EXERCISE 15.1
Introduction - Probability , Mathematics Class 9th
Page No 283:
Question 1:
In a cricket math, a batswoman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary.Answer:
Number of times the batswoman hits a boundary = 6Total number of balls played = 30
∴ Number of times that the batswoman does not hit a boundary = 30 − 6 = 24
$\mathrm{P}(\text { she does not hit a boundary })$
$=\frac{\text { Number of times when she does not hit boundary }}{\text { Total number of balls played }}$
$=\frac{24}{30}=\frac{4}{5}$
Q 2, Ex. 15.1, Page No 283 - Probability - NCERT Class 9th Maths
Question 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
|
(i) 2 girls (ii) 1 girl (iii) No girl
Also check whether the sum of these probabilities is 1.
Answer:
Total number of families = 475 + 814 + 211= 1500
(i) Number of families having 2 girls = 475
$\mathrm{P}_{1}(\text { a randomly chosen family has } 2 \text { girls })=\frac{\text { Number of families having } 2 \text { girls }}{\text { Total number of families }}$
$=\frac{475}{1500}=\frac{19}{60}$
(ii) Number of families having 1 girl = 814
$\left.\mathrm{P}_{2} \text { (a randomly chosen family has } 1 \text { girl }\right)=\frac{\text { Number of families having } 1 \text { girl }}{\text { Total number of families }}$
$=\frac{814}{1500}=\frac{407}{750}$
(iii) Number of families having no girl = 211
$\mathrm{P}_{3}$ (a randomly chosen family has no girl) $=\frac{\text { Number of families having no girl }}{\text { Total number of families }}$
$=\frac{211}{1500}$
Sum of all these probabilities $=\frac{19}{60}+\frac{407}{750}+\frac{211}{1500}$
$=\frac{475+814+211}{1500}$
$=\frac{1500}{1500}=1$
Therefore, the sum of all these probabilities is 1.
Q 3, Ex. 15.1, Page No 283 - Probability - NCERT Class 9th Maths
Question 3:
In a particular section of Class IX, 40 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.
Answer:
Number of students born in the month of August = 6Total number of students = 40
$P\left(\text { Students born in the month of August) }=\frac{\text { Number of students born in August }}{\text { Total number of students }}\right.$
$=\frac{6}{40}=\frac{3}{20}$
Q 4, Ex. 15.1, Page No 283 - Probability - Maths Class 9th
Question 4:
Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes:- Outcome3 heads2 heads1 headNo headFrequency23727728
Answer:
Number of times 2 heads come up = 72
Total number of times the coins were tossed = 200
$\mathrm{P}(2 \text { heads will come up })=\frac{\text { Number of times } 2 \text { heads come up }}{\text { Total number of times the coins were tossed }}$
$=\frac{72}{200}=\frac{9}{25}$
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 vehicle.
(iv) earning Rs 13000 − 16000 per month and owning more than 2 vehicles.
(v) owning not more than 1 vehicle.
(i) Number of families earning Rs 10000 − 13000 per month and owning exactly 2 vehicles = 29
Hence, required probability, P=\frac{29}{2400}
(ii) Number of families earning Rs 16000 or more per month and owning exactly 1 vehicle = 579
Hence, required probability, P=\frac{579}{2400}
(iii) Number of families earning less than Rs 7000 per month and does not own any vehicle = 10
Hence, required probability, $\mathrm{P}=\frac{10}{2400}=\frac{1}{240}$
(iv) Number of families earning Rs 13000 − 16000 per month and owning more than 2 vehicles = 25
Hence, required probability, $P=\frac{25}{2400}=\frac{1}{96}$
(v) Number of families owning not more than 1 vehicle = 10 + 160 + 0 + 305 + 1 + 535 + 2 + 469 + 1 + 579 = 2062
Hence, required probability, $P=\frac{2062}{2400}=\frac{1031}{1200}$
(i) Find the probability that a student obtained less than 20 % in the mathematics test.
(ii) Find the probability that a student obtained marks 60 or above.
(i) Number of students getting less than 20 % marks in the test = 7
Hence, required probability, $P=\frac{7}{90}$
(ii) Number of students obtaining marks 60 or above = 15 + 8 = 23
Hence, required probability, $P=\frac{23}{90}$
(i) likes statistics, (ii) does not like it
(i) Number of students liking statistics = 135
P students liking statistics$=\frac{135}{200}=\frac{27}{40}$
(ii) Number of students who do not like statistics = 65
P(students not liking statistics)$=\frac{65}{200}=\frac{13}{40}$
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?
Number of engineers living less than 7 km from their place of work = 9
Hence, required probability that an engineer lives less than 7 km from her place of work,$P=\frac{9}{40}$
(ii) Number of engineers living more than or equal to 7 km from their place of work = 40 − 9 = 31
Hence, required probability that an engineer lives more than or equal to 7 km from her place of work, $P=\frac{31}{40}$
(iii) Number of engineers living within $\frac{1}{2}$km from her place of work = 0
Hence, required probability that an engineer lives within $\frac{1}{2}$km from her place of work, P = 0
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.
Number of bags containing more than 5 kg of flour = 7
Hence, required probability, $P=\frac{7}{11}$
The above frequency distribution table represents the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days. Using this table, find the probability of the concentration of sulphur dioxide in the interval 0.12 − 0.16 on any of these days.
Total number of days = 30
Hence, required probability, $P=\frac{2}{30}=\frac{1}{15}$
The above frequency distribution table represents the blood groups of 30 students of a class. Use this table to determine the probability that a student of this class, selected at random, has blood group AB.
Total number of students = 30
Hence, required probability, $P=\frac{3}{30}=\frac{1}{10}$
Total number of times the coins were tossed = 200
$\mathrm{P}(2 \text { heads will come up })=\frac{\text { Number of times } 2 \text { heads come up }}{\text { Total number of times the coins were tossed }}$
$=\frac{72}{200}=\frac{9}{25}$
Q 5, Ex. 15.1, Page No 284 - Probability - NCERT Class 9th Maths
Question 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: 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 vehicle.
(iv) earning Rs 13000 − 16000 per month and owning more than 2 vehicles.
(v) owning not more than 1 vehicle.
Answer:
Number of total families surveyed = 10 + 160 + 25 + 0 + 0 + 305 + 27 + 2 + 1 + 535 + 29 + 1 + 2 + 469 + 59 + 25 + 1 + 579 + 82 + 88 = 2400(i) Number of families earning Rs 10000 − 13000 per month and owning exactly 2 vehicles = 29
Hence, required probability, P=\frac{29}{2400}
(ii) Number of families earning Rs 16000 or more per month and owning exactly 1 vehicle = 579
Hence, required probability, P=\frac{579}{2400}
(iii) Number of families earning less than Rs 7000 per month and does not own any vehicle = 10
Hence, required probability, $\mathrm{P}=\frac{10}{2400}=\frac{1}{240}$
(iv) Number of families earning Rs 13000 − 16000 per month and owning more than 2 vehicles = 25
Hence, required probability, $P=\frac{25}{2400}=\frac{1}{96}$
(v) Number of families owning not more than 1 vehicle = 10 + 160 + 0 + 305 + 1 + 535 + 2 + 469 + 1 + 579 = 2062
Hence, required probability, $P=\frac{2062}{2400}=\frac{1031}{1200}$
Q 6, Ex. 15.1, Page No 284 - Probability - Class 9th Maths NCERT
Page No 284:
Question 6:
A teacher wanted to analyse the performance of two sections of students in a mathematics test of 100 marks. Looking at their performances, she found that a few students got under 20 marks and a few got 70 marks or above. So she decided to group them into intervals of varying sizes as follows: 0 − 20, 20 − 30… 60 − 70, 70 − 100. Then she formed the following table:
Marks
|
Number of student
|
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 marks 60 or above.
Answer:
Total number of students = 90(i) Number of students getting less than 20 % marks in the test = 7
Hence, required probability, $P=\frac{7}{90}$
(ii) Number of students obtaining marks 60 or above = 15 + 8 = 23
Hence, required probability, $P=\frac{23}{90}$
Q 7, Ex. 15.1, Page No 284 Probability - NCERT Class IXth Maths
Question 7:
To know the opinion of the students about the subject statistics, a survey of 200 students was conducted. The data is recorded in the following table.- OpinionNumber of studentslikedislike13565
(i) likes statistics, (ii) does not like it
Answer:
Total number of students = 135 + 65 = 200(i) Number of students liking statistics = 135
P students liking statistics$=\frac{135}{200}=\frac{27}{40}$
(ii) Number of students who do not like statistics = 65
P(students not liking statistics)$=\frac{65}{200}=\frac{13}{40}$
Q 8, Ex. 15.1, Page No 284 - Probability - Class 9th NCERT
Question 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 |
(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?Answer:
(i) Total number of engineers = 40Number of engineers living less than 7 km from their place of work = 9
Hence, required probability that an engineer lives less than 7 km from her place of work,$P=\frac{9}{40}$
(ii) Number of engineers living more than or equal to 7 km from their place of work = 40 − 9 = 31
Hence, required probability that an engineer lives more than or equal to 7 km from her place of work, $P=\frac{31}{40}$
(iii) Number of engineers living within $\frac{1}{2}$km from her place of work = 0
Hence, required probability that an engineer lives within $\frac{1}{2}$km from her place of work, P = 0
Page No 285:
Question 9:
Note the frequency of two-wheelers, three-wheelers 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.Answer:
This is an activity based question. Students are advised to perform this activity by yourself.Question 10:
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.Q 11, Ex. 15.1, Page No 285 - Probability - Maths Class 9th
Answer:
This is an activity based question. Students are advised to perform this activity by yourself.Question 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.
Answer:
Number of total bags = 11Number of bags containing more than 5 kg of flour = 7
Hence, required probability, $P=\frac{7}{11}$
Q 12, Ex. 15.1, Page No 285 Probability - NCERT Class 9th Maths
Question 12:
Concentration of SO2 (in ppm)
|
Number of days (frequency )
|
0.00 − 0.04
|
4
|
0.04 − 0.08
|
9
|
0.08 − 0.12
|
9
|
0.12 − 0.16
|
2
|
0.16 − 0.20
|
4
|
0.20 − 0.24
|
2
|
Total
|
30
|
The above frequency distribution table represents the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days. Using this table, find the probability of the concentration of sulphur dioxide in the interval 0.12 − 0.16 on any of these days.
Answer:
Number days for which the concentration of sulphur dioxide was in the interval of 0.12 − 0.16 = 2Total number of days = 30
Hence, required probability, $P=\frac{2}{30}=\frac{1}{15}$
Q 13, Ex. 15.1, Page No 285 Probability - Class 9th
Question 13:
Blood group
|
Number of students
|
A
|
9
|
B
|
6
|
AB
|
3
|
O
|
12
|
Total
|
30
|
Answer:
Number of students having blood group AB = 3Total number of students = 30
Hence, required probability, $P=\frac{3}{30}=\frac{1}{10}$
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