NCERT Exemplar for Class 12 Maths - Probability - Free PDF Download
Free PDF download of NCERT Exemplar for Class 12 Maths Chapter 13 - Probability solved by expert Maths teachers on Vedantu.com as per NCERT (CBSE) Book guidelines. All Chapter 13 - Probability Exercise questions with solutions to help you to revise complete syllabus and score more marks in your Examinations.






Access NCERT Exemplar Solutions for Class 12 Mathematics Unit 13 – Probability
Solved Examples
Short Answer Questions
1. A and B are two candidates seeking admission in a college. The probability that A is selected is
Ans: Let p be the probability that B gets selected.
Than,
And
Thus, the probability that B gets selected is
2. The probability of simultaneous occurrence of at least one of two events
Ans: Given, P (exactly one of A, B occurs) = q, we get
3. 10% of the bulbs produced in a factory are of red colour and
Ans: Let A and B be the events that the bulb is red and defective respectively.
So, the probability of its being defective, if it is red, is
4. Two dice are thrown together. Let
Ans:
So,
5. A committee of 4 students is selected at random from a group consisting 8 boys and 4 girls. Given that there is at least one girl in the committee, calculate the probability that there are exactly 2 girls in the committee.
Ans: Let A denote the event that at least one girl will be chosen, and B the event that exactly 2 girls will be chosen. We require
Because A denotes the event that at least one girl will be chosen, A' denotes that no girl is chosen, i.e., 4 boys are chosen. Then
Than,
Now
So,
6. Three machines
Ans: Let D be the event that the picked up tube is defective.
Let
Given,
Also
And,
Putting these values in (1), we get
7. Find the probability that in 10 throws of a fair die a score which is a multiple of 3 will be obtained in at least 8 of the throws.
Ans: Given that, obtaining multiple of 3 in a throw of a die is a success.
There are total 10 throws of a fair die.
Let
Since there can be a total 6 outcomes in a throw of a die. That is,
And a multiple of 3 in a die is
Then, let
And we know,
Let
Then, the probability of getting r successes out of
Here
Now, put values of
We need to find the probability of getting a multiple of 3 in at least 8 of the throws out of 10 throws of a fair die.
It is given,
Probability
This can be written as,
Just out
8. A discrete random variable
1 | 2 | 3 | 4 | 5 | 6 | 7 | |
Find the value of
Ans: Since
or,
or ,
Either
So, the permissible value of
Mean
Long Answer
9. Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If
Ans: Since 4 balls have to be drawn, therefore,
So, the following is the required probability distribution of
0 | 1 | 2 | 3 | 4 | |
10. Determine variance and standard deviation of the number of heads in three tosses of a coin.
Ans: Let
Sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
So, the probability distribution of
0 | 1 | 2 | 3 | |
Variance of
where
Now
From (1), (2) and (3), we get
And ,standard deviation
11. Refer to Example 6. Calculate the probability that the defective tube was produced} on machine
Ans: Now, we have to find
12. A car manufacturing factory has two plants,
Ans: Let E be the event that the car is of standard quality. Let
And,
Hence, the required probability is
Objective Type Questions
Choose the correct answer from the given four options in each of the Examples 13 to 17.
13. Let
(A)
(B)
(C)
(D) 0
Ans: Correct answer - D
From the given data
This means,
14. Let
Then
(A)
(B)
(C)
(D)
Ans: Correct answer - C
15. If
(A)
(B)
(C)
(D)
Ans: Correct answer - A
We have
Hence
Hence
Multiplying both sides by
Two events
Consider two events
Since
However,
Hence the events are independent but not mutually exclusive. Hence option
Now if
Now we know that
Hence
Since the events
Hence, we have
Hence we have
Hence the events A and B' are independent,
Since
Hence options
Now we know that
Hence, we have
Since
Hence, we have
Hence option d is correct.
Therefore, the option that is incorrect is option A.
16. Let
30 | 10 | -10 | |
Then
(A) 6
(B) 4
(C) 3
(D)
Ans: Correct answer - B
Because we know that
So,
17. Let
(A)
(B)
(C)
(D)
Ans: Correct answer - C
As we know that probability of
The variance of
The standard deviation of the random variable
Fill in the blanks in Examples 18 and 19
18. If
19. If
Ans:
(since
State whether each of the statement in Examples 20 to 22 is True or False
20. Let
Ans: False, because
21. Three events
Ans: False. Reason is that
22. One of the conditions of Bernoulli trials is that the trials are independent of each other.
Ans: True.
Exercise
Short Answer Questions
1. For a loaded die, the probabilities of outcomes are given as under:
The die is thrown two times. Let
Ans: Given
and
Die is thrown two times
So,
Now
And,
Here,
So,
Hence,
2. Refer to Exercise 1 above. If the die were fair, determine whether or not the events
Ans: We have
So,
and
So,
So,
and,
So,
Hence,
3. The probability that at least one of the two events
Ans: We know that,
Thus,
Also,
4. A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red?
Ans: Let
If at least one of the three marbles drawn be black, if the first marble is red.
(i)
(ii)
(iii)
And
So, the probability=
5. Two dice are thrown together and the total score is noted. The events
Ans: Two dice are thrown together
SO,
so,
and
so,
and
so,
Here,we see that
and
and
Since,
Hence, there is no pair which is independent.
6. Explain why the experiment of tossing a coin three times is said to have binomial distribution.
Ans: We know that, a random variable
Where
Similarly, in case tossing a coin 3 times,
Hence, it is said to have a binomial distribution.
7. If
(i)
Ans: We have,
(ii)
Ans: We have,
(iii)
Ans:
(iv)
Ans:
8. Three events A, B and C have probabilities
Ans: We have,
9. Let
Describe in words of the events whose probabilities are:
(i)
Ans: Here,
So,
So,
(ii)
Ans:
So,
(iii)
Ans:
So, either
(iv)
Ans:
So, either
10. A discrete random variable X has the probability distribution given as below:
0.5 | 1 | 1.5 | 2 | |
(i) Find the value of
Ans:
but
(ii) Determine the mean of the distribution.
Ans: Mean of the distribution
11. Prove that
(i)
Ans: To prove
(ii)
Ans: To prove,
12. If
Ans: Given that,
where
and
0 | 1 | 2 | 3 | |
0 | ||||
0 |
We know that,
Where,
And
Standard deviation of
13. In a dice game, a player pays a stake of Re1 for each throw of a die. She receives Rs 5 if the die shows
Ans: Let
-1 | 1 | 4 | |
Since, she loses
so,
Player's expected profit
14. Three dice are thrown at the same time. Find the probability of getting three two's, if it is known that the sum of the numbers on the dice was six.
Ans: The dice is thrown three times
sample space
Let
and
and
Also,
15. suppose 10,000 tickets are sold in a lottery each for Re 1 . First prize is of Rs 3000 and the second prize is of Rs. 2000 . There are three third prizes of Rs. 500 each. If you buy one ticket, what is your expectation?
Ans: Let
0 | 500 | 2000 | 3000 | |
16. A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.
Ans: Let,
And
Let
and
and E = event that the ball drawn from the second bag is white.
and
17. Bag I contains 3 black and 2 white balls, Bag II contains 2 black and 4 white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.
Ans: Given, Bag I
and Bag II
Let
And
18. A box has 5 blue and 4 red balls. one ball is drawn at random and not replaced. Its colour is also not noted. Then another ball is drawn at random. What is the probability of the second ball being blue?
Ans:Given, A box = (5 blue, 4 red)
Let
and
19. Four cards are successively drawn without replacement from a deck of 52 playing cards. What is the probability that all the four cards are king?
Ans: Let's assume
20. A die is thrown 5 times. Find the probability that an odd number will come up exactly three times.
Ans: Here,
21. Ten coins are tossed. What is the probability of getting at least 8 heads?
Ans: Here,
22. The probability of a man hitting a target is
Ans: Here,
23. A lot of 100 watches is known to have 10 defective watches. If 8 watches are selected (one by one with replacement) at random, what is the probability that there will be at least one defective watch?
Ans: Probability (defective watch out of 100 watches)
Here,
24. Consider the probability distribution of a random variable X:
0 | 1 | 2 | 3 | 4 | |
0.1 | 0.25 | 0.3 | 0.2 | 0.15 |
Calculate (i)
Ans: We have
0 | 1 | 2 | 3 | 4 | |
0.1 | 0.25 | 0.3 | 0.2 | 0.15 | |
0 | 0.25 | 0.6 | 0.6 | 0.60 | |
0 | 0.25 | 1.2 | 1.8 | 2.40 |
here,
and
(ii) Variance of
Ans:
25. The probability distribution of a random variable
0 | 1 | 2 | 3 | |
(i) Determine the value of
Ans: We know,
(ii) Determine
Ans:
And
(iii) Find
Ans:
26. For the following probability distribution, determine standard deviation of the random variable
2 | 3 | 4 | |
0.2 | 0.5 | 0.3 |
Ans:
2 | 3 | 4 | |
0.2 | 0.5 | 0.3 | |
0.4 | 1.5 | 1.2 | |
0.8 | 4.5 | 4.8 |
We know that, standard deviation of
where,
27. A biased die is such that
Ans: Here,
So,
0 | 1 | 2 | |
0 | |||
0 |
28. A die is thrown three times. Let
Ans: We have,
Ans: Here, we have
and
Now
Hence, the required expectation is
29. Two biased dice are thrown together. For the first die
Ans: Given,for first die,
[\because P(1)=P(2)=P(3)=P(4)=P(5)]$
For second die,
Let
Hence, the required probability distribution is as below.
0 | 1 | 2 | |
0.54 | 0.42 | 0.04 |
30. Two probability distributions of the discrete random variable
0 | 1 | 2 | 3 | |
0 | 1 | 2 | 3 | |
Prove that,
Ans: We have to prove that,
Hence,
31. A factory produces bulbs. The probability that any one bulb is defective is
(i) none of the bulbs is defective
Ans: Let
Given,
None of the bulbs is defective, i.e.,
(ii) exactly two bulbs are defective
Ans: If exactly two bulbs are defective
(iii) more than 8 bulbs work properly.
Ans: More than 8 bulbs work properly,so, We can say that less than 2 bulbs are only defective
32. Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is a 2-headed coin. If you take one out, toss it and get a head, what is the probability that it was a fair coin?
Ans: Let
and
The probability of a fair coin=
33. Suppose that
Ans: Let
So, from Bayes' Theorem
34. Two natural numbers
Ans: Given,
Hence, the required probability is
35. Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.
Ans: Let
So,
Similarly
and
So, the required distribution:
1 | 2 | 3 | 4 | 5 | 6 | |
Mean
36. The random variable
Ans: Given ,
and
Let
Now we have the following distributions:
0 | 1 | 2 | |
and
Given that:
Hence, the required value of
37. Find the variance of the distribution:
0 | 1 | 2 | 3 | 4 | 5 | |
Ans: Variance (X)
And,
38. A and
Ans: Let
and
Let
and
39. Two dice are tossed. Find whether the following two events
Ans: We have ,
and
So,events
40. An urn contains
Ans: Let A = event of having
And,
Now
So, the probability of drawing a white ball does not depend upon
Long Answer Type Questions
41. Three bags contain a number of red and white balls as follows, Bag I: 3 red balls, Bag II: 2 red balls and 1 white ball and Bag III: 3 white balls. The probability that bag
What is the probability that
(i) a red ball will be selected
Ans: Given,
Bag I=three red balls and no white ball
Bag II=two red balls and one white ball
Bag III=no red ball and three white balls
Let
Let
(ii) a white ball is selected?
Ans: Let
42. Refer to Exercise Q.41 above. If a white ball is selected, what is the probability that it came from
(i) Bag II
Ans: We will use here Bayes' Theorem
(ii) Bag III?
Ans:
43. A shopkeeper sells three types of flower seeds
(i) of a randomly chosen seed to germinate
Ans: Given,
where
Let
(ii) that it will not germinate given that the seed is of type
Ans: From sol.(i)
(iii) that it is of the type
Ans: Using Bayes' Theorem,
44. A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just two consecutive letters TA are visible. What is the probability that the letter came from TATA NAGAR?
Ans: Let
and
(because CALCUTA, the two consecutive letters visible are CA, AL, LC, CU, UT, TT and TA)
Now using Bayes' Theorem,
45. There are two bags, one of which contains 3 black and 4 white balls while the other contains 4 black and 3 white balls. A die is thrown. If it shows up 1 or
Ans: Let
46. There are three urns containing 2 white and 3 black balls, 3 white and 2 black balls and 4 white and 1 black balls, respectively. There is an equal probability of each urn being chosen. A ball is drawn at random from the chosen urn and it is found to be white. Find the probability that the ball drawn was from the second urn.
Ans: Let
Let
So,
47. By examining the chest
Ans: Let
and
48. An item is manufactured by three machines
Ans: Let
Let
Using Bayes' Theorem,
49. Let
where
(i) the value of
Ans: Here,
Than,
Also,
and for otherwise = 0 .
The probability distribution:
1 | 2 | 3 | 4 | 5 | 6 | 7 | otherwise | |
0 |
We know ,
So,
So, the value of
(ii)
Ans: Now the probability distribution:
1 | 2 | 3 | 4 | 5 | 6 | 7 | |
(iii) Standard deviation of
Ans: Standard deviation
Variance
Than, Variance
S.D
50. The probability distribution of a discrete random variable
1 | 2 | 4 | 2A | 3A | 5A | |
Calculate: (i) The value of
Ans: We know,
So,
So,
(ii) Variance of
Ans: Now the distribution:
1 | 2 | 4 | 6 | 9 | 15 | |
So,Variance
51. The probability distribution of a random variable as under:
where
Calculate:
(i)
(ii)
(iii)
Ans:
1 | 2 | 3 | 4 | 5 | 6 | otherwise | |
0 |
We know,
(ii)
Ans: We know,
(iii)
Ans: We know,
52. A bag contains
Ans: Given ,
Let
But
53. Two cards are drawn successively without replacement from a well shuffled deck of cards. Find the mean and standard deviation of the random variable
Ans: Let
and
and
So,
Distribution Table:
0 | 1 | 2 | |
Mean
Standard deviation
54. A die is tossed twice.
Ans: Let
Here
0 | 1 | 2 | |
Variance
55. There are 5 cards numbered 1 to 5 , one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on two cards drawn. Find the mean and variance of X.
Ans: Sample space
So,
Let
So,
Mean,
Variance
Objective Type Questions
Choose the correct answer from the given four options in each of the Exercises from Q. 56 to 82.
56. If
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Because
57. If
(A)
(B)
(C)
(D)
Ans: Correct option is (a)
Because
58. If
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
=
=
59. If
(A)
(B)
(C)
(D) 1
Ans: Correct option is (c)
Because,
And
60. If
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Because,
Now,
61. If
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
62. If
(A)
(B)
(C)
(D)
Ans: Correct option is (b)
Because,if
63. A and B are events such that
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
64. If it is given that
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Because,
And
65. In Exercise 64 above,
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
66. If
(A)
(B)
(C)
(D) 1
Ans: Correct option is (d)
Because,
And
Since,
Also,
And
67. Let
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
68. If
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Because,
69. If
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
70. If two events are independent, then
(A) they must be mutually exclusive
(B) the sum of their probabilities must be equal to 1
(C) (A) and (B) are both are correct
(D) None of the above is correct
Ans: Correct option is (d)
Because,for independent events
So, they will not be mutually exclusive events.
In other words, two independent events having non-zero probabilities of occurrence cannot be mutually exclusive and conversely, two mutually exclusive events having non-zero probabilities of outcome cannot be independent.
71. Let
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
And
72. If the events
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Because,if
73. Two events
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Because,
Let
74. A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement the probability of getting exactly one red ball is
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Probability of getting exactly one red (R) ball
75. Refer to above Question 74 . The probability that exactly two of the three balls were red, the first ball being red, is
(A)
(B)
(C)
(D)
Ans: Correct option is (b)
Let
And
76. Three persons, A, B and C, fire at a target in turn, starting with A. Their probability of hitting the target are
(A)
(B)
(C)
(D)
Ans: Correct option is (b)
Because,
So,Probability of two hits
77. Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
78. A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number on the die and a spade card is
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Because,let
And
Then,
79. A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is
(A)
(B)
(C)
(D)
Ans: Correct option is (a)
Because,Probability of drawing 2 green balls and one blue ball=
80. A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, the probability that both are dead is
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because, probability
81. Eight coins are tossed together. The probability of getting exactly 3 heads is
(A)
(B)
(C)
(D)
Ans: Correct option is (b)
Because, Probability distribution
Here,
82. Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6 , the probability of getting a sum 3 , is
(A)
(B)
(C)
(D)
Ans: Correct option is (c)
Because,if let
And
And
83. Which one is not a requirement of a binomial distribution?
(A) There are 2 outcomes for each trial
(B) There is a fixed number of trials
(C) The outcomes must be dependent on each other
(D) The probability of success must be the same for all the trials
Ans: Correct option is (c)
84. Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability, that both cards are queens, is
(A)
(B)
(C)
(D)
Ans: Correct option is (a)
Because, probability
85. The probability of guessing correctly at least 8 out of 10 answers on a true-false type examination is
(A)
(B)
(C)
(D)
Ans: Correct option is (b)
Because,
and
86. The probability that a person is not a swimmer is
probability that out of 5 persons 4 are swimmers is
(A)
(B)
(C)
(D)
Ans: Correct option is (a)
Because,
Than probability
87. The probability distribution of a discrete random variable
2 | 3 | 4 | 5 | |
The value of
(A) 8
(B) 16
(C) 32
(D) 48
Ans: Correct option is (c)
Because,we know,
88. For the following probability distribution:
-4 | -3 | -2 | -1 | 0 | |
0.1 | 0.2 | 0.3 | 0.2 | 0.2 |
(A) 0
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
89. For the following probability distribution:
1 | 2 | 3 | 4 | |
(A) 3
(B) 5
(C) 7
(D) 10
Ans: Correct option is (d)
Because,
90. Suppose a random variable
(A)
(B)
(C)
(D)
Ans: Correct option is (a)
Because,
and
This is independent of
if
91. In a college
(A)
(B)
(C)
(D)
Ans: Correct option is (b)
Because,
92. A and
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
If,we let
Let
Let
Here,
93. A box has 100 pens of which 10 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement at most one is defective?
(A)
(B)
(C)
(D)
Ans: Correct option is (d)
Because,
Also,
State True or False for the statements in each of the Exercises 94 to 103.
94. Let
Ans: False
For events to be mutually exclusive -
But as per the conditions in question, it is not necessary that they will meet the condition because it might be possible that
For events to be independent-
Again
95. If
Ans: True
Since if
Now we know that
Hence
Since the events
Hence
Hence, we have
We know that
Hence we have
Hence the events
Since
96. If
Ans: False
If
97. Two independent events are always mutually exclusive.
Ans: False
Two events A and B are said to be independent if the occurrence of one does not affect the probability of the probability of the occurrence of the other. Thus if
98. If
Ans: True
Two events A and B are said to be independent if the occurrence of one does not affect the probability of the probability of the occurrence of the other. Thus if
99. Another name for the mean of a probability distribution is expected value.
Ans: True
Mean gives the average of values and if it is related with probability or random variable it is often called expected value.
100. If
Ans: True
101. If
Ans: False
We have,
Also,
102. If
Ans: False
So,
103. If
Ans: True
Fill in the Blanks in Each of the Following Questions:
104. If
Ans: Given,
and
105. If
Ans: Given,
106. If
Ans: because
107. Let
Ans: Because,
108. Let
Ans:
So,
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The NCERT Exemplar for Class 12 Maths Chapter 12 - Probability consists of problems and solutions which is focused on enabling students to get an in-depth understanding of basic concepts of Mathematics. It is recommended that students diligently practise NCERT Class 12 Maths Exemplar Solutions for Chapter 12 - Probability in order to fully understand and grasp complex mathematical problems, and be thoroughly prepared for the Exam. These Exemplar solutions are prepared by our subject experts for Class 12 students.
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3. What are the key differences between NCERT books and NCERT Exemplar problems?
The NCERT textbooks enable students to lay the basic foundation for the subjects. They explain topics in simple language, and consist of easy questions.
While the Exemplar provides complex questions that are of a higher level than that of the NCERT. The questions which are available in NCERT textbook are of basic level meant for understanding the concepts in simple terms. NCERT Exemplar must be referred to by students in order to master the concepts on an advanced level. In addition, the NCERT Exemplar helps students with preparation for competitive Examinations.
4. How does NCERT Exemplar help students for competitive Exams?
The NCERT Exemplar book consists of questions which judge a student’s in-depth understanding of key concepts. While NCERT books help students with their preparation for the Board Examination, the NCERT Exemplar books are considered essential for various competitive Exams such as JEE (Mains and Advanced), AIIMS Exam, NEET etc. As the NCERT Exemplar book contains questions that are of a higher difficulty level, it is considered as an important book for students preparation for the competitive Exams.
5. Is CBSE the same as NCERT?
No, they’re not the same. CBSE refers to the Central Board for Secondary Education. It is one of the most prestigious and prominent education boards in India. It is a government-affiliated education board. The Central Board for Secondary Education aims to holistically educate children and equip them with the skills required in the real world. The Board is responsible for the publication of the National Council of Educational Research and Training (NCERT) books. In addition, the Board is also responsible for creating the syllabus for students affiliated with the CBSE board.











