## 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:

According to the question,

Total number of balls = 30

Numbers of boundary = 6

Number of time bats woman didn’t hit boundary = 30 – 6 = 24

Probability she did not hit a boundary = 24/30 = 4/5

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

**Compute the probability of a family, chosen at random, having**

**(i) 2 girls (ii) 1 girl (iii) No girl
Also check whether the sum of these probabilities is 1.**

Solution:

Total numbers of families = 1500

(i) Numbers of families having 2 girls = 475

Probability = Numbers of families having 2 girls/Total numbers of families

= 475/1500 = 19/60

(ii) Numbers of families having 1 girls = 814

Probability = Numbers of families having 1 girls/Total numbers of families

= 814/1500 = 407/750

(iii) Numbers of families having 2 girls = 211

Probability = Numbers of families having 0 girls/Total numbers of families

= 211/1500

Sum of the probability = (19/60)+(407/750)+(211/1500)

= (475+814+211)/1500

= 1500/1500 = 1

Yes, the sum of these probabilities is 1.

**3. Refer to Example 5, Section 14.4, Chapter 14. Find the probability that a student of the class was born in August.**

Solution:

Total numbers of students in the class = 40

Numbers of students born in August = 6

The probability that a student of the class was born in August, = 6/40 = 3/20

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

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

Solution:

Number of times 2 heads come up = 72

Total number of times the coins were tossed = 200

**∴**, the probability of 2 heads coming up = 72/200 = 9/25

**5. An organisation 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 ₹10000 – 13000 per month and owning exactly 2 vehicles.**

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

**(iii) earning less than ₹7000 per month and does not own any vehicle.**

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

**(v) owning not more than 1 vehicle. **

Solution:

Total number of families = 2400

(i) Numbers of families earning ₹10000 –13000 per month and owning exactly 2 vehicles = 29

**∴**, the probability that the family chosen is earning ₹10000 – 13000 per month and owning exactly 2 vehicles = 29/2400

(ii) Number of families earning ₹16000 or more per month and owning exactly 1 vehicle = 579

**∴**, the probability that the family chosen is earning₹16000 or more per month and owning exactly 1 vehicle = 579/2400

(iii) Number of families earning less than ₹7000 per month and does not own any vehicle = 10

**∴**, the probability that the family chosen is earning less than ₹7000 per month and does not own any vehicle = 10/2400 = 1/240

(iv) Number of families earning ₹13000-16000 per month and owning more than 2 vehicles = 25

**∴**, the probability that the family chosen is earning ₹13000 – 16000 per month and owning more than 2 vehicles = 25/2400 = 1/96

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

= 2062

**∴**, the probability that the family chosen owns not more than 1 vehicle = 2062/2400 = 1031/1200

**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**

**(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.**

Solution:

Total number of students = 90

(i) Number of students who obtained less than 20% in the mathematics test = 7

**∴**, the probability that a student obtained less than 20% in the mathematics test = 7/90

(ii) Number of students who obtained marks 60 or above = 15+8 = 23

**∴**, the probability that a student obtained marks 60 or above = 23/90

**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.**

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) Number of students who like statistics = 135

, the probability that a student likes statistics = 135/200 = 27/40

(ii) Number of students who do not like statistics = 65

**∴**, the probability that a student does not like statistics = 65/200 = 13/40

**8. The distance (in km) of 40 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 7 km from her place of work?**

**(ii) more than or equal to 7 km from her place of work?**

**(iii) Within ½ km from her place of work?**

Solution:

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 3 2

17 16 2 7 9 7 8 3 5 12 15 18 3 12 14 2 9 6

15 15 7 6 12

Total numbers of engineers = 40

(i) Number of engineers living less than 7 km from their place of work = 9

,the probability that an engineer lives less than 7 km from her place of work = 9/40

(ii) Number of engineers living more than or equal to 7 km from their place of work = 40-9 = 31

,probability that an engineer lives more than or equal to 7 km from her place of work = 31/40

(iii) Number of engineers living within ½ km from their place of work = 0

**∴**, the probability that an engineer lives within ½ km from her place of work = 0/40 = 0

**9. Activity : 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.**

Solution:

The question is an activity to be performed by the students.

Hence, perform the activity by yourself and note down your inference.

**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 question is an activity to be performed by the students.

Hence, perform the activity by yourself and note down your inference.

**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 bags present = 11

Number of bags containing more than 5 kg of flour = 7

**∴**, the probability that any of the bags chosen at random contains more than 5 kg of flour = 7/11

**12. In Q.5, Exercise 14.2, you were asked to prepare a frequency distribution table, regarding 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.**

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

Solution:

Total number of days in which the data was recorded = 30 days

Numbers of days in which sulphur dioxide was present in between the interval 0.12-0.16 = 2

**∴**, the probability of the concentration of sulphur dioxide in the interval 0.12-0.16 on any of these days = 2/30 = 1/15

**13. In Q.1, Exercise 14.2, you were asked to prepare a frequency distribution table regarding 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.
The blood groups 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.**

Solution:

Total numbers of students = 30

Number of students having blood group AB = 3

**∴**, the probability that a student of this class, selected at random, has blood group AB = 3/30 = 1/10