vWhere a mathematical reasoning can be had, it is as great a folly to
make use of any other, as to grope for a thing in the dark, when
you have a candle in your hand. – JOHN ARBUTHNOT v
16.1 Introduction
In earlier classes, we studied about the concept of
probability as a measure of uncertainty of various
phenomenon. We have obtained the probability of getting
an even number in throwing a die as
3
6
i.e.,
1
2
. Here the
total possible outcomes are 1,2,3,4,5 and 6 (six in number).
The outcomes in favour of the event of ‘getting an even
number are 2,4,6 (i.e., three in number). In general, to
obtain the probability of an event, we find the ratio of the
number of outcomes favourable to the event, to the total
number of equally likely outcomes. This theory of probability
is known as classical theory of probability.
In Class IX, we learnt to find the probability on the basis of observations and
collected data. This is called statistical approach of probability.
Both the theories have some serious difficulties. For instance, these theories can
not be applied to the activities/experiments which have infinite number of outcomes. In
classical theory we assume all the outcomes to be equally likely. Recall that the outcomes
are called equally likely when we have no reason to believe that one is more likely to
occur than the other. In other words, we assume that all outcomes have equal chance
(probability) to occur. Thus, to define probability, we used equally likely or equally
probable outcomes. This is logically not a correct definition. Thus, another theory of
probability was developed by A.N. Kolmogorov, a Russian mathematician, in 1933. He
16Chapter
PROBABILITY
Kolmogorov
(1903-1987)
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laid down some axioms to interpret probability, in his book ‘Foundation of Probability’
published in 1933. In this Chapter, we will study about this approach called axiomatic
approach of probability. To understand this approach we must know about few basic
terms viz. random experiment, sample space, events, etc. Let us learn about these all,
in what follows next.
16.2 Random Experiments
In our day to day life, we perform many activities which have a fixed result no matter
any number of times they are repeated. For example given any triangle, without knowing
the three angles, we can definitely say that the sum of measure of angles is 180°.
We also perform many experimental activities, where the result may not be same,
when they are repeated under identical conditions. For example, when a coin is tossed
it may turn up a head or a tail, but we are not sure which one of these results will
actually be obtained. Such experiments are called random experiments.
An experiment is called random experiment if it satisfies the following two
conditions:
(i) It has more than one possible outcome.
(ii) It is not possible to predict the outcome in advance.
Check whether the experiment of tossing a die is random or not?
In this chapter, we shall refer the random experiment by experiment only unless
stated otherwise.
16.2.1
Outcomes and sample space A possible result of a random experiment is
called its outcome.
Consider the experiment of rolling a die. The outcomes of this experiment are 1,
2, 3, 4, 5, or 6, if we are interested in the number of dots on the upper face of the die.
The set of outcomes {1, 2, 3, 4, 5, 6} is called the sample space of the experiment.
Thus, the set of all possible outcomes of a random experiment is called the sample
space associated with the experiment. Sample space is denoted by the symbol S.
Each element of the sample space is called a sample point. In other words, each
outcome of the random experiment is also called sample point
.
Let us now consider some examples.
Example 1 Two coins (a one rupee coin and a two rupee coin) are tossed once. Find
a sample space.
Solution
Clearly the coins are distinguishable in the sense that we can speak of the
first coin and the second coin. Since either coin can turn up Head (H) or Tail(T), the
possible outcomes may be
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Heads on both coins = (H,H) = HH
Head on first coin and Tail on the other = (H,T) = HT
Tail on first coin and Head on the other = (T,H) = TH
Tail on both coins = (T,T) = TT
Thus, the sample space is S = {HH, HT, TH, TT}
A
Note The outcomes of this experiment are ordered pairs of H and T. For the
sake of simplicity the commas are omitted from the ordered pairs.
Example 2 Find the sample space associated with the experiment of rolling a pair of
dice (one is blue and the other red) once. Also, find the number of elements of this
sample space.
Solution
Suppose 1 appears on blue die and 2 on the red die. We denote this outcome
by an ordered pair (1,2). Similarly, if ‘3’ appears on blue die and ‘5’ on red, the outcome
is denoted by the ordered pair (3,5).
In general each outcome can be denoted by the ordered pair (x, y), where x is
the number appeared on the blue die and y is the number appeared on the red die.
Therefore, this sample space is given by
S = {(x, y): x is the number on the blue die and y is the number on the red die}.
The number of elements of this sample space is 6 × 6 = 36 and the sample space is
given below:
{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
Example 3 In each of the following experiments specify appropriate sample space
(i) A boy has a 1 rupee coin, a 2 rupee coin and a 5 rupee coin in his pocket. He
takes out two coins out of his pocket, one after the other.
(ii) A person is noting down the number of accidents along a busy highway
during a year.
Solution
(i) Let Q denote a 1 rupee coin, H denotes a 2 rupee coin and R denotes a 5
rupee coin. The first coin he takes out of his pocket may be any one of the three coins
Q, H or R. Corresponding to Q, the second draw may be H or R. So the result of two
draws may be QH or QR. Similarly, corresponding to H, the second draw may be
Q or R.
Therefore, the outcomes may be HQ or HR. Lastly, corresponding to R, the second
draw may be H or Q.
So, the outcomes may be RH or RQ.
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Thus, the sample space is S={QH, QR, HQ, HR, RH, RQ}
(ii) The number of accidents along a busy highway during the year of observation
can be either 0 (for no accident ) or 1 or 2, or some other positive integer.
Thus, a sample space associated with this experiment is S= {0,1,2,...}
Example 4 A coin is tossed. If it shows head, we draw a ball from a bag consisting of
3 blue and 4 white balls; if it shows tail we throw a die. Describe the sample space of
this experiment.
Solution Let us denote blue balls by B
1
, B
2
, B
3
and the white balls by W
1
, W
2
, W
3
, W
4
.
Then a sample space of the experiment is
S = { HB
1
, HB
2
, HB
3
, HW
1
, HW
2
, HW
3
, HW
4
, T1, T2, T3, T4, T5, T6}.
Here HB
i
means head on the coin and ball B
i
is drawn, HW
i
means head on the coin
and ball W
i
is drawn. Similarly, Ti means tail on the coin and the number i on the die.
Example 5 Consider the experiment in which a coin is tossed repeatedly until a head
comes up. Describe the sample space.
Solution In the experiment head may come up on the first toss, or the 2nd toss, or the
3rd toss and so on till head is obtained. Hence, the desired sample space is
S= {H, TH, TTH, TTTH, TTTTH,...}
EXERCISE 16.1
In each of the following Exercises 1 to 7, describe the sample space for the indicated
experiment.
1. A coin is tossed three times.
2. A die is thrown two times.
3. A coin is tossed four times.
4. A coin is tossed and a die is thrown.
5. A coin is tossed and then a die is rolled only in case a head is shown on the coin.
6. 2 boys and 2 girls are in Room X, and 1 boy and 3 girls in Room Y. Specify the
sample space for the experiment in which a room is selected and then a person.
7. One die of red colour, one of white colour and one of blue colour are placed in a
bag. One die is selected at random and rolled, its colour and the number on its
uppermost face is noted. Describe the sample space.
8. An experiment consists of recording boy–girl composition of families with 2
children.
(i) What is the sample space if we are interested in knowing whether it is a boy
or girl in the order of their births?
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(ii) What is the sample space if we are interested in the number of girls in the
family?
9. A box contains 1 red and 3 identical white balls. Two balls are drawn at random
in succession without replacement. Write the sample space for this experiment.
10. An experiment consists of tossing a coin and then throwing it second time if a
head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the
sample space.
11. Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and
classified as defective (D) or non – defective(N). Write the sample space of this
experiment.
12. A coin is tossed. If the out come is a head, a die is thrown. If the die shows up
an even number, the die is thrown again. What is the sample space for the
experiment?
13. The numbers 1, 2, 3 and 4 are written separatly on four slips of paper. The slips
are put in a box and mixed thoroughly. A person draws two slips from the box,
one after the other, without replacement. Describe the sample space for the
experiment.
14. An experiment consists of rolling a die and then tossing a coin once if the number
on the die is even. If the number on the die is odd, the coin is tossed twice. Write
the sample space for this experiment.
15. A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red
and 3 black balls. If it shows head, we throw a die. Find the sample space for this
experiment.
16. A die is thrown repeatedly untill a six comes up. What is the sample space for
this experiment?
16.3 Event
We have studied about random experiment and sample space associated with an
experiment. The sample space serves as an universal set for all questions concerned
with the experiment.
Consider the experiment of tossing a coin two times. An associated sample space
is S = {HH, HT, TH, TT}.
Now suppose that we are interested in those outcomes which correspond to the
occurrence of exactly one head. We find that HT and TH are the only elements of S
corresponding to the occurrence of this happening (event). These two elements form
the set E = { HT, TH}
We know that the set E is a subset of the sample space S . Similarly, we find the
following correspondence between events and subsets of S.
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Description of events Corresponding subset of ‘S’
Number of tails is exactly 2 A = {TT}
Number of tails is atleast one B = {HT, TH, TT}
Number of heads is atmost one C = {HT, TH, TT}
Second toss is not head D = { HT, TT}
Number of tails is atmost two S = {HH, HT, TH, TT}
Number of tails is more than two φ
The above discussion suggests that a subset of sample space is associated with
an event and an event is associated with a subset of sample space. In the light of this
we define an event as follows.
Definition
Any subset E of a sample space S is called an event.
16.3.1
Occurrence of an event Consider the experiment of throwing a die. Let E
denotes the event “ a number less than 4 appears”. If actually ‘1’ had appeared on the
die then we say that event E has occurred. As a matter of fact if outcomes are 2 or 3,
we say that event E has occurred
Thus, the event E of a sample space S is said to have occurred if the outcome
ω
of the experiment is such that
ω
E. If the outcome
ω
is such that
ω
E, we say
that the event E has not occurred.
16.3.2 Types of events Events can be classified into various types on the basis of the
elements they have.
1. 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 the sure event.
To understand these let us consider the experiment of rolling a die. The associated
sample space is
S = {1, 2, 3, 4, 5, 6}
Let E be the event “ the number appears on the die is a multiple of 7”. Can you
write the subset associated with the event E?
Clearly no outcome satisfies the condition given in the event, i.e., no element of
the sample space ensures the occurrence of the event E. Thus, we say that the empty
set only correspond to the event E. In other words we can say that it is impossible to
have a multiple of 7 on the upper face of the die. Thus, the event E =
φ
is an impossible
event.
Now let us take up another event F “the number turns up is odd or even”. Clearly
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F = {1, 2, 3, 4, 5, 6,} = S, i.e., all outcomes of the experiment ensure the occurrence of
the event F. Thus, the event F = S is a sure event.
2. Simple Event
If an event E has only one sample point of a sample space, it is
called a
simple (or elementary) event.
In a sample space containing n distinct elements, there are exactly n simple
events.
For example in the experiment of tossing two coins, a sample space is
S={HH, HT, TH, TT}
There are four simple events corresponding to this sample space. These are
E
1
= {HH}, E
2
={HT}, E
3
= { TH} and E
4
={TT}.
3. Compound Event If an event has more than one sample point, it is called a
Compound event
.
For example, in the experiment of “tossing a coin thrice” the events
E: ‘Exactly one head appeared’
F: ‘Atleast one head appeared’
G: ‘Atmost one head appeared’ etc.
are all compound events. The subsets of S associated with these events are
E={HTT,THT,TTH}
F={HTT,THT, TTH, HHT, HTH, THH, HHH}
G= {TTT, THT, HTT, TTH}
Each of the above subsets contain more than one sample point, hence they are all
compound events.
16.3.3 Algebra of events In the Chapter on Sets, we have studied about different
ways of combining two or more sets, viz, union, intersection, difference, complement
of a set etc. Like-wise we can combine two or more events by using the analogous set
notations.
Let A, B, C be events associated with an experiment whose sample space is S.
1. Complementary Event For every event A, there corresponds another event
A
called the complementary event to A. It is also called the event ‘not A’.
For example, take the experiment ‘of tossing three coins’. An associated sample
space is
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Let A={HTH, HHT, THH} be the event ‘only one tail appears’
Clearly for the outcome HTT, the event A has not occurred. But we may say that
the event ‘not A has occurred. Thus, with every outcome which is not in A, we say
that ‘not A occurs.
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Thus the complementary event ‘not A
to the event A is
A = {HHH, HTT, THT, TTH, TTT}
or A= {ω : ω S and
ω
A} = S – A.
2. The Event ‘A or B’ Recall that union of two sets A and B denoted by A B
contains all those elements which are either in A or in B or in both.
When the sets A and B are two events associated with a sample space, then
‘A B’ is the event ‘either A or B or both’. This event ‘A B’ is also called ‘A or B’.
Therefore Event ‘A or B’ = A B
= {ω : ω ∈ A or
ω
B}
3. The Event ‘A and B’ We know that intersection of two sets A B is the set of
those elements which are common to both A and B. i.e., which belong to both
‘A and B’.
If A and B are two events, then the set A B denotes the event ‘A and B’.
Thus, A
B = {ω : ω ∈ A and ω ∈ B}
For example, in the experiment of ‘throwing a die twice’ Let A be the event
‘score on the first throw is six’ and B is the event ‘sum of two scores is atleast 11’ then
A = {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}, and B = {(5,6), (6,5), (6,6)}
so A B = {(6,5), (6,6)}
Note that the set A B = {(6,5), (6,6)} may represent the event ‘the score on the first
throw is six and the sum of the scores is atleast 11’.
4. The Event ‘A but not B’ We know that A–B is the set of all those elements
which are in A but not in B. Therefore, the set A–B may denote the event ‘A but not
B’.We know that
A – B = A
Example 6 Consider the experiment of rolling a die. Let A be the event ‘getting a
prime number’, B be the event ‘getting an odd number’. Write the sets representing
the events (i) Aor B (ii) A and B (iii) A but not B (iv) ‘not A’.
Solution Here S = {1, 2, 3, 4, 5, 6}, A = {2, 3, 5} and B = {1, 3, 5}
Obviously
(i) ‘A or B’ = A B = {1, 2, 3, 5}
(ii) ‘A and B’ = A B = {3,5}
(iii) ‘A but not B’ = A – B = {2}
(iv) ‘not A = A = {1,4,6}
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