Chapter 16
PROBABILITY
16.1 Overview
Probability is defined as a quantitative measure of uncertainty – a numerical value that
conveys the strength of our belief in the occurrence of an event. The probability of an
event is always a number between 0 and 1 both 0 and 1 inclusive. If an event’s probability
is nearer to 1, the higher is the likelihood that the event will occur; the closer the event’s
probability to 0, the smaller is the likelihood that the event will occur. If the event
cannot occur, its probability is 0. If it must occur (i.e., its occurrence is certain), its
probability is 1.
16.1.1 Random experiment An experiment is random means that the experiment
has more than one possible outcome and it is not possible to predict with certainty
which outcome that will be. For instance, in an experiment of tossing an ordinary coin,
it can be predicted with certainty that the coin will land either heads up or tails up, but
it is not known for sure whether heads or tails will occur. If a die is thrown once, any of
the six numbers, i.e., 1, 2, 3, 4, 5, 6 may turn up, not sure which number will come up.
(i) Outcome A possible result of a random experiment is called its outcome for
example if the experiment consists of tossing a coin twice, some of the outcomes
are HH, HT etc.
(ii) Sample Space A sample space is the set of all possible outcomes of an
experiment. In fact, it is the universal set S pertinent to a given experiment.
The sample space for the experiment of tossing a coin twice is given by
S = {HH, HT, TH, TT}
The sample space for the experiment of drawing a card out of a deck is the set of all
cards in the deck.
16.1.2 Event An event is a subset of a sample space S. For example, the event of
drawing an ace from a deck is
A = {Ace of Heart, Ace of Club, Ace of Diamond, Ace of Spade}
16.1.3 Types of events
(i) Impossible and Sure Events The empty set φ and the sample space S describe
events. In fact φ is called an impossible event and S, i.e., the whole sample
space is called a sure event.
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(ii) Simple or Elementary Event If an event E has only one sample point of a
sample space, i.e., a single outcome of an experiment, it is called a simple or
elementary event. The sample space of the experiment of tossing two coins is
given by
S = {HH, HT, TH, TT}
The event E1 = {HH} containing a single outcome HH of the sample space S is
a simple or elementary event. If one card is drawn from a well shuffled deck,
any particular card drawn like ‘queen of Hearts’ is an elementary event.
(iii) Compound Event If an event has more than one sample point it is called a
compound event, for example, S = {HH, HT} is a compound event.
(iv) Complementary event Given an event A, the complement of A is the event
consisting of all sample space outcomes that do not correspond to the occurrence
of A.
The complement of A is denoted by A′ or A . It is also called the event ‘not A’. Further
P( A ) denotes the probability that A will not occur.
A′ = A = S – A = {w : w ∈ S and w ∉A}
16.1.4 Event ‘A or B’ If A and B are two events associated with same sample space,
then the event ‘A or B’ is same as the event A ∪ B and contains all those elements
which are either in A or in B or in both. Further more, P (A∪B) denotes the probability
that A or B (or both) will occur.
16.1.5 Event ‘A and B’ If A and B are two events associated with a sample space,
then the event ‘A and B’ is same as the event A ∩ B and contains all those elements which
are common to both A and B. Further more, P (A ∩ B) denotes the probability that both
A and B will simultaneously occur.
16.1.6 The Event ‘A but not B’ (Difference A – B) An event A – B is the set of all
those elements of the same space S which are in A but not in B, i.e., A – B = A ∩ B′.
16.1.7 Mutually exclusive Two events A and B of a sample space S are mutually
exclusive if the occurrence of any one of them excludes the occurrence of the other
event. Hence, the two events A and B cannot occur simultaneously, and thus P(A∩B) = 0.
Remark Simple or elementary events of a sample space are always mutually exclusive.
For example, the elementary events {1}, {2}, {3}, {4}, {5} or {6} of the experiment of
throwing a dice are mutually exclusive.
Consider the experiment of throwing a die once.
The events E = getting a even number and F = getting an odd number are mutually
exclusive events because E ∩ F = φ.
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Note For a given sample space, there may be two or more mutually exclusive events.
16.1.8 Exhaustive events If E1, E2, ..., En are n events of a sample space S and if
n
E1 ∪ E2 ∪ E3 ∪ ... ∪ En = ∪ E i = S
i=1
then E1, E2, ..., En are called exhaustive events.
In other words, events E1, E2, ..., En of a sample space S are said to be exhaustive if
atleast one of them necessarily occur whenever the experiment is performed.
Consider the example of rolling a die. We have S = {1, 2, 3, 4, 5, 6}. Define the two
events A : ‘a number less than or equal to 4 appears.’
B : ‘a number greater than or equal to 4 appears.’
Now A : {1, 2, 3, 4}, B = {4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6} = S
Such events A and B are called exhaustive events.
16.1.9 Mutually exclusive and exhaustive events If E1, E2, ..., En are n events of
a sample space S and if Ei ∩ Ej = φ for every i ≠ j, i.e., Ei and Ej are pairwise disjoint
n
and ∪ E i = S , then the events E1, E2, ... , En are called mutually exclusive and exhaustive
i=1
events.
Consider the example of rolling a die.
We have S = {1, 2, 3, 4, 5, 6}
Let us define the three events as
A = a number which is a perfect square
B = a prime number
C = a number which is greater than or equal to 6
Now A = {1, 4}, B = {2, 3, 5}, C = {6}
Note that A ∪ B ∪ C = {1, 2, 3, 4, 5, 6} = S. Therefore, A, B and C are exhaustive
events.
Also A ∩ B = B ∩ C = C ∩ A = φ
Hence, the events are pairwise disjoint and thus mutually exclusive.
Classical approach is useful, when all the outcomes of the experiment are equally
likely. We can use logic to assign probabilities. To understand the classical method
consider the experiment of tossing a fair coin. Here, there are two equally likely
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PROBABILITY
16.1 Overview
Probability is defined as a quantitative measure of uncertainty – a numerical value that
conveys the strength of our belief in the occurrence of an event. The probability of an
event is always a number between 0 and 1 both 0 and 1 inclusive. If an event’s probability
is nearer to 1, the higher is the likelihood that the event will occur; the closer the event’s
probability to 0, the smaller is the likelihood that the event will occur. If the event
cannot occur, its probability is 0. If it must occur (i.e., its occurrence is certain), its
probability is 1.
16.1.1 Random experiment An experiment is random means that the experiment
has more than one possible outcome and it is not possible to predict with certainty
which outcome that will be. For instance, in an experiment of tossing an ordinary coin,
it can be predicted with certainty that the coin will land either heads up or tails up, but
it is not known for sure whether heads or tails will occur. If a die is thrown once, any of
the six numbers, i.e., 1, 2, 3, 4, 5, 6 may turn up, not sure which number will come up.
(i) Outcome A possible result of a random experiment is called its outcome for
example if the experiment consists of tossing a coin twice, some of the outcomes
are HH, HT etc.
(ii) Sample Space A sample space is the set of all possible outcomes of an
experiment. In fact, it is the universal set S pertinent to a given experiment.
The sample space for the experiment of tossing a coin twice is given by
S = {HH, HT, TH, TT}
The sample space for the experiment of drawing a card out of a deck is the set of all
cards in the deck.
16.1.2 Event An event is a subset of a sample space S. For example, the event of
drawing an ace from a deck is
A = {Ace of Heart, Ace of Club, Ace of Diamond, Ace of Spade}
16.1.3 Types of events
(i) Impossible and Sure Events The empty set φ and the sample space S describe
events. In fact φ is called an impossible event and S, i.e., the whole sample
space is called a sure event.
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(ii) Simple or Elementary Event If an event E has only one sample point of a
sample space, i.e., a single outcome of an experiment, it is called a simple or
elementary event. The sample space of the experiment of tossing two coins is
given by
S = {HH, HT, TH, TT}
The event E1 = {HH} containing a single outcome HH of the sample space S is
a simple or elementary event. If one card is drawn from a well shuffled deck,
any particular card drawn like ‘queen of Hearts’ is an elementary event.
(iii) Compound Event If an event has more than one sample point it is called a
compound event, for example, S = {HH, HT} is a compound event.
(iv) Complementary event Given an event A, the complement of A is the event
consisting of all sample space outcomes that do not correspond to the occurrence
of A.
The complement of A is denoted by A′ or A . It is also called the event ‘not A’. Further
P( A ) denotes the probability that A will not occur.
A′ = A = S – A = {w : w ∈ S and w ∉A}
16.1.4 Event ‘A or B’ If A and B are two events associated with same sample space,
then the event ‘A or B’ is same as the event A ∪ B and contains all those elements
which are either in A or in B or in both. Further more, P (A∪B) denotes the probability
that A or B (or both) will occur.
16.1.5 Event ‘A and B’ If A and B are two events associated with a sample space,
then the event ‘A and B’ is same as the event A ∩ B and contains all those elements which
are common to both A and B. Further more, P (A ∩ B) denotes the probability that both
A and B will simultaneously occur.
16.1.6 The Event ‘A but not B’ (Difference A – B) An event A – B is the set of all
those elements of the same space S which are in A but not in B, i.e., A – B = A ∩ B′.
16.1.7 Mutually exclusive Two events A and B of a sample space S are mutually
exclusive if the occurrence of any one of them excludes the occurrence of the other
event. Hence, the two events A and B cannot occur simultaneously, and thus P(A∩B) = 0.
Remark Simple or elementary events of a sample space are always mutually exclusive.
For example, the elementary events {1}, {2}, {3}, {4}, {5} or {6} of the experiment of
throwing a dice are mutually exclusive.
Consider the experiment of throwing a die once.
The events E = getting a even number and F = getting an odd number are mutually
exclusive events because E ∩ F = φ.
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Note For a given sample space, there may be two or more mutually exclusive events.
16.1.8 Exhaustive events If E1, E2, ..., En are n events of a sample space S and if
n
E1 ∪ E2 ∪ E3 ∪ ... ∪ En = ∪ E i = S
i=1
then E1, E2, ..., En are called exhaustive events.
In other words, events E1, E2, ..., En of a sample space S are said to be exhaustive if
atleast one of them necessarily occur whenever the experiment is performed.
Consider the example of rolling a die. We have S = {1, 2, 3, 4, 5, 6}. Define the two
events A : ‘a number less than or equal to 4 appears.’
B : ‘a number greater than or equal to 4 appears.’
Now A : {1, 2, 3, 4}, B = {4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6} = S
Such events A and B are called exhaustive events.
16.1.9 Mutually exclusive and exhaustive events If E1, E2, ..., En are n events of
a sample space S and if Ei ∩ Ej = φ for every i ≠ j, i.e., Ei and Ej are pairwise disjoint
n
and ∪ E i = S , then the events E1, E2, ... , En are called mutually exclusive and exhaustive
i=1
events.
Consider the example of rolling a die.
We have S = {1, 2, 3, 4, 5, 6}
Let us define the three events as
A = a number which is a perfect square
B = a prime number
C = a number which is greater than or equal to 6
Now A = {1, 4}, B = {2, 3, 5}, C = {6}
Note that A ∪ B ∪ C = {1, 2, 3, 4, 5, 6} = S. Therefore, A, B and C are exhaustive
events.
Also A ∩ B = B ∩ C = C ∩ A = φ
Hence, the events are pairwise disjoint and thus mutually exclusive.
Classical approach is useful, when all the outcomes of the experiment are equally
likely. We can use logic to assign probabilities. To understand the classical method
consider the experiment of tossing a fair coin. Here, there are two equally likely
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