Probability & Statistics CQs Part III

1) If (1 – 3p)/2, (1+ 4p)/3, (1 + p)/6 are the probabilities of three mutually exclusive and exhaustive events, then the set of all values of p is
a) (0,1)
b) (-1/4, 1/3)
c) (0,1/3)
d)

2) 100 boys are randomly divided into two subgroups containing 50 boys each. The probability that the two tallest boys are in different groups is
a) 50/99
b) 49/99
c) 25/99
d) none of these

3) One hundred identical coins, each with probability p of showing up heads are tossed once. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of p is
a) 1/2
b) 49/101
c) 50/101
d) 51/101

4) In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs none is defective is
a)
b)
c)
d) none of these

5) A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Two cards are drawn at random. The probability that at least one of them is an ace is
a) 1/5
b) 3/16
c) 9/20
d) 1/9

6) India plays two matches each with Indies and Australia. In any math the probabilities of India getting points 0,1,2 are 0.45 0.05 and 0.50 respectively Assuming that outcomes are independent, the probability of India getting at least 7 points is
a) 0.8750
b) 0.0875
c) 0.0625
d) 0.0250

7) A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to that of getting 9 heads, then probability of getting 3 heads is
a)
b)
c)
d) none of these

8) A speaks truth 60% times and B speaks truth 70% times. The probability that they say same thing while describing a single event is
a) 0.42
b) 0.46
c) 0.54
d) 0.12

9) If two squares are chosen at random on a chessboard, the probability that they have a side in common is
a) 1/9
b) 2/7
c) 1/18
d) none of these

10) A bag contains 10 mangoes out of which 4 are rotten, two mangoes are taken out together. If one of them is found to be good, the probability that other is also good is
a) 5/18
b) 8/13
c) 5/13
d) 2/3

11) A box contain 5 brown and 4 white socks. A man pulls out two socks. The probability that they are of the same colour is
a) 5/108
b) 1/6
c) 5/18
d) 4/9

12) Suppose persons are sitting in a row. Two of them are selected at random. The probability that they are not together is
a) n – 1/n
b) (n – 2)/n
c) 2/n – 1
d) none of these

13) A bag contains (2n+1) coins. It is known that n of these coins have a head on both sides, whereas the remaining (n+1) coins are fair. A coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is 31/42, then n is equal to
a) 10
b) 11
c) 12
d) 13

14) A contest consists of predicting the results win, draw or defeat of 7 football matches. A sent his entry by predicting at random. The probability that his entry will contain exactly 4 correct predictions is
a)
b)
c)
d)

15) Five different objects 1,2,3,4,5 are distributed randomly in 5 places marked 1,2,3,4,5. One arrangement is picked at random. The probability that in the selected arrangement, none of the object occupies the place corresponding to its number is
a) 119/120
b) 1/5
c) 11/30
d) none of these

16) You are given a box with 20 cards in it. 10 of these cards have the letter I printed on them. The other ten have the letter T printed on them. If you pick up 3 cards at random and keep them in the same order, the probability of making the word IIT is
a)
b) 1/7
c) 4/27
d) 5/38

17) For the three events A, B and C, P (exactly one of the events A or B occurs)

= P (exactly one of the events B or C occurs)= P (exactly one of the events C or A occurs)= p and P (all the three events occurs simultaneously)= p2, where 0< p < 1/2, then the probability of at least one of the events A, B and C occurring is

a)
b)
c)
d)

18) The probability that Krishna will be alive 10 years hence is 7/15 and Hari will be alive is 7/10. The probability that both Krishna and Hari will be dead 10 years hence is
a) 21/150
b) 24/150
c) 49/150
d) 56/150

19) A person draws a card from a pack, replaces it shuffles the pack, again draws a card, replaces it and draws again. This he does until he draws a heart. The probability that he will have to make at least four draws is
a) 27/256
b) 175/256
c) 27/64
d) none of these

20) An unbiased dice with face 1,2,3,4,5 and 6 is round 4 times. Out of four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5, is
a) 16/81
b) 1/81
c) 80/81
d) 65/81

21) From a bag containing 9 distinct white and 9 distinct black balls, 9 balls are drawn at random one by one, the drawn balls being replaced each time. The probability that at least four balls of each colour is in the draw, is
a) A little less than 1/2
b) A little greater than 1/2
c) 1/2
d) none of these

22) Cards are drawn one by one at random from a well shuffled pack of 52 playing cards until 2 aces are obtained for the first time. The probability that 18 draws are required for this is
a) 3/34
b) 17/445
c) 561/15925
d) none of these

23) In a multiple-choice question there are four alternative answers of which one or more than one is correct. A candidate will get marks on the question only if he ticks all the correct answers. The candidate decides to tick answers at random. If he is allowed upto three chances to answer the question, the probability that he will get marks on it is given by
a)
b)
c)
d) none of these

24) A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the time on the
draw is

a) 5/64
b) 27/32
c) 5/32
d) 1/2

25) An unbiased die is tossed until a number greater than 4 appear. The probability
that an even number of tosses is needed is

a) 1/2
b) 2/5
c) 1/5
d) 2/3

26) If from each of the three boxes containing 3 white and 1 black, 2 white
and 2 black, 1 white and 3 black balls, one ball is drawn at random, then
the probability that 2 white and 1 black ball will be drawn is

a) 13/32
b) 1/4
c) 1/32
d) 3/6

27) A box contains 2 white and 2 red balls. If the first ball is being with
drawn, then the second ball with drawn is red. The probability of this event
is

a) 8/25
b) 2/5
c) 3/5
d) 21/25

28) If p and q are chosen randomly from the set {1,2,3,4…10} with replacement.
Then the probability that the roots of the equation

are real is equal to

a) 0.62
b) 0.31
c) 0.63
d) none of these

29) Three rifle man take one shot each at the same target. The probability of
the first rifle man hitting the target is 0.4, the probability of the second
rifle man hitting the target is 0.5 and the probability of the third rifle
man hitting the target is 0.8. Then the probability that exactly two of them
hit the target is

a) 0.92
b) 0.44
c) 0.94
d) none of these

30) There are four machines and it is known that exactly two of them are faulty.
They are tested one by one in a random order till both the faulty machines
are identified. Then the probability that only two tests are needed is

a) 1/3
b) 1/6
c) 1/2
d) 1/4

31) A dice is thrown n times. For the probability of a six appearing at least
once to be more than ½ is

a) n < 4
b)
c) n = 4
d) n = 6

32) A fair coin is tossed repeatedly. If tail appears on first four tosses,
then the probability of head appearing on fifth toss equals

a) 1/2
b) 1/32
c) 31/32
d) 1/5

33) If the integers m and n are chosen at random between 1 and 100 then the
probability that a number of the form

is divisible by 5 equals
a) 1/4
b) 1/7
c) 1/8
d) 1/49

34) Five boys and three girls are seated at random in a row. The probability that no boy sits between two girls is
a)
b)
c)
d) none of these

35) In a convex hexagon two diagonals are drawn at random. The probability that the diagonals intersect at an interior point of the hexagon is
a)
b)
c)
d) none of these

36) 4 five-rupee coins, 3 two rupee coins and 2 one-rupee coins are stacked together in a column at random. The probability that the coins of the same denomination are consecutive is
a)
b)
c)
d) none of these

37) Two cards are drawn at random from a pack of 52 cards. The probability of getting at least a spade and an ace is
a)
b)
c)
d)

38) A five-digit number is written down at random. The probability that the
number is divisible by 5 and no two consecutive digits are identical, is

a)
b)
c)
d) none of these

39) If the letters of the word ATTEMPT are written down at random, the chance that all Ts are consecutive is
a)
b)
c)
d) none of these

40) In a single cast with two dice the odds against drawing 7 is
a)
b)
c) 5:1
d) 1:5

41) 7 white balls and 3 black balls are placed in a row at random. The probability that no two black balls are adjacent is
a)
b)
c)
d)

42) 10 apples are distributed at random among 6 persons. The probability that
at least one of them will receive none is

a)
b)
c)
d) none of these

43) 4 gentlemen and 4 ladies take seats at random round a table. The probability
that they are sitting alternately is

a)
b)
c)
d)

44) Let

. The index n is given a positive integral value at random. The probability that the value of x will have 3 in the units place is

a)
b)
c)
d) none of these

45) Three dice are thrown simultaneously. The probability of getting a sum of
15 is

a)
b)
c)
d) none of these

46) Three dice are thrown. The probability of getting a sum which is a perfect
square is

a)
b)
c)
d) none of these

47) The probability of getting a sum of 12 in four throws of an ordinary dice
is

a)
b)
c)
d) none of these

48) Three different numbers are selected at random from the set A = {1, 2, 3,
…, 10}. The probability that the product of two of the numbers is equal to
the third is

a)
b)
c)
d) none of these

49) There are 7 seats in a row. Three persons take seats at random. The probability
that the middle seat is always occupied and no two persons are consecutive
is

a)
b)
c)
d) none of these

50) A second-order determinant is written down at random using the numbers 1,
-1 as elements. The probability that the value of the determinant is nonzero
is

a)
b)
c)
d)

51)
are fifty real numbers such that
for r = 1, 2, 3, …, 49. Five numbers out of these are picked up at random. The
probability that the five numbers have
as the middle number is

a)
b)
c)
d) none of these

52) Numbers 1, 2, 3, …, 100 are written down on each of the cards A, B and C.
One number is selected at random from each of the cards. The probability that
the numbers so selected can be the measures (in cm) of three sides of a right-angled
triangle is

a)
b)
c)
d) none of these

53) Three numbers are chosen at random without replacement from the set
.
The probability that the minimum of the chosen numbers is 3 and maximum is 7,
is

a)
b)
c)
d) none of these

54)

  1. Three natural numbers are taken at random from the set

. The probability that the AM of the numbers taken is 25 is

a)
b)
c)
d) none of these

55) Let S be the universal set and n(X) = k. The probability of selecting two
subsets A and B of the set X such that
is

a)
b)
c)
d)

56) From a group of 10 persons consisting of 5 lawyers, 3 doctors and 2 engineers,
four persons are selected at random. The probability that the selection contains
at least one of each category is

a)
b)
c)
d) none of these

57) 10 different books and 2 different pens are given to 3 boys so that each
gets equal number of things. The probability that the same boy does not receive
both the pens is

a)
b)
c)
d)

58) Two distinct numbers are selected at random from the first twelve natural
numbers. The probability that the sum will be divisible by 3 is

a)
b)
c)
d) none of these

59) The probability of a number n showing in a throw of a dice marked 1 to 6
is proportional to n. Then the probability of the number 3 showing in a throw
is

a)
b)
c)
d)

60) The probability that out of 10 persons, all born in April, at least two
have the same birthday is

a)
b)
c)
d) none of these

61) If one ball is drawn at random from each of the three boxes containing 3
white and 1 black, 2 white and 2 black, 1 white and 3 black balls then the
probability that 2 white and 1 black balls will be drawn is

a)
b)
c)
d)

62) A draws two cards at random from a pack of 52 cards. After returning them
to the pack and shuffling it, B draws two cards at random. The probability
that their draws contain exactly one common card is

a)
b)
c)
d) none of these

63) A and B draw two cards each, one after another, from a pack of well-shuffled
pack of 52 cards. The probability that all the four cards drawn are of the
same suit is

a)
b)
c)
d) none of these

64) Three numbers are chosen at random without replacement from 1, 2, 3, …,
10. The probability that the minimum of the chosen numbers is 4 or their maximum
is 8, is

a)
b)
c)
d) none of these

65) A man draws a card from a pack of 52 cards and then replaces it. After shuffling
the pack, he again draws a card. This he repeats a number of times. The probability
that he will draw a heart for the first time in the third draw is

a)
b)
c)
d) none of these

66) A fair coin is tossed repeatedly. The probability of getting a result in
the fifth toss different from those obtained in the first four tosses is

a)
b)
c)
d)

67) It has been found that if A and B play a game 12 times, A wins 6 times,
B wins 4 times and they draw twice. A and B take part in a series of 3 games.
The probability that they will win alternately is

a)
b)
c)
d) none of these

68) If the probability of A to fail in an examination is
and that of B is
then the probability that either A or B fails is

a)
b)
c)
d) none of these

69) A and B are two events where P(A) = 0.25 and P(B) = 0.5. The probability
of both happening together is 0.14. The probability of both A and B not happening
is

a) 0.39
b) 0.25
c) 0.11
d) none of these

70) Three faces of an ordinary dice are yellow, two faces are red and one face
is blue. The dice is tossed 3 times. The probability that yellow, red and
blue faces appear in the first, second and third tosses respectively is

a)
b)
c)
d) none of these

71) India play two matches each with West Indies and Australia. In any match
the probabilities of India getting 0, 1 and 2 points are 0.45, 0.05 and 0.50
respectively. Assuming that the outcomes are independent, the probability
of India getting at least 7 points is

a) 0.0875
b)
c) 0.1125
d) none of these

72) Let A and B be two independent events such that
a)
b)
c)
d) none of these

73) Let A and B be two independent events such that their probabilities are

and .
The probability of exactly one of the events happening is

a)
b)
c)
d) none of these

74) The probability that at least one of the events A and B occurs is .
If A and B occur simultaneously with probability
then
is

a)
b)
c)
d)

75) A, B, C are three events for which and
then the interval of values of
is

a) [0.2, 0.35]
b) [0.55, 0.7]
c) [0.2, 0.55]
d) none of these

76) A coin is tossed 2n times. The chance that the number of times one gets
head is not equal to the number of times one gets tail is

a)
b)
c)
d) none of these

77) A coin is tossed n times. The probability of getting at least one head
is greater than that of getting at least two tails by .
Then n is

a) 5
b) 10
c) 15
d) none of these

78) A coin is tossed 7 times. Each time a man calls head. The probability
that he wins the toss on more occasions is

a)
b)
c)
d) none of these

79) A bag contains 14 balls of two colours, the number of balls of each colour
being the same. 7 balls are drawn at random one by one. The ball in hand is
returned to the bag before each new draw. If the probability that at least
3 balls of each colour are drawn is p then

a)
b)
c) p < 1
d)

80) From a box containing 20 tickets of value 1 to 20, four tickets are drawn
one by one. After each draw, the ticket is replaced. The probability that
the largest value of tickets drawn is 15 is

a)
b)
c)
d) none of these

81) A dice is thrown 2n + 1 times, .
The probability that faces with even numbers show odd number of times is

a)
b)
c)
d) none of these

82) 6 ordinary dice are rolled. The probability that at least half of them will
show at least 3 is

a)
b)
c)
d) none of these

83) An ordinary dice is rolled a certain number of times. The probability of
getting an odd number 2 times is equal to the probability of getting an even
number 3 times. Then the probability of getting an odd number an odd number
of times is

a)
b)
c)
d) none of these

84) A card is drawn from a pack. The card is replaced and the pack is reshuffled.
If this is done six times, the probability that 2 hearts, 2 diamonds and 2
black cards are drawn is

a)
b)
c)
d) none of these

85) A man firing at a distant target has 10% chance of hitting the target in
one shot. The number of times he must fire at the target to have about 50%
chance of hitting the target is

a) 11
b) 9
c) 7
d) 5

86) There are four machines and it is known that exactly two of them are faulty.
They are tested one by one in a random order till both the faulty machines
are identified. Then the probability that only two tests will be required
is

a)
b)
c)
d)

87) Let A = {2, 3, 4, …, 20}. A number is chosen at random from the set A and
it is found to be a prime number. The probability that it is more than 10
is

a)
b)
c)
d) none of these

88) All the spades are taken out from a pack of cards. From these cards, cards
are drawn one by one without replacement till the ace of spades comes. The
probability that the ace comes in the 4th draw is

a)
b)
c)
d) none of these

89) A point is selected at random from the interior of a circle. The probability
that the point is closer to the center than the boundary of the circle is

a)
b)
c)
d) none of these

90) A, B and C are contesting the election for the post of secretary of a club
which does not allow ladies to become members. The probabilities of A, B and
C winning the election are
respectively. The probabilities of introducing the clause of admitting lady
members to the club by A, B, and C are 0.6, 0.7 and 0.5 respectively. The
probability that ladies will be taken as members in the club after the election
is

a)
b)
c)
d) none of these

91) There are 4 white and 3 black balls in a box. In another box there are 3
white and 4 black balls. An unbiased dice is rolled. If it shows a number
less than or equal to 3 then a ball is drawn from the first box but if it
shows a number more than 3 then a ball is drawn from the second box. If the
ball drawn is black then the probability that the ball was drawn from the
first box is

a)
b)
c)
d)

92) If E and F are two events with
then

a)
b)
c)
d) none of the above implications hold

93) If A and B are two events such that
then

a)
b)
c)
d) none of these

94) If
and
are the complementary events of the events E and F respectively then

a)
b)
c)
d)

95) Given that .
Let A be the event of (x, y) satisfying
and B be the event of (x, y) satisfying

. Then
a)
b) A, B are exhaustive
c) A, B are mutually exclusive
d) A, B are independent

96) Let A and B be two events such that .
Then

a) A, B are independent
b) A, B are mutually exclusive
c) P(A) = P(B)
d)

97) The probability that exactly one of the independent events A and B occurs
is equal to

a)
b)
c) all the above
d) none of these

98) For any two events A and B
a)
b)
c)
d)

99) A coin is tossed repeatedly. A and B call alternately for winning a prize
of Rs. 30. One who calls correctly first wins the prize. A starts the call.
Then the expectation of

a) A is Rs 10, B is Rs 5
b) B is Rs 10, A is Rs 20
c) A is Rs 20, B is Rs 30
d) B is Rs 20, A is Rs 10

100) The probability that a marksman will hit a target is given as
Then the probability of at least one hit in 10 shots is

a)
b)
c)
d)

Answers
Ans 1) b

P (A) = (1 – 3p)/2


P (B) = (1 + 4p)/3


P (C) = (1 + P)/6


Events are mutually exclusive and exhaustive



i.e. 1 = 1. Therefore true for any real p …(1)


Also


And


And

Common solution of (1), (11), (111), (1V) is


Set of values of p is [-1/4, 1/3]



Ans Desc 1)

P (A) = (1 – 3p)/2


P (B) = (1 + 4p)/3


P (C) = (1 + P)/6


Events are mutually exclusive and exhaustive



i.e. 1 = 1. Therefore true for any real p …(1)


Also


And


And

Common solution of (1), (11), (111), (1V) is


Set of values of p is [-1/4, 1/3]


Ans 2) a

Leaving two tallest boys the remaining 98 boys can be divided into two equal groups containing 49 boys each in 98! /(49!)(49!).2 ways. Two tallest boys can be distributed in three groups (one in each) in 2 ways


Therefore Favourable ways


Also 100 boys can be distributed in 2 equal groups of 50 each in


(1)/(2) gives the required probability


 



Ans Desc 2)

Leaving two tallest boys the remaining 98 boys can be divided into two equal groups containing 49 boys each in 98! /(49!)(49!).2 ways. Two tallest boys can be distributed in three groups (one in each) in 2 ways


Therefore Favourable ways


Also 100 boys can be distributed in 2 equal groups of 50 each in


(1)/(2) gives the required probability


 


Ans 3) d



Gives




Therefore p = 51/101.



Ans Desc 3)



Gives




Therefore p = 51/101.


Ans 4) c

Favourable ways of selecting 5 non-defective bulbs =


Total no. of ways of selection =


Therefore P(non-defective bulbs)


 



Ans Desc 4)

Favourable ways of selecting 5 non-defective bulbs =


Total no. of ways of selection =


Therefore P(non-defective bulbs)


 


Ans 5) c

P (both are ace)=


P (one is ace)


Therefore P (at least one is ace)


Ans Desc 5)

P (both are ace)=


P (one is ace)


Therefore P (at least one is ace)

Ans 6) b

Matches played by India = 4


Probability for getting 7 points (3 win and one drawn)=


Probability for getting 8 points (all 4 win)


Therefore required probability (getting at least 7 point, i.e., 7 points or 8 points)= 0.0250 + 0.0625 = 0.0875.



Ans Desc 6)

Matches played by India = 4


Probability for getting 7 points (3 win and one drawn)=


Probability for getting 8 points (all 4 win)


Therefore required probability (getting at least 7 point, i.e., 7 points or 8 points)= 0.0250 + 0.0625 = 0.0875.


Ans 7) a

Let the coin be tossed n times


P (7 heads)


And P (9 heads)


Given, P (7 heads)= P (9 heads)


Therefore P (3 heads)


Ans Desc 7)

Let the coin be tossed n times


P (7 heads)


And P (9 heads)


Given, P (7 heads)= P (9 heads)


Therefore P (3 heads)

Ans 8) c

P (A speaks truth) = 3/5,= 2/5


P (B speaks truth) = 7/10,= 3/10


P (both speak truth)= (3/5). (7/10) = 21/50


P (both speak false)= (2/5). (3/10) = 6/50


P (both say same thing) = 21/50 + 6/50 = 27/50 = 0.54.



Ans Desc 8)

P (A speaks truth) = 3/5,= 2/5


P (B speaks truth) = 7/10,= 3/10


P (both speak truth)= (3/5). (7/10) = 21/50


P (both speak false)= (2/5). (3/10) = 6/50


P (both say same thing) = 21/50 + 6/50 = 27/50 = 0.54.


Ans 9) c

Hints:


Total ways


Favourable ways




Ans Desc 9)

Hints:


Total ways


Favourable ways



Ans 10) c

Let A be event that the first one of the two selected mangoes is good and B is the event that other one is also good. We have to find P (B/A)



 



Ans Desc 10)

Let A be event that the first one of the two selected mangoes is good and B is the event that other one is also good. We have to find P (B/A)



 


Ans 11) d

Number of ways of drawing 2 socks from the box containing


Number of ways of drawing 2 white socks


And number of ways of drawing 2 brown socks


Therefore probability of drawing 2 socks of same colour = probability (2 white socks)+Probability (2 brown socks)




Ans Desc 11)

Number of ways of drawing 2 socks from the box containing


Number of ways of drawing 2 white socks


And number of ways of drawing 2 brown socks


Therefore probability of drawing 2 socks of same colour = probability (2 white socks)+Probability (2 brown socks)



Ans 12) b

. *. *. * *


Let there be n persons (n-2) persons not selected are arranged in places stated above by stars and the selected 2 persons can be arranged at places stated by dots (Dots are n-1 in number). So the favourable ways are and the total ways are so



Ans Desc 12)

. *. *. * *


Let there be n persons (n-2) persons not selected are arranged in places stated above by stars and the selected 2 persons can be arranged at places stated by dots (Dots are n-1 in number). So the favourable ways are and the total ways are so


Ans 13) a

Both heads appear on n coins and head and a tail appear on (n+1) coins so







Ans Desc 13)

Both heads appear on n coins and head and a tail appear on (n+1) coins so






Ans 14) c

P (correct prediction) = 1/3.


P (wrong prediction) = 2/3.


For exactly 4 right predictions



Ans Desc 14)

P (correct prediction) = 1/3.


P (wrong prediction) = 2/3.


For exactly 4 right predictions


Ans 15) c

Let objects 1,2,3,4,5 be placed in places marked 1,2,3,4,5 respectively. Then the number of arrangements in which none of the object occupies its original position is given by =60 -20 + 5 – 1 = 44.


Also total number of arrangements =5! =120


Hence required probability = 44/120 = 11/30



Ans Desc 15)

Let objects 1,2,3,4,5 be placed in places marked 1,2,3,4,5 respectively. Then the number of arrangements in which none of the object occupies its original position is given by =60 -20 + 5 – 1 = 44.


Also total number of arrangements =5! =120


Hence required probability = 44/120 = 11/30


Ans 16) d

Required probability P=(1 in 1st draw) x P(1 in 2nd draw) x P(T in 3rd draw)





Ans Desc 16)

Required probability P=(1 in 1st draw) x P(1 in 2nd draw) x P(T in 3rd draw)




Ans 17) a

Let O (S)=n


For events A, B, C (see the figure)



 


According to problem


a+d+b+f = np…..(1)


b+e+d+c = np….(2)


a+e+c+f = np…..(3)


…(4)


From (1), (2) and (3) we have a + f = b+ d = e + c= np/2





Ans Desc 17)

Let O (S)=n


For events A, B, C (see the figure)



 


According to problem


a+d+b+f = np…..(1)


b+e+d+c = np….(2)


a+e+c+f = np…..(3)


…(4)


From (1), (2) and (3) we have a + f = b+ d = e + c= np/2




Ans 18) b

Required probability = (probability that Krishna will be dead 10 years hence)x(probability that Hari will be dead 10 years hence)



 



Ans Desc 18)

Required probability = (probability that Krishna will be dead 10 years hence)x(probability that Hari will be dead 10 years hence)



 


Ans 19) c

Probability of drawing a heart = 13/52 = 1/4.


P (he requires at least 4 draws for heart)



 



Ans Desc 19)

Probability of drawing a heart = 13/52 = 1/4.


P (he requires at least 4 draws for heart)



 


Ans 20) a

In a single throw the favourable point are 2,3,4 and 5 whose number is 4.


All possible outcomes are 6.


Therefore P = Probability that in a single throw the minimum face values is not less than 2 and the maximum face value is not greater than 5 = 4/6 = 2/3


Since the dice is rolled four times and all the four throws are independent events, therefore the required probability =



Ans Desc 20)

In a single throw the favourable point are 2,3,4 and 5 whose number is 4.


All possible outcomes are 6.


Therefore P = Probability that in a single throw the minimum face values is not less than 2 and the maximum face value is not greater than 5 = 4/6 = 2/3


Since the dice is rolled four times and all the four throws are independent events, therefore the required probability =


Ans 21) a

Out of 9 distinct black and 9 distinct white balls, probability of drawing a white=1/2 and of drawing black is also ½. For at least 4 of each colour in 9 draws with replacement, there are two cases


I P (5 white, 4 black)


II P (4 white, 5 black)


These cases are exclusive so P (at least 4 of each colour)




which is little less than 1/2.


 



Ans Desc 21)

Out of 9 distinct black and 9 distinct white balls, probability of drawing a white=1/2 and of drawing black is also ½. For at least 4 of each colour in 9 draws with replacement, there are two cases


I P (5 white, 4 black)


II P (4 white, 5 black)


These cases are exclusive so P (at least 4 of each colour)




which is little less than 1/2.


 


Ans 22) c
18 draws are required for 2 aces means in the first 17 draws, there is one ace and 16 non-ace and ace. There the required probability x (18th draw is 2nd ace 3/35) = 561/15925.
Ans Desc 22)
18 draws are required for 2 aces means in the first 17 draws, there is one ace and 16 non-ace and ace. There the required probability x (18th draw is 2nd ace 3/35) = 561/15925.

Ans 23) c

The total number of ways of ticking or more alternatives out of 4 is



Out of these 15 combinations, only one combination is correct. The probability of ticking the alternatives correctly at first trial is and same for second and third trial. The probability that the candidate will get marks on the question if he is allowed upto three

chances is



Ans Desc 23)

The total number of ways of ticking or more alternatives out of 4 is



Out of these 15 combinations, only one combination is correct. The probability of ticking the alternatives correctly at first trial is and same for second and third trial. The probability that the candidate will get marks on the question if he is allowed upto three

chances is


Ans 24) c



White ball is drawn,
mean that white ball have been drawn thrice in first six draws and draw.
Therefore required probability



Ans Desc 24)



White ball is drawn,
mean that white ball have been drawn thrice in first six draws and draw.
Therefore required probability


Ans 25) b

Probability of getting greater than 4 from a unbiased die=1/3


Probability




Ans Desc 25)

Probability of getting greater than 4 from a unbiased die=1/3


Probability



Ans 26) a

the
probabilities of drawing one white and one black ball fom the i-th box, where
I=1,2,3.





Two white and one black ball may be drawn from 3 boxes in the following three
ways


Box1 Box2 Box3


Way 1 W W B


Way 2 W B W


Way 3 B W W


Required probability



Ans Desc 26)

the
probabilities of drawing one white and one black ball fom the i-th box, where
I=1,2,3.





Two white and one black ball may be drawn from 3 boxes in the following three
ways


Box1 Box2 Box3


Way 1 W W B


Way 2 W B W


Way 3 B W W


Required probability


Ans 27) b

(red,red) (white, red)




Ans Desc 27)

(red,red) (white, red)



Ans 28) a

Roots of
will be real if .
The possible selections are as follows


p q


1


2 1





    1. 1,2

    2. 1,2,3,4

    3. 1,2,3,4,5,6

    4. 1,2…9


    5. 1,2,3… 10





Total 62


Therefore favourable way =62


Total number of ways


Therefore probability = 62/100 = 0.62.



Ans Desc 28)

Roots of
will be real if .
The possible selections are as follows


p q


1


2 1





    1. 1,2

    2. 1,2,3,4

    3. 1,2,3,4,5,6

    4. 1,2…9


    5. 1,2,3… 10





Total 62


Therefore favourable way =62


Total number of ways


Therefore probability = 62/100 = 0.62.


Ans 29) b

Let A, B, C be the three rifle man. Given that P (A)=0.4, P (B)=0.5 and P(C)
= 0.8


P (exactly two success)




= 0.4X0.5X0.2+0.4X0.5X0.8+0.6X0.5X0.8 = 0.44.



Ans Desc 29)

Let A, B, C be the three rifle man. Given that P (A)=0.4, P (B)=0.5 and P(C)
= 0.8


P (exactly two success)




= 0.4X0.5X0.2+0.4X0.5X0.8+0.6X0.5X0.8 = 0.44.


Ans 30) b

Ans Desc 30)

Ans 31) b


 



Ans Desc 31)


 


Ans 32) a
Since appearance of head on fifth toss does not depend on the outcome of
first four tosses. Hence P (head on 5th toss)=1/2.

Ans Desc 32)
Since appearance of head on fifth toss does not depend on the outcome of
first four tosses. Hence P (head on 5th toss)=1/2.

Ans 33) c


is
divisible by 5 iff is
divisible by 5 and so the unit place of
must be 0 as it cannot be 5. Thus we have

m possible n



    1. 3,7,11,15,….=25

    2. 4,8,12… =25

    3. 1,5,9,……… =25

    4. 2,6,10,……. =25




: : :


Since


Therefore (1,3) and (3,1) are same


Number of favourable cases = 25 x 50



 



 



Ans Desc 33)


is
divisible by 5 iff is
divisible by 5 and so the unit place of
must be 0 as it cannot be 5. Thus we have

m possible n



    1. 3,7,11,15,….=25

    2. 4,8,12… =25

    3. 1,5,9,……… =25

    4. 2,6,10,……. =25




: : :


Since


Therefore (1,3) and (3,1) are same


Number of favourable cases = 25 x 50



 



 


Ans 34) c

Ans Desc 34)

Ans 35) a

n(S) = total number of sections of two diagonals



n(E) = the number of selections of two diagonals which intersect at an interior point.


= the number of selections of four vertices =



 




Ans Desc 35)

n(S) = total number of sections of two diagonals



n(E) = the number of selections of two diagonals which intersect at an interior point.


= the number of selections of four vertices =



 



Ans 36) b




Ans Desc 36)



Ans 37) c

Ans Desc 37)

Ans 38) c

Ans Desc 38)

Ans 39) c

Ans Desc 39)

Ans 40) c

Ans Desc 40)

Ans 41) b

Ans Desc 41)

Ans 42) c

The required probability = 1 – probability of each receiving at least one



Now, the number of integral solutions of


Such that
gives n(E) and the number of integral solutions of
gives n(S).




Ans Desc 42)

The required probability = 1 – probability of each receiving at least one



Now, the number of integral solutions of


Such that
gives n(E) and the number of integral solutions of
gives n(S).



Ans 43) d

Ans Desc 43)

Ans 44) a

Ans Desc 44)

Ans 45) d

Ans Desc 45)

Ans 46) d


Clearly, the sum varies from 3, 18, and among these 4, 9, 16 are perfect squares.


The number of ways to get the sum 4 = the number of integral solutions of



Similarly, the number of ways to get the sum 9


= coefficient of
in



The number of ways to get the sum 16





Ans Desc 46)


Clearly, the sum varies from 3, 18, and among these 4, 9, 16 are perfect squares.


The number of ways to get the sum 4 = the number of integral solutions of



Similarly, the number of ways to get the sum 9


= coefficient of
in



The number of ways to get the sum 16




Ans 47) a

Ans Desc 47)

Ans 48) b

Ans Desc 48)

Ans 49) c




because one has to sit at any one of the two marked seats on the left and the
other has to sit at any one of the two marked seats on the right.




Ans Desc 49)




because one has to sit at any one of the two marked seats on the left and the
other has to sit at any one of the two marked seats on the right.



Ans 50) a

because
each of the four places can be filled in 2 ways.




Ans Desc 50)

because
each of the four places can be filled in 2 ways.



Ans 51) b

Ans Desc 51)

Ans 52) d


For example, when n = 1, sides are 3,4,5; when n = 2, sides are 5,12,13 and
so on.


The number of selections of 3,4,5 from the three cards by taking one from each
is 3!



Ans Desc 52)


For example, when n = 1, sides are 3,4,5; when n = 2, sides are 5,12,13 and
so on.


The number of selections of 3,4,5 from the three cards by taking one from each
is 3!


Ans 53) c

Ans Desc 53)

Ans 54) c


As the AM of three numbers is 25, their sum = 75.




Ans Desc 54)


As the AM of three numbers is 25, their sum = 75.



Ans 55) b

The total number of subsets of X is .
So, n(S) =


n(E) = the number of selections of two nonintersecting


Subsets whose union is X.


(the
number of selections in which one subset has r elements and the rest are in
the other subset =
and every selection appears twice in the total number of selections).




Ans Desc 55)

The total number of subsets of X is .
So, n(S) =


n(E) = the number of selections of two nonintersecting


Subsets whose union is X.


(the
number of selections in which one subset has r elements and the rest are in
the other subset =
and every selection appears twice in the total number of selections).



Ans 56) a

Ans Desc 56)

Ans 57) a

Ans Desc 57)

Ans 58) a

Let E3 = the event of the sum being 3. Similarly,



Alternatively The sum of the numbers for every selection is divisible by
3 or leaves the remainder 1 or leaves the remainder 2. These are equally probable.
So, the required probability = ,
because the sum of the three probabilities is 1.



Ans Desc 58)

Let E3 = the event of the sum being 3. Similarly,



Alternatively The sum of the numbers for every selection is divisible by
3 or leaves the remainder 1 or leaves the remainder 2. These are equally probable.
So, the required probability = ,
because the sum of the three probabilities is 1.


Ans 59) c

=
Probability of the dice showing n = kn.


Clearly,




Ans Desc 59)

=
Probability of the dice showing n = kn.


Clearly,



Ans 60) c

There are 30 days in April.


N(S) = the number of ways in which 10 persons can have birthdays in the month
of April


= …..to
10 times = 3010.


(because each person can have birthday in 30 ways).


N(E) = n(S) – the number of ways in which 10 persons can have different birthdays





Ans Desc 60)

There are 30 days in April.


N(S) = the number of ways in which 10 persons can have birthdays in the month
of April


= …..to
10 times = 3010.


(because each person can have birthday in 30 ways).


N(E) = n(S) – the number of ways in which 10 persons can have different birthdays




Ans 61) a

Let
= the event of drawing a white ball from the first box.


Similarly, E2 and E3.




Ans Desc 61)

Let
= the event of drawing a white ball from the first box.


Similarly, E2 and E3.



Ans 62) b

The probability of both drawing the common card x = P(x) = (probability of
A drawing the card x and any other card y)(probability
of B drawing the card x and a card other than y)




Ans Desc 62)

The probability of both drawing the common card x = P(x) = (probability of
A drawing the card x and any other card y)(probability
of B drawing the card x and a card other than y)



Ans 63) a

The probability of the four cards being spades =


Similarly, for other suits.




Ans Desc 63)

The probability of the four cards being spades =


Similarly, for other suits.



Ans 64) a

The probability of 4 being the minimum number =


(because, after selecting 4 any two can be selected from 5,6,7,8,9,10).


The probability of 8 being the maximum number =


The probability of 4 being the minimum number and 8 being the maximum number
=




Ans Desc 64)

The probability of 4 being the minimum number =


(because, after selecting 4 any two can be selected from 5,6,7,8,9,10).


The probability of 8 being the maximum number =


The probability of 4 being the minimum number and 8 being the maximum number
=



Ans 65) c

Ans Desc 65)

Ans 66) d

Ans Desc 66)

Ans 67) d

Ans Desc 67)

Ans 68) c

Ans Desc 68)

Ans 69) a

Ans Desc 69)

Ans 70) a

Ans Desc 70)

Ans 71) a

Ans Desc 71)

Ans 72) a

Ans Desc 72)

Ans 73) a

Ans Desc 73)

Ans 74) c

Ans Desc 74)

Ans 75) a

Ans Desc 75)

Ans 76) c

Ans Desc 76)

Ans 77) a

Ans Desc 77)

Ans 78) c

The man has to win at least 4 times.




Ans Desc 78)

The man has to win at least 4 times.



Ans 79) a

Ans Desc 79)

Ans 80) b

Ans Desc 80)

Ans 81) d

Ans Desc 81)

Ans 82) a

Ans Desc 82)

Ans 83) c

Ans Desc 83)

Ans 84) c

Ans Desc 84)

Ans 85) c

Ans Desc 85)

Ans 86) b

Ans Desc 86)

Ans 87) c

Ans Desc 87)

Ans 88) a

Ans Desc 88)

Ans 89) c

Ans Desc 89)

Ans 90) a

Ans Desc 90)

Ans 91) d

Ans Desc 91)

Ans 92) d

Ans Desc 92)

Ans 93) a

Ans Desc 93)

Ans 94) a

Ans Desc 94)

Ans 95) c




Ans Desc 95)



Ans 96) a

Ans Desc 96)

Ans 97) b

Ans Desc 97)

Ans 98) c

Ans Desc 98)

Ans 99) b

Ans Desc 99)

Ans 100) a

Ans Desc 100)

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