1. science
2. mathematics
3. algebra

PAGE # 1
CHAPTER EXERCISE – I
1.
A single letter is selected at random from the word ‘PROBABILITY’. The probability that the selected letter
is a vowel is
(a)
2.
1
n
(d) None of these
2/25
(b)
9/100
(c) 11/100
(d) None of these
1/27
(b)
1/9
(c) 4/27
(d) 1/6
1/3
(b)
1/2
(c) 1/4
(d) None of these
1/462
(b)
1/924
(c) 1/2
(d) None of these
4/13
(b)
9/13
(c) 1/4
(d) 13/26
16
81
(b)
1
81
(c)
80
81
(d)
65
81
5
36
(b)
6
36
(c)
7
36
(d)
8
36
1
35
(b)
1
14
(c)
1
15
(d) None of these
The chance of throwing at least 9 in a single throw with two dice, is
(a)
12.
(c) 1
The letter of the word ‘ASSASSIN’ are written down at random in a row. The probability that no two S occur
together is
(a)
11.
1
n!
Two dice are thrown. The probability that the sum of the points on two dice will be 7, is
(a)
10.
1
An unbiased die with faces marked 1, 2, 3, 4, 5 and 6 is rolled four 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)
9.
(b)
A card is drawn at random from a pack of cards. What is the probability that the drawn card is neither a heart
nor a king
(a)
8.
1
n!
Six boys and six girls sit in a row. What is the probability that the boys and girls sit alternatively
(a)
7.
(d) 0
A box contains 10 red balls and 15 green balls. If two balls are drawn in succession then the probability that
one is red and other is green, is
(a)
6.
(c) 4/11
Three letters are to be sent to different persons and addresses on the three envelopes are also written. Without
looking at the addresses, the probability that the letters go into the right envelope is equal to
(a)
5.
3/11
From a book containing 100 page one page is selected randomly. The probability that the sum of the digits of
the page number of the selected page is 11, is
(a)
4.
(b)
There are n letter and n addressed envelopes. The probability that all the letters are not kept in the right
envelope, is
(a)
3.
2/11
1
18
(b)
5
18
(c)
7
18
(d)
11
18
The numbers are selected at random from the numbers 1, 2, …… n. The probability that the difference
between the first and second is not less than m (where 0 < m < n), is
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PAGE # 2
13.
(a)
n m n m 1
n 1
(b)
n m n m 1
2n
(c)
n m n m 1
2n n 1
(d)
n m n m 1
2n n 1
Fifteen persons among whom are A and B, sit down at random at a round table. The probability that there are
4 persons between A and B, is
(a)
14.
10
133
2
7
(d)
1
7
(b)
9
133
(c)
9
1330
(d) None of these
(B) 1/7
(C) 1/12
(D) none
(B) 5/8
(C) 5/9
(D) 7/12
In the French lottery there were 90 tickets bearing numbers 1 to 90. Suppose five tickets are drawn at random.
Then the probability that two of the tickets drawn bear numbers 15 and 89 is :
(A) 2/801
18.
(c)
A drawer contains 5 brown socks & 4 blue socks well mixed up . A man reaches the drawer & pulls out two
socks at random. The probability that they match is :
(A) 4/9
17.
2
3
There are 4 apples & 3 oranges placed at random in a line. Then the chance of the extreme fruits being both
oranges is:
(A) 3/4
16.
(b)
Out of 21 tickets marked with numbers from 1 to 21, three are drawn at random. The chance that the numbers
on them are in A.P., is
(a)
15.
1
3
(B) 2/623
(C) 1/267
(D) none
10 students are seated at random in a row . The probability that two particular students are not seated side by
side is :
(A) 2/3
(B) 3/4
(C) 4/5
(D) none
19.
If two numbers p and q are chosen at random from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with replacement,
determine the probability that the roots of the equation x2 + px + q = 0 are real
20.
Two squares are chosen at random from the small squares drawn on a chessboard. What is the chance that the
two squares chosen have exactly one corner in common.
21.
Five ordinary dice are rolled at random and the sum of the numbers shown on them is 16. What is the
probability that the numbers shown on each is any one from 2, 3, 4 or 5 ?
22.
Out of (2n + 1) tickets numbered consecutively, three are drawn at random. Find the chance that the numbers
on them are in A.P.
23.
A man parks his car among n cars standing in a row, his car not being parked at an end. On his return he finds
that exactly m of the n cars are still there. What is the probability that both the cars parked on two sides of his
car, have left ?
24.
From an ordinary pack of 52 cards an even number of cards are drawn at random. Find the probability of
getting equal number of black and red cards.
25.
An urn contains 3 white and 5 black balls. One ball is drawn. What is the probability that it is black ?
26.
From a pack of 52 cards, four cards are drawn. Find the chance that they will be the four honours of the same
suit.
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PAGE # 3
27.
Six dice are thrown simultaneously. Find the probability that all dice show different faces.
28.
Six boys and six girls sit in a row randomly. Find the probability that
(i)
(ii)
The six girls sit together
The boys and girls sit alternately.
29.
What is the probability that in a group of N people, at least two of them will have the same birthday ?
30.
A group contains 10 men and 4 women. Three member committee is formed which must contain at least one
woman. Find the probability that the committee so formed has more women that men.
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PAGE # 4
CHAPTER EXERCISE – II
1.
The probabilities of three mutually exclusive events are 2/3, 1/4 and 1/6. The statement is
(a)
2.
6.
(d) None of these
0.5
(b)
0.44
(c) 0.6
(d) None of these
0.6
(b)
0.2
(c) 0.21
(d) None of these
P (A ˆ B) = P (A) + P (B) – P (A ‰ B)
(d) P (A ˆ B) = P (A) + P (B) + P (A ‰ B)
§A·
If A and B are two events such that P (A) z 0 and P (B) z 1, then P ¨¨ ¸¸ =
©B¹
§ A·
1 P¨ ¸
©B¹
(b)
§A·
1 P ¨¨ ¸¸
©B¹
(c)
1 P A ‰ B
PB
(d)
PA
PB
The probability of happening an even A in one trial is 0.4. The probability that the event A happens at least
once in three independent trails is
0.936
(b)
0.784
(c) 0.904
(d) 0.216
Two coins are tossed. Let A be the event that the first coin shows head and B be the event that the second coin
shows a tail. Two events A and B are
Mutually exclusive
Independent and mutually exclusive
(b) Dependent
(d) None of these
The probability of happening at least one of the events A and B is 0.6. If the events A and B happens
simultaneously with the probability 0.2, then P A P B =
0.4
(b)
0.8
(c) 1.2
(d) 1.4
The chances to fail in physics are 20% and the chances to fail in mathematics are 10%. What are the chances
to fail in at least one subject
28%
(b)
38%
(c) 72%
(d) 82%
A problem of mathematics is given to three students whose chances of solving the problem are 1/3, 1/4 and
1/5 respectively. The probability that the question will be solved is
(a)
12.
(c) 0.5
(c)
(a)
11.
0.3
(b) P (A ˆ B) is not greater than P (A) + P (B)
(a)
10.
(b)
P (A ˆ B) is not less than P (A) + P (B) – 1
(a)
(c)
9.
0.1
(a)
(a)
8.
(d) Do not know
If A and B are any two events, then the true relation is
(a)
7.
(c) Could be either
The probability of happening an event A is 0.5 and that of B is 0.3. If A and B are mutually exclusive events,
then the probability of happening neither A nor B is
(a)
5.
Wrong
If the probability of X to fail in the examination is 0.3 and that for Y is 0.2, then the probability that either X
or Y fail in the examination is
(a)
4.
(b)
A and B are two events such that and P(A) = 0.4, P(A + B) = 0.7 and P (AB) = 0.2, then P (B) =
(a)
3.
True
2/3
(b)
3/4
(c) 4/5
(d) 3/5
If P (A1 ‰ A2) = 1 – P A 1c P A c2 , where c stands for complement, then the events A1 and A2 are
(a)
Mutually exclusive
(b)
Independent
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(c) Equally likely
(d) None of these
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13.
If A and B are to events such that P A
(a)
14.
0
(b)
1/20
(b)
(a)
PE
(c)
PE
1
,PF
3
1
,PF
6
(d) 1/3
9/20
(c) 11/20
(d) 19/20
1
and the probability
12
1
, then
2
(b)
PE
1
,PF
2
1
6
(d) None of these
p+q
(b)
p + q – 2qp
(c) p + q – pq
(d) p + q + pq
A and B are two independent events. The probability that both A and B occur is
1
and the probability that
6
1
. Then the probability of the two events are respectively.
3
1
1
and
2
3
(b)
1
1
and
5
6
(c)
1
1
and
2
6
(d)
2
1
and
3
4
If A and B are two events, then the probability of the event that at most one of A, B occurs, is
(a)
P A'ˆB P A ˆ B ' P A'ˆ B'
(b) 1 P A ˆ B
(c)
P A' P B' P A ‰ B 1
(d) All of the above
Let A and B be two events such that P (A) = 0.3 and P (A ‰ B) = 0.8. If A and B are independent events, then
P (B) =
(a)
21.
(c) 1/2
The probabilities that A and B will die within a year are p and q respectively, then the probability that only
one of them will be alive at the end of the year is
(a)
20.
1
1
4
1
2
neither of them occurs is
19.
(d) None of these
Let E and F be two independent events. The probability that both E and F happens is
(a)
18.
(c) 23/40
If the probability of a horse A winning a race is 1/4 and the probability of a horse B winning the same race is
1/5, then the probability that either of them will win the race is
that neither E nor F happens is
17.
37/45
§B·
1
, then P ¨¨ ¸¸ =
5
©A¹
§B·
¸=
© A¹
(a)
16.
(b)
1
and p A ˆ B
4
If A and B are two events such that A Ž B, then P ¨
(a)
15.
37/40
1
, P B
3
5
6
(b)
5
7
(c)
3
5
(d)
2
5
If E and F are independent events such that 0 < P (E) < 1 and 0 < P (F) < 1, then
(a)
(b)
E and Fc (the complement of the event F) are independent
Ec and Fc are independent
(c)
§ Ec
§E·
P ¨ ¸ P ¨¨ c
©F¹
©F
·
¸¸ 1
¹
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22.
For two given events A and B, P (A ˆ B) =
(a)
(c)
23.
24.
(b) Not greater than P (A) + P (B)
(d) All of the above
P (A ‰ B) = P (A ˆ B) if an only if the relation between P (A) and P (B) is
(a)
P (A) = P A
(b) P (A ˆ B) = P (Ac ˆ Bc)
(c)
P (A) = P (B)
(d) None of these
The two events A and B have probabilities 0.25 and 0.50 respectively. The probability that both A and B
occur simultaneously is 0.14. Then the probability that neither A nor B occurs is
(a)
25.
Not less than P (A) + P (B) – 1
Equal to P (A) + P (B) – P (A ‰ B)
If
0.39
(b)
0.25
(c) 0.904
(d) None of these
1 3 p 1 4 p
1 p
and
are the probabilities of three mutually exclusive and exhaustive events, then the
,
2
3
6
set of all values of p is
(a)
26.
30.
0.18
(b)
0.23 d x d 0.48
0.35
(b)
ª 1º
«¬0, 3 »¼
(d) (0, f)
(c) 0.10
0.32 d x d 0.84
P (B / A) = P (B) – P (A)
P (A ‰ B)c = P (Ac) P (Bc)
(d) 0.63
(c) 0.25 d x d 0.73
(d) None of these
(b) P (Ac ‰ Bc) = P(Ac) + P (Bc)
(d) P (A / B) = P (A)
The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p, and c respectively. On
these subjects, the student has a 75% chance of passing in at least one, a 50% chance of passing in at least two
and a 40% chance of passing in exactly two. Which of the following relations are true
(a)
pmc
(c)
pmc
19
20
1
10
(b)
pmc
(d)
pmc
27
20
1
4
Two events A & B have probability 0.25 & 0.5 respectively. The probability that both A and B occur
simultaneously is 0.14. Then the probability that neither A nor B occurs is
(A) 0.39
31.
(c)
Let 0 < P (A) < 1, 0 < P (B) < 1 and P (A ‰ B) = P (A) + P (B) – P (A) P (B). Then
(a)
(c)
29.
ª 1 1º
«¬ 4 , 3 »¼
If P (A) = 0.3, P (B) = 0.4, P (C) = 0.8, P (AB) = 0.08, P (AC) = 0.28, P (ABC) = 0.09, P (A + B + C) t 0.75,
and P (BC) = x, then
(a)
28.
(b)
Three groups A, B, C are competing for positions on the Board of Directors of a company. The probabilities
of their winning are 0.5, 0.3, 0.2 respectively. If the group A wins, the probability of introducing a new
product is 0.7 and the corresponding probabilities for group B and C are 0.6 and 0.5 respectively. The
probability that the new product will be introduced, is
(a)
27.
[0, 1]
(B) 0.25
(C) 0.11
(D) none
If M & N are any two events, then which one of the following represents the probability of the occurrence of
exactly one of them ?
(A) P (M) + P (N) - 2 P (M ˆ N)
(C)
P M P N 2P M ˆ N
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(D) P M ˆ N P M ˆ N
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32.
(a)
33.
3
and P A'ˆB
25
If A and B are two independent events such that P A ˆ B'
1
5
(b)
3
8
(c)
2
5
8
, then P A =
25
(d)
4
5
If two events A and B are such that P(Ac) = 0.3, P (B) = 0.4 and P (ABc) = 0.5, then P[B / (A ‰ Bc)] is equal
to
(a)
1
2
(b)
1
3
(c)
1
4
(d) None of these
34.
For the three independent events A, B and C, the probability of exactly one of the events A or B occurring =
the probability of exactly one of the events B or C occurring = the probability of exactly one of the vents C or
A occurring = p. If the probability of all the events occurring simultaneously be p 2 where 0 < p < 0.5 then find
the probability of at least one of the events A, B and C occurring.
35.
Three critics review a book. Odds in favour of the book are 5 : 2, 4 : 3 and 3 : 4 respectively for the three
critics. Find the probability that majority are in favour of the book.
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PAGE # 8
CHAPTER EXERCISE – III
1.
A purse contains 4 copper coins and 3 silver coins, the second purse contains 6 copper coins and 2 silver
coins. If a coin is drawn out of any purse, then the probability that it is a copper coin is
(a)
2.
55%
(c) 35%
(d) 45%
2/5
(b)
8/15
(c) 6/11
(d) 2/3
0.25
(b)
0.21
(c) 0.16
(d) 0.6976
1
2
(b)
3
5
(c)
1
4
(d)
1
3
5
14
(b)
5
16
(c)
5
18
(d)
25
52
32
55
(b)
21
55
(c)
19
55
(d) None of these
A box contains 100 tickets numbered 1, 2 ……… 100. Two tickets are chosen at random. It is given that the
maximum number on the two chosen tickets is not more than 10. The minimum number on them is 5 with
probability
(a)
9.
(b)
Urn A contains 6 red and 4 black balls and urn B contains 4 red and 6 black balls. One ball is drawn at random
from urn A and placed in urn B. Then one ball is drawn at random from urn B and placed in urn A. If one ball
is now drawn at random from urn A, the probability that it is found to be red, is
(a)
8.
5%
A bag A contains 2 white and 3 red balls and bag B contains 4 white and 5 red balls. One ball is drawn at
random from a randomly chosen bag and is found to be red. The probability that it was drawn from bag B was
(a)
7.
(d) None of these
A coin is tossed until a head appears or until the coin has been tossed five times. If a head does not occur on
the first two tosses, then the probability that the coin will be tossed 5 times is
(a)
6.
(c) 37/56
An anti-craft gun take a maximum of four shots at an enemy plane moving away from it. The probability of
hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. The probability
that the gun hits the plane is
(a)
5.
3/4
A bag contains 3 while and 2 black balls and another bag contains 2 white and 4 black balls. A ball is picked
up randomly. The probability of its being black is
(a)
4.
(b)
A speaks truth in 75% cases and B in 80% cases. In what percentage of cases are they likely to contradict each
other in stating the same fact.
(a)
3.
4/7
1
8
(b)
13
15
(c)
1
7
(d) None of these
A bag x contains 3 white balls and 2 black balls and another bag y contains 2 white balls and 4 black balls. A
bag and a ball out of it are picked at random. The probability that the ball is white, is
(a)
3
5
(b)
7
15
(c)
1
2
(d) None of these
10.
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by
adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven
cards numbered 2, 3, 4, ……, 12 is picked and the number on the card is noted. The probability that the noted
number is either 7 or 8, is
(a) 0.24
(b) 0.244
(c) 0.024
(d) None of these
11.
The contents of urn I and II are as follows,
Urn I : 4 white and 5 black balls
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Urn II : 3 white and 6 black balls
One urn is chosen at random and a ball is drawn and its colour is noted and replaced back to the urn . Again a
ball is drawn from the same urn, colour is noted and replaced . The process is repeated 4 times and as a result
one ball of white colour and 3 of black colour are noted . Find the probability the chosen urn was I .
(a)
12.
13.
125/287
(b)
120/203
(c) 115/205
(d) none
There are two urns. There are m white & n black balls in the first urn & p white & q black balls in the second
urn. One ball is taken from the first urn & placed into the second. Now, the probability of drawing a white ball
from the second urn is :
(A)
pm p 1 n
m n p q 1
(B)
p 1 m pn
m n p q 1
(C)
qm q 1 n
m n p q 1
(D)
q 1 m qn
m n p q 1
One bag contains 3 white & 2 black balls, and another contains 2 white & 3 black balls. A ball is drawn from
the second bag & placed in the first, then a ball is drawn from the first bag & placed in the second. When the
pair of the operations is repeated, the probability that the first bag will contain 5 white balls is :
(A) 1/25
(B) 1/125
(C) 1/225
(D) none
1
1
and
8
12
respectively. If the odds against making the same mistake by them be 1000 : 1, find the probability of their
results being correct if they obtain the same result.
14.
Two students A and B attempt to solve the same question. Their chances of solving the question are
15.
Three factories A, B and C produce the same product. The factory A produces twice as many as B produces
while the factories B and C produce in the same quantity. It is known that 2% of the products of A as well as
C are defective while 4% of the products of B are defective. All the products of the three factories are stocked
together. If a product is select at random from the stock, what is the probability that the product is defective?
16.
A card from a pack of 52 cards is lost. From the remaining cards of the pack two cards are drawn and are
found to be spades. Find the probability of the missing card to be a spade.
17.
In a test, an examinee either guesses or copies or knows the answer to a multiple-choice question with four
1
choices, only one answer being correct. The probability that he makes a guess is and the probability that he
3
1
1
copies the answer is . The probability that his answer is correct, given that he copies it, is . Find the
6
8
probability that he knew the answer to the question, given that he correctly answers it.
18.
In a bag there are six balls of unknown colours ; three balls are drawn at random and found to be all black.
Find the probability that no black ball is left in the bag (Or Find the probability that the bag contained exactly
3 black balls.)
19.
An urn contains 6 black balls and unknown number (d 6) of white balls. Three balls are drawn successively
and not replaced and are all found to be white. Prove that the chance that a black ball will be drawn in the
677
.
next draw is
909
20.
There are two bags, one of which contains three black and four white balls while the other contains four black
and three white balls. A dice is cast. If the face 1 or 3 turns up, a ball is taken out from the first bag. But if
any other face turns up, a ball is taken from the second bag. Find the probability of getting a black ball.
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21.
Three urns contain 2 white and 3 black balls, 3 white and 2 black balls ; and 4 white and 1 black ball
respectively. A ball is drawn from an urn chosen at random. What is the probability that a white ball is drawn
if the choices of urns are equiprobable ?
22.
An urn contains two balls each of which is either white or black. A white ball is added to the urn. What is the
probability of drawing a white ball from the urn now ?
23.
The probability that a certain electronic component when first use is 0.10. If it does not fair immediately the
probability that it lasts for one year is 0.99. What is the probability that a new component will last one year ?
24.
A factory A produces 10% defective values and another factory B produces 20% defective valves. A bag
contains 4 valves of factory A and 5 valves of factory B. If two valves are drawn at random from the bag, find
the probability that at least one valve is defective.
25.
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained
by adding the numbers shown on them is noted. If the result is a tail, a card from a well-shuffled pack of
eleven cards numbered 2,3,4, ………, 12 is picked and the number of the card is noted. What is the change
that the noted number is either 7 or 8 ?
26.
A bolt factory has three machines A, B and C manufacturing 25%, 35% and 40% of the total production. Of
these the machines produce 5%, 4% and 2% defective bolts respectively. A bolt is selected at random and it is
found to be defective. Find the probability that it was manufactured by the machine (a) A (b) B (c) C.
27.
A can hit a target 4 times in 5 shots ; B can hit 3 times in 4 shots and C twice in 3 shots. They firs once each.
If two of them hit, what is the chance that C has missed it ?
28.
A bag contains 5 balls of unknown colours. A ball is drawn and replaced twice. On each occasion it is found
to be red. Again, two balls are drawn at a time. What is the probability of both the balls being red ?
29.
A letter is known to have come from either MAHARASTRA or MADRAS. On the postmark only consecutive
letters RA can be read clearly. What is the chance that the letter came from MAHARASTRA ?
30. (a) A bag contains 10 coins of which at least 2 are one-rupee coins. Two coins are drawn and both are found to be
not one-rupee coins. What is the probability of the bag to contain exactly 2 one-rupee coins ?
(b)
A man has three coins A, B, C. The coin A is unbiased. The probability that a head will show when B is
2
1
while it is in case of the coin C. A coin is chosen at random and tossed 3 times giving 2 heads
tossed is
3
3
and 1 tail. Find the probability that the coin A was chosen
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CHAPTER EXERCISE – IV
1.
Two card are drawn successively with replacement from a pack of 52 cards. The probability of drawing tow
aces is
(a)
2.
(b)
1
221
(1/10)5
(b)
(1/5)5
(a)
5
§4· §1·
C1 ¨ ¸ ¨ ¸
©5¹ ©5¹
5
4§1·
C1 ¨ ¸
5©5¹
4
4§1·
(c)
¨ ¸
5 © 5¹
4
(d) None of these
(b)
n2
2 m 1
(c)
m2
2 n1
(d) None of these
15
(b)
14
(c) 12
(d) 7
5%
(b)
55%
(c) 35%
(d) 45%
7
§1· §5·
C4 ¨ ¸ ¨ ¸
©6¹ ©6¹
3
3
(b)
7
§1· §5·
C4 ¨ ¸ ¨ ¸
©6¹ ©6¹
4
4
§1· §5·
(c) ¨ ¸ ¨ ¸
©6¹ ©6¹
3
3
§1· §5·
(d) ¨ ¸ ¨ ¸
©6¹ ©6¹
4
Cards are drawn one by one at random from a well shuffled full pack of 52 cards until two aces are obtained
for the first time. If N is the number of cards required to be drawn, then Pr {N = n}, where 2 d n d 50 , is
(a)
(c)
n 1 52 n 51 n
50 u 49 u 17 u 13
3 n 1 52 n 51 n
50 u 49 u 17 u 13
2 n 1 52 n 51 n
50 u 49 u 17 u 13
4 n 1 52 n 51 n
(d)
50 u 49 u 17 u13
(b)
One hundred identical coins with probability p of showing up heads are tossed once. If 0 < p < 1 and the
probability of heads showing on 50 coins, then the value of p is
(a)
10.
(d) (9/10)5
If a die is thrown 7 times, then the probability of obtaining 5 exactly 4 times is
(a)
9.
(b)
n 1
2 m1
4
8.
(c) (9/5)5
A speaks truth in 75% cases and B in 80% cases. In what percentage of cases are they likely to contradict each
other in stating the same fact.
(a)
7.
4
663
A fair coin is tossed n times. If the probability that head occurs 6 times is equal to the probability that head
occurs 8 times, then n is equal to
(a)
6.
(d)
A coin is tossed m + n times, where m t n. The probability of getting at least m consecutive heads is
(a)
5.
1
2652
If the probability that a student is not a swimmer is 1/5, then the probability that out of 5 students one is
swimmer is
4
4.
(c)
In a box containing 100 eggs, 10 eggs are rotten. The probability that out of a sample of 5 eggs none is rotten
if the sampling is with replacement is
(a)
3.
1
169
1
2
(b)
49
101
(c)
50
101
(d)
51
101
The probability that an event will fail to happen is 0.05. The probability that the event will take place on 4
consecutive occasions is
(a)
(c)
0.00000625
0.00001875
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11.
A fair coin is tossed n times. Let X be the number of times head is observed. If P (X = 4), P (X = 5) and
(X = 6) are in H. P., then n is equal to
(a)
12.
(c) 14
(d) None of these
1
5
(b)
2
5
(c)
4
5
(d) None of these
k
999
(b)
k
1000
(c)
k 1
1000
(d) None of these
The items produced by a firm are supposed to contain 5% defective items. The probability that a sample of 8
items will contain less than 2 defective items, is
(a)
15.
10
A locker can be opened by dialing a fixed three digit code (between 000 and 999). A stranger who does not
know the code tries to open the locker by dialing three digits at random. The probability that the stranger
succeeds at the kth trial is
(a)
14.
(b)
A pair of fair dice is rolled together till a sum of either 5 or 7 is obtained. Then the probability that 5 comes
before 7 is
(a)
13.
7
P
27 § 19 ·
¨ ¸
20 © 20 ¹
7
(b)
533 § 19 ·
¨ ¸
400 © 20 ¹
6
153 § 1 ·
(c)
¨ ¸
20 © 20 ¹
The probability of India winning a test match against West Indies is
7
(d)
35 § 1 ·
¨ ¸
16 © 20 ¹
1
. Assuming independence from match
2
to match, the probability that in a 5 match series India’s second win occurs at the third test, is
(a)
16.
3
4
1/2
(d)
1
8
(b)
1
2
(c)
1
3
(d) None of these
(b)
2/5
(c) 1 / 5
(d) 2 / 3
(B) 1/128
(C) 1/32
(D) none
(B) 1/4
(C) 3/5
(D) 1/3
Two dice are thrown until a 6 appears on atleast one of them. Then the probability that for the first time, a 6
appears in the second throw is :
(A) 175/1296
21.
1
4
A coin is tossed until a head appears or until the coin has been tossed five times. If a head does not occur on
the first two tosses, the probability that the coin will be tossed 5 times is:
(A) 1/2
20.
(c)
If 8 coins are tossed, then the chance that one & only one will turn up head is :
(A) 1/256
19.
1
2
An unbiased die is tossed until a number greater than 4 appears. The probability that an even number of tosses
is needed is
(a)
18.
(b)
A man alternatively tosses a coin and throws a dice beginning with the coin. The probability that he gets a
head in the coin before he gets a 5 or 6 in the dice is
(a)
17.
2
3
(B) 275/1296
(C) 375/1296
(D) none
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 three cards at random & keep them in the same order, the
probability of making the word "IIT" is :
(A) 9/80
(B) 1/8
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22.
In a series of 3 independent trials the probability of exactly 2 success is 12 times as large as the probability of
3 successes. The probability of a success in each trial is :
(A) 1/5
23.
(D) 4/5
(B) 2/3
(C) 3/4
(D) none
Two people take turns tossing a coin . The first person to obtain head is the winner. The probability that the
first player wins the game is :
(A) 1/3
25.
(C) 3/5
A & B having equal skill, are playing a game of best of 5 points . After A has won two points & B has won
one point the probability that A will win the game is :
(A) 1/2
24.
(B) 2/5
(B) 1/2
(C) 2/3
(D) none
Three groups of children contain respectively 3 girls and 1 boy, 2 girls and 2 boys, one girl and 3 boys. One
child is selected at random from each group. The chance that three selected consisting of 1 girl and 2 boys, is
(a)
9
32
(b)
3
32
(c)
13
32
(d) None of these
26.
A coin in tossed (m + n) times, m > n. Show that the probability of getting (at least) m consecutive heads is (n
+ 2)/2 m + 1.
27.
In a multiple–choice question, there are four alternative answer of which one or more answers are correct. A
candidate gets marks if he ticks all the correct answers. The candidate, being ignorant about the answers,
decides to tick at random. How many attempts at least should he be allowed so that the probability of his
1
getting marks in the question may exceed ?
5
28.
In a game A throws two ordinary dice. If he throws 7 or 11 he wins. If he throws 2,3 or 12 he loses. If he
throws any other number, he throws again and continues to throw until either the number he threw first or 7
turns up. In the first case he wins and in the second he loses. Show that the odds against his winning is 251 :
244.
29.
A coin is tossed 10 times. Find the probability of getting (i) exactly six heads (ii) at least six heads (iii) at most
six heads.
30.(a) A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that
at the end of eleven steps he is one step away from the starting point.
1
. How many times at least must he fire at the target in
4
2
order that his chance of hitting the target at least once will exceed ?
3
(b)
The probability of a man hitting a target in one fire is
(c)
In a sequence of independent trials, the probability of success in one trial is
1
. Find the probability that the
4
second success takes place on or after the fourth trial.
(d)
A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the
probability that the lot contains exactly 3 defective articles is 0.6. Articles are drawn from the lot at random
one by one without replacement and test till all the defective articles are found. What is the probability that the
testing procedure ends at the 12th testing.?
(e)
Two persons A and B toss a coin 50 times each together. Find the probability that both of them get tails at the
same time.
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CHAPTER EXERCISE – V
1.
If four persons are chosen at random from a group of 3 men, 2 women and 4 children. Then the probability
that exactly two of them are children, is
(a)
2.
1/7
(b)
2/7
(c) 4/53
(d) 4/49
0.8750
(b)
0.0875
(c) 0.0625
(d) 0.0250
1
1 1 p
n
n
(b)
1 1 p
n
(c)
1
1 1 p
n 1
n
(d) None of these
1/6
(b)
5/6
(c) 1/3
(d) None of these
If n positive integers are taken at random and multiplied together, the probability that the last digit of the
product is 2, 4, 6 or 8, is
(a)
7.
(d) 9/21
There are n different objects 1, 2, 3, …… n distributed at random in n places marked 1, 2, 3, …… n. The
probability that at least three of the objects occupy places corresponding to their number is
(a)
6.
(c) 5/21
Let p denotes the probability that a man aged x years will die in a year. The probability that out of n men A1,
A2, A3 …… An each aged x. A1 will die in a year and will be the first to die, is
(a)
5.
8/63
India plays two matches each with West Indies and Australia. Ina any match the probabilities of India getting
point 0, 1 and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independents, the
probability of India getting at least 7 points is
(a)
4.
(b)
The probability that a leap year selected randomly will have 53 Sundays is
(a)
3.
10/21
4n 2n
5n
(b)
4n u 2n
5n
(c)
4n 2n
5n
(d) None of these
For a biased die the probabilities for different faces to turn up are given below
Face:
1
2
3
4
5
6
Probability:
.1
.32
.21
.15
.05
.17
The die is tossed and you are told that either face 1 or 2 has turned up. Then the probability that it is face 1, is
(a)
5
21
(b)
5
22
(c)
4
21
(d) None of these
8.
A biased die is tossed and the respective probabilities for various faces to turn up are given below
Face:
1
2
3
4
5
6
Probability:
.1
.24
.19
.18
.15
.14
If an even face has turned up, then the probability that it is face 2 or face 4, is
(a) 0.25
(b) 0.42
(c) 0.75
(d) 0.9
9.
A determinant is chosen at random. The set of all determinants of order 2 with elements 0 or 1 only. The
probability that value of the determinant chosen is positive, is
(a)
10.
(b)
3
8
(c)
1
4
(d) None of these
If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the
form 7m + 7n is divisible by 5 equals
(a)
11.
3
16
1
4
(b)
1
7
(c)
1
8
(d)
1
49
The odds against a certain event are 5 : 2 and the odds in favour of another independent event are 6 : 5. Then
the chance that one atleast of the event will happen is :
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(A) 2/3
12.
(C) 52/77
(D) none
If the homework of each one of 4 students can be checked by one of the 7 teachers, then the probability of all
the 4 papers being checked by different teachers is :
(A) 5/343
13.
(B) 8/11
(B) 60/343
(C) 120/343
(D) none
Sixteen players S1, S2, S3…………., S16 play in a tournament. They are dived into eight pairs at random. From
each pair a winner is decided on the basis of a game played between the two players of the pair. Assume that
all the players are of equal strength.
(a)
(b)
Find the probability that the players S1 is among the eight winners.
Find the probability that exactly one of the two players S1 and S2 is among the eight winners.
14.
An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the
urn, otherwise it is replaced along with another ball of the same colour. The process is repeated. Find the
probability that the third ball drawn is black.
15.
A man and a woman appear in an interview for, two vacancies in the same post. The probability of man’s
selection is 1/4 and that of the woman’s selection is 1/3, what is the probability that
(a)
(c)
16.
both of them will be selected
none of them will be selected
(b) only one of them will be selected
If m things are distributed among a men and b women, show that the chance that the number of things
m
1 b a ba
received by men is odd is .
m
2
ba
m
.
17.
If on an average I vessel in every 10 is wrecked, find the chance that out of 5 vessels expected 4 at least will
arrive safely.
18.
In a certain experiment the probability of success is twice the probability of failure. Find the probability of at
least four successes in six trials.
19.
Suppose the probability for A to win a game against B is 0.4. If A has an option of playing either a “best of 3
games’’ or a “best of 5 games” match against B ; which option should A choose so that the probability of his
winning the match is higher ? (No game ends in a draw).
20. (a) A, B, c in order cut a pack of cards, replacing them after each cut, on the condition that the first who cuts a
spade shall win a prize ; find their respective chances.
(b)
Three players, A, B and C, toss a coin cyclically in that order (that is A, B, C, A, B, C, A, B, ….. ) till a head
shows. Let p be the probability that the coin shows a head. Let D, E and J be, respectively, the probabilities
that A, B and C gets the first head. Prove that E = (1 – p)D. Determine D, E and J (in terms of p).
(c)
An employer sends a letter to his employee but he does not receive the reply (It is certain that employee would
have replied if he did receive that letter). It is known that one out of n letters does not reach its destination.
Find the probability that employee does not receive the letter.
(d)
A set A has n elements. A subset P of A is selected at random. Returning the elements of P, the set Q is
formed again and then a subset Q is elected from it. Find the probability that P and Q have no common
elements.
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MISCELLANEOUS EXERCISE – I
1.
A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the
probability that the lot contains exactly 3 defective articles is 0.6. Articles are drawn from the lot at random
one day by one without replacement and are tested till all defective articles are found. What is the probability
that the testing procedure ends at the twelth testing ?
(IIT 1986)
2.
A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at
the end of eleven steps he is one step away from the starting point.
(IIT 1987)
3.
An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is not replaced into the urn.
Otherwise it is replaced along with another all of the same colour. The process is replaced. Find the
probability that the third ball drawn is black.
(IIT 1987)
4.
A box contains 2 fifty paise coins, 5 twenty five paise coins and a certain fixed number N t 2 of ten and five
paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these 5
coins is less than one rupee and fifty paise.
(IIT 1988)
5.
Suppose the probability for A to win a game against B is 0.4. If A has an option of plying either a “best of 3
games” or a “best of 5 games” match against B, which option should choose so that the probability of his
winning the match is higher? (No game ends in a draw).
(IIT 1989)
6.
A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed by replacing
the elements of P. A subset Q of A is again chosen at random. Find the probability that P and Q have no
common elements.
(IIT 1991)
7.
In a test an examinee either guesses or copies of knows the answer to a multiple choice question with four
1
1
choices. The probability that he make a guess is and the probability that he copies the answer is . The
3
6
1
probability that his answer is correct given that he copied it, is . Find the probability that he knew the
8
answer to the question given that he correctly answered it.
(IIT 1991)
8.
A lot contains 50 defective and 50 non-defective bulbs. Two bulbs are drawn at random, one at a time, with
replacement. The events A, B, C are defined as :
A = (the first bulb is defective)
B = (the second bulb is non-defective)
C = (the two bulbs are both defective or both non-defective).
Determine whether
(i) A, B, C are pair wise independent,
(ii) A, B, C are independent.
(IIT 1992)
9.
Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02,.......,99 with
replacement. An event E occurs if and only if the product at the two digits of a selected number is 18. If four
numbers are selected, find probability that the event E occurs at least 3 times.
(IIT 1993)
10.
An unbiased coin is tossed. If the result in a head, a pair of unbiased dice is rolled and the number obtained by
adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven
cards numbered 2, 3, 4......,12 is picked and the number on the card is noted. What is the probability that the
noted number is either 7 or 8?
(IIT 1994)
11.
In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, 3 in the
front and 4 at the back? How many seating arrangements are possible if 3 girls should sit together in a back
row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of 3 girls
sitting together in a back row on adjacent seats?
(IIT 1996)
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12.
Sixteen players S1 ,S2 ,..........,S16 play in a tournament. They are divided into at pairs at random from each pair
a winner is decided on the basis of a game played between the two players of the pair. Assume that all the
players are of equal strength.
(a)
Find the probability that the players S1 is among the eight winners.
(b)
Find the probability that exactly one of the two players S1 and S2 is among the eight winners.
(IIT 1997)
13.
If p and q are chosen randomly from the set { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }, with replacement, determine the
(IIT 1997)
probability that the roots of the equation x 2 px q 0 are real.
14.
Three players A, B. and C, toss a coin cyclically in that order (that is A, B, C, A, B, C, A, B,....) till a head
shows. Let p be the probability that the coin shows a head. Let D, E and J be, respectively that the
probability, the probabilities that A, B, and C gets the first head. Prove that E
D, E and J (in terms of p).
15.
1 p D . Determine
(IIT 1998)
Eight players P1 , P2 ,........, P8 play a knock-out tournament. It is known that whenever the players Pi and Pj
play, the player P1 will win if i < j. Assuming that the players are paired at random in each round, what is the
(IIT 1999)
probability that the player P4 reaches the final?
16.
A coin has probability p of showing head when tossed. It is tossed n times. Let Pn denote the probability that
no two (or more) consecutive heads occur. Prove that p1 1
p2
1 p 2 and p n
1 p .
p n 1 p 1 p p n 2 for all n t 3
(IIT 2000)
17.
An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k
additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the
probability that the ball drawn now is white?
(IIT 2001)
18.
An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6, is thrown n times and the list of n numbers showing up
is noted. What is the probability that among the numbers 1, 2, 3, 4, 5, 6 only three numbers appear in this list?
(IIT 2001)
19.
A box contains N coins, m of which are fair and the rest are biased. The probability of getting a head when a
fair coin is tossed is ½, while it is 2/3 when a biased coin is tossed. A coin is drawn from the box at random
and is tossed twice. The first time it shows head and the second time it shows tail. What is the probability that
the coin drawn is fair?
(IIT 2002)
20.(a) For a student to qualify, he must pass at least two out of three exams. The probability that the will pass the 1st
p
otherwise it
exam is p. If he fails in one of the exams then the probability of his passing in the next exam is
2
remains the same. Find the probability that he will qualify.
(IIT 2003)
(b).
A is targeting to B, B and C are targeting to A. probability of hitting the target by A, B and C are
2 1
1
, and
3 2
3
respectively. If A is hit, then find the probability that B hits the target and C does not.
(IIT 2003)
(c).
If A and B are two independent events, prove that P A ‰ B .P Ac ˆ Bc d P C , where C is an event defined
that exactly one of A and B occurs.
(IIT 2004)
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(d).
(e).
A bag contains 12 red balls and 6 white balls. Six balls are drowns one by one without replacement of which
at least 4 balls are white. Find the probability that in the next two drowns exactly one white ball is drawn.
(Leave the answer in n C r ).
(IIT 2004)
1 3 2
1
, , and
7 7 7
7
respectively. Probability that he reaches offices late, if he takes car, scooter bus or train is
2 1 4
1
, , and respectively. Given that he reached office in time, then what is the probability that he traveled
9 9 9
9
by a car?
(IIT 2005)
A person goes to office either by car, scooter, bus or train probability of which being
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