Probability
Class 9th Mathematics RD Sharma Solution
- Define a trial.
- A coin is tossed 1000 times with the following frequencies: Head: 455, Tail:…
- Two coind are tossed simultaneously 500 times with the following frequencies of…
- Define an elementary event.
- Define an event.
- Three coins are tossed simultaneously 100 times with the following frequencies…
- Define probability of an event.
- 1500 families with 2 children were delected randomly and the following data…
- In a cricket match, a batsman hits a boundry 6 times out of 30 balls he plays.…
- A bag contains 4 white balls and some red balls. If the probability of drawing…
- The percentage of marks obtained by a student in monthly unit tests are given…
- A die is thrown 100 times. If the probability of getting an even number is 2/5…
- Three coins are tossed simulateneously 200 times with the following frequencies…
- To know the opinion of the students about Mathematics, a survey of 200 students…
- The blood groups of 30 students of class iX are recorded as follows : A, B, O,…
- what is the probability of getting at least two heads?
- Eleven bags of wheat flour, each marked 5 kg, actually contained the following…
- Following table shows the birth month of 40 students of class IX. Jan. Feb.…
- Given below is the frequency distribution table regarding the concentration of…
- A company selected 2400 families at random and survey them to determine a…
- The following table gives the life time of 400 neon lamps: Life time (in hrs.)…
- Given below is the frequency distribution of wages (in Rs.) of 30 workers in a…
- The probability of an impossible event isA. 1 B. 0 C. less than 0 D. greater than 1…
- The probability of a certain event isA. 0 B. 1 C. greater than 1 D. less than 0…
- The probability of an event of a trials isA. 1 B. 0 C. less than 1 D. more than 1…
- Which of the following cannot be the probability of an event?A. 1/3 B. 3/5 C. 5/3 D. 1…
- Two coins are tossed simultaneously. The probability of getting atmost one head isA.…
- A coin is tossed 1000 times, if the probability of getting a tail is 3/8, how many…
- A dice is rolled 600 times and the occurrence of the outcomes 1, 2, 3, 4, 5 and 6 are…
- The percentage of attendance of different classes in a year in a school is given below…
- A bag is contains 50 coins and each coin is marked from 51 to 100. One coin is picked…
- In a football match, Ronaldo makes 4 goals from 10 penanlty kicks. The probability of…
Exercise 25.1
Question 1.Define a trial.
Answer:
Any particular performance of a random experiment is called a trial.
By Experiment or Trial in the subject of probability, we mean a Random experiment unless otherwise specified.
Each trial results in one or more outcomes.
Question 2.
A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545
Compute the probability for each event.
Answer:
Given,
The coin tossed = 1000 times
The total number of trial = 1000
Lets assum,
The event of getting head = E
The event of getting tail = F
Then,
Number of trials in which the E happens = 455
So,
Probability of E = 
P (E) = 
Similarly,
The probability of the event getting a tail = 
P (F) = 
Note: we note that P(A)+P(B) = 0.48+0.52
Therefore, A and B are the only two opposite outcomes.
Question 3.
Two coind are tossed simultaneously 500 times with the following frequencies of different outcomes:
Two heads: 95 times
One tail: 290 times
No head: 115 times
Find the probability of occurrence of each of these events.
Answer:
Given,
Total number of trials = 500 times
Probability (E) = 
P (getting two heads) = 
P (getting one tail) = 
P (getting no head) = 
Question 4.
Define an elementary event.
Answer:
In probability theory, an elementary event (also called an atomic event or simple event) is an event which contains only a single outcome in the sample space.
Using set theory terminology, an elementary event is a singleton.
Question 5.
Define an event.
Answer:
When we say "Events" we mean one (or more) outcomes or results of an experiment.
Question 6.
Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes :
If the three coins are simultaneously tossed again, compute the probability of :
(i) 2 heads coming up.
(ii) 3 heads coming up.
(iii) at least one head coming up.
(iv) getting more heads than tails.
(v) getting more tails than heads.
Answer:
Given,
(i) Probability of 2 heads coming up = 
= 
= 0.36
(ii) Probability of 3 heads coming up =
= 
(iii) Probability of atleast one heads coming up =
= 
(iv) Probability of getting more heads than tails =
(v) Probability of getting more tails than heads coming up =
Question 7.
Define probability of an event.
Answer:
The probability of an event E is defined as the number of outcomes favorable to E divided by the total number of equally likely outcomes in the sample space S of the experiment.
Question 8.
1500 families with 2 children were delected randomly and the following data were recorded:
If a family is chosen at random, compute the probability that it has:
(i) No girl
(ii) 1 girl
(iii) 2 girls
(iv) At most one girl
(v) More girls than boys.
Answer:
Given,
Total number of families = 211+814+475 = 1500
(i) Number of families having no girl = 211
Required probability = 
= 
(ii) Number of families having one girl = 
= 
= 0.5426
(iii) Number of families having 2 girls = 
=
0.3166
(iv) Number of families having at most one girl = 
= 
(v) Number of families having more girls thanboys= 
=
Question 9.
In a cricket match, a batsman hits a boundry 6 times out of 30 balls he plays.
Find the probability that on a ball played :
(i) he hits boundry
(ii) he does not hit a boundry.
Answer:
Given,
Number of times batsman hits the boundry = 6
Total balls played = 30
Number of times that batsman does not hit the boundry = 30-6 = 24
Probability (number of times he hits a boundry) = 
Probability (number of times he does not hits a boundry) = 
Question 10.
A bag contains 4 white balls and some red balls. If the probability of drawing a white ball from the bag is 
, find the number of red balls in the bag.
Answer:
let the number of red ball be x.
total number of red balls = 4+x.
probability of drawing a white ball is = 
∴ 
2(4+x) = 4×5
4+x = 2×5 = 10
X = 10-4 = 6
X = 6
Thus the number of red balls are 6.
Question 11.
The percentage of marks obtained by a student in monthly unit tests are given below :
Find the probability that the student gets :
(i)More than 70% marks
(ii) Less than 70% marks
(iii) A distinction.
Answer:
Let,
(i) E be the event of getting more than 70% marks
The number of times E happens = 3
P(A) =
= 0.6
(ii) F be the event of getting less than 70% marks
The number of times F happens = 2
P(B) = 
(iii) G be the event of getting a distinction
The number of times G happens = 1
P(C) = 
Question 12.
A die is thrown 100 times. If the probability of getting an even number is 
. How many times an odd number is obtained?
Answer:
Possible outcomes of a dice = {1, 2, 3, 4, 5, 6}
of the given six possible outcomes,
{1, 3, 5} are 3 odd number and {2, 4, 6} are 3 even so,
here, there is an equal possibility of coming an odd number and even number in every throw.
Given the probability of getting an even number =
∴The probability of getting an odd number = 1 – Probability of getting an even number
= 
Given the dice is thrown 100 times
∴ Number of times an odd number obtained = 
Question 13.
Three coins are tossed simulateneously 200 times with the following frequencies of different outcomes:
Find the probability of getting at most two heads.
Answer:
Probability of getting at most two heads = 
Question 14.
To know the opinion of the students about Mathematics, a survey of 200 students was conducted. The data is recorded in the following tables:
Find the probability that a student chosen at random
(i) likes Mathematics
(ii) does not like it.
Answer:
Given,
(i) Probability that a student likes mathematics = 
(ii) Probability that a student does not like mathematics =
Question 15.
The blood groups of 30 students of class iX 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
A student is selected at random from the class from blood donation. Find the probability that the blood group of the student chosen is :
(i) A
(ii) B
(iii) AB
(iv) O
Answer:
Given,
Probability of a student of blood group A =
Probability of a student of blood group B =
Probability of a student of blood group AB =
Probability of a student of blood group O =
Question 16.
what is the probability of getting at least two heads?
Answer:
Probability of getting at least 2 heads 
Question 17.
Eleven bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg):
Find the probability that any of these bags chosen at random contains more than 5 kg of flour.
Answer:
Given,
Eleven bags of wheat flour, each marked 5 kg, actually contained the different weights of flour
Probability (Bags having more than 5 kg flour) = 
Question 18.
Following table shows the birth month of 40 students of class IX.
Find the probability that a student was born in August.
Answer:
Given,
Probability (students was born in August) =
Question 19.
Given below is the frequency distribution table regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days.
Find the probability of concentration of sulphur dioxide in the interval 0.12-0.16 on any of these days.
Answer:
Given,
The frequency distribution table regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days
Total number of days = 30
Probability of concentration of SO2 in the interval 0.12 – 0.16 is
= 
Question 20.
A company selected 2400 families at random and survey them to determine a relationship between income level and the number of vehicles in a home. The information gathered is listed in the table below :
If a family is chosen find the probability that the family is:
(i) earning Rs. 10,000 – 13,000 per month and owning exactly 2 vehicles.
(ii) earning Rs. 16,000 or more per month and owning exactly 1 vehicles.
(iii) earning less than Rs. 7,000 per month and does not own any vehicles.
(iv) earning Rs. 13,000 – 16,000 per month and owning more than 2 vehicles.
(v) owning not more than 1 vehicle.
(vi)owning at least one vehicle.
Answer:
Given,
(i) The probability that the family is earning Rs. 10,000 – 13,000 per month and owning exactly 2 vehicles = 
(ii) The probability that the family earning Rs. 16,000 or more per month and owning exactly 1 vehicles = 
(iii) The probability that the family earning less than Rs. 7,000 per month and does not own any vehicles = 
(iv) The probability that the family earning Rs. 13,000 – 16,000 per month and owning more than 2 vehicles = 
(v) The probability that the family owning not more than 1 vehicle = 
(vi) The probability that the family owning at least one vehicle =
Question 21.
The following table gives the life time of 400 neon lamps:
A bulb is selected at random. Find the probability that the life time of the selected bulb is:
(i) less than 400
(ii) between 300 to 800 hours
(iii)at least 700 hours.
Answer:
Given,
(i) Probability that the life time of the selected bulb is less than 400 = 
(ii) Probability that the life time of the selected bulb is between 300 – 800 hours = 
(iii) Probability that the life time of the selected bulb is at least 700 hours = 
Question 22.
Given below is the frequency distribution of wages (in Rs.) of 30 workers in a certain factory :
A worker is selected atr random. Find the probability that his wages are :
(i) less than Rs. 150
(ii) at least Rs. 210
(iii) more than or equal to 150 but less than Rs. 210.
Answer:
Given,
Total number of workers = 30
(i) Probability that his wages are less than rs 150 = 
(ii) Probability that his wages are at least rs 210 =
(iii) Probability that his wages are more than or equals to rs 150 but less than rs 200 = 
An Elementary Event is any single outcome or element of a sample space.
Cce - Formative Assessment
Question 1.The probability of an impossible event is
A. 1
B. 0
C. less than 0
D. greater than 1
Answer:
The event is known ahead of time to be not possible, therefore by definition in mathematics, the probability is defined to be 0 which means it can never happen.
Question 2.
The probability of a certain event is
A. 0
B. 1
C. greater than 1
D. less than 0
Answer:
If there is a chance that an event will happen, then, its probability is between zero and 1.
Question 3.
The probability of an event of a trials is
A. 1
B. 0
C. less than 1
D. more than 1
Answer:
The probability of an event of a trials is less than 1.
Question 4.
Which of the following cannot be the probability of an event?
A. 
B. 
C. 
D. 1
Answer:
The most the probability of an event occurring can be is 1
which means the event has a 100% probability of happening. But 5/3 is greater than 1 so it can’t be the probability of an event.
Question 5.
Two coins are tossed simultaneously. The probability of getting atmost one head is
A. 
B. 
C. 
D.
Answer:
When two coins are tossed simultaneously then the possible outcomes obtained are {HH, HT, TH, and TT}.
Here H denotes head and T denotes tail.
Therefore, a total of 4 outcomes obtained on tossing two coins simultaneously.
The favourable outcome of getting at most one head are HT, TH, and TT.
So, the probability of getting at most one head is 3/4
Question 6.
A coin is tossed 1000 times, if the probability of getting a tail is 3/8, how many times head is obtained?
A. 525
B. 375
C. 625
D. 725
Answer:
Possible outcomes of a coin = HT
So here,
there is an equal possibilty of coming of Head and Tail in every toss.
Now it is given that probability of getting a tail = 38
The probability of getting a head = 1 - probability of getting a tail = 
The coin is tossed 1000 times.
Hence number of times a head obtained = 
= 625 times
Question 7.
A dice is rolled 600 times and the occurrence of the outcomes 1, 2, 3, 4, 5 and 6 are given below :
The probability of getting a prime number is
A.
B. 
C. 
D. 
Answer:
Total trials = 600
No. Of getting prime number = 30+120+50 = 200
Let E be the event of getting a prime number so,
P(E) = 
Question 8.
The percentage of attendance of different classes in a year in a school is given below :
What is the probability that the class attendance is more than 75%?
A. 
B.
C. 
D. 
Answer:
Total classes = 6
No. Of class that have more than 75% attendance = 3
Let E be the event of more than 75% attendance
P(E) = 
Question 9.
A bag is contains 50 coins and each coin is marked from 51 to 100. One coin is picked at random. The probability that the number on the coin is not a prime number, is
A. 
B.
C. 
D. 
Answer:
Total number of coins = 50
∴ Total number of outcomes = 50
prime numbers are = 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
P(prime numbers) = 
∴ P(not prime numbers) = 
Question 10.
In a football match, Ronaldo makes 4 goals from 10 penanlty kicks. The probability of converting a penalty kick into a goal by Ronaldo, is
A. 
B. 
C. 
D. 
Answer:
Total penalty kicks = 10
Total goals = 4
Let E be the event of converting a penalty kick into a goal = 