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

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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 are not predictable is called a Random Experiment.. Performing an experiment is called a trial.. Any outcome of an experiment is known as an event.. In n trials of a random experiments, if an event E happens m times, then the probability of happening of E is given by, Number of outcomes favour to E m P( E) Total number of possible outcomes n. For any event E, which is associated to an experiment, we have 0 P( E). 7. If E, E, E,..., E n are n elemantary events associated to a random experiment, then P(E ) + P(E ) + P(E ) P(E n ) = Example. Solution. Example. Solution. ILLUSTRATIVE EXAMPLES In a cricket match, a batswoman hits a boundary times out of 0 balls she plays. Find the probability that she did not hit the boundary. NCERT Since, she hits a boundary times, it means she has missed 0 = times. Required probability = P (she did not hit the boundary) Number of time she did not hit the boundary Total number of balls she plays 0.8 Ans. 0 Three coins are tossed simultaneously 00 times with the following frequencies of different outcomes: Outcome Frequency heads heads 7 heads 77 no heads 88 PROBABILITY MATHEMATICS IX 8 If the three coins are simultaneously tossed again, compute the probability of heads coming up. NCERT Out of total of 00 outcomes, heads come for 7 times. P ( heads coming up) 7 00 Ans.

2 Example. 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 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. 000 or more per month and owning exactly vehicle. (iii) earning less than Rs per month and does not own any vehicle. (iv) earning Rs per month and owning more than vehicles. (v) owning not more than vehicle. NCERT Solution. The total number of families = 00 (i) Number of families earning Rs per month and owning exactly vehicles = Required probability 00 (ii) Number of families earning Rs. 000 or more per month and owning exactly vehicle = 7 Required probability (iii) Number of families earning less than Rs per month and does not own any vehicle = 0 Required probability (iv) Number of families earning Rs per month are owning more than vehicles =. Required probability 00 (v) Number of families owning not more than vehicle = families having no vehicle + families having vehicle = ( ) + ( ) = Required probability Example. A die is throw 00 times and the outcomes are noted in as given below : Outcome Frequency If a die is thrown at random, find the probability of getting. (i) (ii) (iii) (iv) (v) (vi) MATHEMATICS IX PROBABILITY 8

3 Solution. Total number of trials = 00 Example. (i) P (getting ) (ii) P (getting ) (iii) P (getting ) (iv) P (getting ) (v) P (getting ) (iv) P (getting ) The table given below shows the marks obtained by 0 students of a class in a test with maximum marks 00. Marks No. of students PROBABILITY MATHEMATICS IX Above 7 A student of the class is selected at random. Find the probability that student gets : (i) less than 0% marks (ii) 0 or more marks (iii) marks between and 7. (iv) distinction Solution. Total number of students = 0 Example. (i) P (getting less than 0% marks) (ii) P (getting 0 or more marks) (iii) P (getting marks between and 7) (iv) P (getting distinction) 0. 0 A tyre manufacturing company kept a record of the distance covered before a type to be replaced. Following table shows the result of 000 cases. Distance (in km) No. of tyres less than If you buy a tyre of this company, what is the probability that : (i) it will need to be replaced before it has covered 00 km. (ii) it will last more than 00 km? more than 00 0

4 (iii) it will need to be replaced? (iv) it will not need to be replaced at all? (v) it will need to be replaced after it has covered somewhere between 00 km and 00 km? Solution. We have, total number of trials = 000 (i) The number of tyres that needs to be replaced before it has covered 00 km = 00. Required probability (ii) The number of tyres that last more than 00 km = = Required probability (iii) Since, all the tyres we have considered to be replaced. Required probability (iv) The number of tyres that do not need to be replaced at all = 0 Required probability (v) The number of tyres which require replacement after covering somewhere between 00 km and 00 km is + 7 = 70. Required probability Example 7. A bag contains red balls, black balls and white balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is: (i) White (ii) not black (iii) Red or white Solution. Total number of balls = + + =. (i) P (white ball) 7 (ii) P (not black) = P (black) 7 (iii) P (red or white). Example 8. Cards marked with the numbers,,..., 0 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number on the card is: (i) an even number (ii) a number less than (iii) a perfect square number (iv) a prime number less than 0. Solution. Total numbers = 00 (i) Total number of even numbers = 0 P (an even number) 0 00 (ii) P (a number less than ) P(,,..., ) 00 MATHEMATICS IX PROBABILITY

5 Example. Solution. (iii) P (a perfect square number) = P (,,,..., 00) 00 (iv) P (a prime number less than 0) = P (,,, 7,,, 7, ) 8 00 Find the probability that a leap year, selected at random will have sundays. A leap year has days. weeks = 7 = days Days left = = weeks means there will be sundays. From remaining days, the possibilities are as follows : (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday) and (Saturday, Sunday). So, from 7 possibilities, we have cases in which we have rd sunday. Required probability Ans. 7 Example 0. A bag contains 0 white balls and x black balls. If the probability of drawing a black ball is double that of a white ball, find x. Solution. Total number of balls = 0 + x. P (black ball) x 0 x 0 and, P (white ball) 0 x according to given question, P (black ball) = P (white ball) x 0 0 x 0 x x = 0 Ans. ( 0 + x 0) PRACTICE EXERCISE. A number is chosen from to 0. Find the probability that the number chosen is : (i) a prime number (ii) a composite number (iii) a square number (iv) an odd number (v) an even number (vi) number between 7 and. A bag contains red and blue balls. Find the probability that a ball drawn from a bag at random is (i) Red ball (ii) blue ball. In a sample of 00 items, 0 are found to be defective. Find the probability that the item selected at random is (i) defective (ii) non-defective. In a school of 800 students, there are 87 girls. Find the probability that a student chosen at random is (i) a boy (ii) a girl. In a cricket match, a batsman hit a boundary times out balls he plays. Find the probability that he did not hit a boundary. PROBABILITY MATHEMATICS IX

6 . A coin is tossed 700 times and we get head : 8 times; tail : times. When a coin is tossed at random, what is the probability of getting : (i) a head? (ii) a tail? 7. Two coins are tossed 00 times and we get two heads : 8 times, one head : times ; no head : 70 times. When two coins are tossed at random, what is the probability of getting : (i) heads? (ii) head? (iii) no head? 8. Three coins are tossed 0 times and we get: heads : times; heads : times; head : 70 times; 0 head : 78 times. When three coins are tossed at random, what is the probability of getting : (i) heads? (ii) heads? (iii) atleast heads? (iv) atmost heads?. A die is thrown 00 times and the outcomes are noted as given below : Outcomes Frequencies 8 7 When a die is thrown at random, what is the probability of getting a: (i) (ii) (iii) number less than (iv) number which is prime 0. In a survey of 0 ladies, it was found that like coffee, while rest of them dislike it. Find the probability that a lady chosen at random: (i) likes coffee (ii) dislikes coffee.. On one page of a telephone directory, there are 00 phone numbers. The frequency distribution of their units digit is given below : Units digit Frequency 0 0 MATHEMATICS IX PROBABILITY 7 7 One of the numbers is chosen at random from the page. What is the probability that the units digit of the chosen number is : (i) (ii) 8 (iii) an even number (iv) an odd number. The blood groups of 0 students of class IX are recorded as follows : A, B, O, AB, O, A, O, O, B, A, O, A, B, O, O, A, O, B, A, B, O, A, B, AB, O, O, A, A, O, AB A student is selected at random from the class for blood donation. Find the probability that the blood group of the student chosen is: (i) A (ii) B (iii) AB (iv) O. Following are the ages (in years) of 0 patients, getting medical treatment in a hospital. Age (in years) No. of Patients One of the patients is selected at random. Find the probability that his age is : (i) 0 years or more but less than 0 years. (ii) 0 years or less than it. (iii) less than 0 years. (iv) 0 years or more. 0

7 . Given below is the frequency distribution of wages (in Rs.) of 0 workers in a certain factory: Wages (in Rs) No. of workers A workers is selected at random. Find the probability that his wages are: (i) less than Rs. 0. (ii) atleast Rs. 80. (iii) more than or equal to Rs. 0 but less than Rs. 00. (iv) more than Rs The following table gives the life time of 00 neon lamps: Life time (in hours) No. of Lamps A lamp is selected at random. Find the probability that the life time of the selected lamp is: (i) less than 00 hours (ii) atleast 800 times (iii) atmost 00 hours (iv) between 00 hours to 00 hours. PRACTICE TEST MM : Time : hour General Instructions : Each Questions carry marks.. Two coins are tossed simultaneously 0 times with the following frequencies of different outcomes: Two heads : 0 times; One head : times; no head : 0 times. Find the probability of getting : (i) heads (ii) atleast one head. Following table shows the marks scored by 80 students in a mathematics test of 00 marks. Marks No. of students No. Find the probability that a student obtained: (i) less than 0% marks. (ii) 0 or more marks.. 00 families with children were selected at random and the following data were recorded; of girls in a family No. of families 0 0 PROBABILITY MATHEMATICS IX

8 If a family is chosen at random, compute the probability that it has : (i) no girl (ii) girls (iii) at most one girl (iv) at least one girl. The table given below shows the ages of 0 teachers in a school. Age (in years) No. of teachers A teacher from this school is chosen at random. What is the probability that the selected teacher is : (i) 0 or more than 0 years old? (ii) age less than 0 years?. An insurance company selected 000 drivers at random, in a particular city to find out a relationship between age and accidents. The data obtained are given in the following table : Age of drivers Accidents in one year (in years) 0 over above 0 0 Find the probabilities of the following events for a driven chosen at random from the city: (i) being 8- years of age and exactly accidents in one year. (ii) being 0-0 years of age and having one or more accidents in a year. (iii) having no accidents in one year.. (i). (i) 7. (i) 7. (i) 0 8. (i). (i) (i) 70 (ii) 0 (ii) (ii) 7 (ii) 0 8 (ii) 7 (ii) 0 ANSWERS OF PRACTICE EXERCISE (iii). (i). 7. (i) 00 (iii) 0 (iii) 00 0 (iv) 0 (ii) (ii) 00 0 (iv) (iv) 00 (iii) 0 (ii) 70 MATHEMATICS IX PROBABILITY

9 . (i) 8. (i) 0. (i). (i) 0. (i). (i) 0. (i) (i) 80. (i). (i) 00 (ii) 0 (ii) 7 (ii) 70 (ii) 0 (ii) 0 (ii) (ii) 0 (ii) 0 (ii) (ii) 80 0 (iii) 00 (iii) 0 (iv) 00 (iv) 7 (iii) 0 (iv) 70 (iii) (iii) 00 (iv) 0 8 (iv) ANSWERS OF PRACTICE TEST 77 (iii) 0 (iii) 00 (iv) 80 PROBABILITY MATHEMATICS IX

Question Bank In Mathematics Class IX (Term II)

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