MPR – 1
PROBABILITY
There are various phenomena in nature, leading to an outcome, which cannot be predicted correctly e.g. in
tossing of a coin, a head or a tail may result. Probability theory at measuring the uncertainties of such
outcomes.
C1
Important terminology :
(i)
Random Experiment :
It is a process which results in an outcome which is one of the various possible outcomes that are known to
us before hand. For example throwing of a dice is a random experiment as it leads to fall of one of the
outcome from {1, 2, 3, 4, 5, 6}. Similarly taking a card from a pack of 52 cards is also a random experiment.
(ii)
Sample Space :
It is the set of all possible outcome of a random experiment e.g. {H, T} is the sample space associated with
tossing of a coin. In set notation it can be interpreted as the universal set and denoted by S.
(iii)
Event :
It is subset of sample space denoted by E and in set notation E  S . For example getting a head in tossing
a coin or getting a prime number in throwing a die. In general if a sample space consists ‘n’ elements, then
a maximum of 2n events can be associated with it.
(iv)
Simple Event or Elementary Event :
If an event covers only one point of sample space, then it is called a simple event e.g. getting a head
followed by a tail in throwing of a coin two times is a simple event.
(v)
Compound or Composite or Mixed Event :
Where two or more than two events occur simultaneously, the event is said to be a compound event.
Symbolically A  B or AB represent the occurrence of both A and B simultaneously.
“A  B” or A + B represent the occurrence of either A or B.
(vi)
Mutually Exclusive / Disjoint / Incompatible Events :
Two events are said to be mutually exclusive if occurrence of one of them rejects the possibility of occurrence of the other i.e. both cannot occur simultaneoulsy.
In the vain diagram the events A and B are mutually exclusive. Mathematically, we write A  B = 
(vii)
Exhaustive System of Events :
If each outcome of an experiment is associated with at least one of the events E1, E2, E3,.....En, then
collectively the events are said to be exhaustive. Mathematically we write E1  E2  E3........En = S.
(Sample space)
Practice Problems :
1.
Two dice are rolled. A is the event that the sum of the numbers shown on the two dice is 5, and B is the
event that at least one of the dice shows up a 3. Are the two events (i) mutually exclusive,
(ii) exhaustive ? Give arguments in support of your answer.
(viii)
Complement of event :
The complement of an event ‘A’ with respect to a sample space S is the set of all elements of ‘S’ which are
not in A. It is usually denoted by A, A or A C .
(ix)
Equally likely Events :
If events have same chance of occurrence, then they are said to be equally likely
e.g.
(i)
In a single toss of fair coin, the events {H} and {T} are equally likely.
(ii)
In tossing a biased coin the events {H} and {T} are not equally likely.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 2
C2
Classical Definition of Probability :
If an experiment results in a total of (m + n) outcomes which are equally likely and mutually exclusive with
one another and if ‘m’ outcomes are favorable to an event ‘A’ while ‘n’ are unfavourable, then the
probability of occurrence of the event ‘A’ denoted by
P(A), is defined by
m
m
number of favourable outcomes

i.e. P(A) =
mn
mn
total number of outcomes
We say that odds in favour of ‘A’ are m : n, while odds against ‘A’ are n : m.
Note that P( A ) or P( A  ) or P(AC), i.e. probability of non-occurrence of A 
n
 1  P( A )
mn
In the above we shall denote the number of out comes favourable to the event A by n(A) and the total
number of out comes in the sample space S by n(S).
P( A ) 

n( A)
n(S)
Practice Problems :
1.
A coin is tossed once. Find the probability of getting a head.
2.
Two coins are tossed once. Find the probability of
3.
(i)
getting 2 heads
(ii)
getting at least 1 head
(iii)
getting no head
(iv)
getting 1 head and 1 tail
A die is tossed once. What is the probability of getting
(i)
the number 4 ?
(ii)
an even number ?
(iii)
a number less than 5 ?
(iv)
a number greater than 4 ?
(v)
the number 8 ?
(vi)
a number less than 8 ?
4.
A bag containing 9 red and 12 white balls. One ball is drawn at random. Find the probability that the
ball drawnis red.
5.
The odds in favour of occurrence of an event are 5 : 12. Find the probability of the occurrence of this
event.
6.
If the probability of the occurrence of a certain event E is 3/11, find (i) the odds in favour of its
occurrence, and (ii) the odds against its occurrence.
7.
Three dice are thrown together. Find the probability of getting a total of at least 6.
8.
If the odds in favour of an event be 3/5, find the probability of the occurence of the event.
9.
Two dice are thrown. Find (i) the odds in favour of getting the sum 5, and (ii) the odds against getting
the sum 6.
[Answers : (1) 1/2 (2) (i) 1/4 (ii) 3/4 (iii) 1/4 (iv) 1/2 (3) (i) 1/6 (ii) 1/2 (iii) 2/3 (iv) 1/3 (v) 0 (vi) 1
(4) 3/7 (5) 5/17 (6) (i) 3/8 (ii) 8/3 (7) 103/108 (8) 3/8 (9) 31/5]
C3
Addition theorem of probaility :
If ‘A’ and ‘B’ are any two events associated with an experiment, then
P(A  B) = P(A) + P(B) – P(A B)
De Morgan’s Laws :
Distributive Laws :
If A and B are two subsets of a universal set U, then
(a)
(A B)C = AC BC
(b)
(A B)C = AC BC
(a)
A (B C) = (A B) (A C)
(b)
A (B C) = (A B) (A C)
For any three events A, B and C
(i)
P (A or B or C) = P(A) + P(B) + P(C) – P(A B) – P(B C) – P(C A) + P(A B C)
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 3
(ii)
P (at least two of A, B, C occur) = P(B C) + P(C A) + P(A B) – 2P(A B C)
(iii)
P (exactly two of A, B, C occur) = P(B C) + P(C A) + P(A B) – 3P(A B C)
(iv)
P (exactly one of A, B, C occur) =
P(A) + P(B) + P(C) – 2P(B  C) – 2P(C A) – 2P(A B) + 3P(A B C)
If three events A, B and C are pair wise mutually exclusive then they must be mutually exclusive,
i.e. P(A B) = P(B C) = P(C  A) = 0  P (A B C) = 0.
Practice Problems :
1.
If E 1 and E 2 are two events associated with a random experiment such that P(E 2) = 0.35,
P(E1 or E2) = 0.85 and P(E1 and E2) = 0.15, find P(E1).
2.
If E1 and E2 are two events such that P(E1) = 0.5, P(E2) = 0.3 and P(E1 and E2) = 0.1. Find
3.
(i)
P(E1 or E2)
(ii)
P(E1 but not E2)
(iii)
P(E2 but not E1)
(iv)
P(neither E1 nor E2)
The probability that at least one of the events E1 and E2 occurs is 0.6. If the probability of the
simultaneous occurrence of E1 and E2 is 0.2, find P(E1 )  P(E 2 ) .
4.
The probabilities of the occurrences of two events E1 and E2 are 0.25 and 0.50 respectively. The
probability of their simultaneous occurrence is 0.14. Find the probability that neither E1 nor E2
occurs.
5.
A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red
card.
[Answers : (1) 0.65 (2) (i) 0.7 (ii) 0.4 (iii) 0.2 (iv) 0.3 (3) 1.2 (4) 0.39 (5) 7/13]
C4
Independent and dependent events
If two events are such that occurence or non-occurence of one does not affect the chances of occurence or
non-occurence of the other event, then the events are said to be independent. Mathematically :
if P(A  B) = P(A) P(B), then A and B are independent.
(i)
If A and B are independent, then (a) A  and B are independent, (b) A and B are independent
and (c) A  and B are independent.
(ii)
If A and B are independent, then P(A/B) = P(A). If events are not independent then they are said
to be dependent.
Independency of three or more events
Three events A, B and C are independent if and only if all the following conditions hold :
P(A  B) = P(A) . P(B)
;
P(B  C) = P(B) . P(C)
P(C  A) = P(C) . P(A)
;
P(A  B  C) = P(A) . P(B) . P(C)
i.e. they must be independent in pairs as well as mutually independent.
Similarly for n events A1, A2, A3,.........An to be independent, the number of these conditions is equal to
n
C2 + nC3 + ...... + nCn = 2n – n – 1.
Practice Problems :
1.
2.
Let E1 and E2 be two events such that P(E1) = 4/7 and P(E2) = 1/4. Find
(i)
P(E1 or E2), if E1 and E2 are mutually exclusive events
(ii)
P(E1 and E2), if E1 and E2 are independent events
Let E1 and E2 be events such that P(E1) = 0.3, P(E1  E2) = 0.4 and P(E2) = x. Find the value of x such
that
(i)
E1 and E2 are mutually exclusive
(ii)
E1 and E2 are independent
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 4
3.
The probabilities of a specific problem being solved independently by A and B are 1/2 and 1/3
respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved (ii) exactly one of them solves the problame.
4.
A and B appear for an interview for two posts. The probability of A’s selection is (1/3) and that of B’s
selection is (2/5). Find the probability that only one of them will be selected.
5.
Three groups of children contain 3 girls and 1 boy : 2 girls and 2 boys; and 1 girl and 3 boys. One
child is selected at random from each group. Find the chance that the three children selected
comprise 1 girl and 2 boys.
6.
A problem is given to three students whose chances of solving it are 1/3, 2/7 and 3/8. What is the
probability that the problem will be solved ?
7.
A problem in mathematics is given to three students whose chances of solving it correctly are 1/2,
1/3 and 1/4 respectively. What is the probability that only one of them solves it correctly ?
8.
The odds against a man who is 45 years old, living till he is 70 are 7 : 5, and the odds against his wife
who is now 36, living till she is 61 are 5 : 3. Find the probability that
(i)
the couple will be alive 25 years hence
(ii)
at least one of them will be alive 25 years hence.
9.
A, B and C shoot to hit a target. If A hits the target 4 times in 5 trials; B hits it 3 times in 4 trials and
C hits it 2 times in 3 trials, what is the probability that the target is hit by at least 2 persons ?
10.
Two persons A and B throw a coin alternately till one of them gets a ‘head’ and wins the game. Find
their respective probabilities of winning if A starts first.
11.
A and B throw a die alternately till one of them gets a 6 and wins the game. Find their respective
probabilities of winning if A starts first.
12.
Three persons A, B, C throw a die in succession till one gets a ‘six’ and wins the game. Find their
respective probabilities of winning.
[Answers : (1) (i) 23/28 (ii) 1/7 (2) (i) x = 0.1 (ii) x = 1/7 (3) (i) 2/3 (ii) 1/2 (4) 7/15 (6) 13/32 (6) 59/84
(7) 11/24 (8) (i) 5/32 (ii) 61/96 (9) 5/6 (10) 1/3 (11) 5/11 (12) 25/91]
C5
Conditional Probability
If A and B are two events associated with the same random experiment then the probability of occurrence
of A under the condition that B has already occurred, and P(B)  0 is called conditional probability,
denoted by P(A/B). P (A/B) =
P( A  B )
.
P( B )
For mutually exclusive events P(A/B) = 0.
Practice Problems :
1.
A coin is tossed twice and four possible outcomes are assumed to be equally likely. If A is the event
that ‘both head and tail have appeared’ and B is the event that ‘at most one tail has appeared’, find
(i) P(A) (ii) P(B) (iii) P(A/B) (iv) P(B/A)
2.
A die is rolled twice and the sum of the numbers appearing is observed to be 7. What is the
conditional probability that the number 2 has appeared at least once ?
3.
Two unbiased dice are thrown. Find the probability that the sum is 8 or greater if 4 appears on the
first die.
4.
A die is thrown twice and the sum of the numbers appearing is observed tobe 8. What is the
conditional probability that the number 5 has appeared at least once ?
5.
In a class, 40% students study mathematics, 25% study biology and 15% study both mathematics
and biology. One student is selected at random. Find the probability that
(i)
he studies mathematics if it is known that he studies biology
(ii)
he studies biology if it is known that he studies mathematics
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 5
6.
A couple has 2 children. Find the probability that both are boys if it is known that (i) one of the
children is a boy, and (ii) the elder child is boy.
7.
Two numbers are selected at random from the integers 1 through 9. If the sum is even, find the
probability that both the numbers are odd.
[Answers : (1) (i) 1/2 (ii) 3/4 (iii) 2/3 (iv) 1 (2) 1/3 (3) 1/2 (5) (i) 3/5 (ii) 3/8 (6) (i) 1/3 (ii) 1/2 (7) 5/8]
C6
Total Probability Theorem
If an event A can occur with one of the n mutually exclusive and exhaustive events B1, B2,.........,Bn and the
probabilities P(A/B1), P(A/B2)....P(A/Bn) are known, then
n
P( A ) 
 P( B ) . P( A / B )
i
i
i 1
C7
Bayes’ Theorem :
If an event A can occur with one of the n mutually exclusive and exhaustive events B1, B2,.........,Bn and the
probabilities P(A/B1), P(A/B2).....P(A/Bn) are known, then
P( B i / A ) 
P(B i ).P( A / B i )
n
 P(B ).P(A / B )
i
i
i 1
Practice Problems :
1.
A doctor is to visit a patient. From past experience, it is known that the probabilities that he will
come by train, bus, scooter or by car are respectively
3 1 1
2
, , and . The probabilities that he will
10 5 10
5
1 1
1
, if he come by train, bus and scooter respectively; but if he comes by car, he
, and
4 3
12
will not be late. When he arrives, he is late. What is the probability that he has come by train ?
be late are
2.
In an examination, an examiner either guesses or copies or knows the answer to a multiple-choice
question with four choices. The probability that he makes a guess is (1/3) and the probability that he
copies the answer is (1/6). The probability that his answer is correct, given that he copied it, is (1/8).
The probability that his answer is correct, given that he guessed it, is (1/4). Find the probability that
he knew the answer to the question, given that he correctly answered it.
3.
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 one of the bags and it is found to be red. Find the probability that it was drawn from
bag B.
4.
There are 5 bags, each containing 5 white balls and 3 black balls. Also, there are 6 bags, each
containing 2 white balls and 4 black balls. A white ball is drawn at random. Find the probability that
this white ball is from a bag of the first group.
5.
Urn A contains 1 white, 2 black and 3 red balls; urn B contains 2 white, 1 black and 1 red ball; and
urn C contains 4 white, 5 black and 3 red balls. One urn is chosen at random and two balls are
drawn. These happen to be one white and one red. What is the probability that they come from urn
A?
6.
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 both spades. Find the probability of the lost card being a space.
[Answers : (1) 1/2 (2) 24/29 (3) 25/52 (4) 75/123 (5) 33/118 (6) 0.22]
C8
Binomial Probability Theorem
If an experiment is such that the probability of success or failure does not change with trails, then the
probability of getting exactly r success in n trials of an experiment is nCr pr qn – r, where ‘p’ is the probability
of a success and q is the probability of a failure. Note that p + q = 1.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 6
1.
2.
3.
4.
C9
Practice Problems :
A die is thrown 5 times. If getting an odd number is a success, find the probability of getting at least
4 successes.
In 4 throws with a pair of dice, what is the probability of throwing doublets at least twice ?
The bulbs produced in a factory are supposed to contain 5% defective bulbs. What is the probability
that a sample of 10 bulbs will contain not more than 2 defective bulbs ?
If on an average, out of 10 ships, one gets drowned then what is the probability that out of 5 ships at
least 4 reach the shore safely ?
[Answers : (1) 3/16 (2) 171/1296 (3) 99/100 (4) 0.9181]
Value of Testimony
If p1 and p2 are the probabilities of speaking the truth of two independent witnesses A and B then P (their
combined statement is true) =
p 1p 2
.
p 1p 2  (1  p 1 )(1  p 2 )
In this case it has been assumed that we have no knowledge of the event except the statement made by A and
B.
However if p is the probability of the happening of the event before their statement, then P(their combined
statement is true) =
C10
pp 1p 2
pp 1p 2  (1  p )(1  p 1 )(1  p 2 )
Here it has been assumed that the statement given by all the independent witnesses can be given in two
ways only, so that if all the witnesses tell falsehoods they agree in telling the same falsehood. If this is not
the case and c is the chance of their coincidence testimony then the
Probability that the statement is true = P p1 p2
Probability that the statement is false = (1 – p) . c (1 – p1) (1 – p2)
However chance of coincidence testimony is taken only if the joint statement is not contradicted by any
witness.
Expectation :
If a value M1 is associated with a probability of pi, then the expectation is given by
C11
(i)
p M
i
i
Probability Distribution :
A probability distribution spells out how a total probability of 1 is distributed over several values of a
random variable.
If a random variable X takes the values x1, x2,....,xn with respective probabilities p1, p2,....,pn then the
probability distribution of X is given by
X
x1
x2
x3
...
...
xn
P(X)
p1
p2
p3
...
...
pn
Mean of any probability distribution of a random variable is given by :
µ
p x  p x
p
i i
i i
i
(Since
p
i
 1)
(ii)
Variance of a random variable is given by,  2 
(iii)
Standard Deviation = variance
Practice Problems :
Find the mean, variance and standard deviation of the number of trails in two tosses of a coin.
Find the mean, variance and standard deviation of the number of heads when three coins are tossed.
A die is tossed once. If the random variable X is defined as
1.
2.
3.
2
i i
p x
 µ2
1, if the die results in an even number
X
then find the mean and variance of X.
 0, if the die results in an odd number
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 7
4.
5.
6.
7.
8.
9.
Find the mean, variance and standard deviation of the number of sixes in two tosses of a die.
Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Find the
mean and variance of the number of kings.
Two cards are drawn simultaneously (or successively without replacement) from a well-shuffled
pack of 52 cards. Find the mean and variance of the number of aces.
Three defective bulbs are mixed with 7 good ones. Let X be the number of defective bulbs when
3 bulbs are drawn at random. Find the mean and variance of X.
An urn contains 4 white and 3 red balls. Let X be the number of red balls in a random draw of
3 balls. Find the mean and variance of X.
In a game, 3 coins are tossed. A person is paid Rs. 5 if he gets all heads or all tails; and he is supposed
to pay Rs. 3 if he gets one head or two heads. What can he expect to win on an average per game ?
[Answers : (1) 0.707 (2)  = 3/2 (3) 1/4 (4)
5
(5) 24/169 (6) 400/2873 (7) 49/100 (8) 24/49
2
(9) he loses Re 1 per toss]
C12
(i)
Binomial Probability Distribution :
The probability distribution for a binomial variate ‘X’ is given by :
P(X = r) = nCr pr qn – r where P(X = r) is the probability of r successes.
The recurrence formula
(ii)
P(0) is known.
Mean of binomial probability distibution = np ;
Variance of binomial probability distibution = npq.
Standard deviation =
(iii)
1.
2.
3.
4.
5.
6.
7.
8.
9.
P(r  1) n  r p

. , is very helpful for quick computation P(1) . P(2) . P(3) etc. if
P(r )
r1 q
npq
If p represents a person’s chance of success in any venture and ‘M’ the sum of money which he will receive
in case of success, then his expectations or probable value = pM
Practice Problems :
If X follows a binomial distribution with mean 3 and variance (3/2), find (i) P(X  1) (ii) P(X  5).
If X follows a binomial distribution with mean 4 and variance 2, find P(X  5).
Find the binomial distribution for which the mean and variance are 12 and 3 respectively.
If the sum of the mean and variance of a binomial distribution for 5 trials is 1.8, find the distribution.
The sum and the product of the mean and variance of a binomial distribution are 24 and 128
respectively. Find the distribution.
In a binomial distribution, prove that mean > variance.
A die is tossed thrice. Getting an even number is considered a success. What is the variance of the
binomial distribution ?
A die is rolled 20 times. Getting a number greater than 4 is a success. Find the mean and variance of
the number of successes.
A die is tossed 180 times. Find the expected number (µ) of times the face with the number 5 will
appear. Also, find the standard deviation (), and variance (2).
r
 3 1
[Answers : (1) (i) 63/64 (ii) 63/64 (2) 93/256 (3) P( X  r )  16Cr .   .  
 4  4
(16 r )
, wheree
r = 0, 1, 2, 3,..., 15 (4) P(X = r) = nCr . Pr . q(n – r) = 5Cr . (0.2)r . (0.8)(5 – r), where r = 0, 1, 2, 3, 4, 5 (5)
32
C13
(i)
(ii)
1
Cr .  
 2
32
(7) 3/4 (8) 4.44 (9) 5]
Geometrical Applications :
The following statements are axiomatic :
If a point is taken at random on a given straight line segment AB, the chance that it falls on a particular
segment PQ of the line segment is PQ/AB.
If a point is taken at random on the area S which includes an area , the chance that the point falls on  is
/S.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 8
SINGLE CORRECT CHOICE TYPE
1.
The probability of a man hitting a target in one fire
1
is . The number of times at least must he fire at
4
the target in order that his chance of hitting the
target at least once will exceed
2.
6.
2
.
3
(a)
3
(b)
4
(c)
5
(d)
6
7.
If p is chosen at random in the closed interval [0, 5]
then the probability of the equation
1
x 2  px  (p  2)  0 to have real roots is
4
3.
4.
(a)
3
5
(c)
4
5
(b)
2
5
(d)
1
5
8.
If coefficients a, b, c of quadratic equation
ax 2 + bx + c = 0 are chosen at random with
replacement from the set S = {1, 2, 3,. 4, 5, 6}. The
probability that roots of quadratic are real and
distinct is
(a)
17
108
(b)
19
108
(c)
21
108
(d)
23
108
9.
5.
0.2
(b)
0.3
(c)
0.4
(d)
0.5
There are 4 urns. The first urn contains 1 white &
1 black ball, the second urn contains 2 white &
3 black balls, the third urn contains 3 white &
5 black balls & the fourth urn contains 4 white & 7
black balls. The selection of each urn is not equally
(b)
569/1498
(c)
494/569
(d)
none of these
A 5 digit number is formed by using the digits 0, 1,
2, 3, 4 & 5 without repetition. The probability that
the number is divisible by 6 is
(a)
0.08
(b)
0.17
(c)
0.18
(d)
0.36
A bag contains four tickets numbered 00, 01, 10,
11. Four tickets are chosen at random with
replacement the probability that sum of tickets is
23 is
(a)
3
32
(b)
1
64
(c)
5
256
(d)
7
256
Fifteen coupons are numbered 1, 2, ..... 15
respectively. Seven coupons are selected at random
with replacement. The chance that the largest
number appearing on a selected coupon is 9, is
(a)
 9 
 
 16 
(c)
 3
 
5
6
7
(b)
 8 
 
 15 
(d)
none of these
7
A dice is thrown n times. For the probability of a
10.
(a)
n>4
(b)
n4
(c)
n=2
(d)
n=6
Three six-faced dice are thrown together. The
probability that the sum of the numbers appearing
on the dice is k (9  k  14) is
(a)
21k  k 2  83
216
(b)
k 2  3k  2
432
(c)
21k  k 2  83
432
(d)
21k  k 2  80
216
i2  1
34
(i = 1, 2, 3, 4). If we randomly select one of the urns
& draw a ball, then the probability of ball being
white is
likely. The probability of selecting ith urn is
Einstein Classes,
1
we
2
must have
integers. A coin of diameter
(a)
569/1496
six appearing atleast once to be more than
Consider the cartesian plane R2, and let X denote
the subset of points for which both co-ordinates are
1
is tossed randomly
2
onto the plane. The probability that the coin
covers a point of X is
(a)
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 9
EXCERCISE BASED ON NEW PATTERN
COMPREHENSION TYPE
6.
Comprehension-1
A and B play 12 games of chess of which 6 are won
by A, 4 are won by B, and 2 end in a draw. They
agree to play a tournament consisting of 3 games.
1.
3.
4.
(a)
8
81
(c)
 2
 
 3
The probability that A wins all 3 games
(a)
1
8
(c)
5
36
(b)
5
72
(d)
19
27
7.
2.
The experiment is done 5 times. The probability that
a white ball is drawn most of the time is
The probability that 2 games end in a draw
(a)
1
8
(b)
5
72
(c)
5
36
(d)
19
27
1
8
(c)
5
36
(b)
5
72
(d)
19
27
1
8
(b)
5
72
(c)
5
36
(d)
19
27
The experiment is done 10 times. The most
probable number of successes is
(a)
7
(b)
6
(c)
8
(d)
5
8.
If P(ui) is directly proportional to i where i = 1, 2, 3,
...., n then lim P(w) is equal to
n 
9.
(a)
2
3
(b)
3
4
(c)
1
4
(d)
1
u 
If P(ui) = c, a constant then, P n  is equal to
 w 
(a)
1
n1
(b)
2
n1
(c)
n
n1
(d)
1
2
Comprehension-2
In a series of n independent trials for an event of
constant probability p, the most probable number
r of successes is given by (n + 1)p – 1 < r < (n + 1)p.
Hence, the most probable number of successes is
the integral part of (n + 1)p. But if (n + 1)p is an
integer, the chance of r successes is equal to that of
r + 1 successes and both r, r + 1 are most probable
numbers of successes. A bag contains 2 white balls
and 1 black ball. A ball is drawn at random and
returned to the bag.
5.
10.
If n is even and E denotes the event of choosing
even-numbered urn and uis are equiprobable then
w
the value of P  is
E
(a)
1
n1
(b)
n
n1
(c)
n2
2(n  1)
(d)
n 2
2n  1
The experiment is done 10 times. The probability
that a white ball is drawn exactly 5 times is
 2
 
 3
5
(a)
10 !
5!5!
(c)
10 !  2 
 
(5 ! ) 2  9 
(b)
10 !
5!5!
(d)
10 !  2 
 
5!  9 
5
Einstein Classes,
8
27
There are n urns each containing n + 1 balls such
that the ith urn contains i white balls and (n + 1 – i)
red balls. Let ui be the event of selecting ith urn
and w denote the event of getting a white ball.
The probability that B wins at least 1 game
(a)
(d)
6
6
Comprehension-3
The probability that A and B win alternatley
(a)
(b)
 2
9 · 
 3
1
 
 3
5
5
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 10
MATRIX MATCH TYPE
2.
Matching-1
In the game of pocker 5 cards are drawn from a
pack of 52 well-shuffled cards.
Column - A
If A and B are independent events such that
0 < P(A) < 1, 0 < P(B) < 1 then
(a)
A, B are mutually exclusive
(b)
A and B are independent
(c)
A , B are independent
(d)
P(A/B) + P( A /B) = 1
Column - B
(A)
Probability that 4 are aces (p)
1/54,145
(B)
Probability that 4 are aces
and 1 is a king
(q)
1/649,740
Probability that 3 are tens
and 2 are jacks
(r)
1/108,290
Probability that a nine,
ten, jack, queen, king are
obtained in any order
(s)
In throwing a die let A be the event ‘coming up of
an odd number’, B be the event ‘coming up of an
even number’, C be the event ‘coming up of a
number  4’ and D be the event ‘coming up of a
number < 3’, then
64/162,435
(a)
A and B are mutually exclusive and
exhaustive
(b)
A and C are mutually exclusive and
exhaustive
(c)
A, C and D form an exhaustive system
(d)
B, C and D form an exhaustive system
(C)
(D)
3.
Matching-2
Column - A
(A)
(B)
(C)
(D)
Column - B
Five boys and three girls (p)
are seated at random in
a row. The probability
that no boy sits between
two girls is
5/108
Three dice are thrown
simultaneously. The
probability of getting
a sum of 15 is
(q)
3/28
Let S be the universal
set and n(X) = 3. The
probability of selecting
two subsets A and B
of the set X such that
(r)
4.
probability that both E and F happen is
B  A is
10 different books and
(s)
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
5/11
pmc 
(c)
pmc 
19
20
1
10
Einstein Classes,
1
.
2
Then
(a)
P( E ) 
1
1
, P( F ) 
3
4
(b)
P( E ) 
1
1
, P( F ) 
2
6
(c)
P( E ) 
1
1
, P( F ) 
6
2
(d)
P( E ) 
1
1
, P( F ) 
4
3
1/7
5.
The probabilities that a student passes in
mathematics, physics and chemistry are m, p and c
respectively. Of these subjects, a 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 subjects. Which of the
following relations are true ?
(a)
1
and
12
the probability that neither E nor F happens is
MULTIPLE CORRECT CHOICE TYPE
1.
Let E and F be two independent events. The
(b)
pmc 
(d)
pmc 
1
4
27
20
Let P(n) be the probability of getting n heads when
a coin is tossed m times, if P(4), P(5), P(6) are in
A.P., then the possible values of m could be
(a)
10
(b)
11
(c)
7
(d)
14
Assertion-Reason Type
Each question contains STATEMENT-1 (Assertion)
and STATEMENT-2 (Reason). Each question has
4 choices (A), (B), (C) and (D) out of which ONLY
ONE is correct.
(A)
Statement-1 is True, Statement-2 is True;
Statement-2 is a correct explanation
for Statement-1
(B)
Statement-1 is True, Statement-2 is True;
Statement-2 is NOT a correct
explanation for Statement-1
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 11
1.
(C)
Statement-1 is True, Statement-2 is False
(D)
Statement-1 is False, Statement-2 is True
STATEMENT-1 : The probability of getting a tail
most of the time in 10 tosses of a unbiased is
4.
STATEMENT-2 : P( A / B) 
1 
10 ! 
1  10

2  2 5 ! 5 ! 
2.
STATEM ENT-1 : I f P(A/B)  P(A) then
P(B/A)  P(B)
5.
STATEMENT-2 :
2n
C0 + 2nC1 + 2nC2 + .... + 2nCn = 22n – 1, n  N.
STATEMENT-1 : The probability of solving a new
1 1
1
, and
2 3
4
respectively. The probability that the problem will
P( A  B )
P( B )
STATEMENT-1 : Balls are drawn one by one
without replacement from a bag containing a
white and b black balls, then probability that
white balls will be first to exhaust is a/(a + b).
STATEMENT-2 : Balls are drawn ony by one
without replacement from a bag containing a
white and b black balls then probability that third
problem by 3 students are
drawn ball is white is
a
ab
1
.
24
STATEMENT-2 : If A, B and C are three
independent events then the probability of at least
be solved by them is
one of them happenning = 1  P( A ) P( B ) P( C ) .
3.
Let A and B are two independent events.
STATEMENT-1 : If P(A) = 0.3
P( A  B ) = 0.8 then P(B) is
and
2
.
7
STATEMENT-2 : P( E ) = 1 – P(E) where E is any
event.
(Answers) EXCERCISE BASED ON NEW PATTERN
COMPREHENSION TYPE
1.
a
2.
b
3.
c
4.
d
7.
a
8.
a
9.
b
10.
c
2.
[A-q; B-p; C-s; D-r]
5.
c
6.
b
MATRIX MATCH TYPE
1.
[A-p; B-q; C-r; D-s]
MULTIPLE CORRECT CHOICE TYPE
1.
b, c
2.
b, c, d
3.
a, c
4.
a, d
5.
c, d
3.
A
4.
A
5.
D
ASSERTION-REASON TYPE
1.
C
2.
Einstein Classes,
D
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 12
INITIAL STEP EXERCISE
(SUBJECTIVE)
1.
2.
3.
A committee consists of 9 experts taken from three
institutions A, B and C, of which 2 are from A, 3
from B and 4 from C. If three experts resign from
the committee, find the probability of exactly two
of the resigned experts being from the same
institution.
n1
A bag contains ‘W’ white balls and ‘R’ red balls.
Two players P1 and P2 alternately draw a ball from
the bag, replacing the ball each time after the draw,
till one of them draws a white ball and wins the
game. ‘P1’ beings the game. Find the probability of
P2 being the winner
4.
Consider all functions that can be defined from the
set A = {1, 2, 3} to the set B = {1, 2, 3, 4, 5}. A
function f(x) is selected at random from these
functions. Find the probability that, selected
function satisfies f(i)  f(j) for i < j.
5.
A and B play a game in which they alternately toss
a pair of dice. The one who is first to get a total of 7
wins the game. Find the probability that (a) the one
who tosses first will win the game, (b) the one who
tossed second will win the game.
6.
The probabilities that a husband and wife will be
alive 20 years from now are given by 0.8 and 0.9,
respectively. Find the probability that in 20 years
(a) both, (b) neither, (c) at least one, will be alive.
7.
What is the probability of getting a total of 9 (a)
twice, (b) at least twice in 6 tosses of a pair of dice ?
8.
A determinant of the second order is made with the
element 0 and 1. What is the probability that the
determinant made is non-negative ?
9.
A man parks his car among n cars standing in a
row, his car 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 ?
10.
A, B, C are events such that P(A) = 0.3, P(B) = 0.4,
P(C) = 0.8, P(AB) = 0.08, P(AC) = 0.28 and
P(ABC) = 0.09. If P (A  B C)  0.75 then show
that P(BC) lies in the interval 0.23  x  0.48.
Given that x + y = 2a where a is constant and that
all values of x between zero and 2a are equally likely,
show that the chance that xy 
3 2
1
a , is .
4
2
12.
Three points P, Q, R are selected at random from
the circumference of a circle. Find the probability
that the points lie on a semicircle.
13.
Two persons A and B agree to meet at a place
between 11 to 12 noon. The first one to arrive waits
for 20 minutes and then leaves. If the time of their
arrival be independent and at random, what is the
probability that A and B meet ?
14.
Dipesh’s gardener is not dependable. The
probability that he will forget to water the rosebush
is 2/3. The rosebush is in questionable condition any
way hence even if watered, the probability of its
withering is 1/2, if not watered, the probability of
its withering is 3/4. Dipesh goes out of town for a
day and upon returning, he finds that the rosebush
had withered. What is the probability that the
gardener did not water the bush ?
15.
The probability that an entering college student will
graduate is 0.4. Determine the probability that out
of 5 students (a) none, (b) 1, (c) at least 1, will
graduate.
n2
A number of the form 7  7 is formed, by
selecting the numbers n 1 and n 2 from the set
{1, 2, 3,...,99, 100} with replacement. Find the
probability of the formed number being divisible
by 5
Einstein Classes,
11.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 13
FINAL STEP EXERCISE
(SUBJECTIVE)
1.
2.
An inefficient secretary places n different letters
into n differently addressed envelopes at random.
Find the probability that at least one of the letters
will arrive at the proper destination.
(a)
(b)
8.
probability that he makes a guess is
Find the probability that n people
(n  365) selected at random will have n
different birthdays.
The sum of the digits of a seven-digit number is 59.
Find the probability that this number is divisible
by 11.
4.
In a multiple-choice question, there are four
alternative answers 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 getting marks in the question
1
. The
6
probability that his answer is correct, given that he
1
. Find the probability that he knew
8
the answer to the question, given that he correctly
answers it.
copies it, is
9.
6.
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.
Let A, B, C be three events. If the probability of
occurring one event out of A and B is 1 – a, out of B
and C is 1 – 2a, out of C and A is 1 – a and that of
occurring three events simultaneously is a2, then
prove that the probability that at least one out of
A, B, C will occur is greater than or equal to 0.5.
11.
The compressors uses in refrigerators are
manufactured by certain company. at three
factories at Pune, Nasik and Nagpur. It is known
that the Pune factory produces twice as many compressors as the Nasik one, which produces the same
number as the Nagpur one (during the same
period). Experiments show that 0.2% of the
compressors produced at Pune and Nasik are
defective and so are 0.4% of those produced at
Nagpur. A quanlity control engineer while maintaining a refrigerator finds a defective compressor.
What is the probability that Nasik factory is not be
blamed ?
12.
There are 6 red and 8 green balls in a bag. 5 balls
are drawn at random and placed in a red box. The
remaining balls are placed in a green box. What is
the probability that the number of red balls in the
green box plus the number of green balls in the red
box is not a prime number ?
In a series of five one-day cricket matches between
India and Pakistan, the probability of India
1
1
and . If a
3
6
win, loss or draw gives 2, 0 or 1 point respectively
then find the probability that India will score 5
points in the series.
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.
Einstein Classes,
677
.
909
10.
winning or drawing are respectively
7.
An urn contains 6 black and unknown number
( 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 next draw is
1
?
may exceed
5
5.
1
and the
3
probability that he copies the answer is
Determine how may people are required
to make the probability of distinct
birthdays less than 1/2.
3.
In a test, an examinee either guesses or copies or
knows the answer to a multiple-choice question with
four choices, only one answer being correct. The
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPR – 14
ANSWERS (SINGLE CORRECT CHOICE TYPE)
1.
b
2.
a
3.
b
4.
a
5.
a
6.
c
7.
a
8.
d
9.
b
10.
a
ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE)
3.
R
( W  2R )
(b)
5/11
72,689
531,441
8.
13/16
12.
3/4
13.
5/9
(b)
0.26
(c)
0.92
1.
55/84
2.
1/8
4.
7/25
5.
(a)
6/11
6.
(a)
0.72
0.02
(c)
0.98
7.
(a)
61,440
531,441
(b)
9.
(n  m )(n  m  1)
(n  1)(n  2)
14.
15.
3/4
(a)
0.08
(b)
ANSWERS SUBJECTIVE (FINAL STEP EXERCISE)
1 
2  
n 1

1 
 1 
..... 1 

365 
365  
365 

1.
0.6321
2.
(a)
3.
4/21
4.
4
7.
23/30
8.
24/29
11.
0.8
12.
213/1001
Einstein Classes,
6.
(b)
23
1201/7776
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111