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#### Page No 619:

#### Question 1:

A coin is tossed 500 times and we get

head: 285 times, tail: 215 times.

When a coin is tossed at random, what is the probability of getting

(i) a head?

(ii) a tail?

#### Answer:

Total number of tosses = 500

Number of heads = 285

Number of tails = 215

(i) Let E be the event of getting a head.

*P*(getting a head) = *P* (*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{heads}\mathrm{coming}\mathrm{up}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{285}{500}=0.57$

(ii) Let F be the event of getting a tail.

*P*(getting a tail) = *P* (*F*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{tails}\mathrm{coming}\mathrm{up}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{215}{500}=0.43$

#### Page No 619:

#### Question 2:

Two coins are tossed 400 times and we get

two heads: 112 times; one head: 160 times; 0 head: 128 times.

When two coins are tossed at random, what is the probability of getting

(i) 2 heads?

(ii) 1 head?

(iii) 0 head?

#### Answer:

_{ T}otal number of tosses = 400

Number of times 2 heads appear = 112

Number of times 1 head appears = 160

Number of times 0 head appears = 128

*E*

_{1},

*E*

_{2},

*E*

_{3}be the events of getting 2 heads, 1 head and 0 head, respectively. Then,

(i)

*P*(getting 2 heads) =

*P*(

*E*

_{1}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}2\mathrm{heads}\mathrm{appear}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{112}{400}=0.28$

(ii)

*P*( getting 1 head) =

*P*(

*E*

_{2}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}1\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{160}{400}=0.4$

(iii)

*P*( getting 0 head) =

*P*(

*E*

_{3}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}0\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{128}{400}=0.32$

Remark: Clearly, when two coins are tossed, the only possible outcomes are

*E*

_{1},

*E*

_{2}and

*E*

_{3}and

*P*(

*E*

_{1}) +

*P*(

*E*

_{2}) +

*P*(

*E*

_{3}) = (0.28 + 0.4 + 0.32) = 1

#### Page No 619:

#### Question 3:

Three coins are tossed 200 times and we get

three heads: 39 times; two heads: 58 times;

one head: 67 times; 0 head: 36 times.

When three coins are tossed at random, what is the probability of getting

(i) 3 heads?

(ii) 1 head?

(iii) 0 head?

(iv) 2 heads?

#### Answer:

Total number of tosses = 200

Number of times 3 heads appear = 39

Number of times 2 heads appear = 58

Number of times 1 head appears = 67

Number of times 0 head appears = 36

*E*

_{1},

*E*

_{2},

*E*

_{3}and

*E*

_{4}be the events of getting 3 heads, 2 heads, 1 head and 0 head, respectively. Then;

*P*(getting 3 heads) =

*P*(

*E*

_{1}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}3\mathrm{heads}\mathrm{appear}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{39}{200}=0.195$

(ii)

*P*(getting 1 head) =

*P*(

*E*

_{2}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}1\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{67}{200}=0.335$

(iii)

*P*(getting 0 head) =

*P*(

*E*

_{3}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}0\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{36}{200}=0.18$

(iv)

*P*(getting 2 heads) =

*P*(

*E*

_{4}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}2\mathrm{heads}\mathrm{appear}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{58}{200}=0.29$

*E*

_{1},

*E*

_{2},

*E*

_{3}

_{ and }

*E*

_{4}and

*P*(

*E*

_{1}) +

*P*(

*E*

_{2}) +

*P*(

*E*

_{3}) +

*P*(

*E*

_{4}) = (0.195 + 0.335 + 0.18 + 0.29) = 1

#### Page No 620:

#### Question 4:

A dice is thrown 300 times and the outcomes are noted as given below.

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency | 60 | 72 | 54 | 42 | 39 | 33 |

(i) 3?

(ii) 6?

(iii) 5?

(iv) 1?

#### Answer:

Total number of throws = 300

In a random throw of a dice, let E_{1}, E_{2}, E_{3}, E_{4}, be the events of getting 3, 6, 5 and 1, respectively. Then,

(i) *P*(getting 3) = *P*(*E*_{1}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}3\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{tr}\text{ia}\mathrm{ls}}=\frac{54}{300}=0.18$

(ii) *P*(getting 6) = *P*(*E*_{2}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}6\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{tr}\text{ia}\mathrm{ls}}=\frac{33}{300}=0.11$

(iii) *P*(getting 5) = *P*(*E*_{3}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}5\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{tr}\text{ia}\mathrm{ls}}=\frac{39}{300}=0.13$

(iv) *P*(getting 1) = *P*(*E*_{4}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}1\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{tr}\text{ia}\mathrm{ls}}=\frac{60}{300}=0.20$

#### Page No 620:

#### Question 5:

In a survey of 200 ladies, it was found that 142 like coffee, while 58 dislike it.

Find the probability that a lady chosen at random

(i) likes coffee

(ii) dislikes coffee

#### Answer:

Total number of ladies = 200

Number of ladies who like coffee = 142

Let

*E*

_{1}and

*E*

_{2}be the events that the selected lady likes and dislikes coffee, respectively.Then,

(i)

*P*(selected lady likes coffee) =

*P*(

*E*

_{1}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{ladies}\mathrm{who}\mathrm{like}\mathrm{coffee}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{ladies}}=\frac{142}{200}=0.71$

(ii)

*P*(selected lady dislikes coffee) =

*P*(

*E*

_{2}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{ladies}\mathrm{who}\mathrm{dislike}\mathrm{coffee}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{ladies}}=\frac{58}{200}=0.29$

REMARK: In the given survey, the only possible outcomes are

*E*

_{1}and

*E*

_{2}

_{ }and

*P*(

*E*

_{1}) +

*P*(

*E*

_{2}) = (0.71 + 0.29) = 1

#### Page No 620:

#### Question 6:

The percentages of marks obtained by a student in six unit tests are given below.

Unit test | I | II | III | IV | V | VI |

Percentage of marks obtained | 53 | 72 | 28 | 46 | 67 | 59 |

#### Answer:

Total number of unit tests = 6

Number of tests in which the student scored more than 60% marks = 2

*E*be the event that he got more than 60% marks in the unit tests.Then,

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{unit}\mathrm{tests}\mathrm{in}\mathrm{which}\mathrm{he}\mathrm{got}\mathrm{more}\mathrm{than}60\%\mathrm{marks}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{unit}\mathrm{tests}}=\frac{2}{6}=\frac{1}{3}$

#### Page No 620:

#### Question 7:

On a particular day, in a city, 240 vehicles of various types going past a crossing during a time interval were observed, as shown:

Types of vehicle | Two-wheelers | Three-wheelers | Four-wheelers |

Frequency | 84 | 68 | 88 |

#### Answer:

Total number of vehicles going past the crossing = 240

Number of two-wheelers = 84

*E*be the event that the selected vehicle is a two-wheeler. Then,

*P*(

*E*) = $\frac{84}{240}=0.35$

#### Page No 620:

#### Question 8:

On one page of a telephone directory, there are 200 phone numbers. The frequency distribution of their units digits is given below:

Units digit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Frequency | 19 | 22 | 23 | 19 | 21 | 24 | 23 | 18 | 16 | 15 |

(i) 5?

(ii) 8?

#### Answer:

Total phone numbers on the directory page = 200

(i) Number of numbers with units digit 5 = 24

*E*

_{1}be the event that the units digit of selected number is 5.

*P*(

*E*

_{1}) = $\frac{24}{200}=0.12$

(ii) Number of numbers with units digit 8 = 16

*E*

_{2}be the event that the units digit of selected number is 8.

*P*(

*E*

_{2}) = $\frac{16}{200}=0.08$

#### Page No 620:

#### Question 9:

The following table shows the blood groups of 40 students of a class.

Blood group | A | B | O | AB |

Number of students | 11 | 9 | 14 | 6 |

(i) O?

(ii) AB?

#### Answer:

Total number of students = 40

(i) Number of students with blood group O = 14

*E*

_{1}be the event that the selected student's blood group is O.

*P*(

*E*

_{1}) = $\frac{14}{40}=0.35$

(ii) Number of students with blood group AB = 6

*E*

_{2}be the event that the selected student's blood group is AB.

*P*(

*E*

_{2}) = $\frac{6}{40}=0.15$

#### Page No 621:

#### Question 10:

The table given below shows the marks obtained by 30 students in a test.

Marks (Class interval) |
Number of students (Frequency) |

1−10 | 7 |

11−20 | 10 |

21−30 | 6 |

31−40 | 4 |

41−50 | 3 |

#### Answer:

Total number of students = 30

Number of students whose marks lie in the interval 21-30 = 6

*E*be the event that the selected student's marks lie in the interval 21- 30 .

∴ Required probability = *P*(*E*) = $\frac{6}{30}=\frac{1}{5}=0.2$

#### Page No 621:

#### Question 11:

Following are the ages (in years) of 360 patients, getting medical treatment in a hospital:

Age (in years) | 10−20 | 20−30 | 30−40 | 40−50 | 50−60 | 60−70 |

Number of patients | 90 | 50 | 60 | 80 | 50 | 30 |

Find the probability that his age is

(i) 30 years or more but less than 40 years

(ii) 50 years or more but less than 70 years

(iii) less than 10 years

(iv) 10 years or more

#### Answer:

Total number of patients = 360

(i) Number of patients whose age is 30 years or more but less than 40 years = 60

*E*

_{1}be the event that the selected patient's age is in between 30 - 40.

*P*(patient's age 30 is years or more but less than 40 years) =

*P*(

*E*

_{1}) = $\frac{60}{360}=\frac{1}{6}$

(ii) Number of patients whose age is 50 years or more but less than 70 years = (50 +30) = 80

*E*

_{2}be the event that the selected patient's age is in between 50 - 70.

∴

*P*(patient's age is 50 years or more but less than 70 years) =

*P*(

*E*

_{2}) = $\frac{80}{360}=\frac{2}{9}$

(iii) Number of patients whose age is less than 10 years = 0

*E*

_{3}be the event that the selected patient's age is less than 0.

*P*(patient's age is less than 10 years)=

*P*(

*E*

_{3}) = $\frac{0}{360}=0$

(iv) Number of patients whose age is 10 years or more = 90 + 50 + 60 + 80 + 50 + 30 = 360

*E*

_{4}be the event that the selected patient's age is 10 years or more. Then

∴

*P*(patient's age is 10 years or more) =

*P*(

*E*

_{4}) = $\frac{360}{360}=1$

#### Page No 623:

#### Question 1:

A coin is tossed 100 times with the following outcomes:

head 43 times and tail 57 times.

In a single throw of a coin, what is the probability of getting a head?

(a) $\frac{43}{57}$

(b) $\frac{57}{43}$

(c) $\frac{43}{100}$

(d) $\frac{7}{50}$

#### Answer:

(c) $\frac{43}{100}$

Explanation:

Total number of trials = 100

Number of heads = 43

Number of tails = 57

Let E be the event of getting a head.

*P*(getting a head) = *P* (*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{heads}\mathrm{coming}\mathrm{up}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{43}{100}$

#### Page No 623:

#### Question 2:

A coin is tossed 200 times with the following outcomes:

head 112 times and tail 88 times.

What is the probability of getting a tail in a single throw of a coin ?

(a) $\frac{11}{25}$

(b) $\frac{14}{25}$

(c) $\frac{11}{14}$

(d) $\frac{14}{11}$

#### Answer:

(a) $\frac{11}{25}$

Explanation:

Total number of trials = 200

Number of heads = 112

Number of tails = 88

Let *E* be the event of getting a tail.

*P*(getting a tail) = *P* (*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{tails}\mathrm{coming}\mathrm{up}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{88}{200}=\frac{11}{25}$

#### Page No 623:

#### Question 3:

A survey of 200 persons of a locality shows the like and dislike for tea.

No. of persons who like tea | 148 |

No. of persons who dislike tea | 52 |

(a) $\frac{13}{37}$

(b) $\frac{37}{13}$

(c) $\frac{13}{50}$

(d) $\frac{37}{50}$

#### Answer:

(d) $\frac{37}{50}$

Explanation:

Total number of persons = 200

Number of persons who like tea = 148

*E*be the event that the selected person likes tea. Then,

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{persons}\mathrm{who}\mathrm{like}\mathrm{tea}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{persons}}=\frac{148}{200}=\frac{37}{50}$

#### Page No 623:

#### Question 4:

In a locality, 100 families were chosen at random and the following data was collected.

Number of children in each family | 0 | 1 | 2 | 3 | 4 or more |

Number of families | 6 | 184 | 672 | 127 | 11 |

(a) $\frac{1}{336}$

(b) $\frac{84}{125}$

(c) $\frac{41}{125}$

(d) $\frac{164}{375}$

#### Answer:

(b) $\frac{84}{125}$

Explanation:

Total number of families = 1000

Number of families with 2 children = 672

*E*be the event that the chosen family has 2 children.Then,

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{families}\text{with}2\mathrm{children}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{persons}}=\frac{672}{1000}=\frac{84}{125}$

#### Page No 624:

#### Question 5:

The table given below shows the month of birth of 36 students of a class.

Month of birth | Jan | Feb | Mar | Apr | May | June | July | Aug | Sept | Oct | Nov | Dec |

No. of students | 4 | 3 | 5 | 0 | 1 | 6 | 1 | 3 | 4 | 3 | 4 | 2 |

(a) $\frac{1}{3}$

(b) $\frac{2}{3}$

(c) $\frac{1}{4}$

(d) $\frac{1}{12}$

#### Answer:

(d) $\frac{1}{12}$

Explanation:

Total number of students = 36

Number of students born in October = 3

*E*be the event that the chosen student was born in October. Then,

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{students}\mathrm{born}\mathrm{in}\mathrm{October}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{students}}=\frac{3}{36}=\frac{1}{12}$

#### Page No 624:

#### Question 6:

In 50 tosses of a coin, tail appears 32 times. If a coin is tossed at random, what is the probability of getting a head?

(a) $\frac{1}{32}$

(b) $\frac{1}{18}$

(c) $\frac{16}{25}$

(d) $\frac{9}{25}$

#### Answer:

(d) $\frac{9}{25}$

Explanation:

Total number of tosses = 50

Number of times tail appears = 32

So, the number of times head appears = 50 − 32 = 18

Let E be the event of getting a head.

*P*(getting a head) = *P* (*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{heads}\mathrm{coming}\mathrm{up}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{18}{50}=\frac{9}{25}$

#### Page No 624:

#### Question 7:

In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays. What is the probability that in a given delivery, the ball does not hit the boundary?

(a) $\frac{1}{4}$

(b) $\frac{1}{5}$

(c) $\frac{4}{5}$

(d) $\frac{3}{4}$

#### Answer:

(c) $\frac{4}{5}$

Explanation:

Total number of balls faced = 30

Number of times the ball hits the boundary = 6

Number of times the ball does not hit the boundary = (30 − 6 )= 24

*E*be the event that the ball does not hit the boundary. Then,

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{ball}\mathrm{does}\mathrm{not}\mathrm{hit}\mathrm{the}\mathrm{boundary}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{balls}}=\frac{24}{30}=\frac{4}{5}$

#### Page No 624:

#### Question 8:

A dice is thrown 40 times and each time the number on the uppermost face is noted as shown:

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Number of times | 5 | 6 | 8 | 10 | 7 | 6 |

(a) $\frac{5}{7}$

(b) $\frac{7}{5}$

(c) $\frac{1}{8}$

(d) $\frac{7}{40}$

#### Answer:

(d) $\frac{7}{40}$

Explanation:

Total number of trials = 40

Number of times 5 appears = 7

*P*(getting 5) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}5\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{tr}\text{ia}\mathrm{ls}}=\frac{7}{40}$

#### Page No 624:

#### Question 9:

In 50 throws of a dice, the outcomes were noted as shown below:

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Number of times | 8 | 9 | 6 | 7 | 12 | 8 |

(a) $\frac{12}{25}$

(b) $\frac{3}{50}$

(c) $\frac{1}{8}$

(d) $\frac{1}{2}$

#### Answer:

(a) $\frac{12}{25}$

Explanation:

Total number of trials = 50

Let E be the event of getting an even number.

Then, E contains 2, 4 and 6, i.e. 3 even numbers.

∴ *P*(getting an even number) = *P*(*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{even}\mathrm{number}s\mathrm{appear}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{throws}}=\frac{(9+7+8)}{50}=\frac{24}{50}=\frac{12}{25}$

#### Page No 625:

#### Question 10:

The outcomes of 65 throws of a dice were noted as shown below:

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Number of times | 8 | 10 | 12 | 16 | 9 | 10 |

(a) $\frac{3}{35}$

(b) $\frac{3}{5}$

(c) $\frac{31}{65}$

(d) $\frac{36}{65}$

#### Answer:

(c) $\frac{31}{65}$

Explanation:

Total number of throws = 65

Let E be the event of getting a prime number.

Then, E contains 2, 3 and 5, i.e. three numbers.

∴ *P*(getting a prime number) = *P*(*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{prime}\mathrm{number}\text{s}\mathrm{occur}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{throws}}=\frac{(10+12+9)}{65}=\frac{31}{65}$

#### Page No 625:

#### Question 11:

On one page of a directory, there are 160 telephone numbers. The frequency distribution of the unit place digit is shown below:

Unit place digit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Frequency | 19 | 16 | 18 | 21 | 14 | 11 | 15 | 16 | 13 | 17 |

(a) $\frac{2}{5}$

(b) $\frac{3}{32}$

(c) $\frac{3}{80}$

(d) $\frac{29}{32}$

#### Answer:

(b) $\frac{3}{32}$

Explanation:

Total phone numbers on a page of the directory = 160

Numbers with units digit 6 = 15

*E*be the event that the units digit of selected number is 6.

*P*(

*E*) = $\frac{15}{160}=\frac{3}{32}$

#### Page No 625:

#### Question 12:

Two coins are tossed 1000 times and the outcomes are recorded as shown below:

Number of heads | 2 | 1 | 0 |

Frequency | 266 | 540 | 194 |

(a) $\frac{403}{500}$

(b) $\frac{27}{50}$

(c) $\frac{367}{500}$

(d) $\frac{97}{500}$

#### Answer:

(c)$\frac{367}{500}$

Explanation:

Total number of tosses = 1000

Number of times 2 heads appear = 266

Number of times 1 head appears = 540

Number of times 0 head appears = 194

*P*(getting at most 1 head) $=\frac{734}{1000}=\frac{367}{500}$

#### Page No 625:

#### Question 13:

80 bulbs are selected at random from a lot and their lifetime is recorded in the form of a frequency table given below:

Lifetime (in hours) | 300 | 500 | 700 | 900 | 1100 |

Frequency | 10 | 15 | 23 | 25 | 7 |

(a) $\frac{73}{80}$

(b) $\frac{3}{5}$

(c) $\frac{5}{16}$

(d) $\frac{23}{80}$

#### Answer:

(b) $\frac{3}{5}$

Explanation:

Total number of bulbs in the lot = 80

Number of bulbs with life time of less than 900 hours = (10 + 15 + 23) = 48

*E*be the event that the chosen bulb's life time is less than 900 hours.

*P*(

*E*) = $\frac{48}{80}=\frac{3}{5}$

#### Page No 626:

#### Question 14:

In a medical examination of 40 students of a class, the following blood groups were recorded:

Blood group | A | B | AB | O |

No. of students | 11 | 15 | 9 | 5 |

(a) $\frac{3}{5}$

(b) $\frac{5}{8}$

(c) $\frac{3}{5}$

(d) $\frac{8}{3}$

#### Answer:

(a) $\frac{3}{8}$

Explanation:

Total number of students = 40

Number of students with blood group B = 15

*E*be the event that the selected student's blood group is B.

*P*(

*E*) = $\frac{15}{40}=\frac{3}{8}$

#### Page No 626:

#### Question 15:

In a group of 60 persons, 35 like coffee. Out of this group, if one person is chosen at random, what is the probability that he or she does not like coffee?

(a) $\frac{7}{12}$

(b) $\frac{5}{12}$

(c) $\frac{5}{7}$

(d) $\frac{3}{12}$

#### Answer:

(b) $\frac{5}{12}$

Explanation:

Total number of persons = 60

Number of persons who like coffee = 35

Let

*E*be the event that the selected person does not like coffee.Then,

*P*(selected person does not like coffee) =

*P*(

*E*) = $\frac{25}{60}=\frac{5}{12}$

#### Page No 626:

#### Question 16:

A dice is thrown 50 times and the outcomes are recorded as shown below.

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

No. of times | 11 | 9 | 8 | 5 | 7 | 10 |

(a) $\frac{1}{5}$

(b) $\frac{3}{50}$

(c) $\frac{1}{3}$

(d) 0

#### Answer:

(d) 0

Explanation:

Total number of trials = 50

Number of times 8 appears = 0

*P*(getting 8) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}8\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{tr}\text{ia}\mathrm{ls}}=\frac{0}{50}$ = 0

(Remark: A dice is marked from 1 to 6. So, it is impossible to get 8 and probability of an impossible event is 0.)

#### Page No 626:

#### Question 17:

It is given that the probability of winning a game is 0.7. What is the probability of losing the game?

(a) 0.8

(b) 0.3

(c) 0.7

(d) 0.07

#### Answer:

(b) 0.3

Explanation:

Let E be the event of winning the game. Then,

*P*(E) = 0.7

*P*(not E) = *P*(losing the game) = 1− *P*(E) ⇒ 1− 0.7 = 0.3

#### Page No 626:

#### Question 18:

A coin is tossed 60 times and the tail appears 35 times. What is the probability of getting a head?

(a) $\frac{7}{12}$

(b) $\frac{12}{7}$

(c) $\frac{5}{12}$

(d) $\frac{1}{25}$

#### Answer:

(c) $\frac{5}{12}$

Explanation:

Total number of trials = 60

Number of times tail appears = 35

*E*be the event of getting a head.

∴

*P*(getting a head) =

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{25}{60}=\frac{5}{12}$

#### Page No 627:

#### Question 19:

**Assertion:** In a cricket match, a batsman hits 9 boundaries out of 45 balls he plays. The probability that in a given throw he does not hit the boundary is $\frac{4}{5}$.

**Reason:** $P\left(E\right)+P(\mathrm{not}E)=1$.

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.

(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.

(c) Assertion is true and Reason is false.

(d) Assertion is false and Reason is true.

#### Answer:

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.

Explanation:

Total number of balls = 45

Number of times the ball hits the boundary = 9

Number of times the ball does not hit the boundary = (45 − 9 ) = 36

*E*be the event that the ball hits the boundary.Then,

*P*(*E*) = $\frac{9}{45}=\frac{1}{5}$

*P*(not

*E*) =

*P*( he does not hit the boundary) = $\left(1-\frac{1}{5}\right)=\frac{4}{5}$ [

*P*(not

*E*) = $\frac{36}{45}=\frac{4}{5}$]

Also,

*P*(

*E*) +

*P*(not

*E*) = 1 is true.

Thus, Assertion ( A) and Reason (R) are true and Reason ( R) is a correct explanation of Assertion (A).

So, the correct answer is (a).

#### Page No 627:

#### Question 20:

**Assertion:** The probability of a sure event is 1.

**Reason:** Let *E* be an event. Then, $0\le P\left(E\right)\le 1$.

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.

(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.

(c) Assertion is true and Reason is false.

(d) Assertion is false and Reason is true.

#### Answer:

(b) Both Assertion and Reason are true and Reason is not a correct explanation of Assertion.

Explanation:

Hence, the correct answer is (b).

#### Page No 627:

#### Question 21:

**Fill in the blanks**

(i) Probability of an impossible event = ...... .

(ii) Probability of a sure event = ...... .

(iii) Let *E* be an event. Then, *P*(not *E*) = ...... .

(iv) *P*(*E*) + *P*(not *E*) = ...... .

(v) *P*(*E*) lies between ..... and ...... .

#### Answer:

(i) 0

(ii) 1

(iii) 1− *P*(*E*)

(iv) 1

(v) 0, 1

#### Page No 627:

#### Question 22:

The marks obtained by 90 students in mathematics out of 100 are given below.

Marks | 0−20 | 20−30 | 30−40 | 40−50 | 50−60 | 60−70 | 70 and above |

No. of students | 7 | 8 | 12 | 25 | 19 | 10 | 9 |

What is the probability that the chosen student

(i) gets 20% or less marks?

(ii) gets 60% or more marks?

#### Answer:

Total number of students = 90

(i) Number of students who scored 20% marks or less = 7.

*E*

_{1}be the event that selected student got 20% marks or less.Then,

*P*(

*E*

_{1}) = $\frac{7}{90}$

(ii) Number of students who scored 60% marks or more = (10 + 9 ) = 19.

*E*

_{2}be the event that selected student got 60% marks or more.Then,

*P*(

*E*

_{2}) = $\frac{19}{90}$

#### Page No 627:

#### Question 23:

It is known that a box of 800 electric bulbs contains 36 defective bulbs. One bulb is taken at random out of the box. What is the probability that the bulb chosen is non-defective?

#### Answer:

Total number of bulbs in the box = 800

Number of defective bulbs = 36

Number of non-defective bulbs = 800 − 36 = 764

*E*be the event that the chosen bulb is non-defective.

*P*(

*E*) = $\frac{764}{800}=\frac{191}{200}$

#### Page No 627:

#### Question 24:

The table given below shows the ages of 75 teachers in a school.

Age (in years) | 18−29 | 30−39 | 40−49 | 50−59 |

No. of teachers | 5 | 25 | 35 | 10 |

A teacher from the school is chosen at random. What is the probability that the teacher chosen is

(i) 40 or more than 40 years old?

(ii) 49 or less than 49 years old?

(iii) 60 or more than 60 years old?

#### Answer:

Total number of teachers = 75

(i) Number of teachers who are 40 or more than 40 years old = 35 + 10 = 45

*E*

_{1}be the event that the selected teacher's age is 40 years or more.

*P*(

*E*

_{1}) = $\frac{45}{75}=\frac{3}{5}$

(ii) Number of teachers who are 49 or less than 49 years old = (35 + 25 + 5) = 65

*E*

_{2}be the event that the selected teacher's age is 49 years or less.

∴ Required probability =

*P*(

*E*

_{2}) = $\frac{65}{75}=\frac{13}{15}$

(iii) Number of teachers who are 60 or more than 60 years old = 0

*E*

_{3}be the event that the selected teacher's age is more than 60 years.

*P*(

*E*

_{3}) = $\frac{0}{75}=0$

#### Page No 631:

#### Question 1:

There are 600 electric bulbs in a box, out of which 20 bulbs are defective. If one bulb is chosen at random from the box, what is the probability that it is defective?

(a) $\frac{1}{19}$

(b) $\frac{1}{20}$

(c) $\frac{1}{30}$

(d) $\frac{29}{30}$

#### Answer:

(c) $\frac{1}{30}$

Explanation:

Total number of bulbs in the box = 600

Number of defective bulbs = 20

*E*be the event that the chosen bulb is defective.

*P*(

*E*) = $\frac{20}{600}=\frac{1}{30}$

#### Page No 631:

#### Question 2:

A bag contains 5 red, 8 black and 7 white balls. One ball is chosen at random. What is the probability that the chosen ball is black?

(a) $\frac{2}{3}$

(b) $\frac{2}{5}$

(c) $\frac{3}{5}$

(d) $\frac{1}{3}$

#### Answer:

(b) $\frac{2}{5}$

Explanation:

Total number of balls in the bag = 5 + 8 + 7 = 20

Number of black balls = 8

*E*be the event that the chosen ball is black.

*P*(

*E*) = $\frac{8}{20}=\frac{2}{5}$

#### Page No 631:

#### Question 3:

A bag contains 16 cards bearing numbers 1, 2, 3, ..., 16 respectively. One card is chosen at random. What is the probability that the chosen card bears a number divisible by 3?

(a) $\frac{3}{16}$

(b) $\frac{5}{16}$

(c) $\frac{11}{16}$

(d) $\frac{13}{16}$

#### Answer:

(b) $\frac{5}{16}$

Explanation:

Total number of cards in the bag = 16

Numbers on the cards that are divisible by 3 are 3, 6, 9, 12 and 15.

*E*be the event that the chosen card bears a number divisible by 3.

*P*(

*E*) = $\frac{5}{16}$

#### Page No 631:

#### Question 4:

In a cricket match, a batsman hits a boundary 4 times out of the 32 balls he faces. In a given ball, what is the probability that he does not hit a boundary?

(a) $\frac{1}{8}$

(b) $\frac{7}{8}$

(c) $\frac{1}{7}$

(d) $\frac{6}{7}$

#### Answer:

(b) $\frac{7}{8}$

Total number of balls that the batsman faces = 32

Number of times the batsman hits a boundary = 4

Number of times the batsman does not hit a boundary = (32 − 4 ) = 28

*E*be the event that the batsman does not hit a boundary.

*P*(

*E*) = $\frac{28}{32}=\frac{7}{8}$

#### Page No 631:

#### Question 5:

Define the probability of an event E.

#### Answer:

The probability of an event E is the ratio of number of trials in which event E happened to total number of trials.

*P*(*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{trials}\mathrm{in}\mathrm{which}\mathrm{event}\mathrm{happened}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}$

Thus, $0\le P\left(E\right)\le 1$.

#### Page No 631:

#### Question 6:

A coin is tossed 60 times with the following outcomes:

Head 28 times and Tail 32 times.

Find the probability of getting a head in a single throw of the coin.

#### Answer:

Total number of tosses = 60

Number of times tail appears = 32

*E*be the event of getting a head. Then,

*P*(getting a head) =

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{28}{60}=\frac{7}{15}$

#### Page No 631:

#### Question 7:

Write down all the possible outcomes when a dice is thrown.

#### Answer:

When a dice is thrown, the possible outcomes are 1, 2, 3 ,4, 5 and 6.

#### Page No 631:

#### Question 8:

Fill in the blanks:

(i) Probability of a sure event = ..... .

(ii) Probability of an impossible event = ..... .

(iii) If *E* be an event, then *P*(*E*) + *P*(not *E*) = ..... .

(iv) If *E* is an event, then .......$\le P\left(E\right)\le $ ...... .

#### Answer:

(i) 1

(ii) 0

(iii) 1

(iv) 0, 1

#### Page No 632:

#### Question 9:

A die was thrown 80 times and the outcomes were noted as shown:

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency | 14 | 11 | 9 | 13 | 18 | 15 |

#### Answer:

Total number of trials = 80

Number of times 2 appears = 11

Let E be the event of getting 2 on the die.

∴* P*( getting 2) = *P*(*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}2\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{11}{80}$

#### Page No 632:

#### Question 10:

The probability of losing a game is 0.6. What is the probability of winning the game?

#### Answer:

Let E be the event of losing the game. Then,

*P*(E) = 0.6

*P*( not E) = *P*(winning the game) = 1− *P*(E)

⇒ 1 − 0.6 = 0.4

Hence, probability of winning the game is 0.4.

#### Page No 632:

#### Question 11:

A coin is tossed 50 times and the tail appears 28 times. In a single throw of a coin, what is the probability of getting a head?

#### Answer:

Total number of trials = 50

Number of times the tail appears = 28

*E*be the event of getting a head. Then,

*P*(getting a head) =

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{22}{50}=\frac{11}{25}$

#### Page No 632:

#### Question 12:

In a one-day cricket match, a batsman hits the boundary 8 times out of the 48 balls that he faced. Find the probability that he does not hit the boundary.

#### Answer:

Total number of balls faced by the batsman = 48

Number of times the batsman hits the boundary = 8

Number of times the he does not hit the boundary = (48 − 8 ) = 40

*E*be the event that the batsman does not hit the boundary.

*P*(

*E*) = $\frac{40}{48}=\frac{5}{6}$

#### Page No 632:

#### Question 13:

Two coins were tossed simultaneously 80 times and the outcomes were recorded as under:

Outcome | Two heads | One head | No head |

Frequency | 24 | 36 | 20 |

(i) two heads

(ii) one head

(iii) no head

#### Answer:

Total number of trials = 80

Number of times 2 heads appear = 24

Number of times 1 head appears = 36

Number of times 0 head appears = 20

*E*

_{1},

*E*

_{2},

*E*

_{3}be the events of getting 2 heads, 1 head and 0 head, respectively. Then,

(i)

*P*(getting 2 heads) =

*P*(

*E*

_{1}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}2\mathrm{heads}\mathrm{appear}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{24}{80}=\frac{3}{10}=0.3$

(ii)

*P*(getting 1 head) =

*P*(

*E*

_{2}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}1\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{36}{80}=\frac{9}{20}=0.45$

(iii)

*P*(getting 0 head) =

*P*(

*E*

_{3}) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}0\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{20}{80}=\frac{1}{4}=0.25$

Clearly, when two coins are tossed, the only possible outcomes are

*E*

_{1},

*E*

_{2}and

*E*

_{3}and

*P*(

*E*

_{1}) +

*P*(

*E*

_{2}) +

*P*(

*E*

_{3}) = (0.3 + 0.45 + 0.25) = 1

#### Page No 632:

#### Question 14:

Marks obtained by 90 students of Class IX in a test are given below:

Marks | 0−20 | 20−40 | 40−60 | 60−80 | 80−100 |

No. of students | 8 | 15 | 32 | 26 | 9 |

(i) less than 20% marks

(ii) 80% or more marks

(iii) 60% or more marks

#### Answer:

Total number of students = 90

(i) Number of students who scored less than 20% marks = 8

*E*

_{1}be the event that selected student got less than 20% mark .

∴

*P*(

*E*

_{1}) = $\frac{8}{90}=\frac{4}{45}$

*E*

_{2}be the event that selected student got 60% marks or more .

*P*(

*E*

_{2}) = $\frac{9}{90}=\frac{1}{10}$

(iii) Number of students who scored 60% marks or more = (26 + 9 ) = 35

*E*

_{3}be the event that selected student got 60% marks or more.

*P*(

*E*

_{3}) = $\frac{35}{90}=\frac{7}{18}$

#### Page No 632:

#### Question 15:

The blood groups of 30 students of a class were recorded as under:

Blood Group | A | B | AB | O |

No. of students | 9 | 11 | 4 | 6 |

(i) O?

(ii) A?

(iii) AB?

#### Answer:

Total number of students = 30

(i) Number of students with blood group O = 6

*E*

_{1}be the event that the selected student has blood group O.

*P*(

*E*

_{1}) = $\frac{6}{30}=\frac{1}{5}$

(ii) Number of students with blood group A = 9

*E*

_{2}be the event that the selected student has blood group A.

*P*(

*E*

_{2}) = $\frac{9}{30}=\frac{3}{10}$

(iii) Number of students with blood group AB = 4

*E*

_{3}be the event that the selected student has blood group AB.

*P*(

*E*

_{3}) = $\frac{4}{30}=\frac{2}{15}$

#### Page No 633:

#### Question 16:

In a survey of 100 families with 2 or less boys, the following data were obtained:

Number of boys in a family | 0 | 1 | 2 |

Number of families | 18 | 46 | 36 |

(i) 1 boy?

(ii) 2 boys?

(iii) no boys?

#### Answer:

Total number of families = 100

(i) Number of families with 1 boy = 46

*E*

_{1}be the event that the selected family has 1 boy.

*P*(

*E*

_{1}) = $\frac{46}{100}=\frac{23}{50}$

(ii) Number of families with 2 boys = 36

*E*

_{2}be the event that the selected families have 2 boys.

*P*(

*E*

_{2}) = $\frac{36}{100}=\frac{9}{25}$

(iii) Number of families without boys = 18

*E*

_{3}be the event that the selected families have no boys.

*P*(

*E*

_{3}) = $\frac{18}{100}=\frac{9}{50}$

#### Page No 633:

#### Question 17:

A die was thrown 100 times and the following observations were recorded:

Outcome | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency | 12 | 18 | 14 | 26 | 14 | 16 |

(i) a number less than 3

(ii) a number greater than 4

(iii) an even number

#### Answer:

Total number of trials = 100

(i) Let E_{1} be the event of getting a number less than 3. Then,

* P*(getting a number less than 3) = P(*E*_{1}) = $\frac{\left(12+18\right)}{100}=\frac{30}{100}=\frac{3}{10}$

(ii) Let E_{2} be the event of getting a number greater than 4. Then,

* P*(getting a number greater than 4) = P(*E*_{2}) = $\frac{\left(14+16\right)}{100}=\frac{30}{100}=\frac{3}{10}$

(iii) Let E_{3} be the event of getting an even number. Then,

* P*(getting an even number) = *P*(*E*_{3}) = $\frac{\left(18+26+16\right)}{100}=\frac{60}{100}=\frac{3}{5}$

Number of times 1 appearsTotal number of trails = 60300 = 0.2

#### Page No 633:

#### Question 18:

A survey of 600 students on their preference for coffee was conducted and recorded as shown:

Opinion | Like | Dislike |

No. of students | 360 | 140 |

(i) likes coffee?

(ii) dislikes coffee?

#### Answer:

Total number of students = 600

(i) Number of students who like coffee = 360

Let *E*_{1} be the event that the selected person likes coffee.Then,

* P *( selected person likes coffee) = *P*(*E*_{1}) = $\frac{360}{600}=\frac{3}{5}$

(ii) Number of students who dislike coffee = 140

Let *E*_{2} be the event that the selected person dislikes coffee.Then,

* ** P* ( selected person dislike coffee) = *P*(*E _{2}*) = $\frac{140}{600}=\frac{7}{30}$

#### Page No 633:

#### Question 19:

Two coins are tossed 1000 times and the outcomes are recorded as given below:

Number of heads | 0 | 1 | 2 |

Frequency | 240 | 450 | 310 |

(i) at most one head?

(ii) at least one head?

#### Answer:

Total number of tosses = 1000

Number of times 2 heads appear = 310

Number of times 1 head appears = 450

Number of times 0 head appears = 240

(i) Let

*E*

_{1}be the event of getting at most 1 head.

*E*

_{1} (either 1 head or 0 head) = (450 + 240) = 690

*P*(getting at most 1 head) =

*P*(

*E*

_{1}) = $\frac{(450+240)}{1000}=\frac{690}{1000}=\frac{69}{100}=0.69$

(ii) Let

*E*

_{2}be the event of getting at least 1 head.

*E*

_{2}

_{ }(either 1 head or 2 heads appear) = 450 + 310 = 760

*P*( getting at least 1 head) =

*P*(

*E*

_{2}) = $\frac{760}{1000}=\frac{19}{25}=0.76$

#### Page No 633:

#### Question 20:

A coin is tossed 80 times with the following outcomes:

Head: 35 times and Tail: 45 times.

In a single throw of a coin, if *E* be the event of getting a head, verify that *P*(*E*) + *P*(not *E*) = 1.

#### Answer:

Total number of trials = 80

Number of times the tail appears = 45

*E*be the event of getting a head. Then,

*P*(getting a head) =

*P*(

*E*) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{head}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{35}{80}=\frac{7}{16}$

*P*(not

*E*) =

*P*(getting a tail) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{times}\mathrm{tail}\mathrm{appears}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{45}{80}=\frac{9}{16}$

*P*(

*E*) +

*P*(not

*E*) = $\frac{7}{16}+\frac{9}{16}=\frac{16}{16}=1$

Hence, for an event

*E*of getting a head,

*P*(

*E*) +

*P*(not

*E*) = 1

View NCERT Solutions for all chapters of Class 9