CBSE Class 9 Mathematics Probability Worksheet Set 02

Question. Can the experimental probability of an event be a negative number ? Justify your answer.
Answer: No, the experimental probability of an event cannot be a negative number.
Justification: The experimental (or empirical) probability of an event \( E \) is given by: \[ P(E) = \frac{\text{Number of trials in which the event happened}}{\text{Total number of trials}} \] Since both the number of trials in which the event happened and the total number of trials represent frequencies/counts, they can never be negative. Moreover, the number of favorable trials is always less than or equal to the total number of trials, which implies that: \[ 0 \le P(E) \le 1 \] Hence, probability can never be a negative number.

Question. In a single throw of two dice , what is the probability of getting a sum of 9.
Answer: When two dice are thrown together, the total number of possible outcomes is: \[ n(S) = 6 \times 6 = 36 \] Let \( E \) be the event of getting a sum of 9.
The favorable outcomes are: \[ (3, 6), (4, 5), (5, 4), (6, 3) \] Number of favorable outcomes, \( n(E) = 4 \)
Therefore, the probability of getting a sum of 9 is: \[ P(E) = \frac{n(E)}{n(S)} = \frac{4}{36} = \frac{1}{9} \]

Question. In a survey of 364 children , 91 liked to eat potato chips. If a child is selected at random, find the probability that he/ she does not like to eat potato chips.
Answer: Total number of children surveyed \( = 364 \)
Number of children who liked to eat potato chips \( = 91 \)
Number of children who do not like to eat potato chips \( = 364 - 91 = 273 \)
Therefore, the probability that the selected child does not like to eat potato chips is: \[ P(\text{does not like potato chips}) = \frac{273}{364} = \frac{3}{4} = 0.75 \]

Question. A die was rolled 100 times and the no. of times 6 came up was noted. If the experimental probability calculated from this information is \( \frac{2}{5} \), then how many times 6 came up ?
Answer: Total number of trials (times the die was rolled) \( = 100 \)
Let the number of times 6 came up be \( x \).
Experimental probability is given by: \[ P(\text{getting 6}) = \frac{\text{Number of times 6 came up}}{\text{Total number of trials}} \] \[ \frac{2}{5} = \frac{x}{100} \] \[ x = \frac{2}{5} \times 100 = 40 \] Hence, 6 came up 40 times.

Question. Two dice are thrown 500 times. Each time the sum of two numbers appearing is recorded.

If the dice are thrown once more, what is the probability of getting a sum
(i) more than 10
(ii) less than or equal to 5
(iii) between 8 and 12 ?
Answer: Total number of trials \( = 500 \)

(i) More than 10:
Sum can be 11 or 12.
Number of favorable outcomes \( = 28 + 15 = 43 \)
\[ P(\text{sum more than 10}) = \frac{43}{500} = 0.086 \]

(ii) Less than or equal to 5:
Sum can be 2, 3, 4, or 5.
Number of favorable outcomes \( = 14 + 30 + 42 + 55 = 141 \)
\[ P(\text{sum } \le 5) = \frac{141}{500} = 0.282 \]

(iii) Between 8 and 12:
Sum can be 9, 10, or 11.
Number of favorable outcomes \( = 53 + 46 + 28 = 127 \)
\[ P(\text{sum between 8 and 12}) = \frac{127}{500} = 0.254 \]

Question. A recent survey found that the ages of workers in a factory are distributed as follows:

If a person is selected at random, find the probability that the person is
(i) 40 years or more
(ii) under 40 years
(iii) under 60 but over 39 years
Answer: Total number of workers \( = 38 + 27 + 86 + 46 + 3 = 200 \)
(i) 40 years or more:
Number of workers aged 40 years or more \( = 86 + 46 + 3 = 135 \)
\[ P(\text{40 years or more}) = \frac{135}{200} = \frac{27}{40} = 0.675 \]

(ii) Under 40 years:
Number of workers aged under 40 years \( = 38 + 27 = 65 \)
\[ P(\text{under 40 years}) = \frac{65}{200} = \frac{13}{40} = 0.325 \]

(iii) Under 60 but over 39 years:
Workers in the age groups 40–49 and 50–59 \( = 86 + 46 = 132 \)
\[ P(\text{under 60 but over 39 years}) = \frac{132}{200} = \frac{33}{50} = 0.66 \]

Question. The percentage of marks obtained by student in monthly unit tests are

Find the probability that in the next test he gets
(i) more than 70% marks
(ii) at least 60% marks.
Answer: Total number of unit tests \( = 6 \)
(i) More than 70% marks:
Tests in which the score was more than 70% (Tests IV, V, VI) \( = 3 \)
\[ P(\text{more than 70\% marks}) = \frac{3}{6} = \frac{1}{2} = 0.5 \]

(ii) At least 60% marks:
Tests in which the score was 60% or more (Tests II, III, IV, V, VI) \( = 5 \)
\[ P(\text{at least 60\% marks}) = \frac{5}{6} \]

Question. A company selected 4000 households at random and surveyed to find out relationship between income level and number of television sets in a home.

Find the probability:
(i) of a household earning ₹ 10000 – ₹ 14999 per year and having exactly one television.
(ii) of a household earning ₹ 25000 and more per year and owning 2 televisions.
(iii) of a household not having any television.
Answer: Total number of households surveyed \( = 4000 \)
(i) Earning ₹ 10000 – ₹ 14999 and having exactly 1 television:
Number of households \( = 240 \)
\[ P = \frac{240}{4000} = \frac{6}{100} = \frac{3}{50} = 0.06 \]

(ii) Earning ₹ 25000 and more and owning 2 televisions:
Number of households \( = 760 \)
\[ P = \frac{760}{4000} = \frac{19}{100} = 0.19 \]

(iii) Not having any television:
Total households with 0 televisions \( = 20 + 10 + 0 + 0 + 0 = 30 \)
\[ P = \frac{30}{4000} = \frac{3}{400} = 0.0075 \]

Question. Given below is the frequency distribution of wages (in ₹) of 30 workers in a certain factory:

A worker is selected at random. Find the probability that his wages are:
(i) less than ₹ 150
(ii) at least ₹ 210
(iii) more than or equal to 150 but less than ₹ 210
Answer: Total number of workers \( = 30 \)
(i) Less than ₹ 150:
Number of workers having wages less than ₹ 150 (intervals 110–130 and 130–150) \( = 3 + 4 = 7 \)
\[ P(\text{wages } < ₹ 150) = \frac{7}{30} \]

(ii) At least ₹ 210:
Number of workers having wages of ₹ 210 or more (intervals 210–230 and 230–250) \( = 4 + 3 = 7 \)
\[ P(\text{wages } \ge ₹ 210) = \frac{7}{30} \]

(iii) More than or equal to 150 but less than ₹ 210:
Number of workers having wages in the intervals 150–170, 170–190, and 190–210 \( = 5 + 6 + 5 = 16 \)
\[ P = \frac{16}{30} = \frac{8}{15} \]

Question. The given table shows the ages (in years) of 360 patients, getting medical treatment in a hospital.

One of the patients is selected at random. The probability that the selected patient’s age is 30 years or more but less than 40 years, is
(a) 1/6
(b) 2/9
(c) 0
(d) 1
Answer : A

Question. The following data shows the relation between the number of families and number of children they have. What is the probability of a family chosen at random having at least two children?

(a) 33/53
(b) 35/53
(c) 34/53
(d) 19/53
Answer : C

Question. If 10/100, 13/100, 15/100, 18/100, x/100, 30/100 are the probabilities of 6 observations of an experiment, then find the value of x.
(a) 12
(b) 13
(c) 14
(d) 15
Answer : C

Question. In a one day match, a player played 40 balls. The runs scored are as follows :

Find the probability that player hits a four or a six.
(a) 3/20
(b) 1/2
(c) 1/4
(d) 9/20
Answer : A

Question. 100 plants each were sown in six different colonies A, B, C, D, E and F. After 31 days, the number of plants survived are as follows:

Find the probability that more than 80 plants survived in a colony?
(a) 2/3
(b) 1/3
(c) 5/6
(d) 1/6
Answer : A

Question. The given table shows the number of students participating in various activities in a school.

One student is chosen at random. Find the probability that the student participates in games.
(a) 3/10
(b) 1/2
(c) 1/7
(d) 1/10
Answer : A

Question. On a particular day, the number of vehicles passing through a crossing is given below:

A particular vehicle is chosen at random. What is the probability that it is not a four-wheeler?
(a) 21/80
(b) 21/40
(c) 3/4
(d) 5/4
Answer : B

Question. Two sections of class IX having 27 students in each section appeared for Mathematics olympiad. The marks obtained by them are shown below.
46, 31, 74, 68, 42, 54, 14, 61, 48, 37, 26, 8, 64, 57,
93, 72, 53, 59, 38, 16, 88, 56, 46, 66, 45, 61, 54, 27,
27, 44, 63, 58, 43, 81, 64, 36, 49, 50, 76, 38, 47, 77,
62, 53, 40, 71, 60, 45, 42, 34, 46, 40, 59, 42
One student is selected at random. Find the probability that selected student is having marks more than 49.
(a) 1/3
(b) 1/2
(c) 1/4
(d) 7/9
Answer : B

Case Based MCQs

Case I : Read the following passage and answer the questions from 46 to 50.
On one day in school, teacher performed an activity with the students. She tells every student to write their weights on their notebook. After that she makes the following frequency distribution table from the collected data and asks some questions

1. Find the probability of a randomly chosen student of weight atleast 50 kg.
(a) 11/50
(b) 13/50
(c) 21/50
(d) 7/50
Answer : A

2. The probability that the selected student has atmost 65 kg weight is
(a) 0
(b) 23/41
(c) 16/17
(d) 1
Answer : D

3. The probability that the weight of the student is not more than or equal to 45 kg is
(a) 15/31
(b) 1/2
(c) 27/50
(d) 2/3
Answer : C

4. The probability that weight of student is less than 60 kg but more than or equal to 45 kg is
(a) 23/50
(b) 13/50
(c) 11/50
(d) 19/50
Answer : D

5. If an experiment is performed, then the probability of an event (E) is
(a) P(E) = No. of trials in which the event has happened/Total no. of trials
(b) P(E) = Total no. of trials/No. of trials in which the event has happened
(c) P(E) = No. of trials in which the event has not happened/Total no. of trials
(d) none of these
Answer : A

Case II : Read the following passage and answer the questions from 51 to 55.
In a factory, the workers are paid on daily basis. The new manager wants to know the salary slab of the workers and finds the data as given below .

Now, if a worker is chosen at random, then :

1. Probability that the worker is getting atmost ₹ 500 is
(a) 3/80
(b) 9/80
(c) 1/80
(d) 4/25
Answer : B

2. Probability that the worker is getting atleast ₹ 701 is
(a) 23/80
(b) 43/80
(c) 41/80
(d) 61/80
Answer : B

3. Probability that the worker is getting ₹ 901 - ₹ 1100 is
(a) 3/40
(b) 1/40
(c) 9/40
(d) 2/23
Answer : C

4. Probability that the worker is getting atmost ₹ 900 is
(a) 7/40
(b) 3/31
(c) 31/40
(d) 4/31
Answer : C

5. Probability that the worker is getting less than ₹ 300 is
(a) 0
(b) 1
(c) 1/2
(d) 2/3
Answer : A

Assertion & Reasoning Based MCQs

Directions (Q.56 to 60) : In these questions, a statement of Assertion is followed by a statement of Reason is given.
Choose the correct answer out of the following choices :
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

1. Assertion : A coin is tossed 30 times and head appears 18 times. Then the probability of getting a tail is 12/30 .
Reason : Probability of happening of an event
= Number of trials in which the event happened/Total number of trials.
Answer : A

2. Assertion : A die is thrown, if probability of getting an even number is 1/2, then the probability of getting an odd number is also 1/2 .
Reason : P(E₁) + P(E₂) + P(E₃) + ... + P(E_n) = 1, where E₁, E₂, E₃, ..., E_n are n elementary eventsassociated to a random experiment.
Answer : A

3. Assertion : Three coins are tossed. Numberof elements in the sample space is 8.
Reason : When a coin is tossed n times,then the number of elements in its sample spaceis 2n.
Answer : A

4 Assertion : In a survey of 365 studentsaged 13-14 years, it was found that 130 likedto opt French as third language. If a studentselected at random, then the probability that he/she does not like to opt French as third languageis 47/73.
Reason : The probability of an event
E = Number of trials in which E is not happened/Total number of trials
Answer : C

5. Assertion : In a class of 45 students, 39 passed an examination. If a student is selected at random, then the probability that the student has not passed the examination is 13/15
Reason : P(E) + P(not E) = 1
Answer : D