# NCERT Textbook: Probability Notes | Study NCERT Textbooks (Class 6 to Class 12) - UPSC

## UPSC: NCERT Textbook: Probability Notes | Study NCERT Textbooks (Class 6 to Class 12) - UPSC

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PROBABILLITY 271
CHAPTER 15
PROBABILITY
It is remarkable that a science, which began with the consideration of
games of chance, should be elevated to the rank of the most important
subject of human knowledge. —Pierre Simon Laplace
15.1 Introduction
In everyday life, we come across statements such as
(1) It will probably rain today.
(2) I doubt that he will pass the test.
(3) Most probably, Kavita will stand first in the annual examination.
(4) Chances are high that the prices of diesel will go up.
(5) There is a 50-50 chance of India winning a toss in today’s match.
The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the
statements above involve an element of uncertainty. For example, in (1), ‘probably
rain’ will mean it may rain or may not rain today. We are predicting rain today based
on our past experience when it rained under similar conditions. Similar predictions are
also made in other cases listed in (2) to (5).
The uncertainty of ‘probably’ etc can be measured numerically by means of
‘probability’ in many cases.
Though probability started with gambling, it has been used extensively in the fields
of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather
Forecasting, etc.
2020-21
Page 2

PROBABILLITY 271
CHAPTER 15
PROBABILITY
It is remarkable that a science, which began with the consideration of
games of chance, should be elevated to the rank of the most important
subject of human knowledge. —Pierre Simon Laplace
15.1 Introduction
In everyday life, we come across statements such as
(1) It will probably rain today.
(2) I doubt that he will pass the test.
(3) Most probably, Kavita will stand first in the annual examination.
(4) Chances are high that the prices of diesel will go up.
(5) There is a 50-50 chance of India winning a toss in today’s match.
The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the
statements above involve an element of uncertainty. For example, in (1), ‘probably
rain’ will mean it may rain or may not rain today. We are predicting rain today based
on our past experience when it rained under similar conditions. Similar predictions are
also made in other cases listed in (2) to (5).
The uncertainty of ‘probably’ etc can be measured numerically by means of
‘probability’ in many cases.
Though probability started with gambling, it has been used extensively in the fields
of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather
Forecasting, etc.
2020-21
272 MATHEMA TICS
15.2 Probability – an Experimental Approach
In earlier classes, you have had a glimpse of probability when you performed
experiments like tossing of coins, throwing of dice, etc., and observed their outcomes.
You will now learn to measure the chance of occurrence of a particular outcome in an
experiment.
The concept of probability developed in a very
strange manner. In 1654, a gambler Chevalier
de Mere, approached the well-known 17th
century French philosopher and mathematician
Blaise Pascal regarding certain dice problems.
Pascal became interested in these problems,
studied them and discussed them with another
French mathematician, Pierre de Fermat. Both
Pascal and Fermat solved the problems
independently. This work was the beginning
of Probability Theory.
The first book on the subject was written by the Italian mathematician, J.Cardan
(1501–1576). The title of the book was ‘Book on Games of Chance’ (Liber de Ludo
Aleae), published in 1663. Notable contributions were also made by mathematicians
J. Bernoulli (1654–1705), P . Laplace (1749–1827), A.A. Markov (1856–1922) and A.N.
Kolmogorov (born 1903).
Activity 1 : (i) Take any coin, toss it ten times and note down the number of times a
head and a tail come up. Record your observations in the form of the following table
Table 15.1
Number of times Number of times Number of times
the coin is tossed head comes up tail comes up
10 — —
Write down the values of the following fractions:
Number of times a head comes up
Total number of times the coin is tossed
and
Number of times a tail comes up
Total number of times the coin is tossed
Blaise Pascal
(1623–1662)
Fig. 15.1
Pierre de Fermat
(1601–1665)
Fig. 15.2
2020-21
Page 3

PROBABILLITY 271
CHAPTER 15
PROBABILITY
It is remarkable that a science, which began with the consideration of
games of chance, should be elevated to the rank of the most important
subject of human knowledge. —Pierre Simon Laplace
15.1 Introduction
In everyday life, we come across statements such as
(1) It will probably rain today.
(2) I doubt that he will pass the test.
(3) Most probably, Kavita will stand first in the annual examination.
(4) Chances are high that the prices of diesel will go up.
(5) There is a 50-50 chance of India winning a toss in today’s match.
The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the
statements above involve an element of uncertainty. For example, in (1), ‘probably
rain’ will mean it may rain or may not rain today. We are predicting rain today based
on our past experience when it rained under similar conditions. Similar predictions are
also made in other cases listed in (2) to (5).
The uncertainty of ‘probably’ etc can be measured numerically by means of
‘probability’ in many cases.
Though probability started with gambling, it has been used extensively in the fields
of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather
Forecasting, etc.
2020-21
272 MATHEMA TICS
15.2 Probability – an Experimental Approach
In earlier classes, you have had a glimpse of probability when you performed
experiments like tossing of coins, throwing of dice, etc., and observed their outcomes.
You will now learn to measure the chance of occurrence of a particular outcome in an
experiment.
The concept of probability developed in a very
strange manner. In 1654, a gambler Chevalier
de Mere, approached the well-known 17th
century French philosopher and mathematician
Blaise Pascal regarding certain dice problems.
Pascal became interested in these problems,
studied them and discussed them with another
French mathematician, Pierre de Fermat. Both
Pascal and Fermat solved the problems
independently. This work was the beginning
of Probability Theory.
The first book on the subject was written by the Italian mathematician, J.Cardan
(1501–1576). The title of the book was ‘Book on Games of Chance’ (Liber de Ludo
Aleae), published in 1663. Notable contributions were also made by mathematicians
J. Bernoulli (1654–1705), P . Laplace (1749–1827), A.A. Markov (1856–1922) and A.N.
Kolmogorov (born 1903).
Activity 1 : (i) Take any coin, toss it ten times and note down the number of times a
head and a tail come up. Record your observations in the form of the following table
Table 15.1
Number of times Number of times Number of times
the coin is tossed head comes up tail comes up
10 — —
Write down the values of the following fractions:
Number of times a head comes up
Total number of times the coin is tossed
and
Number of times a tail comes up
Total number of times the coin is tossed
Blaise Pascal
(1623–1662)
Fig. 15.1
Pierre de Fermat
(1601–1665)
Fig. 15.2
2020-21
PROBABILLITY 273
(ii) Toss the coin twenty times and in the same way record your observations as
above. Again find the values of the fractions given above for this collection of
observations.
(iii) Repeat the same experiment by increasing the number of tosses and record
the number of heads and tails. Then find the values of the corresponding
fractions.
You will find that as the number of tosses gets larger, the values of the fractions
come closer to 0.5. To record what happens in more and more tosses, the following
group activity can also be performed:
Acitivity 2 : Divide the class into groups of 2 or 3 students. Let a student in each
group toss a coin 15 times. Another student in each group should record the observations
regarding heads and tails. [Note that coins of the same denomination should be used in
all the groups. It will be treated as if only one coin has been tossed by all the groups.]
Now, on the blackboard, make a table like Table 15.2. First, Group 1 can write
down its observations and calculate the resulting fractions. Then Group 2 can write
down its observations, but will calculate the fractions for the combined data of Groups
1 and 2, and so on. (We may call these fractions as cumulative fractions.) We have
noted the first three rows based on the observations given by one class of students.
Table 15.2
Group
Number Number
Cumulative number of heads Cumulative number of tails
of of T otal number of times T otal number of times
heads tails the coin is tossed the coin is tossed
(1) (2) (3) (4) (5)
1 3 12
3
15
12
15
2 7 8
7 3 10
15 15 30
+
=
+
8 12 20
15 15 30
+
=
+
3 7 8
7 10 17
15 30 45
+
=
+
8 20 28
15 30 45
+
=
+
4
   
What do you observe in the table? You will find that as the total number of tosses
of the coin increases, the values of the fractions in Columns (4) and (5) come nearer
and nearer to 0.5.
Activity 3 : (i) Throw a die* 20 times and note down the number of times the numbers
*A die is a well balanced cube with its six faces marked with numbers from 1 to 6, one number
on one face. Sometimes dots appear in place of numbers.
2020-21
Page 4

PROBABILLITY 271
CHAPTER 15
PROBABILITY
It is remarkable that a science, which began with the consideration of
games of chance, should be elevated to the rank of the most important
subject of human knowledge. —Pierre Simon Laplace
15.1 Introduction
In everyday life, we come across statements such as
(1) It will probably rain today.
(2) I doubt that he will pass the test.
(3) Most probably, Kavita will stand first in the annual examination.
(4) Chances are high that the prices of diesel will go up.
(5) There is a 50-50 chance of India winning a toss in today’s match.
The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the
statements above involve an element of uncertainty. For example, in (1), ‘probably
rain’ will mean it may rain or may not rain today. We are predicting rain today based
on our past experience when it rained under similar conditions. Similar predictions are
also made in other cases listed in (2) to (5).
The uncertainty of ‘probably’ etc can be measured numerically by means of
‘probability’ in many cases.
Though probability started with gambling, it has been used extensively in the fields
of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather
Forecasting, etc.
2020-21
272 MATHEMA TICS
15.2 Probability – an Experimental Approach
In earlier classes, you have had a glimpse of probability when you performed
experiments like tossing of coins, throwing of dice, etc., and observed their outcomes.
You will now learn to measure the chance of occurrence of a particular outcome in an
experiment.
The concept of probability developed in a very
strange manner. In 1654, a gambler Chevalier
de Mere, approached the well-known 17th
century French philosopher and mathematician
Blaise Pascal regarding certain dice problems.
Pascal became interested in these problems,
studied them and discussed them with another
French mathematician, Pierre de Fermat. Both
Pascal and Fermat solved the problems
independently. This work was the beginning
of Probability Theory.
The first book on the subject was written by the Italian mathematician, J.Cardan
(1501–1576). The title of the book was ‘Book on Games of Chance’ (Liber de Ludo
Aleae), published in 1663. Notable contributions were also made by mathematicians
J. Bernoulli (1654–1705), P . Laplace (1749–1827), A.A. Markov (1856–1922) and A.N.
Kolmogorov (born 1903).
Activity 1 : (i) Take any coin, toss it ten times and note down the number of times a
head and a tail come up. Record your observations in the form of the following table
Table 15.1
Number of times Number of times Number of times
the coin is tossed head comes up tail comes up
10 — —
Write down the values of the following fractions:
Number of times a head comes up
Total number of times the coin is tossed
and
Number of times a tail comes up
Total number of times the coin is tossed
Blaise Pascal
(1623–1662)
Fig. 15.1
Pierre de Fermat
(1601–1665)
Fig. 15.2
2020-21
PROBABILLITY 273
(ii) Toss the coin twenty times and in the same way record your observations as
above. Again find the values of the fractions given above for this collection of
observations.
(iii) Repeat the same experiment by increasing the number of tosses and record
the number of heads and tails. Then find the values of the corresponding
fractions.
You will find that as the number of tosses gets larger, the values of the fractions
come closer to 0.5. To record what happens in more and more tosses, the following
group activity can also be performed:
Acitivity 2 : Divide the class into groups of 2 or 3 students. Let a student in each
group toss a coin 15 times. Another student in each group should record the observations
regarding heads and tails. [Note that coins of the same denomination should be used in
all the groups. It will be treated as if only one coin has been tossed by all the groups.]
Now, on the blackboard, make a table like Table 15.2. First, Group 1 can write
down its observations and calculate the resulting fractions. Then Group 2 can write
down its observations, but will calculate the fractions for the combined data of Groups
1 and 2, and so on. (We may call these fractions as cumulative fractions.) We have
noted the first three rows based on the observations given by one class of students.
Table 15.2
Group
Number Number
Cumulative number of heads Cumulative number of tails
of of T otal number of times T otal number of times
heads tails the coin is tossed the coin is tossed
(1) (2) (3) (4) (5)
1 3 12
3
15
12
15
2 7 8
7 3 10
15 15 30
+
=
+
8 12 20
15 15 30
+
=
+
3 7 8
7 10 17
15 30 45
+
=
+
8 20 28
15 30 45
+
=
+
4
   
What do you observe in the table? You will find that as the total number of tosses
of the coin increases, the values of the fractions in Columns (4) and (5) come nearer
and nearer to 0.5.
Activity 3 : (i) Throw a die* 20 times and note down the number of times the numbers
*A die is a well balanced cube with its six faces marked with numbers from 1 to 6, one number
on one face. Sometimes dots appear in place of numbers.
2020-21
274 MATHEMA TICS
1, 2, 3, 4, 5, 6 come up. Record your observations in the form of a table, as in Table 15.3:
Table 15.3
Number of times a die is thrown Number of times these scores turn up
1 2 3 4 5 6
20
Find the values of the following fractions:
Number of times 1 turned up
Total number of times the die is thrown
Number of times 2 turned up
Total number of times the die is thrown


Number of times 6 turned up
Total number of times the die is thrown
(ii) Now throw the die 40 times, record the observations and calculate the fractions
as done in (i).
As the number of throws of the die increases, you will find that the value of each
fraction calculated in (i) and (ii) comes closer and closer to
1
6
.
To see this, you could perform a group activity, as done in Activity 2. Divide the
students in your class, into small groups. One student in each group should throw a die
ten times. Observations should be noted and cumulative fractions should be calculated.
The values of the fractions for the number 1 can be recorded in Table 15.4. This
table can be extended to write down fractions for the other numbers also or other
tables of the same kind can be created for the other numbers.
2020-21
Page 5

PROBABILLITY 271
CHAPTER 15
PROBABILITY
It is remarkable that a science, which began with the consideration of
games of chance, should be elevated to the rank of the most important
subject of human knowledge. —Pierre Simon Laplace
15.1 Introduction
In everyday life, we come across statements such as
(1) It will probably rain today.
(2) I doubt that he will pass the test.
(3) Most probably, Kavita will stand first in the annual examination.
(4) Chances are high that the prices of diesel will go up.
(5) There is a 50-50 chance of India winning a toss in today’s match.
The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the
statements above involve an element of uncertainty. For example, in (1), ‘probably
rain’ will mean it may rain or may not rain today. We are predicting rain today based
on our past experience when it rained under similar conditions. Similar predictions are
also made in other cases listed in (2) to (5).
The uncertainty of ‘probably’ etc can be measured numerically by means of
‘probability’ in many cases.
Though probability started with gambling, it has been used extensively in the fields
of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather
Forecasting, etc.
2020-21
272 MATHEMA TICS
15.2 Probability – an Experimental Approach
In earlier classes, you have had a glimpse of probability when you performed
experiments like tossing of coins, throwing of dice, etc., and observed their outcomes.
You will now learn to measure the chance of occurrence of a particular outcome in an
experiment.
The concept of probability developed in a very
strange manner. In 1654, a gambler Chevalier
de Mere, approached the well-known 17th
century French philosopher and mathematician
Blaise Pascal regarding certain dice problems.
Pascal became interested in these problems,
studied them and discussed them with another
French mathematician, Pierre de Fermat. Both
Pascal and Fermat solved the problems
independently. This work was the beginning
of Probability Theory.
The first book on the subject was written by the Italian mathematician, J.Cardan
(1501–1576). The title of the book was ‘Book on Games of Chance’ (Liber de Ludo
Aleae), published in 1663. Notable contributions were also made by mathematicians
J. Bernoulli (1654–1705), P . Laplace (1749–1827), A.A. Markov (1856–1922) and A.N.
Kolmogorov (born 1903).
Activity 1 : (i) Take any coin, toss it ten times and note down the number of times a
head and a tail come up. Record your observations in the form of the following table
Table 15.1
Number of times Number of times Number of times
the coin is tossed head comes up tail comes up
10 — —
Write down the values of the following fractions:
Number of times a head comes up
Total number of times the coin is tossed
and
Number of times a tail comes up
Total number of times the coin is tossed
Blaise Pascal
(1623–1662)
Fig. 15.1
Pierre de Fermat
(1601–1665)
Fig. 15.2
2020-21
PROBABILLITY 273
(ii) Toss the coin twenty times and in the same way record your observations as
above. Again find the values of the fractions given above for this collection of
observations.
(iii) Repeat the same experiment by increasing the number of tosses and record
the number of heads and tails. Then find the values of the corresponding
fractions.
You will find that as the number of tosses gets larger, the values of the fractions
come closer to 0.5. To record what happens in more and more tosses, the following
group activity can also be performed:
Acitivity 2 : Divide the class into groups of 2 or 3 students. Let a student in each
group toss a coin 15 times. Another student in each group should record the observations
regarding heads and tails. [Note that coins of the same denomination should be used in
all the groups. It will be treated as if only one coin has been tossed by all the groups.]
Now, on the blackboard, make a table like Table 15.2. First, Group 1 can write
down its observations and calculate the resulting fractions. Then Group 2 can write
down its observations, but will calculate the fractions for the combined data of Groups
1 and 2, and so on. (We may call these fractions as cumulative fractions.) We have
noted the first three rows based on the observations given by one class of students.
Table 15.2
Group
Number Number
Cumulative number of heads Cumulative number of tails
of of T otal number of times T otal number of times
heads tails the coin is tossed the coin is tossed
(1) (2) (3) (4) (5)
1 3 12
3
15
12
15
2 7 8
7 3 10
15 15 30
+
=
+
8 12 20
15 15 30
+
=
+
3 7 8
7 10 17
15 30 45
+
=
+
8 20 28
15 30 45
+
=
+
4
   
What do you observe in the table? You will find that as the total number of tosses
of the coin increases, the values of the fractions in Columns (4) and (5) come nearer
and nearer to 0.5.
Activity 3 : (i) Throw a die* 20 times and note down the number of times the numbers
*A die is a well balanced cube with its six faces marked with numbers from 1 to 6, one number
on one face. Sometimes dots appear in place of numbers.
2020-21
274 MATHEMA TICS
1, 2, 3, 4, 5, 6 come up. Record your observations in the form of a table, as in Table 15.3:
Table 15.3
Number of times a die is thrown Number of times these scores turn up
1 2 3 4 5 6
20
Find the values of the following fractions:
Number of times 1 turned up
Total number of times the die is thrown
Number of times 2 turned up
Total number of times the die is thrown


Number of times 6 turned up
Total number of times the die is thrown
(ii) Now throw the die 40 times, record the observations and calculate the fractions
as done in (i).
As the number of throws of the die increases, you will find that the value of each
fraction calculated in (i) and (ii) comes closer and closer to
1
6
.
To see this, you could perform a group activity, as done in Activity 2. Divide the
students in your class, into small groups. One student in each group should throw a die
ten times. Observations should be noted and cumulative fractions should be calculated.
The values of the fractions for the number 1 can be recorded in Table 15.4. This
table can be extended to write down fractions for the other numbers also or other
tables of the same kind can be created for the other numbers.
2020-21
PROBABILLITY 275
Table 15.4
Group T otal number of times a die Cumulative number of times 1 turned up
is thrown in a group T otal number of times the die is thrown
(1) (2) (3)
1 — —
2 — —
3 — —
4 — —
The dice used in all the groups should be almost the same in size and appearence.
Then all the throws will be treated as throws of the same die.
What do you observe in these tables?
You will find that as the total number of throws gets larger, the fractions in
Column (3) move closer and closer to
1
6
.
Activity 4 : (i) Toss two coins simultaneously ten times and record your
observations in the form of a table as given below:
Table 15.5
Number of times the Number of times Number of times Number of times
10 — — —
Write down the fractions:
A =
Number of times no head comes up
Total number of times two coins are tossed
B =
Number of times one head comes up
Total number of times two coins are tossed
C =
Number of times two heads come up
Total number of times two coins are tossed
Calculate the values of these fractions.
2020-21
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