
≠ ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻܲሺ‫ܥ‬ሻ.
Thus ‫,ܣ‬ ‫ܤ‬ and ‫ ܥ‬are not independent. ▄
Theorem 3.5
Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫ܣ‬ and ‫ܤ‬ be independent events (‫,ܣ‬ ‫ܤ‬ ∈ ℱ).Then
(i) ‫ܣ‬௖
and ‫ܤ‬ are independent events;

33. 33
(ii) ‫ܣ‬ and ‫ܤ‬௖
are independent events;
(iii) ‫ܣ‬௖
and ‫ܤ‬௖
are independent events.
Proof. We have
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ = ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻ.
(i) Since ‫ܤ‬ = ሺ‫ܣ‬ ∩ ‫ܤ‬ሻ ∪ ሺ‫ܣ‬௖
∩ ‫ܤ‬ሻ and ሺ‫ܣ‬ ∩ ‫ܤ‬ሻ ∩ ሺ‫ܣ‬௖
∩ ‫ܤ‬ሻ = ߶, we have
ܲሺ‫ܤ‬ሻ = ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ + ܲሺ‫ܣ‬௖
∩ ‫ܤ‬ሻ
⇒ ܲሺ‫ܣ‬௖
∩ ‫ܤ‬ሻ = ܲሺ‫ܤ‬ሻ − ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ
= ܲሺ‫ܤ‬ሻ − ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻ
= ൫1 − ܲሺ‫ܣ‬ሻ൯ܲሺ‫ܤ‬ሻ
= ܲሺ‫ܣ‬௖
ሻܲሺ‫ܤ‬ሻ,
i.e., ‫ܣ‬௖
and ‫ܤ‬ are independent events.
(ii) Follows from (i) by interchanging the roles of ‫ܣ‬ and ‫.ܤ‬
(iii) Follows on using (i) and (ii) sequentially. ▄
The following theorem strengthens the results of Theorem 3.5.
Theorem 3.6
Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫ܨ‬ଵ, … , ‫ܨ‬௡ሺ݊ ∈ ℕ, ݊ ≥ 2ሻ be independent events in
ℱ. Then, for any ݇ ∈ ሼ1, 2, … , ݊ − 1ሽ and any permutationሺߙଵ, … , ߙ௡ሻ of ሺ1, … , ݊ሻ, the events
‫ܨ‬ఈଵ
, … , ‫ܨ‬ఈ௞
, ‫ܨ‬ఈೖశభ
௖
, … , ‫ܨ‬ఈ೙
௖
are independent. Moreover the events ‫ܨ‬ଵ
௖
, … , ‫ܨ‬௡
௖
are independent.
Proof. Since the notion of independence is symmetric in the events involved, it is enough to
show that for any ݇ ∈ ሼ1, 2, … , ݊ − 1ሽ the events ‫ܨ‬ଵ, … , ‫ܨ‬௞, ‫ܨ‬௞ାଵ
௖
, … , ‫ܨ‬௡
௖
are independent. Using
backward induction and symmetry in the notion of independence the above mentioned
assertion would follow if, under the hypothesis of the theorem, we show that the events
‫ܨ‬ଵ, … , ‫ܨ‬௡ିଵ, ‫ܨ‬௡
௖
are independent. For this consider a sub collection ൛‫ܨ‬௜ଵ
, … , ‫ܨ‬௜௠
, ‫ܩ‬ൟ of
‫ܨ‬ଵ, … , ‫ܨ‬௡ିଵ, ‫ܨ‬௡
௖ሺሼ݅ଵ, … , ݅௠ሽ ⊆ ሼ1, … , ݊ − 1ሽሻ, where ‫ܩ‬ = ‫ܨ‬௡
௖
or ‫ܩ‬ = ‫ܨ‬௝, for some ݆ ∈ ሼ1, … , ݊ −
1ሽ − ሼ݅ଵ, … , ݅௠ሽ, depending on whether or not ‫ܨ‬௡
௖
is a part of sub collection ൛‫ܨ‬௜ଵ
, … , ‫ܨ‬௜௠
, ‫ܩ‬ൟ .
Thus the following two cases arise:
۱‫.۷ ܍ܛ܉‬ ‫ܩ‬ = ‫ܨ‬௡
௖
Since ‫ܨ‬ଵ, … , ‫ܨ‬௡ are independent, we have

34. 34
ܲ ቌሩ ‫ܨ‬௜௝
௠
௝ୀଵ
ቍ = ෑ ܲ
௠
௝ୀଵ
ቀ‫ܨ‬௜௝
ቁ,
and
ܲ ൮ቌሩ ‫ܨ‬௜௝
௠
௝ୀଵ
ቍ ∩ ‫ܨ‬௡൲ = ቎ෑ ܲ ቀ‫ܨ‬௜௝
ቁ
௠
௝ୀଵ
቏ ܲሺ‫ܨ‬௡ሻ
= ܲ ቌሩ ‫ܨ‬௜௝
௠
௝ୀଵ
ቍ ܲሺ‫ܨ‬௡ሻ
⇒ events ሩ ‫ܨ‬௜௝
௠
௝ୀଵ
and ‫ܨ‬௡ are independent
⇒ events ⋂ ‫ܨ‬௜௝
௠
௝ୀଵ and ‫ܨ‬௡
௖
are independent ሺTheorem 3.5 ሺiiሻ)
⇒ ܲ ൮ቌሩ ‫ܨ‬௜௝
௠
௝ୀଵ
ቍ ∩ ‫ܨ‬௡
௖
൲ = ܲ ቌሩ ‫ܨ‬௜௝
௠
௝ୀଵ
ቍ ܲሺ‫ܨ‬௡
௖ሻ
= ቎ෑ ܲ ቀ‫ܨ‬௜௝
ቁ
௠
௝ୀଵ
቏ ܲሺ‫ܨ‬௡
௖ሻ
⇒ ܲ൫‫ܨ‬௜ଵ
∩ ⋯ ∩ ‫ܨ‬௜௠
∩ ‫ܩ‬൯ = ቎ෑ ܲ ቀ‫ܨ‬௜௝
ቁ
௠
௝ୀଵ
቏ ܲሺ‫ܩ‬ሻ.
Case II. ‫ܩ‬ = ‫ܨ‬௝, for some ݆ ∈ ሼ1, … , ݊ − 1ሽ − ሼ݅ଵ, … , ݅௠ሽ.
In this case ൛‫ܨ‬௜ଵ
, … , ‫ܨ‬௜௠
, ‫ܩ‬ൟ is a sub collection of independent events ‫ܨ‬ଵ, … , ‫ܨ‬௡ and therefore
ܲ൫‫ܨ‬௜ଵ
∩ ⋯ ∩ ‫ܨ‬௜௠
∩ ‫ܩ‬൯ = ቎ෑ ‫ܨ‬௜௝
௠
௝ୀଵ
቏ ܲሺ‫ܩ‬ሻ.
Now the result follows on combining the two cases. ▄

35. 35
When we say that two or more random experiments are independent (or that two or more
random experiments are performed independently) it simply means that the events associated
with the respective random experiments are independent.
4. Continuity of Probability Measures
We begin this section with the following definition.
Definition 4.1
Let ሺߗ, ℱ, ܲሻ be a probability space and let ሼ‫ܣ‬௡: ݊ = 1, 2, … ሽ be a sequence of events in ℱ.
(i) We say that the sequence ሼ‫ܣ‬௡: ݊ = 1, 2, … ሽ is increasing (written as ‫ܣ‬௡ ↑) if
‫ܣ‬௡ ⊆ ‫ܣ‬௡ାଵ, ݊ = 1,2, … ;
(ii) We say that the sequence ሼ‫ܣ‬௡: ݊ = 1, 2, … ሽ is decreasing (written as ‫ܣ‬௡ ↓) if
‫ܣ‬௡ାଵ ⊆ ‫ܣ‬௡, ݊ = 1,2, … ;
(iii) We say that the sequence ሼ‫ܣ‬௡: ݊ = 1, 2, … ሽ is monotone if either ‫ܣ‬௡ ↑ or ‫ܣ‬௡ ↓;
(iv) If ‫ܣ‬௡ ↑ we define the limit of the sequence ሼ‫ܣ‬௡: ݊ = 1, 2, … ሽ as ⋃ ‫ܣ‬௡
ஶ
௡ୀଵ and write
Lim௡→ஶ ‫ܣ‬௡ = ⋃ ‫ܣ‬௡
ஶ
௡ୀଵ ;
(v) If ‫ܣ‬௡ ↓ we define the limit of the sequence ሼ‫ܣ‬௡: ݊ = 1, 2, … ሽ as ⋂ ‫ܣ‬௡
ஶ
௡ୀଵ and write
Lim௡→ஶ ‫ܣ‬௡ = ⋂ ‫ܣ‬௡
ஶ
௡ୀଵ . ▄
Throughout we will denote the limit of a monotone sequence ሼ‫ܣ‬௡: ݊ = 1, 2, … ሽ of events by
Lim௡→ஶ ‫ܣ‬௡ and the limit of a sequence ሼܽ௡: ݊ = 1, 2, … ሽ of real numbers (provided it exists) by
lim௡→ஶ ܽ௡.
Theorem 4.1 (Continuity of Probability Measures)
Let ሼ‫ܣ‬௡: ݊ = 1, 2, … ሽ be a sequence of monotone events in a probability spaceሺߗ, ℱ, ܲሻ. Then
ܲ ቀLim
௡→ஶ
‫ܣ‬௡ቁ = lim
௡→ஶ
ܲሺ‫ܣ‬௡ሻ.
Proof.
Case I. ‫ܣ‬௡ ↑
In this case, Lim௡→ஶ ‫ܣ‬௡ = ⋃ ‫ܣ‬௡
ஶ
௡ୀଵ . Define ‫ܤ‬ଵ = ‫ܣ‬ଵ, ‫ܤ‬௡ = ‫ܣ‬௡ − ‫ܣ‬௡ିଵ, ݊ = 2, 3, ….

36. 36
Figure 4.1
Then‫ܤ‬௡ ∈ ࣠, ݊ ൌ 1, 2 … , ‫ܤ‬௡s are mutually exclusive and⋃ ‫ܤ‬௡
ஶ
௡ୀଵ ൌ ⋃ ‫ܣ‬௡
ஶ
௡ୀଵ ൌ Lim௡→ஶ ‫ܣ‬௡ .
Therefore,
ܲ ቀLim
௡→ஶ
‫ܣ‬௡ቁ ൌ ܲ ൭ራ ‫ܤ‬௡
ஶ
௡ୀଵ
൱
ൌ ෍ ܲሺ‫ܤ‬௡ሻ
ஶ
௡ୀଵ
ൌ lim
௡→ஶ
෍ ܲሺ‫ܤ‬௞ሻ
௡
௞ୀଵ
ൌ lim
௡→ஶ
൥ܲሺ‫ܣ‬ଵሻ ൅ ෍ ܲሺ‫ܣ‬௞ െ ‫ܣ‬௞ିଵሻ
௡
௞ୀଶ
൩
ൌ lim
௡→ஶ
൥ܲሺ‫ܣ‬ଵሻ ൅ ෍൫ܲሺ‫ܣ‬௞ሻ െ ܲሺ‫ܣ‬௞ିଵሻ൯
௡
௞ୀଶ
൩
(using Theorem 2.1 (iv) since ‫ܣ‬௞ିଵ ⊆ ‫ܣ‬௞, ݇ ൌ 1, 2, …)
ൌ lim
௡→ஶ
൥ܲሺ‫ܣ‬ଵሻ ൅ ෍ ܲ
௡
௞ୀଶ
ሺ‫ܣ‬௞ሻ െ ෍ ܲ
௡
௞ୀଶ
ሺ‫ܣ‬௞ିଵሻ൩
ൌ lim
௡→ஶ
ሾܲሺ‫ܣ‬ଵሻ ൅ ܲሺ‫ܣ‬௡ሻ െ ܲሺ‫ܣ‬ଵሻሿ
ൌ lim
௡→ஶ
ܲሺ‫ܣ‬௡ሻ.

37. 37
Case II. ‫ܣ‬௡ ↓
In this case, Lim
௡→ஶ
‫ܣ‬௡ = ⋂ ‫ܣ‬௡
ஶ
௡ୀଵ and ‫ܣ‬௡
௖
↑. Therefore,
ܲ ቀLim
௡→ஶ
‫ܣ‬௡ቁ = ܲ ൭ሩ ‫ܣ‬௡
ஶ
௡ୀଵ
൱
= 1 − ܲ ൭൭ሩ ‫ܣ‬௡
ஶ
௡ୀଵ
൱
௖
൱
= 1 − ܲ ൭ራ ‫ܣ‬௡
௖
ஶ
௡ୀଵ
൱
= 1 − ܲሺLim
௡→ஶ
‫ܣ‬௡
௖
ሻ
= 1 − lim
௡→ஶ
ܲሺ‫ܣ‬௡
௖ ሻ ሺusing Case I, since ‫ܣ‬௡
௖
↑ሻ
= 1 − lim
௡→ஶ
൫1 − ܲሺ‫ܣ‬௡ሻ൯
= lim
௡→ஶ
ܲሺ‫ܣ‬௡ሻ. ▄
Remark 4.1
Let ሺߗ, ℱ, ܲሻ be a probability space and let ሼ‫ܧ‬௜: ݅ = 1, 2, … ሽ be a countably infinite collection of
events in ℱ. Define
‫ܤ‬௡ = ራ ‫ܧ‬௜
௡
௜ୀଵ
and ‫ܥ‬௡ = ሩ ‫ܧ‬௜
௡
௜ୀଵ
, ݊ = 1,2, …
Then ‫ܤ‬௡ ↑, ‫ܥ‬௡ ↓, Lim
௡→ஶ
‫ܤ‬௡ = ⋃ ‫ܤ‬௡
ஶ
௡ୀଵ = ⋃ ‫ܧ‬௜
ஶ
௜ୀଵ and Lim
௡→ஶ
‫ܥ‬௡ = ⋂ ‫ܧ‬௜
ஶ
௜ୀଵ . Therefore
ܲ ൭ራ ‫ܧ‬௜
ஶ
௜ୀଵ
൱ = ܲ ቀLim
௡→ஶ
‫ܤ‬௡ቁ
= lim
௡→ஶ
ܲሺ‫ܤ‬௡ሻ ሺusing Theorem 4.1ሻ
= lim
௡→ஶ
ܲ ൭ራ ‫ܧ‬௜
௡
௜ୀଵ
൱
= lim
௡→ஶ
ൣܵଵ,௡ + ܵଶ,௡ + ⋯ + ܵ௡,௡൧,

38. 38
where S୩,୬s are as defined in Theorem 2.2.
Moreover,
ܲ ൭ሩ ‫ܧ‬௜
ஶ
௜ୀଵ
൱ = ܲ ቀLim
௡→ஶ
‫ܥ‬௡ቁ
= lim
௡→ஶ
ܲሺ‫ܥ‬௡ሻ ሺusing Theorem 4.1ሻ
= lim
௡→ஶ
ܲሺ⋂ ‫ܧ‬௜
௡
௜ୀଵ ሻ.
Similarly, if ሼ‫ܧ‬௜: ݅ = 1, 2, ⋯ ሽ is a collection of independent events, then
ܲ ൭ሩ ‫ܧ‬௜
ஶ
௜ୀଵ
൱ = lim
௡→ஶ
ܲ ൭ሩ ‫ܧ‬௜
௡
௜ୀଵ
൱
= lim
௡→ஶ
൥ෑ ܲ
௡
௜ୀଵ
ሺ‫ܧ‬௜ሻ൩
= ෑ ܲ
ஶ
௜ୀଵ
ሺ‫ܧ‬௜ሻ. ▄
Problems
1. Let ߗ = ሼ1, 2, 3, 4ሽ. Check which of the following is a sigma-field of subsets of ߗ:
(i) ℱଵ = ൛߶, ሼ1, 2ሽ, ሼ3, 4ሽൟ;
(ii)ℱଶ = ൛߶, ߗ, ሼ1ሽ, ሼ2, 3, 4ሽ, ሼ1, 2ሽ, ሼ3, 4ሽൟ;
(iii) ℱଷ = ൛߶, ߗ, ሼ1ሽ, ሼ2ሽ, ሼ1, 2ሽ, ሼ3, 4ሽሼ2, 3, 4ሽ, ሼ1, 3, 4ሽൟ.
2. Show that a class ℱ of subsets of ߗ is a sigma-field of subsets of ߗ if, and only if, the
following three conditions are satisfied: (i) ߗ ∈ ℱ; (ii) ‫ܣ‬ ∈ ℱ ⇒ ‫ܣ‬஼
= ߗ − ‫ܣ‬ ∈ ℱ;
(iii) ‫ܣ‬௡ ∈ ℱ, n = 1, 2, ⋯ ⇒ ⋂ ‫ܣ‬௡ ∈ஶ
௡ୀଵ ℱ.
3. Let ሼℱఒ: ߣ ∈ ߉ሽ be a collection of sigma-fields of subsets of ߗ.
(i) Show that ⋂ ℱఒఒ∈௸ is a sigma-field;
(ii) Using a counter example show that ∪ఒ∈௸ ℱఒ may not be a sigma-field;

39. 39
(iii) Let ࣝ be a class of subsets of ߗ and let ሼℱఒ: ߣ ∈ ߉ሽ be a collection of all sigma-fields
that contain the class ࣝ. Show that ߪሺࣝሻ = ⋂ ℱఒఒ∈௸ , where ߪሺࣝሻ denotes the
smallest sigma-field containing the class ࣝ (or the sigma-field generated by class ࣝ).
4. Let ߗ be an infinite set and let ࣛ = ሼ‫ܣ‬ ⊆ ߗ: ‫ ܣ‬is finite or ‫ܣ‬஼
is finiteሽ.
(i) Show that ࣛ is closed under complements and finite unions;
(ii) Using a counter example show that ࣛ may not be closed under countably infinite
unions (and hence ࣛ may not be a sigma-field).
5. (i) Let ߗ be an uncountable set and let ℱ = ሼ‫ܣ‬ ⊆ ߗ: ‫ ܣ‬is countable or‫ܣ‬஼
is countableሽ.
(a) Show that ℱ is a sigma-field;
(b) What can you say about ℱwhen ߗ is countable?
(ii) Let Ω be a countable set and let ࣝ = ሼሼ߱ሽ: ߱ ∈ Ωሽ. Show that ߪሺࣝሻ = ࣪ሺߗሻ.
6. Let ℱ = ࣪ሺߗሻ =the power set of ߗ = ሼ0, 1, 2, … ሽ. In each of the following cases, verify
if ሺߗ, ℱ, ܲሻ is a probability space:
(i) ܲሺ‫ܣ‬ሻ = ∑ ݁ିఒ
௫ ∈஺ ߣ௫
‫!ݔ‬⁄ , ‫ ܣ‬ ∈ ℱ, ߣ > 0;
(ii) ܲሺ‫ܣ‬ሻ = ∑ ‫݌‬ሺ1 − ‫݌‬ሻ௫
௫ ∈஺ , ‫ ܣ‬ ∈ ℱ, 0 < ‫݌‬ < 1;
(iii) ܲሺ‫ܣ‬ሻ = 0, if ‫ܣ‬ has a finite number of elements, and ܲሺ‫ܣ‬ሻ = 1, if ‫ܣ‬ has infinite
number of elements, ‫ ܣ‬ ∈ ℱ.
7. Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫,ܣ‬ ‫,ܤ‬ ‫,ܥ‬ ‫ܦ‬ ∈ ℱ . Suppose that ܲሺ‫ܣ‬ሻ =
0.6, ܲሺ‫ܤ‬ሻ = 0.5, ܲሺ‫ܥ‬ሻ = 0.4, ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ = 0.3, ܲሺ‫ܣ‬ ∩ ‫ܥ‬ሻ = 0.2, ܲሺ‫ܤ‬ ∩ ‫ܥ‬ሻ = 0.2,
ܲሺ‫ܣ‬ ∩ ‫ܤ‬ ∩ ‫ܥ‬ሻ = 0.1, ܲሺ‫ܤ‬ ∩ ‫ܦ‬ሻ = ܲሺ‫ܥ‬ ∩ ‫ܦ‬ሻ = 0, ܲሺ‫ܣ‬ ∩ ‫ܦ‬ሻ = 0.1 and ܲሺ‫ܦ‬ሻ = 0.2.
Find:
(i) ܲሺ‫ܣ‬ ∪ ‫ܤ‬ ∪ ‫ܥ‬ሻand ܲሺ‫ܣ‬஼
∩ ‫ܤ‬஼
∩ ‫ܥ‬஼ሻ;
(ii) ܲሺሺ‫ܣ‬ ∪ ‫ܤ‬ሻ ∩ ‫ܥ‬ሻand ܲሺ‫ܣ‬ ∪ ሺ‫ܤ‬ ∩ ‫ܥ‬ሻሻ;
(iii) ܲሺሺ‫ܣ‬஼
∪ ‫ܤ‬஼
ሻ ∩ ‫ܥ‬஼ሻand ܲሺሺ‫ܣ‬஼
∩ ‫ܤ‬஼
ሻ ∪ ‫ܥ‬஼ሻ;
(iv) ܲሺ‫ܤ‬ ∩ ‫ܥ‬ ∩ ‫ܦ‬ሻand ܲሺ‫ܣ‬ ∩ ‫ܥ‬ ∩ ‫ܦ‬ሻ;
(v) ܲሺ‫ܣ‬ ∪ ‫ܤ‬ ∪ ‫ܦ‬ሻand ܲሺ‫ܣ‬ ∪ ‫ܤ‬ ∪ ‫ܥ‬ ∪ ‫ܦ‬ሻ;
(vi) ܲ൫ሺ‫ܣ‬ ∩ ‫ܤ‬ሻ ∪ ሺ‫ܥ‬ ∩ ‫ܦ‬ሻ൯.
8. Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫ܣ‬ and ‫ ܤ‬be two events (i.e., ‫,ܣ‬ ‫ܤ‬ ∈ ℱ).
(i) Show that the probability that exactly one of the events ‫ܣ‬ or ‫ܤ‬ will occur is given by
ܲሺ‫ܣ‬ሻ + ܲሺ‫ܤ‬ሻ − 2ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ;
(ii) Show that ܲሺ‫ܣ‬ ∩ ‫ܤ‬ሻ − ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬ሻ = ܲሺ‫ܣ‬ሻܲሺ‫ܤ‬஼ሻ − ܲሺ‫ܣ‬ ∩ ‫ܤ‬஼ሻ = ܲሺ‫ܣ‬஼ሻܲሺ‫ܤ‬ሻ −
ܲሺ‫ܣ‬஼
∩ ‫ܤ‬ሻ = ܲሺሺ‫ܣ‬ ∪ ‫ܤ‬ሻ஼ሻ − ܲሺ‫ܣ‬஼ሻܲሺ‫ܤ‬஼ሻ.

40. 40
9. Suppose that ݊ ሺ≥ 3ሻ persons ܲଵ, … , ܲ௡ are made to stand in a row at random. Find the
probability that there are exactly ‫ݎ‬ person between ܲଵand ܲଶ; here ‫ݎ‬ ∈ ሼ1, 2, … , ݊ − 2ሽ.
10. A point ሺܺ, ܻሻ is randomly chosen on the unit square ܵ = ሼሺ‫,ݔ‬ ‫ݕ‬ሻ: 0 ≤ ‫ ݔ‬ ≤ 1, 0 ≤ ‫ ݕ‬ ≤
1ሽ (i.e., for any region ܴ ⊆ ܵ for which the area is defined, the probability that ሺܺ, ܻሻ
lies on ܴ is
ୟ୰ୣୟ ୭୤ ோ
ୟ୰ୣୟ ୭୤ ௌ
ሻ ⋅ Find the probability that the distance from ሺܺ, ܻሻ to the nearest
side does not exceed
ଵ
ଷ
units.
11. Three numbers ܽ, ܾ and ܿ are chosen at random and with replacement from the set
ሼ1, 2, … ,6ሽ. Find the probability that the quadratic equation ܽ‫ݔ‬ଶ
+ ܾ‫ݔ‬ + ܿ = 0 will have
real root(s).
12. Three numbers are chosen at random from the set ሼ1, 2, … ,50ሽ. Find the probability that
the chosen numbers are in
(i) arithmetic progression;
(ii) geometric progression.
13. Consider an empty box in which four balls are to be placed (one-by-one) according to
the following scheme. A fair die is cast each time and the number of dots on the upper
face is noted. If the upper face shows up 2 or 5 dots then a white ball is placed in the
box. Otherwise a black ball is placed in the box. Given that the first ball placed in the box
was white find the probability that the box will contain exactly two black balls.
14. Let ൫ሺ0, 1ሿ, ℱ, ܲ൯ be a probability space such that ℱ is the smallest sigma-field
containing all subintervals of ߗ = ሺ0, 1ሿ and ܲሺሺܽ, ܾሿሻ = ܾ − ܽ, where 0 ≤ ܽ < ܾ ≤ 1
(such a probability measure is known to exist).
(i) Show that ሼܾሽ = ⋂ ቀܾ −
ଵ
௡ାଵ
, ܾቃஶ
௡ୀଵ , ∀ܾ ∈ ሺ0, 1ሿ;
(ii) Show that ܲሺሼܾሽሻ = 0, ∀ܾ ∈ ሺ0, 1ሿand ܲ൫ሺ0, 1ሿ൯ = 1(Note that here ܲሺሼܾሽሻ = 0
but ሼܾሽ ≠ ߶ and ܲ൫ሺ0, 1ሻ൯ = 1 but ሺ0, 1ሻ ≠ Ω) ;
(iii) Show that, for any countable set ‫ܣ‬ ∈ ℱ, ܲሺ‫ܣ‬ሻ = 0;
(iv) For ݊ ∈ ℕ, let ‫ܣ‬௡ = ቀ0,
ଵ
௡
ቃ and ‫ܤ‬௡ = ቀ
ଵ
ଶ
+
ଵ
௡ାଶ
, 1ቃ . Verify that ‫ܣ‬௡ ↓, ‫ܤ‬௡ ↑,
ܲሺLim௡→ஶ ‫ܣ‬௡ሻ = lim௡→ஶ ܲሺ‫ܣ‬௡ሻ and ܲሺLim௡→ஶ ‫ܤ‬௡ሻ = lim௡→ஶ ܲሺ‫ܤ‬௡ሻ.
15. Consider four coding machines ‫ܯ‬ଵ, ‫ܯ‬ଶ, ‫ܯ‬ଷand ‫ܯ‬ସ producing binary codes 0 and 1. The
machine ‫ܯ‬ଵ produces codes0 and 1 with respective probabilities
ଵ
ସ
and
ଷ
ସ
. The code
produced by machine ‫ܯ‬௞ is fed into machine ‫ܯ‬௞ାଵሺ݇ = 1, 2, 3ሻ which may either leave

41. 41
the received code unchanged or may change it. Suppose that each of the machines
‫ܯ‬ଶ, ‫ܯ‬ଷ and‫ܯ‬ସ change the received code with probability
ଷ
ସ
. Given that the machine ‫ܯ‬ସ
has produced code 1, find the conditional probability that the machine ‫ܯ‬ଵ produced
code 0.
16. A student appears in the examinations of four subjects Biology, Chemistry, Physics and
Mathematics. Suppose that probabilities of the student clearing examinations in these
subjects are
ଵ
ଶ
,
ଵ
ଷ
,
ଵ
ସ
and
ଵ
ହ
respectively. Assuming that the performances of the students
in four subjects are independent, find the probability that the student will clear
examination(s) of
(i) all the subjects; (ii) no subject; (iii) exactly one subject;
(iv) exactly two subjects; (v) at least one subject.
17. Let ‫ܣ‬ and ‫ ܤ‬be independent events. Show that
maxሼܲሺሺ‫ܣ‬ ∪ ‫ܤ‬ሻ௖ሻ, ܲ ሺ‫ܣ‬ ∩ ‫ܤ‬ሻ, ܲ ሺ‫ ܣ‬Δ ‫ܤ‬ሻሽ ≥
4
9
,
where ‫ ܣ‬Δ ‫ܤ‬ = ሺ‫ܣ‬ − ‫ܤ‬ሻ ∪ ሺ‫ܤ‬ − ‫ܣ‬ሻ.
18. For independent events ‫ܣ‬ଵ, … , ‫ܣ‬௡, show that:
ܲ ൭ሩ ‫ܣ‬௜
௖
௡
௜ୀଵ
൱ ≤ ݁ି ∑ ௉ሺ஺೔ሻ೙
೔సభ .
19. Let ሺߗ, ℱ, ܲሻ be a probability space and let ‫ܣ‬ଵ, ‫ܣ‬ଶ, … be a sequence of events
ሺi. e. , ‫ܣ‬௜ ∈ ℱ, ݅ = 1, 2, … ሻ . Define ‫ܤ‬௡ = ⋂ ‫ܣ‬௜
ஶ
௜ୀ௡ , ‫ܥ‬௡ = ⋃ ‫ܣ‬௜, ݊ = 1,2, … ,ஶ
௜ୀ௡ ‫ܦ‬ =
⋃ ‫ܤ‬௡
ஶ
௡ୀଵ and ‫ܧ‬ = ⋂ ‫ܥ‬௡
ஶ
௡ୀଵ . Show that:
(i) ‫ܦ‬ is the event that all but a finite number of ‫ܣ‬௡s occur and ‫ܧ‬ is the event that
infinitely many ‫ܣ‬௡s occur;
(ii) ‫ܦ‬ ⊆ ‫;ܧ‬
(iii) ܲሺ‫ܧ‬௖ሻ = lim௡→ஶ ܲሺ‫ܥ‬௡
௖ሻ = lim௡→ஶ lim௠→ஶ ܲሺ⋂ ‫ܣ‬௞
௖௠
௞ୀ௡ ሻ and ܲሺ‫ܧ‬ሻ = lim௡→ஶ ܲሺ‫ܥ‬௡ሻ;
(iv) if ∑ ܲሺ‫ܣ‬௡ሻஶ
௡ୀଵ < ∞ then, with probability one, only finitely many ‫ܣ‬௡s will occur;
(v) if ‫ܣ‬ଵ, ‫ܣ‬ଶ, … are independent and ∑ ܲሺ‫ܣ‬௡ሻஶ
௡ୀଵ < ∞ then, with probability one,
infinitely many ‫ܣ‬௡‫ݏ‬ will occur.

42. 42
20. Let ‫,ܣ‬ ‫ ܤ‬and ‫ܥ‬ be three events such that ‫ ܣ‬and ‫ܤ‬ are negatively (positively) associated
and ‫ܤ‬ and ‫ܥ‬ are negatively (positively) associated. Can we conclude that, in general, ‫ܣ‬
and ‫ܥ‬ are negatively (positively) associated?
21. Let ሺߗ, ℱ, ܲሻ be a probability space and let A and B two eventsሺi. e., ‫,ܣ‬ ‫ ܤ‬ ∈ ℱሻ. Show
that if ‫ܣ‬ and ‫ܤ‬ are positively (negatively) associated then ‫ܣ‬ and ‫ܤ‬௖
are negatively
(positively) associated.
22. A locality has ݊ houses numbered 1, … . , ݊ and a terrorist is hiding in one of these
houses. Let ‫ܪ‬௝ denote the event that the terrorist is hiding in house numbered
݆, ݆ = 1, … , ݊ and let ܲ൫‫ܪ‬௝൯ = ‫݌‬௝ ∈ ሺ0,1ሻ, ݆ = 1, … , ݊. During a search operation, let ‫ܨ‬௝
denote the event that search of the house number ݆ will fail to nab the terrorist there
and let ܲ൫‫ܨ‬௝|‫ܪ‬௝൯ ൌ ‫ݎ‬௝ ∈ ሺ0,1ሻ, ݆ = 1, … , ݊. For each ݅, ݆ ∈ ሼ1, … , ݊ሽ, ݅ ≠ ݆, show that
‫ܪ‬௝ and ‫ܨ‬௝ are negatively associated but ‫ܪ‬௜ and ‫ܨ‬௝ are positively associated. Interpret
these findings.
23. Let ‫,ܣ‬ ‫ ܤ‬and ‫ܥ‬ be three events such that ܲሺ‫ܤ‬ ∩ ‫ܥ‬ሻ > 0. Prove or disprove each of the
following:
(i) ܲሺ‫ܣ‬ ∩ ‫ܥ|ܤ‬ሻ ൌ ܲሺ‫ܤ|ܣ‬ ∩ ‫ܥ‬ሻܲሺ‫ܥ|ܤ‬ሻ; (ii) ܲሺ‫ܣ‬ ∩ ‫ܥ|ܤ‬ሻ ൌ ܲሺ‫ܥ|ܣ‬ሻܲሺ‫ܥ|ܤ‬ሻ if ‫ ܣ‬and ‫ܤ‬
are independent events.
24. A ݇-out-of-݊ system is a system comprising of ݊ components that functions if, and only
if, at least ݇ ሺ݇ ∈ ሼ1,2, … , ݊ሽሻ of the components function. A1-out-of-݊ system is called
a parallel system and an݊-out-of-݊ system is called a series system. Consider ݊
components ‫ܥ‬ଵ, … , ‫ܥ‬௡ that function independently. At any given time ‫ݐ‬ the probability
that the component ‫ܥ‬௜ will be functioning is ‫݌‬௜ሺ‫ݐ‬ሻ൫∈ ሺ0,1ሻ൯ and the probability that it
will not be functioning at time ‫ݐ‬ is 1 − ‫݌‬௜ሺ‫ݐ‬ሻ, ݅ = 1, … , ݊.
(i) Find the probability that a parallel system comprising of components ‫ܥ‬ଵ, … , ‫ܥ‬௡ will
function at time ‫;ݐ‬
(ii) Find the probability that a series system comprising of components ‫ܥ‬ଵ, …,‫ܥ‬௡ will
function at time ‫;ݐ‬
(iii) If ‫݌‬௜ሺ‫ݐ‬ሻ = ‫݌‬ሺ‫ݐ‬ሻ, ݅ = 1, … , ݊, find the probability that a ݇-out-of-݊ system comprising
of components ‫ܥ‬ଵ, … , ‫ܥ‬௡ will function at time ‫.ݐ‬