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

Above 2

Less than 7000

7000 – 10000

10000 – 13000

13000 – 16000

16000 or more

160

305

535

469

579

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

Total

(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

Dislike

135

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

Total

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)

7 – 8

8 – 9

9 – 10

10 – 11

11 – 12

100

130

250

100

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

Total

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)