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15. Probability

Class 9 Mathematics Chapter 15. Probability

Chapter 15. Probability

Important Note :

1. An event for an experiment is the collection of some outcomes of the experiment.

2. A trial is an action which results in one or several outcomes .
3. The empirical (or experimental) probability  of an event  is given by
 

4. (i) For one coin :

  The sample space  H , T

(ii) For two coins :

 The sample space  HH , HT , TH , TT  

(iii) For three coins :

The sample space  HHH , HTT , HHT , HTH , THT , THH , TTH , TTT  

(iv) For one dice :

The sample space  1 , 2 , 3 , 4 , 5 , 6  

(v) For two dice : 

The sample space (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)  

5. The playing card consists of 52 cards which are divided into 4 suits of 13 cards each :
(i) Spades (black colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2

(ii­) Clubs  (black colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2

(iii) Hearts (Red colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2

(iv) Diamonds (Red colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2

(v)  Kings, queens and jacks are called face cards.

6. Prime number 1 to 100 are :

   2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 .

7. The Probability of an event lies between 0 and 1 (0 and 1 inclusive).

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EXERCISE 15.1

1. In a cricket match, a batswoman hits a boundary 6 times out of 30 balls she plays . Find the probability that she did not hit a boundary .

Solution : Total numbers of balls  

The number of ball did not hit a boundary

P(did not hit a boundary)

2. 1500 families with 2 children were selected randomly, and the following data were recorded :

   Number of girls in a family

     2

    1

    0

   Number of families

  475

   814

  211

Compute the probability of a family , choose at random, having

(i) 2 girls       (ii) 1 girl               (iii) No girl

 Also check whether the sum of these probabilities is 1 .

Solution: Total number of a family

(i) Number of 2 girls in a family  

P(2 girls in a family)

(ii) Number of 1 girls in a family  

P(1 girls in a family)

(iii) Number of no girls in a family  

P(2 girls in a family)

  Therefore, P(2 girls in a family) +  P(1 girls in a family) + P(No girls in a family)

3. In a particular section of Class IX , 40 students were asked about  the month of their birth and the following  graph was prepared for the data so obtained :

Observe the bar graph given above and answer the following questions :

Find the probability that a student of the class was born in August .

Solution:  Total number of students in class IX = 40 .

The number of a student of the class was born in August = 6

P(Born in August)

4. Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes :

    Outcome

   3 heads

   2 heads

    1 head

     No head

   Frequency

        23

       72

       77

         28

If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up .

Solution:   Total number of tosses

The number of two head come up  

   P( getting 2 heads)

5. An organization selected 2400 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 (in Rs.)

                Vehicles per family

      0

      1

     2

   Above 2

    Less than 7000

       7000 – 10000

     10000 – 13000

     13000 – 16000

      16000 or more

      10

       0

       1

       2

       1

    160

    305

    535

    469

    579

     25

     27

     29

     59

     82

        0

        2

        1

      25

      88

Suppose a family is chosen . Find the probability that the family chosen is

(i) earning Rs 10000 – 13000 per month and owning exactly 2 vehicles .

(ii) earning Rs 16000 or more per month and owning exactly 1 vehicle .

(iii) earning less than Rs 7000 per month and does not own any vehicles .

(iv) earning Rs 13000 – 16000 per month and owning more than 2 vehicles .

(v) owning not more than 1 vehicle .

Solution:  Total number of families = 2400

(i) The number of families earning Rs 10000 – 13000 per month and owning exactly 2 vehicles = 29

P(exactly 2 vehicles)

(ii)  The number of families earning Rs 16000 or more per month and owning  exactly 1 vehicles = 579

P(exactly 1 vehicles)

(iii) The number of families earning less than Rs 7000 per month and does not own any vehicles = 10  

P(does not own any vehicles)

(iv)  The number of families earning Rs 13000 – 16000 per month and owning more than 2 vehicles = 25

P(more than 2 vehicles)

(v) The number of families owning not more than 1 vehicle = 10+0+1+2+1+160+305+535+469+579 = 2062 .

P(owning not more than 1 vehicle)

6. A teacher wanted to analyse the performance of two sections of students in a mathematics test of 100 marks .

       Marks

      Number of students

        0 – 20

      20 – 30

      30 – 40

      40 – 50

     50 – 60

     60 – 70

   70 – above

                    7

                  10

                  10

                  20

                  20

                 15

                   8

         Total

                 90

(i) Find the probability that a student obtained less than 20% in the mathematics test .

(ii) Find the probability that a student obtained .

Solution:  Total number of students = 90 .

 (i) The number of a student who obtained less than 20% in the mathematics test  = 7

P(less than 20%)

(ii) The number of a student who obtained marks 60 or above = 15 + 8 = 23 .

P(getting a student obtained marks 60 or above )

7. To know the opinion of the students about the subject statistics , a survey of 200 students was conducted . The data is recording in the following table .

     Opinion

     Number of students

       Like

     Dislike

                 135

                   65

Find the probability that a student chosen at random (i) likes statistics    (ii) does not like it .

Solution:  Total number of students = 135 + 65 = 200

(i)  The number of students likes statistics = 135

P(likes statistics)

(ii) The number of students does not likes statistics = 65

P(does not like statistics)

8. The distance (in km) of 40 engineers from their residence to their place of work were found as follows :  5     3    10    20   25    11  13    7    12   31  19    10   12   17   18   11   32   17   16    2       7     9    7       8      3     5    12   15   18   3    12    14    2     9     6     15   15    7      6   12

Construct a grouped frequency distribution table with class size 5 for the data given above taking the first interval as 0 – 5 (5 not included) . What is the empirical probability that an engineer lives :    (i) less than 7 km from her place of work ?   (ii) more than or equal to 7 km from her place of work ?   (iii)  within  km from her place of work ?

Solution:  We construct the table :

Distances (in km)

Frequency

          0 – 5

          5 – 10

         10 – 15

         15 – 20

         20 – 25

         25 – 30

         30 – 35

         5

       11

       11

        9

       1

       1

       2

         Total

         40

Total number of engineers  = 40 .

(i) The number  of engineers less than 7 km from her place of work = 9 .

P( less than 7 km)

(ii) The number of engineers more than or equal to 7 km from her place of work = 40 – 9 = 31

P( more than or equal to 7 km)

(iii) The number of engineers within km from her place of work = 0

P(within km from her place of work)

9. Activity : Note the frequency of two-wheelers, three-wheeler and four-wheelers  going past during a time interval , in front of your school gate . Find the probability that any one vehicle out of the total vehicles you have observed is a two-wheeler .

Solution:  Do yourself .

[   Hint:    We construct the table :

Time

(Hours)

No.  Vehicles (Wheelers)

   2W

    3W

   4W

  7 – 8

  8 – 9

  9 – 10

  10 – 11

  11 – 12

   100

   130

   250

    50

    45

   10

   15

   50

  100

   80

   150

   100

   300

  150

  100

Total number of vehicles = 1630 .

The number of 2-wheelers = 575 .

P( getting a two-wheeler )

10. Activity : Ask all the students in your class to write a 3-digit number . Choose any student from the room at random . What is the probability that the number written by her/him is divisible by 3 ? Remember that a number is divisible by 3 , if the sum of its digits is divisible by 3 .

Solution:   The list of three digit number are : 100 , 101 , 102 , 103 ,………, 999 .

The list of number divisible by 3 are : 102 , 105 , 108 , 111 , 114 , ………., 999 .

[Here,  ,  ,  and 

 ]

Total  three digit  number  = 900 .

The number of  three digit number divisible by 3 = 300 .

The probability that the number written by her/him is divisible by 3

11. Eleven bags of wheat flour, each marked 5 kg , actually contained the following weights of flour (in kg) :

   4.97    5.05    5.08    5.03   5.00    5.06    5.08     4.98    5.04    5.07     5.00

Find the probability that any of these bags chosen at random contains more than 5 kg of flour .

Solution:  Total number of the bags = 11 .

The number of bags of the flour more than 5 kg = 7

P(getting more than 5 kg bag of flour )

12. A study was conducted to find out the concentration of sulphur dioxide in the air in parts per million (ppm)of a certain city .The data obtained for 30 days is as follows:  0.03         0.08      0.08      0.09     0.04     0.17     0.16        0.05      0.02      0.06     0.18      0.20        0.11        0.08      0.12     0.13      0.22      0.07       0.08       0.01       0.10     0.06      0.09      0.18      0.11      0.07        0.05     0.07      0.01     0.04

Using this table , find the probability of the concentration of sulphur dioxide in the interval 0.12 – 0.16 on any of these days .

Solution:  We construct the table :

Concentration of Sulphur dioxide

  (in ppm)

  Frequency

     0.00 – 0.04

     0.04 – 0.08

     0.08 – 0.12

     0.12 – 0.16

     0.16 – 0.20

     0.20 – 0.24

        4

        9

       9

       2

       4

       2

        Total

       30

Total number of the day = 30

The number of day in which the concentration of sulphur dioxide in the interval 0.12 – 0.16  = 2

P (The concentration of sulphur dioxide in the interval 0.12 – 0.16)

13. The blood group of 30 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.

Use this table to determine the probability that a student of this class, selected at random, has blood group AB .

Solution:   We construct the table :

Blood group

 No. of Students

            O

            12

            A

             9

           B

            6

          AB

            3

          Total

            30

Total number of  students of class VIII = 30 .

The number of students whose  blood groups AB = 3  .

P(getting a blood group)