Topic
Page No.
Theory
01 - 05
Exercise - 1
06 - 13
Exercise - 2
14 - 16
Exercise - 3
17 - 23
Exercise - 4
24 - 25
Answer Key
26 - 27
Contents
ETOOS ACADEMY Pvt. Ltd
F-106, Road No.2 Indraprastha Industrial Area, End of Evergreen Motor, BSNL Lane,
Jhalawar Road, Kota, Rajasthan (324005)
Tel. : +91-744-242-5022, 92-14-233303
PROBABILITY
Syllabus
Addition and multiplication rules of probability, conditional probability,
independence of events, computation of probability of events using
permuta-tions and combinapermuta-tions.
KEY CONCEPTS
THINGS TO REMEMBER :
RESULT - 1
(i) SAMPLE - SPACE : The set of all possible outcomes of an experiment is called the
SAMPLE-SPACE (S)
(ii) EVENT : A sub set of sample-space is called an EVENT.
(iii) COMPLEMENT OF AN EVENT A : The set of all out comes which are in S but not in
A is called the COMPLEMENT OF THE EVENT A DENOTED A OR AA
C.
(iv) COMPOUND EVENT : If A & B are two given events then A
B is called COMPOUND
EVENT and is denoted by A
B or AB or A & B.
(v) MUTUALLY EXCLUSIVE EVENTS : Two events are said to be MUTUALLY EXCLU
SIVE (or disjoint or incompatible ) if the occurence of one precludes (rules out) the
simultaneous occurence of the other .If A & B are two mutually exclusive events then
P (A & B) = 0.
(vi) EQUALLY LIKELY EVENTS : Events are said to be EQUALLY LIKELY when each
event is as likely to occur as any other event.
(vii) EXHAUSTIVE EVENTS : Events A,B,C...L are said to be EXHAUSTIVE EVENTS
if no event outside this set can result as an outcome of an experiment. For example,
if A & B are two events defined on a sample space S, then A & B are exhaustive
A
B = S
P (A
B) = 1.
(viii) CLASSICAL DEF. OF PROBABILITY : If n represents the total number of equally
likely, mutually exclusive and exhaustive outcomes of an experiment and m of them
are favourable to the happening of the event A, then the probability of happening of
the event A is given by P (A) = m/n.
Note : (1) 0
P (A)
1 (2)
P (A) + P ( A ) = 1, where A = Not A . .
(3) If x cases are favourable to A & y cases are favouable to A then P (A) =
(
x
x
y
)
and P( A ) =
(
x
y
y
)
We say that
ODDS IN FAVOUR OF A
A are x : y & odds
against A are y : x
PROBLEMS ON EQUALLY lIKELY, MUTUALLY EXCLUSIVE AND EXHAUSTIVE EVENTS.
Experiment
Events
E/L
M/E Exhaustive
1.
Throwing of a die
A: throwing an odd face {1, 3, 5}
No
Yes
No
B : throwing a composite {4,.6}
2.
A ball is drawn from
E
1: getting a W ball
an urn containing 2W,
E
2: getting a R ball
No
Yes
Yes
3R and 4G balls
E
3: getting a G ball
3.
Throwing a pair of
A : throwing a doublet
dice
{11, 22, 33, 44, 55, 66}
B : throwing a total of 10 or more
Yes
No
No
{ 46, 64, 55, 56, 65, 66 }
4.
From a well shuffled
E
1: getting a heart
pack of cards a card is
E
2: getting a space
Yes
Yes
Yes
drawn
E
3: getting a diamond
E
4: getting a club
5.
From a well shuffled
A = getting a heart
pack of cards a card is
B = getting a face card
No
No
No
drawn
RESULT - 2
AUB = A + B = A or B denotes occurence of at least A or B. For 2 events A & B :
(See fig.1)
(i) P (A
B) = P (A) + P (B) - P (A
B) =
P (A. B ) + P ( A .B) + P ( A.B) = 1 - P ( A . B )
(ii) Opposite of “ atleast A or B ” is
NEITHER
A NOR B i.e.
A
B
= 1 - (A or B) = A
B
Note that P (A+B) + P ( A
B ) = 1.
(iii) If A & B are mutually exclusive then P(A
B) = P(A) + P(B).
(iv) For any two events A & B, P (exactly one of A , B occurs )
= P (A
B ) + P (B
A ) = P ( A ) + P (B) – 2P ( A
( A
B)
= P (A
B) – P (A
B) = P ( AA
C
B
C) – P (A
C
B
C)
(v) If A & B are any two events P (A
B) = P (A) .P (B / A ) = P ( B). P ( A / B ), Where P
(B / A ) means conditional probability of B given A & P( A / B ) means conditional
probability of A given B. (This can be easily seen from the figure).
B
A AB BA
B A
(vi)
DE MORGAN’S LAW : - If A & B are two subsets of a universal set U , then
(a) (A
B)
C= AA
C
B
C&
(b)
( A
B)
C= AA
C
B
C(vii) A
(B
C) = ( AA
B)
( A
A
C) & A
(B
C) = ( AA
B)
( A
A
C)
RESULT - 3
For any three events A, B and C we have
(See Fig.2 )
(i) P (A) or B or C) = P(B)
+ P (C) – P (A
B) – P (B
C) –
P (C
A) + P ( A
B
C)
(ii) P ( at least two of A,B,C occur) =
P (B
C) + P ( C
A) +
P ( A
B) – 2P ( AA
B
C)
(iii)
P ( exactly two of A, B, C occur) =
P (B
C) + P ( C
A) +
P (A
B) – 3 P ( AA
B
C)
(iv)
P (exactly one of A, B, C occurs ) =
P (A) + P (B) + P (C) – 2P (B
C) – 2P (C
A) – 2P (A
B) + 3P (A
B
C)
NOTE : 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. However the converse of this is not true.
RESULT - 4
INDEPENDENT EVENTS : Two events A & B are said to be independent if occurence or non
occurence of one does not effect the probability of the occurence or non occurence of other.
(i) If the occurence of one event affects the probability of the occurence of the other event then
the events are said to be
DEPENDENT or CONTINGENT. For two independent events
A and B : P ( A
B) = P (A) . P(B). Often this is taken as the definition of independent events.
(ii) Three events A, B & C are independent if & 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 pairwise as well as mutually independent .
Similarly for n events A
1, A
2, A
3,... A
nto be independent, the number of these
conditions is equal to
nc
2
+
nc
3+ ...+
nc
n= 2
n– n – 1.
(iii) The probability of getting exactly r success in n idenpendent trials is given by
P (r) =
nC
r
p
rq
n–rwhere : p = probability of sucess in a single trial.
q = probability of failure in a single trial. note : p + q = 1.
Note : Independent events are not in general mutually exclusive & vice versa. Mutually
exclusiveness can be used when the events are taken from the same experiment
& independence can be used when the events are taken from different experiments.
A B U A B C A B C B A C A B C A C B C A B A B C C Fig.2 C A B
RESULT – 5 : TOTAL PROBABILITY THEOREM AND BAYE’S THEOREM :
(i) Total probability theorem : If an event A can occur only with one of the n mutually
exclusive and exhaustive events B
1, B
2...B
nthen
P(A) =
n i i i 1 P(B ).P(A /B )(ii) Baye’s Theorem : If an event A can occur only with one of the n mutually exclusive and
exhaustive events B
1, B
2...B
nthen,
)
B
/
A
(
P
).
B
(
P
)
B
/
A
(
P
).
B
(
P
)
A
/
B
(
P
i i n 1 i i i i
PROOF :
The events A occurs with one of the n mutually exclusive & exhaustive events B
1, B
2,
B
3...B
nA = AB
1+ AB
2+ AB
3+ ...+ AB
nP (A) = P (AB
1) + P (AB
2) +...+ P(AB
n) =
P
(
AB
i)
n1 i
NOTE :
A
event what we have ; B
i
event what we want ;
B
1, B
2, B
3...B
nare alternative evnet .
Now , P (AB
i) = P (A) . P (B
i/A ) = P (B
i) . P (A /B
i)
)
AB
(
P
)
B
/
A
(
P
).
B
(
P
)
A
(
P
)
B
/
A
(
P
).
B
(
P
)
A
/
B
(
P
i n i i i i i i i
)
B
/
A
(
P
).
B
(
P
)
B
/
A
(
P
).
B
(
P
)
A
/
B
(
P
i i i i i
B1 B2 B3 Bn Bn1 AFig.3
RESULT – 6
If p
1and p
2are the probabilties of speaking the truth of two independent witnesses A and B then
P ( their combined statement is true)
p
p
(
1
p
p
p
)(
1
p
)
2 1 2 1 2 1
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 probabilty of the happening of the event
before their statement then
P ( their combined statement is true )
p
p
p
(
1
p
p
p
)(
p
1
p
)(
1
p
)
2 1 2 1 2 1
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
Pr. that the statement is true = P p
1p
2Pr. that the statement is false = ( 1 – p) . c (1– p
1)(1– p
2)
However chance of coincidence testimony is taken only if the joint statement is not
contradicted by any witness.
ONLY IN C.B.S.E
RESULT - 7
(i)
A PROBABILITY DISTRIBUTION spells out how a total probability of 1 is distributed
over several values of a random variable.
(ii) Mean of any probability distribution of a random variable is given by :
i i i i i
p
x
p
x
p
( Since
p
1
i
)
(iii) Variance of a random variable is given by,
2 i i 2
(
x
)
.
p
2 i 2 i 2
p
x
( Note that SD =
2)
(iv) The probability distrubution for binomial variate ‘ X ‘ is given by ; P (X = r )=
nC
rp
rq
n– rwhere all symbols have the same meaning as given in results 4.
The recurrence formula
P
P
r(
r(
)
1
)
n
r
1
r
.
p
q
, is very helpful for quickly computing P (1), P
(2)
P (3) etc. if P (0) is known.
(v) Mean of BPD = np ; variance of BPD = npq.
(vi) If p represents a persons 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. expectations = pM
RESULT – 8 : GEMOMETRICAL APPICATIONS :
The following statements are axiomatic :
(i) If a point is taken at random on a given staright line AB, the chance that it falls on a
particular segment PQ of the line is PQ/AB.
(ii) If a point is taken at random on the area S which inclues an area
, the chance that
the point falls on
is
/S.
PART - I : OBJECTIVE QUESTIONS
* Marked Questions are having more than one correct option.
Section (A) : Problems based on Classical definition of Probability (PRCD)
A-1*. 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 (A) A and B are mutually exclusive and exhautive
(B) A and C are mutually exclusive and exhautive (C) A, C and D form an exhautive system
(D) B, C and D form an exhautive system
A-2. In drawing of a card from a well shuffled ordinary deck of playing cards the events ‘card drawn is spade’ and ‘card drawn is an ace’ are
(A) mutually exclusive (B) equally likely
(C) forming an exhaustive system (D) none of these
A-3. A 9 digit number using the digits 1, 2, 3, 4, 5, 6, 7, 8 & 9 is written randomly without repetetion. The probability that the number will be divisible by 9 is:
(A) 1/9 (B) 1/2 (C) 1 (D) 9!/99
A-4. An integer x is chosen from the first 100 positive integers. The probability that, x +
100
x
> 50 is:(A)
10
1
(B)11
20
(C)1
2
(D) noneA-5. Two dies are rolled simultaneously. The probability that the sum of the two numbers on the top faces will be atleast 10 is:
(A) 1/6 (B) 1/12 (C) 1/18 (D) none
A-6. 2n boys are randomly divided into two subgroups containing n boys each. The probability that the two tallest boys are in different groups is
(A) 1 n 2 n (B) 2n 1 1 n (C) 24nn21 (D) none of these
A-7. The chance that a 13 card combination from a pack of 52 playing cards is dealt to a player in a game of bridge, in which 9 cards are of the same suit, is
(A) 13 52 4 39 9 13 C C . C . 4 (B) 13 52 4 39 9 13 C C . C . ! 4 (C) 13 52 4 39 9 13 C C . C (D) none of these
A-8. Out of 13 applicants for a job, there are 5 women and 8 men. It is desired to select 2 persons for the job. The probability that at least one of the selected persons will be a woman is
(A) 25/39 (B) 14/39 (C) 5/13 (D) 10/13
A-9. 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:
Section (B) : Problems based on venn diagram & set theory (PRVD) B-1. If A and B are any event of an experiment, then AC – B =
(A) (A B)C (B) AC – (A B) (C) (A B)C (D) A – B
B-2. A die is thrown. Let A be the event ‘an odd number turns up’ and B be the event ‘a number divisible by 3 turns up’. Write which of the following are true.
(i) B AA (ii) B – A AA (iii) B – A AAC (iv) A – B = A BC
(A) (i) and (ii) (B) (i), (iii) (C) (ii) and (iii) (D) (iii) and (iv)
B-3. An experiment results in four possible out comes S1, S2, S3 & S4 with probabilities p1, p2, p3 & p4 respectively.
Which one of the following probability assignment is possbile. [ Assume S1 S2 S3 S4 are pair wise exclusive]
(A) p1 = 0.25, p2 = 0.35, p3 = 0.10, p4 = 0.05
(B) p1 = 0.40, p2 =0.20, p3 = 0.60, p4 = 0.20
(C) p1 = 0.30, p2 = 0.60, p3 = 0.10, p4 = 0.10
(D) p1 = 0.20, p2 = 0.30, p3 = 0.40, p4 = 0.10
B-4*. If M & N are any two events, then which one of the following represents the probability of the occurance of
exactly one of them ?
(A) P (M) + P (N) 2 P (M N) (B) P (M) + P (N) P (M N) (C) P
M
+ P
N
2 P
M
N
(D) P
M
N
+ P
M
N
B-5*. Let 0 < P(A) < 1, 0 < P(B) < 1 & P(A B) = P(A) + P(B) P(A). P(B), then: (A) P(B/A) = P(B) P(A) (B) P(AC BC) = P(AC) + P(BC)
(C) P((A B)C) = P(AC). P(BC) (D) P(A/B) = P(A)
B-6*. If M & N are independent events such that 0 < P(M) < 1 & 0 < P(N) < 1, then:
(A) M & N are mutually exclusive (B) M &
N
are independent (C)M
&N
are independent (D) P
M N
+ P
M N
= 1Section (C) : Problems based on Conditional Probability & Dependent & Independent event (PRCP/PRDI) C-1. In throwing a pair of dice, the events ‘coming up of 6 on Ist die’ and ‘a total of 7 on both the dies’ are
(A) mutually exclusive (B) forming an exhaustive system
(C) independent (D) dependent
C-2. A fair die is tossed. If the number is odd, find the probability that it is prime is
(A) 32 (B) 21 (C) 1 (D) 31
C-3. A pair of dice is thrown. If total of numbers turned up on both the dies is 8, then the probability that the number then up on the second die is 5’ is
Section (D) : Problem based on total Probability & Baye's Theorem (PRBA/PRTP)
D-1. A card is drawn from a well shuffled ordinary deck of 52-playing cards. The card drawn is found to be a spade. Then the probability that the card is an ace, is
(A) 131 (B) 521 (C)
4
1 (D) none of these
D-2. Odds in favour of A’s speaking the truth are 1 : 2 and odds against B’s speaking the truth are 1 : 3. A die is thrown. Both A and B assert that on the die 4 has turned up. Then the probability of the truth of their assertion is
(A) 301 (B) 241 (C) 36011 (D) 1715
Section (E) : Problem based on Testimony/Geometry (PRTT/PRGE)
E-1. Odds in favour of A’s speaking the truth are 1 : 2 and odds against B’s speaking the truth are 1 : 3. A die is thrown. Both A and B assert that on the die 4 has turned up. Then the probability of the truth of their assertion is
(A) 301 (B) 241 (C) 36011 (D) 1715
Section (F) : Problem Based on Binomial/Distribution/Expectation/Mean & Variance (PRBT/PRPD) F-1. 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 (B) 2/5 (C) 3/5 (D) 4/5
F-2. A bag contains 2 white & 4 black balls. A ball is drawn 5 times, each being replaced before another is drawn. The probability that atleast 4 of the balls drawn are white is:
(A) 4/81 (B) 10/243 (C) 11/243 (D) none
F-3. From an urn containing 3 red & 2 white balls, a man is to draw 2 balls at random without replacement, being promised 20 paise for each red ball he draws & 10 paise for each white one. His expectation is:
(A) 24 paise (B) 30 paise (C) 32 paise (D) 35 paise
F-4. A & B throw with one dice for a stake of Rs. 99/- which is to be won by the player who first throws 4. If A has the first throw then their respective expectations of rupees are:
(A) 50 & 49 (B) 54 & 45 (C) 45 & 54 (D) none
F-5. A fair coin is tossed 99 times. If X is the number of times heads occur, then P (X = r) is maximum when r is
PART - II : SUBJECTIVE QUESTIONS
Section (A) : Problems based on Classical definition of Probability (PRCD) A-1. Write the sample space of the following experiment
(i) ‘Three coins are tossed'
(ii) ‘Selection of two children from a group of 3 boys and 2 girls without replacement’.
A-2. There are three events A, B, C, one of which must, and only one can, happen; the odds are 8 to 3 against A, 5 to 2 against B : find the odds against C.
A-3. Two balls are drawn in succession from a box containing 4 red, 3 white and 5 blue balls. Find the probabil-ity of the event ‘one ball is red and other ball is white’.
A-.4 The digits 1, 2, 3, ..., 9 are arranged in a random order, find the probability that 1, 2, 3 will appear as neighbours in the order mentioned
A-5. If 6 boys and 6 girls sit in a row randomly, find the probability that all the 6 girls sit together
A-6. Three persons A, B and C speak at a function along with 5 other persons. If the persons speak at random, find the probability that A speaks before B and B speaks before C
A-7. A divisor of 1200 is selected at random. Find the probability that it is even.
A-8. (i) In a two child family, one child is a boy. What is the probability that the other child is a girl? (ii) If the older child is a boy, then probability that the second child is a girl is
A-9. (i) A rectangle is randomly selected from the grid of equally spaced squares as shown.
Find the probability that the rectangle is a square.
(ii) Letter of the word ‘VECTOR’ are arranged in the grid as shown
Find the chance that no row remains empty.
A-10. In throwing a pair of dice, find whether the two events
(i) E1 : ‘coming up of an odd number on first die’ and E2 : ‘coming up of a total of 8’. (ii) E1 : ‘coming up of 4 on first die’ and E2 : ‘coming up of 5 on the second die’. are mutually exclusive or not
A-11. (i) In throwing of a pair of dice, find the probability of the event : total is ‘not 8’ and ‘not 11’. (ii) In throwing a pair of dies find the probability of the event ‘Getting an odd number on the first die and a total 8’.
A-12. A bag contains 6 white, 7 red and 5 blue balls. Three balls are drawn at random. Find the probability of the
Section (B) : Problems based on venn diagram & set theory (PRVD)
B-1. If P(A) = 0.4, P(B) = 0.48 and P(A B) = 0.16, then find the value of each of the following :
(i) P(A B) (ii) P(A / B) (iii) P(A B)
B-2. A card is drawn from a well shuffled ordinary deck of 52 playing cards. Find the probability that the card drawn is :
(i) A king or a queen (ii) A king or a spade
Section (C) : Problems based on Conditional Probability & Dependent & Independent event (PRCP/PRDI) C-1. The odds against a certain event are 5 to 2, and the odds in favor of another event independent of the former are
6 to 5 : find the chance that one at least of the events will happen.
C-2. A, B, C in order draws a card from a pack of cards, replacing them after each draw, on condition that the first who draws a spade shall win a prize : find their respective chances.
Section (D) : Problem based on total Probability & Baye's Theorem (PRBA/PRTP)
D-1. There are 5 brilliant students in class XI and 8 brilliant students in class XII. Each class has 50 students. The odds in favour of choosing the class XI are 2 : 3. One of the classes is chosen randomly and then a student is randomly selected. Find the probability of selecting a brilliant student.
D-2. Box – I contains 5 red and 2 blue balls while box – II contains 2 red and 6 blue balls. A fair coin is tossed. If it turns up head, a ball is drawn from box–I, else a ball is drawn from box–II. Find the probability of each of the following :
(i) A red ball is drawn (ii) Ball drawn is from box–I if it is blue
D-3. Two cards are drawn successively from a well-shuffled ordinary deck of 52-playing cards without replacement and is noted that the second card is a king. Find the probability of the event ‘first card is also a king’.
Section (E) : Problem based on Testimony/Geometry (PRTT/PRGE)
E-1. A speaks the truth 2 out of 3 times, and B 4 times out of 5; they agree in the assertion that from a bag containing 6 balls of different colours a red ball has been drawn : find the probability that the statement is true.
E-2. A parallelogram is inscribed inside a circle of radius 10 cm. One side of parallelogram being 12 cms. Find the probability that a point inside the circle also lies inside the parallelogram.
Section (F) : Problem Based on Binomial/Distribution/Expectation/Mean & Variance (PRBT/PRPD) F-1. If on an average 1 vessel in every 10 is wrecked, find the chance that out of 5 vessels expected 4 at least will
arrive safely.
F-2. There are 9 coins in a bag, 5 of which are Rupee and the rest are unknown coins of equal value; find what they must be if the probable value of a draw is 60 paise.
F-3. A had in his pocket a Rupee and four 10 paise coins ; taking out two coins at random he promises to give them to B and C. What is the worth of C’s expectation?
F-4. A box contains 2 red and 3 blue balls. Two balls are drawn successively. If getting ‘a red ball on first draw and a blue ball on second draw’ is considered a success, then write the probability distribution of suc-cesses. It is given that the above experiment is performed 3-times, with replacement
F-5. A coin is tossed 5-times. Find the mean and variance of the probability distribution of appearance of heads on the tosses.
PART - III : MISCELLANEOUS OBJECTIVE QUESTIONS
(MATCH THE COLUMN)
1. Column – Column –
(A) If the probability that units digit in square of an even integer is 4 (p) 1 is p, then the value of 5p is
(B) If A and B are independent events and P(A B) = 61, (q) 2 P( A ) = 32, then 6P(B/A) =
(C) Among 2 children, a child may equally be a boy or a girl (r) 3 if the probability that exactly one of them is a boy is p,
then 6p =
(D) A boy has 20% chance of hitting at a target. Let p denote (s) 4 the probability of hitting the target for the first time at the nth
trial. If p satisfies the inequality 625p2 – 175p + 12 0, then
value of n is
2. Column – Column –
(A) A pair of dice is thrown. If total of numbers turned up (p) 5/16 on both the dies is 8, then the probability that the
number then up on the second die is 5’ is
(B) A box contains 4 white and 3 black balls. Two balls are (q) 1/3 drawn successively and is found that second ball is
white, then the probability that Ist ball is also white is
(C) A biased coin with probability p, 0 < p < 1 of heads is (r) 21 tossed until a head appears for the first time. If the
probability that the number of tosses required is even is 2/5, then p equals
(D) A coin whose faces are marked 3 and 5 is tossed 4 times : what (s) 51 is the probability that the sum of the numbers thrown being less,
(COMPREHENSION) Comprehension #1
If A and B are two events, then probability that atleast one of them is selected is P(A B) = P(A) + P(B) – P(A B). For three events A, B, C the probability that atleast one of them is seleted is P(A B C) = P(A) + P(B) + P(C) – P(A B) – P(B C) – P(C A) + P(A B C).
3. Probability that none of the three events (A, B, C) occurs
(A) P(ABC) – P(A) – P(B) – P(C) + P(A B) + P(B C) + P(C A)
(B) P(A B C) – P(A) – P(B) – P(C) + P(A B) + P(B C) + P(C A) – P(A B C) (C) P( A ) + P(B) + P(C) – P(A B) – P(B C) – P(C A) + P(A B C)
(D) none of these
4. Probability that exactly two events occurs
(A) P(A) + P(B) + P(C) – P(A B) – P(B C) – P(C A) (B) 2(P(A) + P(B) + P(C)) – P(A B) – P(B C) – P(C A) (C) P(A B) + P(B C) + P(C A) – 3 × P(A B C) (D) P(AB) + P(BC) + P(CA)
5. Probability that atmost two events happen
(A) P(A B) + P(B C) + P(C A) – 3 × P(A B C) (B) P(A) + P(B) + P(C) – P(A B) – P(B C) – P(C A) (C) P(A) +P(B) + P(C)
(D) P(A) + P(B) + P(C) – P(A B) – P(B C) – P(C A) + P(A B C)C Comprehension#2
Consider the experiment of distribution of balls among urns. Suppose we are given M urns, numbered 1 to M, among which we are to distribute n balls (n < M). Let P(A) denote the probability that each of the urns numbered 1 to n will contain exactly one ball. Then answer the following questions.
6. If the balls are different and any number of balls can go to any urns, then P(A) is equal to
(A) nMM! (B) Mnn! (C) n MP ! n (D) M1n
7. If the balls are identical and any number of balls can go to any urns, then P(A) equals (A) Mn 1 (B) 1 M 1 n M C 1 (C) 1 n 1 n M C 1 (D) 1 M 1 n M P 1
8. If the balls are identical but atmost one ball can be put in any box, then P(A) is equal to (A) n MP 1 (B) M nC ! n (C) n MC ! n (D) n MC 1
(ASSERTION/REASON)
9. Statement 1 : If P is chosen at random in the closed interval [0, 5], then the probability that the equation
x2+ px +
4 1
(P + 2) = 0 has real not is 53.
Statement 2 : If discriminant 0 then roots of the quadratic equation are always real.
(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 (C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
10. STATEMENT-1 : Since sample space of the experiment 'A coin is tossed if it turns up head, a die is thrown'
is {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), T}.
Prob. of the event {(H, 1), (H, 2), (H, 5)} is 73.
STATEMENT-2 : If all the sample points in the sample space of an experiment are pair wise mutually
exclusive, equally likely and exhaustive, then probability of an event E is defined as P(E) = Totalnumbernumberof sampleof samplepointspofavourableintsinthetosampletheeventspaceE
(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 (C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
11. STATEMENT-1 : If A and B are two independent events such that P(A) 0, P(B) 0, then A and B can not be mutually exclusive.
STATEMENT-2 : For independent events A and B, we have P(A/B) = P(A) which is not so for mutually
exclusive events.
(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 (C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
12. Statement 1 : If 144P, 14P, 142P are probabilities of three pair-wise mutually exclusive events, then the possible values of P belong to the set 41,21.
Statement 2 : If three events are pair wire mutually exclusive and exhaustive then sum of there probability is
equal to 1.
(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 (C) Statement-1 is True, Statement-2 is False
PART - I : OBJECTIVE QUESTIONS
1. A local post office is to send M telegrams which are distributed at random over N communication channels, (N > M). Each telegram is sent over any channel with equal probability. Chance that not more than one telegram will be sent over each channel is:
(A) N MM N ! M . C (B) N MN M ! N . C (C) 1 N MN M ! M . C (D) 1 N MM N ! N . C
2. A bag contains 7 tickets marked with the numbers 0, 1, 2, 3, 4, 5, 6 respectively. A ticket is drawn & replaced. Then the chance that after 4 drawings the sum of the numbers drawn is 8 is:
(A) 165/2401 (B) 149/2401 (C) 3/49 (D) none
3. A car is parket by an owner amongst 25 cars in a row, not at either end. On his return he finds that exactly 15 placed are still occupied. The probability that both the neighboring places are empty is
(A)27691 (B) 18415 (C) 1592 (D) none of these
4. Two whole numbers are randomly selected & multiplied. If the probability that the units place in their product is "Even" is p & the probability that the units place in their product is "Odd" is q, then p/q is:
(A) 4 (B) 3 (C) 2 (D) 1
5. A card is drawn from a pack, the card is replaced & the pack shuffled. If this is done 6 times, the probability that the cards drawn are 2 hearts, 2 diamonds & 2 black cards is:
(A) 90/1024 (B) 45/1024 (C) 1/1024 (D) none
6. A has 3 tickets in a lottery containing 3 prizes and 9 blanks; B has 2 tickets in a lottery containing 2 prizes and 6 blanks : compare their chances of success.
(A) 952 to 715 (B) 950/952 (C) 952/710 (D) None of these
7. In a purse there are 10 coins, all 5 paise except one which is a rupee. In another purse there are 10 coins all 5 paise. 9 coins are taken out from the former purse & put into the latter & then 9 coins are taken out from the latter & put into the former. Then the chance that the rupee is still in the first purse is:
(A) 9/19 (B) 10/19 (C) 4/9 (D) none
8. Suppose that of all used cars of a particular year, 30% have bad brakes. You are considering buying a used car of that year. You take the car to a mechanic to have the brakes checked. The chance that the mechanic will give you wrong report is 20%. Assuming that the car you take to the mechanic is selected " at random" from the population of cars of that year. The chance that the car's brakes are good given that the mechanic says its brakes are good
is-(A) 2831 (B) 2431 (C) 2831 (D) None of these
9. A bag contains (n + 1) coins. It is known that one of these coins has a head on both sides, whereas the other coins are normal. One of these coins is selected at random & tossed. If the probability that the toss results in head, is 7/12, then the value of n is.
(A) 5 (B) 6 (C) 4 (D) 3
10. If P (A B) = 3 , P (A1 B) = 56 and P(A) = 12, then which one of the following is correct? (A) P(A) = P (B) (B) A and B are mutually exclusing events (C) A and B are independent events (D) P(A) > P(B)
11. 2/3rd of the students in a class are boys & the rest girls. It is known that probability of a girl getting a first
class is 0.25 & that of a boy is 0.28. The probability that a student chosen at random will get a first class is:
12. 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 likely. The probability of selecting ith urn is
34 1 i2
(i = 1, 2, 3, 4). If we randomly select one of the urns & draw a ball, then the probability of ball being white is :
(A) 1496569 (B) 5627 (C) 738 (D) none of these
13. Two whole numbers are randomly selected & multiplied. The probability that the unit's place in their product is 0 or 5 is:
(A) 1/3 (B) 16/25 (C) 9/25 (D) none
14. The contents of urn I and II are as follows, Urn I: 4 white and 5 black balls
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) 287125 (B)
127 64
(C) 28725 (D) 19279
15. A letter is known to have come either from "KRISHNAGIRI" or "DHARMAPURI". On the post mark only the two consecutive letters "RI" are visible. Then the chance that it came from Krishnagiri is:
(A) 3/5 (B) 2/3 (C) 9/14 (D) none
16. Three six faced dice are tossed together, then the probability that exactly two of the three numbers are equal to
(A) 165216 (B) 177216 (C) 512216 (D) 21690
17. A die is weighted so that the probability of different faces to turn up is as given
Number 1 2 3 4 5 6
Probability 0.2 0.1 0.1 0.3 0.1 0.2
If P(A/B) = p1 and P(B/C) = p2 and P(C/A) = p3 then the values of p1, p2, p3 respectively are -Take the events A, B & C as A = {1, 2, 3}, B = {2, 3, 5} and C = {2, 4, 6}
(A) 32, 31, 41 (B) 31, 31, 61 (C) 41, 31, 61 (D) 32, 61, 41
18. Mean and variance of a Binomial variate are in the ratio of 3 : 2. The most probable number of happening of the variable in 10 trials of the experiment is
(A) 2 (B) 3 (C) 4 (D) 5
19. The sides of a rectangle are chosen at random, each less than 10 cm, all such lengths being equally likely. The chance that the diagonal of the rectangle is less than 10 cm is
(A) 1/10 (B) 1/20 (C)/4 (D)/8
Multiple choice
20. A student appears for tests I, II & III. The student is successful if he passes either in tests I & II or tests I & III. The probabilities of the student passing in the tests I, II & III are p, q &
1/2 respectively. If the probability that the student is successful is 1/2, then:
(A) p = 1, q = 0 (B) p = 2/3, q = 1/2
(C) p = 3/5, q = 2/3 (D) there are infinitely many values of p & q.
21. A student has to mathc thre hisotircal events i.e. Dandi March, Quit India Movement and Mahatma Gnadhi's assasination with the years 1948, 1930 and 1942 and each event happens in different years. The student has no knowledge of the correct answers and decides to match the events and years randomly. Let Ei: (o
i 3) denote the event that the student gets exactly i correct answer, then (A) P(E0) + P (E3) = P (E1) (B) P(E0) . P (E1) = P(E3)
22. For any two events A & B defined on a sample space, (A)
P A B
P A
P B
P B
( )
( )
( )
1
, P (B) 0 is always true (B) P
A
B
= P (A) - P (A B)(C) P (A B) = 1 - P (Ac). P (Bc), if A & B are independent
(D) P (A B) = 1 - P (Ac). P (Bc), if A & B are disjoint
23. An unbiased coin is tossed n times. Let X denote the number of times head occurs. If P(X = 4), P (X = 5) and P(X = 6) are in AP, then the value of n can be
(A) 7 (B) 10 (C) 12 (D) 14
PART - II : SUBJECTIVE QUESTIONS
1. A cube painted red on all sides, is cut into 125 equal small cubes. A small cube when picked up is found to show red colour on one of its faces. Find the Probability that two more faces also show red colour.
2. let X be a set containing n elements. Two subsets A nad B of X are chosen at random. Find the pbobility that AB = X.
3. An urn contains 'm' green and 'n' red balls. K (, m, n) balls are drawn and laid aside, their colour being ignored. Then one more ball is drawn. Find the probability that it is green.
4. 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 brobability that, among the numbers 1, 2, 3, 4, 5, 6, only three numbers appear in this list?
5. There is a group of k targets, each of which independently of the other targets, can be detected by a radar unit with probability p. Each of 'm' radar units detectes the targets independently of other units. Find the probability that not all the targets in the group will be detedted.
6. 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?
7. In a multiple choice question there are 4 alternative answers of which 1, 2, 3 or all may be correct. A candidate will get marks in the question only if he ticks all the correct answer. The candidate decides to tick answers at random. If he is allowed upto 5 chances to answer the question, find the probability that he will get the marks in the question.
8. 3 firemen X, Y and Z shoot at a common target. The probabilities that X and Y can hit the target are 2/3 and 3/4 respectively. If the probability that exactly two bullets are found on the target is 11/24, then find the probability of Z to hit the target.
9. In a Nigerian hotel, among the english speaking people 40% are English & 60% Americans. The English & American spellings are"RIGOUR" & "RIGOR" respectively. An English speaking person in the hotel writes this word. A letter from this word is chosen at random & found to be a vowel. Find the probability that the writer is an Englishman.
10. A man has 10 coins and one of them is known to have two heads. He takes one at random and tosses it 5 times and it always falls head : what is the chance that it is the coins with two heads?
11. A cubical die with faces marked 1, 2, 3, ..., 6 is loaded such that the probability of throwing the number is proportional to t2. Find the probability that the number 5 has appeared given that when the die is rolled the numbe
turned up is not even.
12. 2 hunters A & B shot at a bear simultaneously. The bear was shot dead with only one hole in its hide. Probability of A shooting the bear 0.8 & that of B shooting the bear is 0.4. The hide was sold for Rs. 280/-. If this sum of money is divided between A & B in a fair way, then find their respective shares.
13. A coin has probability ‘ p ‘ of showing head when tossed. If is tossed ‘ n’ times. Let pn denote the
probability that no two ( or more ) consecutive heads occur. Prove that , p1 = 1, p2 = 1 – p2 & pn =
PART-I IIT-JEE (PREVIOUS YEARS PROBLEMS)
1. 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 1/2, 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 (Mains), 5]
2. If P (B) =
3
4
, P (A
B
C
) =1
3
and P(
A B C)
=1
3
, then P (B
C) is: [IIT - 2003, 3 + 3](A) 1/12 (B) 1/6 (C) 1/15 (D) 1/9
3. Two numbers are selected randomly from the set S = {1, 2, 3, 4, 5, 6} without replacement one by one. The probability that minimum of the two numbers is less than 4 is: [IIT - 2003, 3 + 3]
(A) 1/15 (B) 14/15 (C) 1/5 (D) 4/5
4. A person has to go through three successive tests. Probability of his passing first exam is P. Probability of passing successive test is P or P/2 according as he passed the last test or not. He is selected if he passes atleast two tests. Find the probability of his selection. [IIT - 2003 (Mains), 2 + 2 ] 5. In a combat, A targets B, and both B and C target A, The probabilities of A, B, C hitting their targets are 2/
3, 1/2 and 1/3 respectively. They shoot simultaneously and A is hit. Find the probability that B hits his
target whereas C does not. [IIT - 2003 (Mains), 2 + 2 ]
6. Three distinct numbers are selected from first 100 natural numbers. The probability that all the three
numbers are divisible by 2 and 3 is [IIT - 2004]
(A) 254 (B) 354 (C) 554 (D) 11554
7. Prove that P(A U B) P
AB
P (C) where A and B are independent events and P(C) is the probability ofexactly one of A or B occurs. [IIT - 2004, 2]
8. A box contains 6 white and 12 red balls. 6 balls are drawn without replacement, in which at least 4 balls are white, find the probability that exactly one of the ball in next two draws, is white. [IIT - 2004, 4] 9. A six faced fair die is thrown until 1 comes, then the probability that 1 comes in even number of trials is
[IIT - 2005]
(A) 115 (B) 65 (C) 116 (D) 61
10. A person goes to office either by car, scooter, bus or train, the probability of which being 7
1
, 73, 72 and 71 respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is 92, 91, 94 and 91 respectively. Given that he reached office in time, then what is the probability that he
travelled by a car. [IIT - 2005, 2]
Comprehension [IIT - 2006]
There are n urns numbered 1 to n. The ith urn contains i white and (n + 1 – i) black balls. Let Ei denote the event
11. If P(Ei) i for i = 1, 2, ..., n, then nlim P(W) is [IIT - 2006 (5, –1)]
(A) 32 (B) 31 (C) 43 (D) 41
12. If P(Ei) = c, a constanti, then P(En
|
W) is [IIT - 2006 (5, –1)](A) n21 (B) n11 (C) nn1 (D) 21
13. If n is even and P(Ei) = n
1
n and E denotes the event of choosing even numbered urn, then P(W
|
E) is[IIT - 2006 (5, –2)]
(A) 21n (B) 2n(n41) (C) 2n(n21) (D) n2n1
14. One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated
adjacent to his wife is [IIT - JEE 2007]
(A) 2 1
(B) 31 (C) 52 (D) 51
15. Let H1, H2, ... Hn be mutually exclusive and exhaustive events with P(Hi) > 0 i = 1, 2, ..., n. Let E be any other
event with 0 < P(E) < 1. [IIT - JEE 2007]
STATEMENT-1 : P(Hi/ E) > P (E / Hi) . P(Hi) for i = 1, 2, ..., n. because STATEMENT-2 :
n 1 i i ) H ( P = 1.(A) Statement-1 is True, Statement-2 is True ; Statement-2is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True ; Statement-2is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
16. Let Ec denote the complement of an event E. Let E, F, G be pairwise independent events with P(G) > 0 and P(E
F G) = 0. Then P (Ec Fc|G) equals [IIT - JEE 2007]
(A) P(Ec) + P(Fc) (B) P(Ec) – P(Fc) (C) P(Ec) – P(F) (D) P(E) – P(Fc)
17. Consider the system of equations [IIT-JEE 2008, Paper-1, (3, –1), 81]
ax + by = 0 ; cx + dy = 0, where a, b, c, d {0, 1}
STATEMENT-1 : The probability that the system of equations has a unique solution is 83.
and
STATEMENT-2 : The probability that the system of equations has a solution is 1.
(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-2is NOT a correct explanation for
STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True
18. An experiment has 10 equally likely outcomes. Let A and B be non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is
[IIT-JEE 2008, Paper-1, (3, –1), 81]
(A) 2, 4 or 8 (B) 3, 6 or 9 (C) 4 or 8 (D) 5 or 10
Comprehension
A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.
19. The probability that X = 3 equals [IIT-JEE 2009, Paper-1, (4, –1), 80]
(A) 21625 (B) 3625 (C) 365 (D) 216125
20. The probability that X 3 equals [IIT-JEE 2009, Paper-1, (4, –1), 80]
(A) 216125 (B) 3625 (C) 365 (D) 21625
21. The conditional probability that X 6 given X > 3 equals [IIT-JEE 2009, Paper-1, (4, –1), 80]
(A) 216125 (B) 21625 (C) 365 (D) 3625
22. Let be a complex cube root of unity with 1. A fair die is thrown three times. If r1, r2 and r3 are the numbers
obtained on the die, then the probability that r1r2r3 = 0 is[IIT-JEE 2010, Paper-1, (3, –1), 84]
(A) 181 (B) 91 (C) 92 (D) 361
23. A signal which can be green or red with probability 54 and 51 respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is 43. If the signal received at station B is green, then the probability that the original signal was green is
[IIT-JEE 2010, Paper-2, (5, –2), 79] Paragraph for Qeustion Nos. 24 and 25
Let U1 and U2 be two urns such that U1 contains 3 white and 2 red balls, and U2 contains only 1 white ball. A
fair coin is tossed. If head appears then 1 ball is drawn at random from U1 and put into U2. However, if tail
appears then 2 balls are drawn at random from U1 and put into U2. Now 1 ball is drawn at random from U2. 24. The probability of the drawn ball from U2 being white is
(A) 1330 (B) 2330 (C) 1930 (D) 3011
25. Given that the drawn ball from U2 is white, the probability that head appeared on the coin is
(A) 1723 (B) 1123 (C) 1523 (D) 1223
26. Let E and F be two independent events. The probability that exactly one of them occurs is 25 and the11 probability of none of them occurring is 25 . If P(T) denotes the probability of occurrence of the event T,,2 then
(A) P(E) = 45 , P(F) = 35 (B) P(E) = 15 , P(F) = 25 (A) P(E) = 5 , P(F) =2 51 (B) P(E) = 35 , P(F) = 45
27*. A ship is fitted with three engines E1, E2 and E3, The engines function independently of each other with
respective probabilities 12, 14 and 41. For the ship to be operational at least two of its engines must function. Let X denote the event the the ship is operational and let X1, X2 and X3 denote respectively the events that
the engines E1 E2 and E3 are functioning. Which of the following is (are) true ? [JEE-2012]
(A) P X | X 1c 163 (B) P[Exactly two engines of the ship are functioning|X] = 78
(C) P X | X
2
165 (D) P X | X
1
16728. Four fair dice D1, D2, D3 and D4 each having six faces numbered 1,2, 3, 4, 5, and 6, are rolled simultaneously.
The probability that D4 shows a number appearing on one of D1, D2 and D3 is [JEE-2012]
(A) 21691 (B) 108216 (C) 125216 (D) 127216
29*. Let X and Y be two events such that P(X|Y) = 1
2, P(Y|X) = 1
3 and P(XY) = 1
6 , Which of the following
is (are) correct? [JEE-2012]
(A) P(X Y) = 23 (B) X and Y are independent
(C) X and Y are not independent (D) P (XC Y) = 1
3
30. Four persons independently solve a certain problem correctly with probabilities 1 3 1 1, , ,
2 4 4 8 . Then the probability that the problem is solved correctly by at least one or them is
[JEE- Advanced- 2013]
(A) 235256 (B) 25621 (C) 2563 (D) 253256
31. Of the three independent events E1, E2 and E3, the probability that only E1 occurs is
, only E2occurs is
andonly E3 occurs is
. Let the probability p that none of events E1, E2 or E3 occurs satisfy the equations (
– 2
)p =
and (
– 3
) p = 2
All the given probabilities are assumed to lie in the interval (0, 1).
Then 1
3
Probability of occurrence of E
Pr obability of occurrence of E [JEE- Advanced- 2013]
Paragraph for Question 32 and 33
A box B1 contains 1 white ball, 3 red balls and 2 black balls. Another box B2 contains 2 white balls, 3 red balls
and 4 black balls. A third box B3 contains 3 white balls, 4 red balls and 5 black balls.[JEE- Advanced- 2013] 32. If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the
other ball is red, the probability that these 2 balls are drawn from box B2 is
(A)116181 (B)126181 (C)18165 (D)18155
33. If 1 ball is drawn from each of the boxes B1, B2 and B3 the probability that all 3 drawn balls are of the same colour
is
PART-II AIEEE (PREVIOUS YEARS PROBLEMS)
1. A problem in mathematics is given to three students A, B, C and their respectively probability of solving the problem is ,31
2 1
and 4
1. Probability that the problem is solved, is [AIEEE 2002]
(A) 3/4 (B) 1/2 (C) 2/3 (D) 1/3
2. A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial, is [AIEEE 2002]
(A) 251 (B) 2524 (C) 252 (D) None of these
3. If A and B are two mutually exclusive events, then [AIEEE 2002]
(A) P(A) < P(B) (B) P(A) > P(B) (C) P(A) < P(B) (D) None of these
4. The probability of India winning a test match against West-Indies is 1/2 assuming independence from match to match. The probability that in a match series India's second win occurs at the third test is [AIEEE 2002]
(A) 81 (B) 41 (C) 21 (D) 32
5. A biased coin with probability p, 0 < p < 1, of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even, is 2/5, then p equals [AIEEE 2002]
(A) 31 (B) 32 (C) 52 (D) 53
6. A fair die is tossed eight times. The probability that a third six is observed on the eight throw, is[AIEEE 2002]
(A) 7C26755 (B) 7C62855 (C) 7C26655 (D) None of these
7. In an experiment with 15 observations on x, the following results were available x2 = 2830, x = 170.
One observation that was 20, was found to be wrong and was replaced by the correct value 30. Then the
corrected variance is : [AIEEE 2003]
(A) 78.00 (B) 188.66 (C) 177.33 (D) 8.33
8. Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr.
A selected the winning horse, is : [AIEEE 2003]
(A) 54 (B) 53 (C) 51 (D) 52
9. Events A, B, C are mutually exclusive events such that P(A) = 3x31, P(B) = 14x and P(C) = 122x . The set
of possible values of x are in the interval : [AIEEE 2003]
(A) 31, 21 (B) 31, 32 (C) 31, 133 (D) [0, 1]
10. The mean and variance of a random variable X having a binomial distribution are 4 and 2 respectively, then
P(X = 1) is : [AIEEE 2003]
11. The probability that A speaks truth is 4/5 while this probability for B is 3/4. The probability that they contradict
each other when asked to speak on a fact, is : [AIEEE 2004]
(A) 3/20 (B) 1/5 (C) 7/20 (D) 4/5
12. A random variable X has the probability distribution : [AIEEE 2004]
X : 1 2 3 4 5 6 7 8
P(X) : 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05
For the events E = {X is a prime number} and F = {X < 4}, the probability P(E F) is :
(A) 0.87 (B) 0.77 (C) 0.35 (D) 0.50
13. The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2
suc-cesses is : [AIEEE 2004]
(A) 25637 (B) 256219 (C) 256128 (D) 25628
14. Let A and B be two events such that P(AB) 61, P(AB) = 4 and 4 1 ) A (
P , where A stands for
complement of event A. Then events A and B are : [AIEEE 2005]
(A) mutually exclusive and independent (B) independent but not equally likely (C) equally likely but not independent (D) equally likely and mutually exclusive
15. Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others, The probability that all the three apply for the same house, is : [AIEEE 2005]
(A) 7/9 (B) 8/9 (C) 1/9 (D) 2/9
16. A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice
is [AIEEE 2007]
(A) 1/729 (B) 8/9 (C) 8/729 (D) 8/243
17. Two aeroplanes and bomb a target in succession. The probability of and scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is
hit by the second plane is [AIEEE 2007]
(A) 0.06 (B) 0.14 (C) 0.32 (D) 0.7
18. It is given that the events A and B are such that P(A) = 41, PBA = 21 and PAB = 32. Then, P(B) is
[AIEEE 2008]
(A) 61 (B) 31 (C) 32 (D) 21
19. A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number
obtained is less than 5. Then P(A B) is [AIEEE 2008]
(A) 53 (B) 0 (C) 1 (D) 52 20. In a binomial distribution B 4 1 p ,
n , if the probability of at least one success is greater than or equal to 109 ,
then n is greater than : [AIEEE 2009]
(A) log 41log 3
10 10 (B) log 4 log 3 9 10 10 (C) log 4 log 3 4 10 10 (D) log 4 log 3 1 10 10
21. One ticket is selected at random from 50 tickets numbered 00, 01, 02, ..., 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equal :
[AIEEE 2009]
(A) 71 (B) 145 (C) 501 (D) 141
22. Four numbers are chosen at random (without replacement) from the set {1,2,3,...,20}.
Statement -1 : The probability that the chosen numbers when arranged in some order will form an AP is 851 .
[AIEEE 2010] Statement -2 : If the four chosen numbers form an AP, then the set of all possible values of common
difference is {±1, ±2, ±3, ±4, ±5}
(A) Statement -1 is true, Statement-2 is true ; Statement -2 is not a correct explanation for Statement -1. (B) Statement-1 is true, Statement-2 is false.
(C) Statement -1 is false, Statement -2 is true.
(D) Statement -1 is true, Statement -2 is true; Statement-2 is a correct explanation for Statement-1.
23. An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is
(A) 72 (B) 211 (C) 232 (D) 31
24. Consider 5 independent Bernoulli trials each with probability of success p. If the probability of at least one failure is greater than or equal to 32 , then p lies in the interval:31 [AIEEE 2011]
(A)2 41 3, (B)3 114 12, (C) 1, 21 (D)11,112
25. If C and D are two events such tha CD and P(D)0, then the correct statement among the following is : (A) p(C|D) = P(C) (B) P(C|D)P(C) (C) P(C|C) < P(C) (D) P(C|D) =P(C)P(D)
26. Let A, B, C be pariwise independent events with P(C) > 0 and P (A AC) = 0. The P(ACBC/C).
(A) P(1) – P(BC) = (B) P(Ac) + P(Bc) (C) P(Ac) – P(Bc) (D) P(Ac) – P(B)
27. Three numbers are chosen at random without replacement from {1, 2, 3 .... 8}. The probability that their
mini-mum is 3, given that their maximini-mum is 6, is [AIEEE 2012]
(A) 38 (B) 15 (C) 14 (D) 52
28. A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is:
(A) 5 17 3 (B) 5 13 3 (C) 5 11 3 (D) 5 10 3 [JEE-Mains 2013]
NCERT BOARD QUESTIONS
Short Answer
1. The probability that at least one of the two events a and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, evaluate P(A) P(B) .
2. A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marbles is red?
3. Two dice are thrown together an the total score is noted. The events E, F and G are 'a total of 4', 'a total of 9 or more' , and 'a total divisible by 5', respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.
4. A and B are two events such that P(A) = 1,P(B) 1
2 3 and P(A B) 14, Find :
(i) P(A|B) (ii) P(B|A) (iii) P(A'|B) (iv) P(A'|B')
5. Three events A, B and C have probabilities 2 1,
5 3 and 12, respectively. Given that P (A C) = 1
5 and P (B C) 1
4, find the values of P(C|B) and P (A' C").
6. Let E1 and E2 be two independent events such that p(E1) = p1 and P(E2) = p2. Describe in words of the events whose probabilities are :
(i) p1p2 (ii) (1 – p1) p2 (iii) 1 – (1 – p1) (1 – p2) (iv) p1 + p2 – 2p1p2 7. A discreate random variable X has the probability distribution given as below :
2 2
X 0.5 1 1.5 2
P(X) k k 2k k
(i) Find the value of k
(ii) Determine the mean of the distribution.
8. If X is the number of tails in three tosses of a coin, determine the standard deviation of X.
9. In a dice game, a player pays a stake of Re1 for each throw of a die. She receives Rs 5 if the die shows a 3, Rs 2 if the die shows a 1 or 6, and nothing otherwise. What is the player's expected profit per throw over a long series of throws?
Long Answer
10. Three bags contain a number of red and white balls as follows: Bag 1 : 3 red balls, Bag 2 : 2 red balls and 1 white balls Bag 3 : 3 white balls.
The probability that bag i will be chosen and a ball is selectd from it is 6 , i = 1, 2, 3. What is the probability thati (i) a red ball will be selected? (ii) a white balls is selectecd?
11. Refer to Question 41 above. If a white balls is selected, what is the probability that it came from
(i) Bag 2 (ii) Bag 3
12. A shopkeeper sells three types of flower seeds A1, A2 and A3. They are sold as a mixture where the proportions are 4 : 4 : 2 respectively. The germination rates of the three tyeps of seeds are 45%, 60% and 35%. Calculate the probability
(i) of a randomly chosen seed to germinate
(ii) that it will not germinate given that the seed is of type A3,
