Keyy To Success – An I I T Alumnus Est.

997-111-6074
PROBABILITY
CLASSWORK – 1 5 6 7
P(A) = , P(B) = and P(A  B)  .
11 11 11
Q1. A die is rolled. If the outcome is an odd number, what is the
Find (i) P(A  B) (ii) P(A/B) (iii) P(B/A) (iv) P(A / B)
probability that it is prime?
Q2. Ten cards numbered 1 to 10 are placed in a box, mixed up Q12. Let A and B be the events such that
thoroughly and then one card is drawn randomly. If it is 3 1 2
P(A) = , P(B) = and P(B/A) = .
known that the number on the drawn card is more than 3, 10 2 5
what is the probability that it is an even number? Find (i) P(A  B) (ii) P(A  B) (iii) P(A/B)
Q3. A die is thrown twice and the sum of the numbers Q13. Let A and B be the events such that
appearing is observed to be 6. What is the conditional
5 2
probability that the number 4 has appeared at least once? 2P(A) = P(B) = and P(A/B) = .
Q4. Assume that each born child is equally likely to be a boy or 13 5
a girl. If a family has two children, what is the conditional Find (i) P(A  B) (ii) P(A  B) .
probability that both are girls given that (i) the youngest Q14. A die is rolled. If the outcome is an even number, what is
child is a girl, (ii) at least one of the children is a girl? the probability that it is a number greater than 2?
Q5. An instructor has a question bank consisting of 300 easy Q15. A coin is tossed twice. If the outcome is at most one tail,
true/false questions; 200 difficult true/false questions; what is the probability that both head and tail have
500 easy multiple-choice questions and 400 difficult appeared?
multiple-choice questions. If a question is selected at Q16. Three coins are tossed simultaneously. Find the
random from the question bank, what is the probability probability that all coins show heads if at least one of the
that it will be an easy question given that it is a multiple- coins shows a head.
choice question? Q17. Two unbiased dice are thrown. Find the probability that
Q6. Two numbers are selected at random from the integers 1 the sum of the numbers appearing is 8 or greater, if 4
through 9. If the sum is even, find the probability that both appears on the first die.
the numbers are odd. Q18. A die is thrown twice and the sum of the numbers
3 appearing is observed to be 8. What is the conditional
Q7. If A and B are the two events such that P(A) = , P(B) = probability that the number 5 has appeared at least once?
5
Q19. Two dice were thrown and it is known that the numbers
7 9
and P(A  B)  then find (i) P(A  B) (ii) which come up were different. Find the probability that the
10 10 sum of the two numbers was 5.
P(A/B) (iii) P(B/A). Q20. A coin is tossed and then a die is thrown. Find the
6 1 probability of obtaining a 6, given that a head came up.
Q8. Evaluate P(A  B) , if 2P(A) = P(B) = and P(A/B) =
13 3 Q21. A couple has 2 children. Find the probability that both are
1 1 boys if it is known that (i) one of the children is a boy, and
Q9. Let A and B be the events such that P(A) = , P(B) = (ii) the elder child is a boy.
3 4
Q22. In a class, 40% students study mathematics; 25% study
1 biology and 15% study both mathematics and biology. One
and P(A  B)  .
5 student is selected at random. Find the probability that
Find: (i) P(A/B) (ii) P(B/A) (iii) P(A  B) (iv) P(B / A) (i) he studies mathematics if it is known that he studies
biology
ANSWERS TO CLASSWORK – 1 (ii) he studies biology if it is known that he studies
mathematics.
Q23. The probability that a student selected at random from a
2 4
A1. A2. 4
3 7 class will pass in Hindi is and the probability that he
5
2 1 1
A3. A4. (i) (ii) 1
5 2 3 passes in Hindi and English is . What is the probability
2
5 5
A5. A6. that he will pass in English if it is known that he has passed
9 8 in Hindi?
2 4 2 7 Q24. The probability that a certain person will buy a shirt is 0.2,
A7. (i) (ii) (iii) A8.
5 7 3 13 the probability that he will buy a coat is 0.3 and the
4 3 23 37 probability that he will buy a shirt given that he buys a
A9. (i) (ii) (iii) (iv) coat is 0.4. Find the probability that he will buy both a
5 5 60 40
shirt and a coat.
Q25. In a hostel, 60% of the students read Hindi newspaper,
HOMEWORK – 1 40% read English newspaper and 20% read both Hindi
and English newspapers. A student is selected at random.
Q10. Let A and B be the events such that (i) Find the probability that he reads neither Hindi nor
7 9 4 English newspaper.
P(A) = , P(B) = and P(A  B)  .
13 13 13 (ii) If he reads Hindi newspaper, what is the probability
Find (i) P(A/B) (ii) P(B/A) (iii) P(A  B) (iv) P(B / A) that he reads English newspaper?
Q11. Let A and B be the events such that
Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 1
 Keyy To Success – An I I T Alumnus Est. 997-111-6074
(iii) If he reads English newspaper, what is the probability them will solve a problem selected at random from the
that he reads Hindi newspaper? book?
Q26. Two integers are selected at random from integers 1 Q35. The probability that A hits a target is (1/3) and the
through 11. If the sum is even, find the probability that probability that B hits it is (2/5). What is the probability
both the numbers selected are odd. that the target will be hit if both A and B shoot at it?
Q36. A and B appear for an interview for two posts. The
ANSWERS TO HOMEWORK – 1 probability of A‘s selection is (1/3) and that of B‘s
selection is (2/5). Find the probability that only one of
4 4 12 1 them will be selected.
A10. (i) (ii) (iii) (iv) Q37. A speaks the truth in 60% of the cases, and B in 90% of the
9 7 13 6
cases. In what percentage of cases are they likely to
4 2 4 4
A11. (i) (ii) (iii) (iv) contradict each other in stating the same fact?
11 3 5 5 Q38. The probabilities of a specific problem being solved
3 17 6 2 11 independently by A and B are 1/2 and 1/3 respectively. If
A12. (i) (ii) (iii) A13. (i) (ii)
25 25 25 13 26 both try to solve the problem independently, find the
2 2 probability that
A14. A15. (i) the problem is solved
3 3
(ii) exactly one of them solves the problem.
1 1 Q39. Amit and Nisha appear for an interview for two vacancies
A16. A17.
7 2 in a company. The probability of Amit’s selection is 1/5
2 2 and that of Nisha’s selection is 1/6. What is the probability
A18. A19.
5 15 that
1 1 1 (i) both of them are selected?
A20. A21. (i) (ii) (ii) only one of them is selected?
6 3 2 (iii) none of them is selected?
3 3 5 Q40. Three groups of children contain 3 girls and 1 boy; 2 girls
A22. (i) (ii) A23.
5 8 8 and 2 boys; and 1 girl and 3 boys. One child is selected at
1 1 1 random from each group. Find the chance that the three
A24. 0.12 A25. (i) (ii) (iii) children selected comprise 1 girl and 2 boys.
5 3 2
Q41. A problem is given to three students whose chances of
3 solving it are 1/3, 2/7 and 3/8. What is the probability
A26.
5 that the problem will be solved?
Q42. A problem in mathematics is given to three students whose
CLASSWORK –2 chances of solving it correctly are 1/2, 1/3 and 1/4
respectively. What is the probability that only one of them
Q27. An urn contains 8 white and 4 red balls. Two balls are solves it correctly?
drawn from the urn one after the other without Q43. Three critics review a book. For the three critics, the odds
replacement. What is the probability that both drawn balls in favour of the book are (5 : 2), (4 : 3) and ( 3 : 4)
are white? respectively. Find the probability that the majority is in
Q28. Three cards are drawn successively without replacement favour of the book.
from a pack of 52 well-shuffled cards. What is the Q44. The odds against a man who is 45 years old, living till he is
probability that first two cards are queens and the third 70 are 7 : 5, and the odds against his wife who is now 36,
card drawn is a king? living till she is 61 are 5 : 3.
Q29. Let E1 and E2 be two events such that P(E1 ) = 0.3, Find the probability that
P(E1  E2 ) = 0.4 and P(E2 ) = x. Find the value of x such (i) the couple will be alive 25 years hence
(ii) at least one of them will be alive 25 years hence.
that
Q45. A, B and C shoot to hit a target. If A hits the target 4 times
(i) E1 and E2 are mutually exclusive,
in 5 trials; B hits it 3 times in 4 trials and C hits it 2 times in
(ii) E1 and E2 are independent. 3 trials, what is the probability that the target is hit by at
Q30. Let E1 and E2 are the two independent events such that least 2 persons?
P(E1 ) = 0.35 and P(E1  E2 ) = 0.60, find P(E2 ) .
ANSWERS TO CLASSWORK – 2
Q31. A coin is tossed thrice. Let the event E be ‘the first throw
results in a head’, and the event F be ‘the last throw results
14 2
in a tail’. Find whether the events E and F are independent. A27. A28.
Q32. An unbiased die is tossed twice. Find the probability of 33 5525
getting a 4, 5 or 6 on the first toss and a 1, 2, 3 or 4 on the 1 5
A29. (i) 0.1 (ii) A30.
second toss. 7 13
Q33. Ramesh appears for an interview for two posts, A and B, 1
for which the selection is independent. The probability for A31. Independent events A32.
3
his selection for Post A is (1/6) and for Post B, it is (1/7).
Find the probability that Ramesh is selected for at least 2
A33. A34. 0.97
one post. 7
Q34. A can solve 90% of the problems given in a book, and B 3 7
can solve 70%. What is the probability that at least one of A35. A36.
5 15

Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 2

 Keyy To Success – An I I T Alumnus Est. 997-111-6074
2 1 Q56. Kamal and Vimal appeared for an interview for two
A37. 42% A38. (i) (ii) vacancies. The probability of Kamal’s selection is 1/3 and
3 2
that of Vimal’s selection is 1/5. Find the probability that
1 3 2 13
A39. (i) (ii) (iii) A40. only one of them will be selected.
30 10 3 32 Q57. Arun and Ved appeared for an interview for two vacancies.
59 11 The probability of Arun’s selection is 1/4 and that of Ved’s
A41. A42.
84 24 rejection is 2/3. Find the probability that at least one of
209 5 61 them will be selected.
A43. A44. (i) (ii) Q58. A and B appear for an interview for two vacancies in the
343 32 96
same post. The probability of A’s selection is 1/6 and that
5 of B’s selection is 1/4. Find the probability that
A45.
6 (i) both of them are selected
(ii) only one of them is selected
HOMEWORK – 2 (iii) none is selected
(iv) at least one of them is selected.
Q46. A bag contains 17 tickets, numbered from 1 to 17. A ticket Q59. Given the probability that A can solve a problem is 2/3,
is drawn and then another ticket is drawn without and the probability that B can solve the same problem is
replacing the first one. Find the probability that both the 3/5, find the probability that
tickets may show even numbers. (i) at least one of A and B will solve the problem
Q47. Two marbles are drawn successively from a box containing (ii) none of the two will solve the problem.
3 black and 4 white marbles. Find the probability that both Q60. A problem is given to three students whose chances of
the marbles are black, if the first marble is not replaced solving it are 1/4, 1/5 and 1/6 respectively. Find the
before the second draw. probability that the problem is solved.
Q48. A card is drawn from a well-shuffled deck of 52 cards and Q61. The probabilities of A, B, C solving a problem are 1/3, 1/4
without replacing this card, a second card is drawn. Find and 1/6 respectively. If all the three try to solve the
the probability that the first card is a club and the second problem simultaneously, find the probability that exactly
card is a spade. one of them will solve it.
Q49. There is a box containing 30 bulbs of which 5 are defective. Q62. A can hit a target 4 times in 5 shots, B can hit 3 times in 4
If two bulbs are chosen at random from the box in shots, and C can hit 2 times in 3 shots. Calculate the
succession without replacing the first, what is the probability that
probability that both the bulbs chosen are defective? (i) A, B and C all hit the target
Q50. A bag contains 10 white and 15 black balls. Two balls are (ii) B and C hit and A does not hit the target.
drawn in succession without replacement. What is the
probability that the first ball is white and the second is Q63. Neelam has offered physics, chemistry and mathematics in
black? Class XII. She estimates that her probabilities of receiving a
Q51. An urn contains 5 white and 8 black balls. Two successive grade A in these courses are 0.2, 0.3 and 0.9 respectively.
drawings of 3 balls at a time are made such that the balls Find the probabilities that Neelam receives
drawn in the first draw are not replaced before the second (i) all A grades (ii) no A grade (iii) exactly 2 A grades.
draw. Find the probability that the first draw gives 3 white Q64. An article manufactured by a company consists of two
balls and the second draw gives 3 black balls. parts X and Y. In the process of manufacture of part X, 8 out
1 of 100 parts may be defective. Similarly, 5 out of 100 parts
Q52. Let E1 and E2 be two events such that P(E1 ) = and of Y may be defective. Calculate the probability that the
3
assembled product will not be defective.
3
P(E2 ) = . Find Q65. A town has two fire-extinguishing engines, functioning
5 independently. The probability of availability of each
(i) P(E1  E2 ) , when E1 and E2 are mutually exclusive, engine when needed is 0.95. What is the probability that
(ii) P(A  B) , when E1 and E2 are independent. (i) neither of them is available when needed?
(ii) an engine is available when needed?
1
Q53. If E1 and E2 are the two events such that P(E1 ) = , Q66. A machine operates only when all of its three components
4 function. The probabilities of the failures of the first,
1 1 second and third components are 0.14, 0.10 and 0.05
P(E2 ) = and P(E1  E2 ) = , show that E1 and E2
3 2 respectively. What is the probability that the machine will
are independent events. fail?
Q54. If E1 and E2 are independent events such that P(E1 ) = Q67. An anti-aircraft gun can take a maximum of 4 shots at an
0.3 and P(E2 ) = 0.4, enemy plane moving away from it. The probabilities of
hitting the plane at the first, second, third and fourth shots
find (i) P(A  B) (ii) P(E1  E2 ) (iii) P(E1  E2 ) (iv) are 0.4, 0.3, 0.2 and 0.1 respectively. What is the
P(E1  E2 ) probability that at least one shot hits the plane?
Q68. Let S1 and S2 be the two switches and let their
1 1
Q55. Let A and B be the events such that P(A) = , P(B) = probabilities of working be given by P(S1 ) = 4/5 and
2 2
1 P(S2 ) = 9/10. Find the probability that the current flows
and P(not A or not B) = .
4 from the terminal A to terminal B when S1 and S2 are
State whether A and B are installed in series, shown as follows:
(i) mutually exclusive, (ii) independent.

Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 3

 Keyy To Success – An I I T Alumnus Est. 997-111-6074
defective. What is the probability that this defective bolt
has been produced by the machine X?
Q69. Let S1 and S2 be two the switches and let their Q73. In a bolt factory, three machines, A, B, C, manufacture 25%,
35% and 40%of the total production respectively. Of their
probabilities of working be given by P(S1 ) = 2/3 and
respective outputs, 5%, 4% and 2% are defective. A bolt is
P(S2 ) = 3/4. Find the probability that the current flows drawn at random from the total product and it is found to
from terminal A to terminal B, when S1 and S2 are be defective. Find the probability that it was manufactured
installed in parallel, as shown below: by the machine C.
Q74. A company has two plants to manufacture bicycles. The
first plant manufactures 60% of the bicycles and the
second plant, 40%. Also, 80% of the bicycles are rated of
standard quality at the first plant and 90% of standard
quality at the second plant. A bicycle is picked up at
Q70. A coin is tossed. If a head comes up, a die is thrown but if a
random and found to be of standard quality. Find the
tail comes up, the coin is tossed again. Find the probability
probability that it comes from the second plant.
of obtaining
Q75. An insurance company insured 2000 scooter drivers, 4000
(i) two tails (ii) a head and the number 6 car drivers and 6000 truck drivers. The probability of an
(iii) a head and an even number.
1 3
accident involving a scooter, a car and a truck is ,
ANSWERS TO HOMEWORK – 2 100 100
3
and respectively. One of the insured persons meets
7 1 20
A46. A47.
34 7 with an accident. What is the probability that he is a
13 2 scooter driver?
A48. A49. Q76. A doctor is to visit a patient. From past experience, it is
204 87
known that the probabilities that he will come by train,
1 7 3 1 1 2
A50. A51.
4 429 bus, scooter or by car are respectively , , and .
10 5 10 5
14 1 1 1 1
A52. (i) (ii)
15 5 The probabilities that he will be late are , and , if
4 3 12
A54. (i) 0.12 (ii) 0.58 (iii) 0.42 (iv) 0.28
he comes by train, bus and scooter respectively; but if he
2 comes by car, he will not be late. When he arrives, he is
A55. (i) No (ii) No A56.
5 late. What is the probability that he has come by train?
1 Q77. A man is known to speak the truth 3 out of 4 times. He
A57. throws a die and reports that it is a six. Find the probability
2
1 1 5 3 that it is actually a six.
A58. (i) (ii) (iii) (iv) Q78. In an examination, an examinee either guesses or copies or
24 3 8 8 knows the answer to a multiple-choice question with four
13 2 1 choices. The probability that he makes a guess is (1/3) and
A59. (i) (ii) A60.
15 15 2 the probability that he copies the answer is (1/6). The
31 2 1 probability that his answer is correct, given that he copied
A61. A62. (i) (ii) it, is (1/8). The probability that his answer is correct, given
72 5 10
that he guessed it, is (1/4). Find the probability that he
A63. (i) 0.054 (ii) 0.056 (iii) 0.348
knew the answer to the question, given that he correctly
437 1 19 answered it.
A64. A65. (i) (ii)
500 400 200 Q79. By examining the chest X-ray, the probability that a person
A66. 0.2647 A67. 0.6976 is diagnosed with TB when he is actually suffering from it,
18 11 is 0.99. The probability that the doctor incorrectly
A68. A69.
25 12 diagnoses a person to be having TB, on the basis of X-ray
1 1 3 reports, is 0.001. In a certain city, 1 in 1000 persons suffers
A70. (i) (ii) (iii) from TB. A person is selected at random and is diagnosed
8 8 8 to have TB. What is the chance that he actually has TB?
Q80. Bag A contains 2 white and 3 red balls, and bag B contains
CLASSWORK – 3 4 white and 5 red balls. One ball is drawn at random from
one of the bags and it is found to be red. Find the
Q71. There are three urns containing 3 white and 2 black balls; probability that it was drawn from bag B.
2 white and 3 black balls; 1 black and 4 white balls Q81. There are 5 bags, each containing 5 white balls and 3 black
respectively. There is equal probability of each urn being balls. Also, there are 6 bags, each containing 2 white balls
chosen. One ball is drawn from an urn chosen at random. and 4 black balls. A white ball is drawn at random. Find the
What is the probability that a white ball is drawn? probability that this white ball is from a bag of the first
Q72. A factory has three machines, X, Y and Z, producing 1000, group.
2000 and 3000 bolts per day respectively. The machine X Q82. Urn A contains 1 white, 2 black and 3 red balls; urn B
produces 1% defective bolts, Y produces 1.5% defective contains 2 white, 1 black and 1 red ball; and urn C contains
bolts and Z produces 2%defective bolts. At the end of the 4 white, 5 black and 3 red balls. One urn is chosen at
day, a bolt is drawn at random and it is found to be

Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 4

 Keyy To Success – An I I T Alumnus Est. 997-111-6074
random and two balls are drawn. These happen to be one Q90. A bag A contains 1 white and 6 red balls. Another bag
white and one red. What is the probability that they come contains 4 white and 3 red balls. One of the bags is selected
from urn A? at random and a ball is drawn from it, which is found to be
Q83. A card from a pack of 52 cards is lost. From the remaining white. Find the probability that the ball drawn is from the
cards of the pack, two cards are drawn and are found to be bag A.
both spades. Find the probability of the lost card being a Q91. There are two bags I and II. Bag I contains 3 white and 4
spade. black balls, and bag II contains 5 white and 6 black balls.
One ball is drawn at random from one of the bags and is
found to be white. Find the probability that it was drawn
ANSWERS TO CLASSWORK – 3 from bag I.
Q92. A box contains 2 gold and 3 silver coins. Another box
3 contains 3 gold and 3 silver coins. A box is chosen at
A71. A72. 0.1 random, and a coin is drawn from it. If the selected coin is a
5
gold coin, find the probability that it was drawn from the
16 3
A73. A74. second box.
69 7 Q93. Three urns A, B and C contain 6 red and 4 white; 2 red and
1 1 6 white; and 1 red and 5 white balls respectively. An urn is
A75. A76.
52 2 chosen at random and a ball is drawn. If the ball drawn is
3 24 found to be red, find the probability that the ball was
A77. A78. drawn from the urn A.
8 29
Q94. Three urns contain 2 white and 3 black balls; 3 white and 2
110 25 black balls, and 4 white and 1 black ball respectively. One
A79. A80.
221 52 ball is drawn from an urn chosen at random and it was
75 33 found to be white. Find the probability that it was drawn
A81. A82. from the first urn.
123 118
A83. 0.22 Q95. There are three boxes, the first one containing 1 white, 2
red and 3 black balls; the second one containing 2 white, 3
HOMEWORK – 3 red and 1 black ball and the third one containing 3 white, 1
red and 2 black balls. A box is chosen at random and from
Q84. In a bulb factory, three machines, A, B, C, manufacture it two balls are drawn at random. One ball is red and the
60%, 25% and 15% of the total production respectively. Of other, white. What is the probability that they come from
their respective outputs, 1%, 2% and 1% are defective. A the second box?
bulb is drawn at random from the total product and it is Q96. Urn A contains 7 white and 3 black balls; urn B contains 4
found to be defective. Find the probability that it was white and 6 black balls; urn C contains 2 white and 8 black
manufactured by machine C. balls. One of these urns is chosen at random with
Q85. A company manufactures scooters at two plants, A and B. probabilities 0.2, 0.6 and 0.2 respectively. From the chosen
Plant A produces 80% and plant B produces 20% of the urn, two balls are drawn at random without replacement.
total product. 85% of the scooters produced at plant A and Both the balls happen to be white. Find the probability that
65% of the scooters produced at plant B are of standard the balls drawn are from the urn C.
quality. A scooter produced by the company is selected at Q97. There are 3 bags, each containing 5 white and 3 black balls.
random and it is found to be of standard quality. What is Also, there are 2 bags, each containing 2 white and 4 black
the probability that it was manufactured at plant A? balls. A white ball is drawn at random. Find the probability
Q86. In a certain college, 4% of boys and 1% of girls are taller that this ball is from a bag of the first group.
than 1.75 metres. Furthermore, 60% of the students are Q98. There are four boxes, A, B, C and D, containing marbles. A
girls. If a student is selected at random and is taller than contains 1 red,6 white and 3 black marbles; B contains 6
1.75 metres, what is the probability that the selected red, 2 white and 2 black marbles; C contains 8 red, 1 white
student is a girl? and 1 black marbles; and D contains 6 white and 4 black
Q87. In a class, 5% of the boys and 10% of the girls have an IQ of marbles. One of the boxes is selected at random and a
more than 150. In this class, 60% of the students are boys. single marble is drawn from it. If the marble is red, what is
If a student is selected at random and found to have an IQ the probability that it was drawn from the box A?
of more than 150, find the probability that the student is a Q99. A car manufacturing factory has two plants X and Y. Plant X
boy. manufactures 70% of the cars and plant Y manufactures
Q88. Suppose 5% of men and 0.25% of women have grey hair. A 30%. At plant X, 80% of the cars are rated of standard
grey-haired person is selected at random. What is the quality and at plant Y, 90% are rated of standard quality. A
probability of this person being male? Assume that there car is picked up at random and is found to be of standard
are equal number of males and females. quality. Find the probability that it has come from plant X.
Q89. Two groups are competing for the positions on the board Q100. An insurance company insured 2000 scooters and 3000
of directors of a corporation. The probabilities that the first motorcycles. The probability of an accident involving a
and the second groups will win are 0.6 and 0.4 scooter is 0.01 and that of a motorcycle is 0.02. An insured
respectively. Further, if the first group wins, the vehicle met with an accident. Find the probability that the
probability of introducing a new product is 0.7 and when accidented vehicle was a motorcycle.
the second group wins, the corresponding probability is Q101. In a bulb factory, machines A, B and C manufacture 60%,
0.3. Find the probability that the new product introduced 30% and 10% bulbs respectively. Out of these bulbs 1%,
was by the second group. 2% and 3% of the bulbs produced respectively by A, B and
C are found to be defective. A bulb is picked up at random

Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 5

 Keyy To Success – An I I T Alumnus Est. 997-111-6074
from the total production and found to be defective. Find 1 1
the probability that this bulb was produced by the machine A104. Mean = , variance =
2 4
A.
1 5 1 5
A105. Mean = , variance = , standard deviation = .
ANSWERS TO HOMEWORK – 3 3 18 3 2
2 24
A106. Mean = , variance =
2 68 13 169
A84. A85.
25 81 2 400
A107. Mean = , variance =
3 3 13 2873
A86. A87.
11 7 9 49
A108. Mean = , variance =
2 2 10 100
A88. A89.
3 9 9 24
A109. Mean = , variance =
1 33 7 49
A90. A91.
5 68 A110.loses 1 per toss
5 36
A92. A93.
9 61 HOMEWORK – 4
2 6
A94. A95.
9 11 Q111. Find the mean ( ) , variance (2 ) and standard deviation
1 45
A96. A97. () for each of the following probability distributions:
40 61
(i)
1 56 x 0 1 2 3
A98. A99.
15 83 p(x) 1 1 3 1
3 2 6 2 10 30
A100. A101.
4 5
(ii)
CLASSWORK – 4 xi 1 2 3 4

Q102. Find the mean, variance and standard deviation of the

pi 0.4 0.3 0.2 0.1
number of tails in two tosses of a coin.
Q103. Find the mean, variance and standard deviation of the (iii)
number of heads when three coins are tossed. xi ‒3 ‒1 0 2
Q104. A die is tossed once. If the random variable X is defined as pi 0.2 0.4 0.3 0.1
1, if the die results in an even number
X  
0, if the die results in an odd number (iv)
then find the mean and variance of X. xi ‒2 ‒1 0 1 2
Q105. Find the mean, variance and standard deviation of the pi 0.1 0.2 0.4 0.2 0.1
number of sixes in two tosses of a die.
Q112. Find the mean and variance of the number of heads when
Q106. Two cards are drawn successively with replacement from
two coins are tossed simultaneously.
a well-shuffled pack of 52 cards. Find the mean and
Q113. Find the mean and variance of the number of tails when
variance of the number of kings.
three coins are tossed simultaneously.
Q107. Two cards are drawn simultaneously (or successively
Q114. A die is tossed twice. ‘Getting an odd number on a toss’ is
without replacement) from a well-shuffled pack of 52
considered a success. Find the probability distribution of
cards. Find the mean and variance of the number of aces.
number of successes. Also, find the mean and variance of
Q108. Three defective bulbs are mixed with 7 good ones. Let X
the number of successes.
be the number of defective bulbs when 3 bulbs are drawn
Q115. A die is tossed twice. ‘Getting a number greater than 4’ is
at random. Find the mean and variance of X.
considered a success. Find the probability distribution of
Q109. An urn contains 4 white and 3 red balls. Let X be the
number of successes. Also, find the mean and variance of
number of red balls in a random draw of 3 balls. Find the
the number of successes.
mean and variance of X.
Q116. A pair of dice is thrown 4 times. If getting a doublet is
Q110. In a game, 3 coins are tossed. A person is paid 5 if he
considered a success, find the probability distribution of
gets all heads or all tails; and he is supposed to pay 3 if he
number of successes. Also, find the mean and variance of
gets one head or two heads. What can he expect to win on
number of successes.
an average per game?
Q117. A coin is tossed 4 times. Let X denote the number of
heads. Find the probability distribution of X. Also, find the
ANSWERS TO CLASSWORK – 4
mean and variance of X.
Q118. Let X denote the number of times ‘a total of 9’ appears in
1
A102. Mean = 1, variance = , standard deviation = 0.707 two throws of a pair of dice. Find the probability
2 distribution of X. Also, find the mean, variance and
3 3 3 standard deviation of X.
A103. Mean = , variance = , standard deviation =
2 4 2

Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 6

 Keyy To Success – An I I T Alumnus Est. 997-111-6074
Q119. There are 5 cards, numbered 1 to 5, one number on each A117. Mean = 2, variance = 1
card. Two cards are drawn at random without X  xi 0 1 2 3 4
replacement. Let X denote the sum of the numbers on the pi 1 1 3 1 1
two cards drawn. Find the mean and variance of X.
Q120. Two cards are drawn from a well-shuffled pack of 52 16 4 8 4 16
cards. Find the probability distribution of number of kings. 2 16 4
A118. Mean = , variance = , SD =
Also, compute the variance for the number of kings. 9 81 9
Q121. A box contains 16 bulbs, out of which 4 bulbs are X  xi 0 1 2
defective. Three bulbs are drawn at random from the box.
Let X be the number of defective bulbs drawn. Find the pi 64 16 1
mean and variance of X. 81 81 81
Q122. 20% of the bulbs produced by a machine are defective. A119. Mean = 6, variance = 3
Find the probability distribution of the number of defective X  xi 3 4 5 6 7 8 9
bulbs in a sample of 4 bulbs chosen at random.
pi 1 1 1 1 1 1 1
Q123. Four bad eggs are mixed with 10 good ones. Three eggs
are drawn one by one without replacement. Let X be the 10 10 5 5 5 10 10
number of bad eggs drawn. Find the mean and variance of 400
A120. variance =
X. 2873
Q124. Four rotten oranges are accidentally mixed with 16 good X  xi 0 1 2
ones. Three oranges are drawn at random from the mixed
lot. Let X be the number of rotten oranges drawn. Find the pi 188 32 1
mean and variance of X. 221 221 221
Q125. Three balls are drawn simultaneously from a bag 3 39
containing 5 white and 4 red balls. Let X be the number of A121. Mean = , variance =
4 80
red balls drawn. Find the mean and variance of X.
6 30
Q126. Two cards are drawn without replacement from a well- A123. Mean = , variance =
shuffled deck of 52 cards. Let X be the number of face cards 7 49
drawn. Find the mean and variance of X. 3 68
A124. Mean = , variance =
Q127. Two cards are drawn one by one with replacement from 5 125
a well-shuffled deck of 52 cards. Find the mean and 4 5
variance of the number of aces. A125. Mean = , variance =
3 9
Q128. Three cards are drawn successively with replacement
from a well-shuffled deck of 52 cards. A random variable X 20 1000
A126. Mean = , variance =
denotes the number of hearts in the three cards drawn. 13 2873
Find the mean and variance of X. 2 24
Q129. Five defective bulbs are accidently mixed with 20 good A127. Mean = , variance =
13 169
ones. It is not possible to just look at a bulb and tell
3 9
whether or not it is defective. Find the probability A128. Mean = , variance =
distribution from this lot. 4 16
A129.
ANSWERS TO HOMEWORK – 4 X 0 1 2 3 4
P(X) 969 114 38 4 1
A111. (i) Mean = 1.2, variance = 0.56, SD = 0.74 2530 253 253 253 2530
(ii) Mean = 2, variance = 1, SD = 1
(iii) Mean = -0.8, variance = 2.6, SD = 1.612
(iii) Mean = 0, variance = 1.2, SD = 1.095 CLASSWORK – 5
A112. Mean = 1, variance = 0.5
A113. Mean = 1.5, variance = 0.75 Q130. A coin is tossed 4 times. If X is the number of heads
A114. Mean = 1, variance = 0.5 observed, find the probability distribution of X.
X  xi 0 1 2 Q131. Find the probability distribution of the number of sixes in
pi 1 1 1 three tosses of a die.
4 2 4 Q132. Find the probability distribution of the number of
doublets in four throws of a pair of dice.
2 4
A115. Mean = , variance = Q133. An unbiased coin is tossed 6 times. Find, using binomial
3 9 distribution, the probability of getting at least 5 heads.
X  xi 0 1 2 Q134. An unbiased coin is tossed 6 times. Find, using binomial
pi 4 4 1 distribution, the probability of getting at least 5 heads.
Q135. Six coins are tossed simultaneously. Find the probability
9 9 9
of getting
2 5 (i) 3 heads (ii) no head (iii) at least one head
A116. Mean = , variance =
3 9 (iv) not more than 3 heads.
X  xi 0 1 2 3 4 Q136. A die is thrown 5 times. If getting an odd number is a
success, find the probability of getting at least 4 successes.
pi 625 125 25 5 1
Q137. In 4 throws with a pair of dice, what is the probability of
1296 324 216 324 1296 throwing doublets at least twice?

Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 7

 Keyy To Success – An I I T Alumnus Est. 997-111-6074
Q138. The bulbs produced in a factory are supposed to contain A147. Mean = 6.67 and variance = 4.44
5% defective bulbs. What is the probability that a sample A148. Variance = 25, SD = 5
of 10 bulbs will contain not more than 2 defective bulbs?
Q139. If on an average, out of 10 ships, one gets drowned then HOMEWORK – 5
what is the probability that out of 5 ships at least 4 reach
the shore safely? Q149. A coin is tossed 6 times. Find the probability of getting at
Q140. If X follows a binomial distribution with mean 3 and least 3 heads.
variance (3/2), find Q150. A coin is tossed 5 times. What is the probability that a
(i) P(X  1) (ii) P(X  5) . head appears an even number of times?
Q141. If X follows a binomial distribution with mean 4 and Q151. 7 coins are tossed simultaneously. What is the probability
variance 2, find P(X  5) . that a tail appears an odd number of times?
Q152. A coin is tossed 6 times. Find the probability of getting
Q142. Find the binomial distribution for which the mean and
(i) exactly 4 heads (ii) at least 1 head (iii) at most 4 heads.
variance are 12 and 3 respectively.
Q153. 10 coins are tossed simultaneously. Find the probability
Q143. If the sum of the mean and variance of a binomial
of getting
distribution for 5 trials is 1.8, find the distribution.
(i) exactly 3 heads (ii) not more than 4 heads
Q144. The sum and the product of the mean and variance of a
(iii) at least 4 heads.
binomial distribution are 24 and 128 respectively. Find the
Q154. A die is thrown 6 times. If ‘getting an even number’ is a
distribution.
success, find the probability of getting
Q145. In a binomial distribution, prove that mean > variance.
(i) exactly 5 successes (ii) at least 5 successes
Q146. A die is tossed thrice. Getting an even number is
(iii) at most 5 successes.
considered a success. What is the variance of the binomial
Q155. A die is thrown 4 times. ‘Getting a 1 or a 6’ is considered a
distribution?
success. Find the probability of getting
Q147. A die is rolled 20 times. Getting a number greater than 4
(i) exactly 3 successes (ii) at least 2 successes
is a success. Find the mean and variance of the number of
(iii) at most 2 successes.
successes.
Q156. Find the probability of a 4 turning up at least once in two
Q148. A die is tossed 180 times. Find the expected number ( ) tosses of a fair die.
of times the face with the number 5 will appear. Also, find Q157. A pair of dice is thrown 4 times. If ‘getting a doublet’ is
the standard deviation () , and variance ( 2 ) . considered a success, find the probability of getting 2
successes.
Q158. A pair of dice is thrown 7 times. If ‘getting a total of 7’ is
ANSWERS TO CLASSWORK – 5
considered a success, find the probability of getting
(i) no success (ii) exactly 6 successes
 X: 0 1 2 3 4  (iii) at least 6 successes (iv) at most 6 successes.
A130.  
 P(X): 1 1 3 1 1  Q159. There are 6% defective items in a large bulk of items.
 16 4 8 4 16  Find the probability that a sample of 8 items will include
 X: 0 1 2 3  not more than one defective item.
A131.   Q160. In a box containing 60 bulbs, 6 are defective. What is the
 P(X): 125 25 5 1 
 probability that out of a sample of 5 bulbs (i) none is
 216 72 72 216  defective, (ii) exactly 2 are defective?
 X: 0 1 2 3 4  Q161. The probability that a bulb produced by a factory will
  fuse after 6 months of use is 0.05. Find the probability that
 P(X): 625 125 25 5 1 
A132.
 out of 5 such bulbs
 1296 324 216 324 1296 
(i) none will fuse after 6 months of use
7 219 (ii) at least one will fuse after 6 months of use
A133. A134.
64 256 (iii) not more than one will fuse after 6 months of use.
5 1 63 21 Q162. In the items produced by a factory, there are 10%
A135. (i) (ii) (iii) (iv)
16 64 64 32 defective items. A sample of 6 items is randomly chosen.
Find the probability that this sample contains (i) exactly 2
3 171
A136. A137. defective items, (ii) not more than 2 defective items, (iii) at
16 1296 least 3 defective items.
99 Q163. Assume that on an average one telephone number out of
A138. A139. 0.9181
100 15, called between 3 p.m. and 4 p.m. on weekdays, will be
63 63 93 busy. What is the probability that if six randomly selected
A140. (i) (ii) A141. telephone numbers are called, at least 3 of them will be
64 64 256
busy?
r (16  r)
3 1 Q164. Three cars participate in a race. The probability that any
A142. 16 Cr .   .   , where r = 0, 1, 2, 3, 4 …..15
4 4 one of them has an accident is 0.1. Find the probability that
all the cars reach the finishing line without any accident.
A143. 5 Cr . (0.2)r . (0.8)(5  r) , where r = 0, 1, 2, 3, 4, 5 Q165. Past records show that 80% of the operations performed
32 by a certain doctor were successful. If he performs 4
1
A144. 32 Cr .   operations in a day, what is the probability that at least 3
 2 operations will be successful?
3
A146.
4

Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 8

 Keyy To Success – An I I T Alumnus Est. 997-111-6074
Q166. The probability of a man hitting a target is (1/4). If he 7
5 35 1 1
fires 7 times, what is the probability of his hitting the A158. (i)   (ii) (iii) (iv) 1 
6 67 65 67
target at least twice?
Q167. In a hurdles race, a player has to cross 10 hurdles. The 7
 47   71 
probability that he will clear each hurdle is (5/6). What is A159.     
the probability that he will knock down fewer than 2  50   50 
5
hurdles?  9  729
Q168. A man can hit a bird, once in 3 shots. On this assumption A160. (i)   (ii)
 10  10000
he fires 3 shots. What is the chance that at least one bird is
5 5 4
hit?  19   19   19 
6
A161. (i)   (ii) 1    (iii)  
  
Q169. If the probability that a man aged 60 will live to be 70 is 20
  20
   20 
5
0.65, what is the probability that out of 10 men, now 60, at 4 4
3  9  3  19   3  19 4 
least 8 will live to be 70? A162.(i)    (ii)    (iii) 1      
Q170. A bag contains 5 white, 7 red and 8 black balls. If four 20  10  2  20   2  20  
balls are drawn one by one with replacement, what is the  
4
probability that (i) none is white, (ii) all are white, (iii) at  14   59  729
least one is white? A163. 1    .   A164.
 15   45  1000
Q171. A policeman fires 6 bullets at a burglar. The probability
that the burglar will be hit by a bullet is 0.6.What is the 512 4547
A165. A166.
probability that the burglar is still unhurt? 625 8192
Q172. A die is tossed thrice. A success is 1 or 6 on a toss. Find 510 19
the mean and variance of successes. A167. A168.
9 27
Q173. A die is thrown 100 times. Getting an even number is (2  6 )
considered a success. Find the mean and variance of A169. 0.2615
successes. 81 1 175
A170. (i) (ii) (iii)
Q174. Determine the binomial distribution whose mean is 9 and 256 256 256
variance is 6. A171. 0.004096
Q175. Find the binomial distribution whose mean is 5 and 2
variance is 2.5. A172.   1, 2  A173.   50, 2  25
Q176. The mean and variance of a binomial distribution are 4 3
r (27  r)
and (4/3) respectively. Find P(X  1) 1 2
A174. 27 Cr .   .   , where r = 0, 1, 2, 3, ….., 27
Q177. For a binomial distribution, the mean is 6 and the 3 3
standard deviation is 2 . Find the probability of getting 5 r (10  r)
1 1
successes. A175. 10 Cr .   .   , 0  r  10
Q178. In a binomial distribution, the sum and the product of the
2
  2
mean and the variance are (25/3) and (50/3) respectively. 728
A176.
Find the distribution. 729
Q179. Obtain the binomial distribution whose mean is 10 and 5 4
2 1
standard deviation is 2 2 . A177. 9 C5 .   .  
Q180. Bring out the fallacy, if any, in the following statement: 3 3
r (15  r)
‘The mean of a binomial distribution is 6 and its variance is 1 2
9’. A178. 15 Cr .   .  
3
  3
r (50  r)
ANSWERS TO HOMEWORK – 5 1 4
A179. 50 Cr .   .   , 0  r  50
5
  5
21 1 A180. The probability of getting a failure (i.e., q) cannot be
A149. A150.
32 2 greater than 1.
1
A151.
2
15 63 57
A152. (i) (ii) (iii)
64 64 64
15 193 53
A153. (i) (ii) (iii)
128 512 64
3 7 63
A154. (i) (ii) (iii)
32 64 64
8 11 8
A155. (i) (ii) (iii)
81 27 9
11
A156.
36
25
A157.
216

Prepared By: Praful Agarwaal (B. Tech I I T - Roorkee) Page 9