ICAI Notes- Probability | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


PROBABILITY
15
CHAPTER
Concept of probability is used in accounting and finance to understand the likelihood
of occurrence or non-occurrence of a variable. It helps in developing financial forecasting  in
which you need to develop expertise at an advanced stage of chartered accountancy course.
Introduction
Random
Experimental
Event
Definition of
probability
Random
Variable
Simple Compound
Mutually
Exclusive
Events
Exhaustive
Events
Statistical definition
of probability
Set theoretic approach
to Probability
Modern definition
of Probability
Expected value
of Random
Variable
Probability
Distribution
Compound theorem of
probability
Conditional
probability
Addition
theorems
Compound Probability
Equally Likely
Events
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
Page 2


PROBABILITY
15
CHAPTER
Concept of probability is used in accounting and finance to understand the likelihood
of occurrence or non-occurrence of a variable. It helps in developing financial forecasting  in
which you need to develop expertise at an advanced stage of chartered accountancy course.
Introduction
Random
Experimental
Event
Definition of
probability
Random
Variable
Simple Compound
Mutually
Exclusive
Events
Exhaustive
Events
Statistical definition
of probability
Set theoretic approach
to Probability
Modern definition
of Probability
Expected value
of Random
Variable
Probability
Distribution
Compound theorem of
probability
Conditional
probability
Addition
theorems
Compound Probability
Equally Likely
Events
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
15 .2
STATISTICS
The terms 'Probably' 'in all likelihood', 'chance', 'odds in favour', 'odds against' are too familiar
nowadays and they have their origin in a branch of Mathematics, known as Probability. In recent
time, probability has developed itself into a full-fledged subject and become an integral part of
statistics. The theories of Testing Hypothesis and Estimation are based on probability.
It is rather surprising to know that the first application of probability was made by a group of
mathematicians in Europe about three hundreds years back to enhance their chances of winning
in different games of gambling. Later on, the theory of probability was developed by Abraham
De Moicere and Piere-Simon De Laplace of France, Reverend Thomas Bayes and R. A. Fisher of
England, Chebyshev, Morkov, Khinchin, Kolmogorov of Russia and many other noted
mathematicians as well as statisticians.
Two broad divisions of probability are Subjective Probability and Objective Probability. Subjective
Probability is basically dependent on personal judgement and experience and, as such, it may be
influenced by the personal belief, attitude and bias of the person applying it. However in the
field of uncertainty, this would be quite helpful and it is being applied in the area of decision
making management. This Subjective Probability is beyond the scope of our present discussion.
We are going to discuss Objective Probability in the remaining sections.
In order to develop a sound knowledge about probability, it is necessary to get ourselves familiar
with a few terms.
Experiment: An experiment may be described as a performance that produces certain results.
Random Experiment: An experiment is defined to be random if the results of the experiment
depend on chance only. For example if a coin is tossed, then we get two outcomes—Head (H)
and Tail (T). It is impossible to say in advance whether a Head or a Tail would turn up when we
toss the coin once. Thus, tossing a coin is an example of a random experiment. Similarly, rolling
a dice (or any number of dice), drawing items from a box containing both defective and non—
defective items, drawing cards from a pack of well shuffled fifty two cards etc. are all random
experiments.
Events: The results or outcomes of a random experiment are known as events. Sometimes events
may be combination of outcomes. The events are of two types:
(i) Simple or Elementary,
(ii) Composite or Compound.
An event is known to be simple if it cannot be decomposed into further events. Tossing a coin
once provides us two simple events namely Head and Tail. On the other hand, a composite
event is one that can be decomposed into two or more events. Getting a head when a coin is
tossed twice is an example of composite event as it can be split into the events HT and TH which
are both elementary events.
© The Institute of Chartered Accountants of India
Page 3


PROBABILITY
15
CHAPTER
Concept of probability is used in accounting and finance to understand the likelihood
of occurrence or non-occurrence of a variable. It helps in developing financial forecasting  in
which you need to develop expertise at an advanced stage of chartered accountancy course.
Introduction
Random
Experimental
Event
Definition of
probability
Random
Variable
Simple Compound
Mutually
Exclusive
Events
Exhaustive
Events
Statistical definition
of probability
Set theoretic approach
to Probability
Modern definition
of Probability
Expected value
of Random
Variable
Probability
Distribution
Compound theorem of
probability
Conditional
probability
Addition
theorems
Compound Probability
Equally Likely
Events
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
15 .2
STATISTICS
The terms 'Probably' 'in all likelihood', 'chance', 'odds in favour', 'odds against' are too familiar
nowadays and they have their origin in a branch of Mathematics, known as Probability. In recent
time, probability has developed itself into a full-fledged subject and become an integral part of
statistics. The theories of Testing Hypothesis and Estimation are based on probability.
It is rather surprising to know that the first application of probability was made by a group of
mathematicians in Europe about three hundreds years back to enhance their chances of winning
in different games of gambling. Later on, the theory of probability was developed by Abraham
De Moicere and Piere-Simon De Laplace of France, Reverend Thomas Bayes and R. A. Fisher of
England, Chebyshev, Morkov, Khinchin, Kolmogorov of Russia and many other noted
mathematicians as well as statisticians.
Two broad divisions of probability are Subjective Probability and Objective Probability. Subjective
Probability is basically dependent on personal judgement and experience and, as such, it may be
influenced by the personal belief, attitude and bias of the person applying it. However in the
field of uncertainty, this would be quite helpful and it is being applied in the area of decision
making management. This Subjective Probability is beyond the scope of our present discussion.
We are going to discuss Objective Probability in the remaining sections.
In order to develop a sound knowledge about probability, it is necessary to get ourselves familiar
with a few terms.
Experiment: An experiment may be described as a performance that produces certain results.
Random Experiment: An experiment is defined to be random if the results of the experiment
depend on chance only. For example if a coin is tossed, then we get two outcomes—Head (H)
and Tail (T). It is impossible to say in advance whether a Head or a Tail would turn up when we
toss the coin once. Thus, tossing a coin is an example of a random experiment. Similarly, rolling
a dice (or any number of dice), drawing items from a box containing both defective and non—
defective items, drawing cards from a pack of well shuffled fifty two cards etc. are all random
experiments.
Events: The results or outcomes of a random experiment are known as events. Sometimes events
may be combination of outcomes. The events are of two types:
(i) Simple or Elementary,
(ii) Composite or Compound.
An event is known to be simple if it cannot be decomposed into further events. Tossing a coin
once provides us two simple events namely Head and Tail. On the other hand, a composite
event is one that can be decomposed into two or more events. Getting a head when a coin is
tossed twice is an example of composite event as it can be split into the events HT and TH which
are both elementary events.
© The Institute of Chartered Accountants of India
15 .3 PROBABILITY
Mutually Exclusive Events or Incompatible Events: A set of events A1, A2, A3, …… is known
to be mutually exclusive if not more than one of them can occur simultaneously. Thus occurrence
of one such event implies the non-occurrence of the other events of the set. Once a coin is tossed,
we get two mutually exclusive events Head and Tail.
Exhaustive Events: The events A1, A2, A3, ………… are known to form an exhaustive set if one
of these events must necessarily occur. As an example, the two events Head and Tail, when a
coin is tossed once, are exhaustive as no other event except these two can occur.
Equally Likely Events or Mutually Symmetric Events or Equi-Probable Events: The events of
a random experiment are known to be equally likely when all necessary evidence are taken into
account, no event is expected to occur more frequently as compared to the other events of the set
of events. The two events Head and Tail when a coin is tossed is an example of a pair of equally
likely events because there is no reason to assume that Head (or Tail) would occur more frequently
as compared to Tail (or Head).
Let us consider a random experiment that result in n finite elementary events, which are assumed
to be equally likely. We next assume that out of these n events, n
A
 ( ? n) events are favourable to
an event A. Then the probability of occurrence of the event A is defined as the ratio of the
number of events favourable to A to the total number of events. Denoting this by P(A), we have
P(A)
A
n
n
?
= 
No. of equally likely events favourable to A
Total no. of equally likely events
……………. (15.1)
However if instead of considering all elementary events, we focus our attention to only those
composite events, which are mutually exclusive, exhaustive and equally likely and if m( ? n)
denotes such events and is furthermore m
A
( ? n
A
) denotes the no. of mutually exclusive, exhaustive
and equally likely events favourable to A, then we have
P(A) = 
A
m
m
  = 
No. of mutually exclusive, exhaustive and equally likely events favourable to A
Total no.of mutually exclusive, exhaustive andequally likely events
………………   (15.2)
For this definition of probability, we are indebted to Bernoulli and Laplace. This definition is
also termed as a priori definition because probability of the event A is defined on the basis of
prior knowledge.
© The Institute of Chartered Accountants of India
Page 4


PROBABILITY
15
CHAPTER
Concept of probability is used in accounting and finance to understand the likelihood
of occurrence or non-occurrence of a variable. It helps in developing financial forecasting  in
which you need to develop expertise at an advanced stage of chartered accountancy course.
Introduction
Random
Experimental
Event
Definition of
probability
Random
Variable
Simple Compound
Mutually
Exclusive
Events
Exhaustive
Events
Statistical definition
of probability
Set theoretic approach
to Probability
Modern definition
of Probability
Expected value
of Random
Variable
Probability
Distribution
Compound theorem of
probability
Conditional
probability
Addition
theorems
Compound Probability
Equally Likely
Events
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
15 .2
STATISTICS
The terms 'Probably' 'in all likelihood', 'chance', 'odds in favour', 'odds against' are too familiar
nowadays and they have their origin in a branch of Mathematics, known as Probability. In recent
time, probability has developed itself into a full-fledged subject and become an integral part of
statistics. The theories of Testing Hypothesis and Estimation are based on probability.
It is rather surprising to know that the first application of probability was made by a group of
mathematicians in Europe about three hundreds years back to enhance their chances of winning
in different games of gambling. Later on, the theory of probability was developed by Abraham
De Moicere and Piere-Simon De Laplace of France, Reverend Thomas Bayes and R. A. Fisher of
England, Chebyshev, Morkov, Khinchin, Kolmogorov of Russia and many other noted
mathematicians as well as statisticians.
Two broad divisions of probability are Subjective Probability and Objective Probability. Subjective
Probability is basically dependent on personal judgement and experience and, as such, it may be
influenced by the personal belief, attitude and bias of the person applying it. However in the
field of uncertainty, this would be quite helpful and it is being applied in the area of decision
making management. This Subjective Probability is beyond the scope of our present discussion.
We are going to discuss Objective Probability in the remaining sections.
In order to develop a sound knowledge about probability, it is necessary to get ourselves familiar
with a few terms.
Experiment: An experiment may be described as a performance that produces certain results.
Random Experiment: An experiment is defined to be random if the results of the experiment
depend on chance only. For example if a coin is tossed, then we get two outcomes—Head (H)
and Tail (T). It is impossible to say in advance whether a Head or a Tail would turn up when we
toss the coin once. Thus, tossing a coin is an example of a random experiment. Similarly, rolling
a dice (or any number of dice), drawing items from a box containing both defective and non—
defective items, drawing cards from a pack of well shuffled fifty two cards etc. are all random
experiments.
Events: The results or outcomes of a random experiment are known as events. Sometimes events
may be combination of outcomes. The events are of two types:
(i) Simple or Elementary,
(ii) Composite or Compound.
An event is known to be simple if it cannot be decomposed into further events. Tossing a coin
once provides us two simple events namely Head and Tail. On the other hand, a composite
event is one that can be decomposed into two or more events. Getting a head when a coin is
tossed twice is an example of composite event as it can be split into the events HT and TH which
are both elementary events.
© The Institute of Chartered Accountants of India
15 .3 PROBABILITY
Mutually Exclusive Events or Incompatible Events: A set of events A1, A2, A3, …… is known
to be mutually exclusive if not more than one of them can occur simultaneously. Thus occurrence
of one such event implies the non-occurrence of the other events of the set. Once a coin is tossed,
we get two mutually exclusive events Head and Tail.
Exhaustive Events: The events A1, A2, A3, ………… are known to form an exhaustive set if one
of these events must necessarily occur. As an example, the two events Head and Tail, when a
coin is tossed once, are exhaustive as no other event except these two can occur.
Equally Likely Events or Mutually Symmetric Events or Equi-Probable Events: The events of
a random experiment are known to be equally likely when all necessary evidence are taken into
account, no event is expected to occur more frequently as compared to the other events of the set
of events. The two events Head and Tail when a coin is tossed is an example of a pair of equally
likely events because there is no reason to assume that Head (or Tail) would occur more frequently
as compared to Tail (or Head).
Let us consider a random experiment that result in n finite elementary events, which are assumed
to be equally likely. We next assume that out of these n events, n
A
 ( ? n) events are favourable to
an event A. Then the probability of occurrence of the event A is defined as the ratio of the
number of events favourable to A to the total number of events. Denoting this by P(A), we have
P(A)
A
n
n
?
= 
No. of equally likely events favourable to A
Total no. of equally likely events
……………. (15.1)
However if instead of considering all elementary events, we focus our attention to only those
composite events, which are mutually exclusive, exhaustive and equally likely and if m( ? n)
denotes such events and is furthermore m
A
( ? n
A
) denotes the no. of mutually exclusive, exhaustive
and equally likely events favourable to A, then we have
P(A) = 
A
m
m
  = 
No. of mutually exclusive, exhaustive and equally likely events favourable to A
Total no.of mutually exclusive, exhaustive andequally likely events
………………   (15.2)
For this definition of probability, we are indebted to Bernoulli and Laplace. This definition is
also termed as a priori definition because probability of the event A is defined on the basis of
prior knowledge.
© The Institute of Chartered Accountants of India
1 5 .4
STATISTICS
This classical definition of probability has the following demerits or limitations:
(i) It is applicable only when the total no. of events is finite.
(ii) It can be used only when the events are equally likely or equi-probable. This assumption is
made well before the experiment is performed.
(iii) This definition has only a limited field of application like coin tossing, dice throwing, drawing
cards etc. where the possible events are known well in advance. In the field of uncertainty or
where no prior knowledge is provided, this definition is inapplicable.
In connection with classical definition of probability, we may note the following points:
(a) The probability of an event lies between 0 and 1, both inclusive.
i.e. 0 ? P(A) ? 1 …….  (15.3)
When P(A) = 0, A is known to be an impossible event and when P(A) = 1, A is known to be
a sure event.
(b) Non-occurrence of event A is denoted by A’ or A
C
 or 
?
? and it is known as complimentary
event of A. The event A along with its complimentary A’ forms a set of mutually exclusive
and exhaustive events.
i.e. P(A) + P (A’) = 1
? P(A’) = 1 ? P(A)
1 ? ?
A
m
m
= 
?
A
m m
m
       …………… (15.4)
(c) The ratio of no. of favourable events to the no. of unfavourable events is known as odds in
favour of the event A and its inverse ratio is known as odds against the event A.
i.e. odds in favour of A = m
A
 : (m – m
A
) ……………… (15.5)
and odds against A = (m – m
A
) : m
A
……………… (15.6)
 ILLUSTRATIONS:
Example 15.1: A coin is tossed three times. What is the probability of getting:
(i) 2 heads
(ii) at least 2 heads.
© The Institute of Chartered Accountants of India
Page 5


PROBABILITY
15
CHAPTER
Concept of probability is used in accounting and finance to understand the likelihood
of occurrence or non-occurrence of a variable. It helps in developing financial forecasting  in
which you need to develop expertise at an advanced stage of chartered accountancy course.
Introduction
Random
Experimental
Event
Definition of
probability
Random
Variable
Simple Compound
Mutually
Exclusive
Events
Exhaustive
Events
Statistical definition
of probability
Set theoretic approach
to Probability
Modern definition
of Probability
Expected value
of Random
Variable
Probability
Distribution
Compound theorem of
probability
Conditional
probability
Addition
theorems
Compound Probability
Equally Likely
Events
CHAPTER OVERVIEW
© The Institute of Chartered Accountants of India
15 .2
STATISTICS
The terms 'Probably' 'in all likelihood', 'chance', 'odds in favour', 'odds against' are too familiar
nowadays and they have their origin in a branch of Mathematics, known as Probability. In recent
time, probability has developed itself into a full-fledged subject and become an integral part of
statistics. The theories of Testing Hypothesis and Estimation are based on probability.
It is rather surprising to know that the first application of probability was made by a group of
mathematicians in Europe about three hundreds years back to enhance their chances of winning
in different games of gambling. Later on, the theory of probability was developed by Abraham
De Moicere and Piere-Simon De Laplace of France, Reverend Thomas Bayes and R. A. Fisher of
England, Chebyshev, Morkov, Khinchin, Kolmogorov of Russia and many other noted
mathematicians as well as statisticians.
Two broad divisions of probability are Subjective Probability and Objective Probability. Subjective
Probability is basically dependent on personal judgement and experience and, as such, it may be
influenced by the personal belief, attitude and bias of the person applying it. However in the
field of uncertainty, this would be quite helpful and it is being applied in the area of decision
making management. This Subjective Probability is beyond the scope of our present discussion.
We are going to discuss Objective Probability in the remaining sections.
In order to develop a sound knowledge about probability, it is necessary to get ourselves familiar
with a few terms.
Experiment: An experiment may be described as a performance that produces certain results.
Random Experiment: An experiment is defined to be random if the results of the experiment
depend on chance only. For example if a coin is tossed, then we get two outcomes—Head (H)
and Tail (T). It is impossible to say in advance whether a Head or a Tail would turn up when we
toss the coin once. Thus, tossing a coin is an example of a random experiment. Similarly, rolling
a dice (or any number of dice), drawing items from a box containing both defective and non—
defective items, drawing cards from a pack of well shuffled fifty two cards etc. are all random
experiments.
Events: The results or outcomes of a random experiment are known as events. Sometimes events
may be combination of outcomes. The events are of two types:
(i) Simple or Elementary,
(ii) Composite or Compound.
An event is known to be simple if it cannot be decomposed into further events. Tossing a coin
once provides us two simple events namely Head and Tail. On the other hand, a composite
event is one that can be decomposed into two or more events. Getting a head when a coin is
tossed twice is an example of composite event as it can be split into the events HT and TH which
are both elementary events.
© The Institute of Chartered Accountants of India
15 .3 PROBABILITY
Mutually Exclusive Events or Incompatible Events: A set of events A1, A2, A3, …… is known
to be mutually exclusive if not more than one of them can occur simultaneously. Thus occurrence
of one such event implies the non-occurrence of the other events of the set. Once a coin is tossed,
we get two mutually exclusive events Head and Tail.
Exhaustive Events: The events A1, A2, A3, ………… are known to form an exhaustive set if one
of these events must necessarily occur. As an example, the two events Head and Tail, when a
coin is tossed once, are exhaustive as no other event except these two can occur.
Equally Likely Events or Mutually Symmetric Events or Equi-Probable Events: The events of
a random experiment are known to be equally likely when all necessary evidence are taken into
account, no event is expected to occur more frequently as compared to the other events of the set
of events. The two events Head and Tail when a coin is tossed is an example of a pair of equally
likely events because there is no reason to assume that Head (or Tail) would occur more frequently
as compared to Tail (or Head).
Let us consider a random experiment that result in n finite elementary events, which are assumed
to be equally likely. We next assume that out of these n events, n
A
 ( ? n) events are favourable to
an event A. Then the probability of occurrence of the event A is defined as the ratio of the
number of events favourable to A to the total number of events. Denoting this by P(A), we have
P(A)
A
n
n
?
= 
No. of equally likely events favourable to A
Total no. of equally likely events
……………. (15.1)
However if instead of considering all elementary events, we focus our attention to only those
composite events, which are mutually exclusive, exhaustive and equally likely and if m( ? n)
denotes such events and is furthermore m
A
( ? n
A
) denotes the no. of mutually exclusive, exhaustive
and equally likely events favourable to A, then we have
P(A) = 
A
m
m
  = 
No. of mutually exclusive, exhaustive and equally likely events favourable to A
Total no.of mutually exclusive, exhaustive andequally likely events
………………   (15.2)
For this definition of probability, we are indebted to Bernoulli and Laplace. This definition is
also termed as a priori definition because probability of the event A is defined on the basis of
prior knowledge.
© The Institute of Chartered Accountants of India
1 5 .4
STATISTICS
This classical definition of probability has the following demerits or limitations:
(i) It is applicable only when the total no. of events is finite.
(ii) It can be used only when the events are equally likely or equi-probable. This assumption is
made well before the experiment is performed.
(iii) This definition has only a limited field of application like coin tossing, dice throwing, drawing
cards etc. where the possible events are known well in advance. In the field of uncertainty or
where no prior knowledge is provided, this definition is inapplicable.
In connection with classical definition of probability, we may note the following points:
(a) The probability of an event lies between 0 and 1, both inclusive.
i.e. 0 ? P(A) ? 1 …….  (15.3)
When P(A) = 0, A is known to be an impossible event and when P(A) = 1, A is known to be
a sure event.
(b) Non-occurrence of event A is denoted by A’ or A
C
 or 
?
? and it is known as complimentary
event of A. The event A along with its complimentary A’ forms a set of mutually exclusive
and exhaustive events.
i.e. P(A) + P (A’) = 1
? P(A’) = 1 ? P(A)
1 ? ?
A
m
m
= 
?
A
m m
m
       …………… (15.4)
(c) The ratio of no. of favourable events to the no. of unfavourable events is known as odds in
favour of the event A and its inverse ratio is known as odds against the event A.
i.e. odds in favour of A = m
A
 : (m – m
A
) ……………… (15.5)
and odds against A = (m – m
A
) : m
A
……………… (15.6)
 ILLUSTRATIONS:
Example 15.1: A coin is tossed three times. What is the probability of getting:
(i) 2 heads
(ii) at least 2 heads.
© The Institute of Chartered Accountants of India
1 5 .5 PROBABILITY
Solution: When a coin is tossed three times, first we need enumerate all the elementary events.
This can be done using 'Tree diagram' as shown below:
Hence the elementary events are
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Thus the number of elementary events (n) is 8.
(i) Out of these 8 outcomes, 2 heads occur in three cases namely HHT, HTH and THH. If we
denote the occurrence of 2 heads by the event A and if assume that the coin as well as
performer of the experiment is unbiased then this assumption ensures that all the eight
elementary events are equally likely. Then by the classical definition of probability, we
have
P (A) =
A
n
n
=
3
8
= 0.375
(ii) Let B denote occurrence of at least 2 heads i.e. 2 heads or 3 heads. Since 2 heads occur in 3
cases and 3 heads occur in only 1 case, B occurs in 3 + 1 or 4 cases. By the classical definition
of probability,
P(B) =
4
8
= 0.50
Example 15.2: A dice is rolled twice. What is the probability of getting a difference of 2 points?
Solution: If an experiment results in p outcomes and if the experiment is repeated q times, then
the total number of outcomes is p
q
. In the present case, since a dice results in 6 outcomes and the
dice is rolled twice, total no. of outcomes or elementary events is 6
2
 or 36. We assume that the
dice is unbiased which ensures that all these 36 elementary events are equally likely. Now a
difference of 2 points in the uppermost faces of the dice thrown twice can occur in the following
cases:
Start
T
H
T
H
T
H
T
H
H
H
T
H
T
T
© The Institute of Chartered Accountants of India