Question Bank In Mathematics Class IX (Term II)

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1 Question Bank In Mathematics Class IX (Term II) PROBABILITY A. SUMMATIVE ASSESSMENT. PROBABILITY AN EXPERIMENTAL APPROACH. The science which measures the degree of uncertainty is called probability.. In the experimental approach to probability, we find the probability of the occurrence of an event by actually performing the experiment a number of times and record the happening of an event.. The observations of an experiment are called outcomes.. A trial is an action which results in one or several outcomes.. An event of an experiment is the collection of some outcomes of the experiment. 6. The emperical (or experimental) probability P(E) of an event E is given by : Number of trials in which E has happened P E Total number of trials. The probability of an event lies between 0 and (0 and inclusive). 8. A die is a well balanced cube with its six faces marked with numbers to 6, one number on one face. Sometimes dots appear in place of numbers. TEXTBOOK S EXERCISE. Q.. In a cricket match, a batswoman hits a boundary 6 times out of 0 balls she plays. Find the probability that she did not hit a boundary. [0 (T-II)] Sol. Total number of balls played by the batswoman 0, Boundaries hit 6 No. of balls in which she did not hit any boundary 0 6 No. of balls in which she did not hit any boundary P (she did not hit a boundary) Total number of balls played 0 Q.. 00 families with children were selected randomly, and the following data were recorded. : Number of girls in a family 0 [0 (T-II)] Number of families 8 Compute the probability of a family, chosen at random, having (i) girls (ii) girl (iii) No girl Also check whether the sum of these probabilities is. Sol. (i) P (a family having girls) No. of families having girls Total no. of families 00 60

2 (ii) P (a family having girl) No. of families having girl Total no. of families (iii) P (a family having no girl) No. of families having no girl Total no. of families Sum of the probabilities in all three cases Q.. In a particular section of Class IX, 0 students were asked about the months of their birth and the following graph was prepared for the data so obtained. Find the probability that a student of the class was born in August. Sol. Total number of students considered 0 No. of students born in August 6 P (a student was born in August) No. of students born in August Total no. of students considered Q.. Three coins are tossed simultaneously 00 times with the following frequencies of different outcomes : Outcome heads heads head No head Frequency 8 If the three coins are simultaneously tossed again, compute the probability of heads coming up. [0 (T-II)]

3 Sol. Total number of tosses 00 No. of times heads occur No. of times heads occur P ( heads coming up) Total no. of tosses 00 Q.. An organisation selected 00 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The information gathered is listed in the table below : Monthly income Vehicles per family in (Rs) 0 Above Less than or more 8 88 Suppose a family is chosen. Find the probability that the family chosen is (i) earning Rs per month and owning exactly vehicles. (ii) earning Rs 6000 or more per month and owning exactly vehicle. (iii) earning less than Rs 000 per month and does not own any vehicle. (iv) earning Rs per month and owning more than vehicles. (v) owning not more than vehicle. Sol. Total no. of families considered 00 (i) P(a family earning Rs per month and owning exactly vehicles) No. of families earning Rs per month and owning vehicles Total no. of families (ii) P (a family earning Rs 6000 or more per month and owning exactly vehicle) No. of families earning Rs 6000 or more per month and owning vehicle Total no. of families (iii) P(a family earning less than Rs 000 per month and does not own any vehicle) No. of families earning less than Rs 000 per month and does not own any vehicle Total no. of families (iv) P(a family earning Rs per month and owing more than vehicles) No. of families earning Rs per month and owning more than vehicles Total no. of families 00 6

4 (v) P (a family owning 0 vehicle or vehicle) P (a family not owning more than vehicle) Q.6. Following table shows the performance of two sections of students in Mathematics test of 00 marks. Marks Number of students above 8 Total 0 (i) Find the probability that a student obtained less than 0% in the mathematics test. (ii) Find the probability that a student obtained marks 60 or above. Sol. (i) Total no. of students 0 No. of students who obtained less than 0% P (a student obtained less than 0%) Total no. of students 0 (ii) P (a student obtained 60 marks or above) No. of students who obtained 60 marks or more + 8 Total number of students 0 0 Q.. To know the opinion of the students about the subject statistics, a survey of 00 students was conducted. The data is recorded in the following table. [00] Opinion Number of students like dislike 6 Sol. Find the probability that a student chosen at random (i) likes statistics, (ii) does not like it. (i) P (a student likes statistics) No. of students who like statistics Total no. of students 00 0

5 Q.8. No. of students who do not like statistics (ii) P (a student does not like statistics) Total no. of students The distance (in km) of 0 engineers from their residence to their place of work were found as follows : What is the empirical probability that an engineer lives : (i) less than km from her place of work? (ii) more than or equal to km from her place of work? (iii) within km from her place of work? Sol. Total no. of engineers 0 Let us arrange the data in ascending order as follows : Q..,,,,,,, 6, 6,,,,, 8,,, 0, 0,,,,,,,,,,,,, 6,,, 8, 8,, 0,,,. (i) P (an engineer lives less than km from her place of work) No. of engineers who live less than km from their place of work Total no. of engineers 0 (ii) P (an engineer lives more than or equal to km from her work place) No. of engineers who live more than or equal to km from their work place Total no. of engineers 0 (iii) P (an engineer lives within km from her place of work) No. of engineers who live within km from their place of work 0 Total no. of engineers 0 0 Questions and 0 are activities, so students should perform these activities on their own. Eleven bags of wheat flour, each marked kg, actually contained the following weights of flour (in kg) : [0 (T-II)] Find the probability that any of these bags chosen at random contains moer than kg of flour. Sol. Total no. of bags examined P (a bag weighing more than kg) No. of bags which weigh more than kg Total no. of bags Q.. A study was conducted to find out the concentration of sulphur dioxide in the air parts per million (ppm) of a certain city. The data obtained for 0 days is as follows :

6 Using this table, find the probability of the concentration of sulphur dioxide in the interval on any of these days. Sol. Total no. of days 0 P(concentration of sulphur dioxide in the interval in a day) No. of days on which the concentration was in the interval Total no. of days Q.. The blood groups of 0 students of Class VIII are recorded as follows : A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O, A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O 0 Use this table to determine the probability that a student of this class, selected at random, has blood group AB. [00] Sol. Total no. of students 0 No. of students which have the blood group AB P (a student has blood group AB) Total no. of students 0 0 Q.. OTHER IMPORTANT QUESTIONS In an experiment a coin is tossed 00 times. If the head turns up 80 times, the experimental probability of getting a head is : [00, 0 (T-II)] no. of heads 80 P a head total no. of tosses 00 Sol. Q.. In a One-Day cricket match, a batsman played 0 balls. The runs scored were as follows : Runs scored No. of balls 0 6 The probability that the batsman scored no run is : 6

7 0 Sol. P (the batsman scored no run) Q.. 0 number of times he score no run total number of balls played 0 A coin is tossed 00 times and head appeared 00 times. The sum of the probability of getting a head and the probability of getting a tail is : [0 (T-II)] Sol P a head, P a tail P a head + P a tail 00 Q.. packets of salt, each marked kg, actually contained the following weights (in kg) of salt :.80,.000,.0,.80,.0,.00,.0,.00,.060,.80,.00,.0 Out of these packets one packet is chosen at random. The probability that the chosen packet contains less than kg of salt is : Sol. No. of packets which contains less than kg of salt is 6. P (a packet contains less than kg of salt) Q.. Number of packets which contains less than kg of salt 6 Total number of packets A die is tossed 00 times simultaneously, and the frequencies of various outcomes are given below : Outcomes 6 Frequencies The probability of getting is : 0 0 Sol. 0 number of times occurs 6 P total number of tosses 00 0 Q.6. A die is thrown times and the results were as follows : Outcomes 6 Frequencies [0 (T-II)] The probability of getting a prime number is : 8 [0 (T-II)]

8 no. of times a prime number occurs Sol. P a prime number total no. of tosses Q.. A machine generated these 0 codes : {0A, AAA, ABC, B, B, BB, AC,,, }. [00] A code is drawn at random to allot an employee. The probability that the code have at least two digits is : none of these Sol. The codes generated by the machine having at least digits are : 0A, B B,,,, i.e., there are 6 such codes. no. of codes having at least digits 6 P the code has at least digits total no. of codes 0 Q.8. A die is tossed 0 times and the results were as follows : Outcomes 6 Frequencies The probability of getting either or is : Sol. no. of times either or occurs P getting either or total no. of tosses 0 Q.. Three coins are thrown simultaneously 60 times, with the following frequencies : Heads : 0 times, Heads : times, Head : 8 times, No Head : times P (getting heads) + P (getting no head) is equal to : [Imp.] Sol. P getting heads 60 6 P getting no head P getting heads + P getting no head Q.0. Two coins are tossed simultaneously. List all possible outcomes. Sol. All possible outcomes are : HH, HT, TH, TT. Q.. Two coins are tossed 00 times and the outcomes are recorded as below: 0 [Imp.] No. of heads 0 Frequency Based on this information, find the probability for at most one head. [0 (T-II)] 8

9 no. of times either head or 0 head occurs Sol. P at most one head total no. of trials Q.. A student opens his book and notes down the units digit on the right hand page of his book. He repeats the process for 0 times. The outcomes are recorded as below : Digit Frequency Based on the above information, find the probability of occurrence of : (i) as units digit. (ii) or as the units digit. (iii) at least as the units digit. no. of times has occurred as units digit 0 P as units digit total no. of trials 0 Sol. (i) no. of times or has occurred as units digits (ii) P or as units digit total no. of trials no. of times the units digit has occurred at least (iii) P at least as the units digit total no. of trials Q.. Over the past 00 working days, the number of defective parts produced by a machine is given below: No. of defective parts Days Determine the probability that tomorrow s output will have : (i) no defective part (ii) not more than defective parts (iii) more than defective parts? Sol. (i) P (the output will have no defective part) no. of days on which the output has no defective part 0 total no. of days 00 (ii) P the output has not more than defective parts no. of days on which the output has not more than defective parts total no. of days [HOTS]

10 (iii) P the output has more than defective parts no. of days on which the output has more than defective parts 0 0 total no. of days 00 PRACTICE EXERCISE.A Choose the correct option (Q. ) : Mark Questions. In a cricket match, a batsman hits a boundary times out of balls he plays. The probability that he hits a boundary is : [0 (T-II)]. A coin is tossed 00 times and head appears 6 times. The probability of getting a tail is : 8 0. Three coins are tossed simultaneously 00 times with following outcomes : Outcomes heads heads head No head Frequency 8 The probability of getting two heads is : 8. Following are the ages (in years) of 60 patients, getting medical treatment in a hospital : Age in years No. of patients One of the patients is selected at random. The probability that his age is 0 years or more but less than 0 years is : 6 0. Marks obtained by 80 students of a class in a test of maximum marks 00 are given below : Marks and above No. of students 6 6 A student of the class is selected at random. The probability that he gets less than % marks is : [0 (T-II)]

11 6. 80 bulbs are selected at random from a lot and their life time (in hrs) is recorded in the form of a frequency table given below : Life time (in hours) Frequency 0 0 One bulb is selected at random from the lot. The probability that its life is 0 hours, is : 0 [Imp.] In a medical examination of students of a class, the following blood groups are recorded : Blood group A AB B O Number of students 0 A student is selected at random from the class. The probability that he/she has blood group B, is : Games played by 00 students of a school are recorded as below : Games Cricket Football Tennis Badminton No. of students 6 A student is chosen at random. The probability that he plays neither cricket nor football is : 0 0. Marks obtained by 0 students in a class test of 00 marks are given below : Marks No. of students 8 6 The probability that a student obtains less than 0% marks is : 0. A coin is tossed 00 times with the following frequencies : Head : 6, Tail : 6 6 The ratio of probabilities for each event is : 6 : : 6 : : 8 [Imp.]

12 . A coin is tossed 0 times and head appears 8 times. If we toss the coin randomly, the probability of getting neither a head nor a tail is : [0 (T-II)] 6 0. In a cricket match, a batsman hits a boundary 6 times out of 6 balls he plays. The probability that he did not hit a boundary is: [0 (T-II)] The salaries of 0 employees in an office are given below : Salary (in Rs.) or above No. of employees 6 8 An employee is selected at random. The probability that his salary is Rs 6000 or more but less than Rs 000 is : 0 0. The following table shows the blood groups of 60 students of a class: Blood Groups A B O AB No. of Students 6 One student of the class is chosen at random. What is the probability that the chosen student has either blood group A or B? 8 [V. Imp.] 0 0. Two coins are tossed 000 times and the outcomes are recorded as below : [0(T-II)] Number of heads 0 Frequency Based on this information, the probability for at most one head is : Marks Questions 6. Can the experimental probability of an event be greater than? If not, why? [Imp.]. Can the experimental probability of an event be zero? If not, why? 8. As the number of tosses increases, the ratio of the number of heads to the total number of tosses will be. Is it correct? if not, write the correct one. [Imp.]

13 . In a sample study of 0 people, it was found that 0 people were government employees. If a person is selected at random, find the probability that the person is not a government employee? 0. A dice is tossed 00 times and the outcomes are recorded as below : [0 (T-II)] Outcome Even number less than 6 Odd no. greater than 6 Frequency 0 0 Find the probability of getting the number 6 even number less than 6. Following table shows the marks obtained by 0 students in a class test : [0 (T-II)] Marks obtained Number of Students 6 Find the probability that a student secures 60 marks less than 60 marks. A survey of 00 students is done regarding the likes and dislikes about the subject mathematics. The data are given below : [0 (T-II)] Views Number of students Likes 80 Dislikes 0 Find the probability that student chosen randomly like mathematics does not like mathematics Marks Questions. Two dice are thrown simultaneously 00 times. Each time the sum of numbers appearing on their top is noted and recorded as below: Sum Frequency If the dice are thrown once more, what is the probability of getting a sum : (i)? (ii) more than 0? (iii) less than or equal to? [Imp.]. The ages (in years) of workers of a factory are as follows : Age (in years) and above No. of workers If a worker is selected at random, find the probability that the person is : (i) 0 years or more (ii) below 0 years (iii) having age from 0- years

14 . The percentage of marks obtained by a student in monthly unit test are given below : Unit test I II III IV V % of marks obtained Find the probability that the student gets : (i) more than 0% marks (ii) less than 0% marks (iii) more than 0% marks. 6. Bulbs are packed in cartons, each containing 0 bulbs. Seven hundered cartons were examined for defective bulbs and the results are given in the following table : No. of defective bulbs 0 6 more than 6 Frequency One carton was selected at random, what is the probability that it has : (i) no defective bulbs? (ii) defective bulbs from to 6? (iii) defective bulbs less than?. In a One Day International, a batsman played 0 balls. The runs scored are as follows : Runs scored 0 6 No. of balls Find the probability that the batsman will score : (i) 6 runs? (ii) a four or a six run? (iii) 0 or or 6 runs? [V. Imp.] 8. The records of a weather station shows that out of the 0 consecutive days, its weather forecast were correct times. [0 (T-II)] (i) What is the probability that on a given day it was correct? (ii) What is the probability that it was not correct on a given day?. Marks obtained by students of class IX in Mathematics are given in the table : [0 (T-II)] Marks Number of students (i) Find the probability that a student gets less than 0% in a test. (ii) Find the probability that a student gets more than 80%. 0. A bag contains tickets which are numbered from to 00. Find the probability that a ticket number picked up at random [0 (T-II)] (i) is a multiple of (ii) is not a multiple of. Two coins are tossed simultaneously. Find the probability of getting [0 (T-II)] (i) atleast one head (ii) both heads.. If a person is selected at random, find the probability that the person is : [0 (T-II)] (i) under 0 years of age (ii) having age from 0 years to years

15 B. FORMATIVE ASSESSMENT Group Activity Objective : To find experimental probability when a die is tossed different number of times by different persons. Materials Required : Dice, geometry box, etc. Procedure :. Divide the class into groups of or students. Let a student in each group throw a die 0 times. Another student in each group should note down the number of times the numbers,,,,, 6 come up. Following table can be used to record the observations. No. of times a die is thrown Table No. of times these numbers turn up Now throw the die 60 times and record the number of times the numbers,,,,, 6 come up. No. of times a die is thrown Table. Again throw the die 0 times and record the observations. No. of times a die is thrown Table No. of times these numbers turn up No. of times these numbers turn up Repeat the above steps by throwing the die 0, 0,... times and record your observations. Observations :. For table :

16 . For table :. Similarly, calculate the fractions as done above for other tables.. You will observe that as the number of throws of the die increases, the value of each fraction Conclusions : calculated for Table, Table,... comes closer and closer to.. The thoretical probability of getting, or or or or or 6 is. Clearly, this is not true in case of experimental probability.. As the number of throws of the die increases, then the experimental probability of getting or or or or or 6 comes closer and closer to. Note : The above data is for your comprehension. The students must do the Activity and collect the data. Group Activity This is an experiment to find out if you can see into the future. You need to work in pairs and you need one coin. One of you is the tosser and recorder and the other is the guesser. 6

17 . The guesser predicts whether the coin will land head up or tail up. The tosser then tosses the coin. Practice Exercise.A ANSWERS No. yes 8. No (i). (i). (i) 0. (i) When this experiment is repeated 00 times, about how many times do you expect the guesser to predict the actual outcome.. Now perform the experiment described at least 00 times and record each result as right or wrong as appropriate. Use an observation sheet in the form of a tally chart.. Compare what you expected to happen with what did happen, using appropriate diagrams as illustrations. Comment on the likelihood of the guesser being able to predict which way the coin will land.. State how could you make your results more reliable.. Suggest other experiments that you could perform to test whether someone can see into the future (ii) (ii) (ii) 00 0 (ii) 0 (iii) (iii) 0 0 (iii) 0 6. (i) 8. (i). (i) 0. (i) 0 (ii) (ii) (ii) (ii). (iii) (iii) (i) (ii). (i) 0 (ii) 00

6. For any event E, which is associated to an experiment, we have 0 P( 7. If E 1

6. For any event E, which is associated to an experiment, we have 0 P( 7. If E 1 CHAPTER PROBABILITY Points to Remember :. An activity which gives a result is called an experiment.. An experiment which can be repeated a number of times under the same set of conditions, and the outcomes