Page 1 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 5050 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. 202021 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 5050 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. 202021 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 wellknown 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 202021 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 5050 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. 202021 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 wellknown 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 202021 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. 202021 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 5050 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. 202021 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 wellknown 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 202021 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. 202021 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. 202021 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 5050 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. 202021 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 wellknown 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 202021 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. 202021 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. 202021 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 two coins are tossed no head comes up one head comes up two heads come up 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. 202021Read More
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