syllabus (free sample) foundation & olympiad explorer class - x 8. probability 190 ©brain...

CLASS - X

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Published by:

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ISBN: 978-81-907285-5-3

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Preface

Speed and accuracy play an important role in climbing the competitive ladder. Students

have to integrate the habit of being able to calculate and function quickly as well as efficiently

in order to excel in the learning culture. They need to think on their feet, understand basic

requirements, identify appropriate information sources and use that to their best advantage.

The preparation required for the tough competitive examinations is fundamentally different

from that of qualifying ones like the board examinations. A student can emerge successful in

a qualifying examination by merely scoring the minimum percentage of marks, whereas in a

competitive examination, he has to score high and perform better than the others taking the

examination.

This book provides all types of questions that a student would be required to tackle at the

foundation level. The questions in the exercises are sequenced as Basic Practice, Further Practice,

Brainworks, Multiple Answer Questions and Paragraph Questions. Simple questions involving

a direct application of the concepts are given in Basic Practice. More challenging questions

on direct application are given in Further Practice. Questions involving higher order thinking

or an open-ended approach to problems are given in Brainworks. These questions encourage

students to think analytically, to be creative and to come up with solutions of their own.

Constant practice and familiarity with these questions will not only make him/her

conceptually sound, but will also give the student the confidence to face any entrance

examination with ease.

Valuable suggestions as well as criticism from the teacher and student community are most

welcome and will be incorporated in the ensuing edition.

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1. Mathematical Induction ........................... 01

2. Progressions .............................................. 32

3. Functions ................................................... 61

This page is intentionally left blank. 97

Limits ......................................................... 98

5. Polynomials - III ......................................... 121

6. Quadratic Equations - II ............................. 142


7. Inequalities - III .......................................... 167

8. Probability ........................................................ 188

9. Plane Geometry - III ................................... 212

10. Co-ordinate Geometry - III ........................ 250


11. Trigonometry - II ........................................ 286

12. Binomial Theorem ..................................... 320


13. Combinatorics ........................................... 343

CONTENTS

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IIT Foundation & Olympiad Explorer Class - X

© Brain Mapping Academy8. Probability 188

SYNOPSIS

Probabilities of Simple Combined Events

x

xx

x

x xxx

x

sample space S

event E

outcome x

A random experiment is a process involving chance that generates a result called anoutcome. In general, there are two or more possible outcomes in a random experiment.The probability that an outcome will occur is a measure of the change of the occurrenceof the outcome.

The set of the possible outcomes is called the sample space, usually denoted by S. A setof some of the possible outcomes is called an event, usually denoted by E. In otherwords, an event E is a subset of the sample space S.

For a finite sample space with equally likely outcomes, the probability of an event E,denoted by P(E), is given by

P(E) = Number of outcomes favourable to the event E

Total number of possible outcomes

i.e.,n(E)

P(E)n(S)

= .

A summary of the basic properties of probability is given below.

1. 0 P(E) 1≤ ≤ for any event E.

2. P (certain event) = 1, P (sample space) = 1.

3. P (impossible event) = 0

4. P(E') = 1 � P(E)

1

Chapter

8 ProbabilityProbability

Chapter

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In this chapter, we will learn how to solve more complicated probability problems. Inorder to help us visualize the sample space and the events in such problems, we usuallydraw a diagram called a possibility diagram or a tree diagram.

A. Possibility Diagram:

When a random experiment involves two stages, we can use a rectangular grid, called apossibility diagram, to represent the sample space.

Ex: An unbiased coin is tossed and a letter is selected at random from the word �SMART�.Find the probability of getting a head on the tossed coin and the letter �M� from theword.

Sol: Here, the tossing of a coin is a stage; and the selection of a letter is another stage. Thisexperiment involves two stages. The outcomes can be represented by the crosses in thediagram below. This diagram is called a possibility diagram.

x x x x x

x x x x x

T

H

S M A

Letter

R T

Coin

The sample space S = {HS, HM, HA, HR, HT, TS, TM, TA, TR, TT}, where the first letterstands for the result of tossing the coin (H for Head and T for Tail), and the second letterstands for the letter selected from the word �SMART�.

The favourable outcome of getting a head and the letter �M� is HM.

∴ P (getting a head and the letter �M�) 110

= .

B. Tree Diagram:

If a random experiment consists of two or more stages, we can use a tree diagram torepresent the process. Let us study some examples.

Ex: Three unbiased coins are tossed. Find the probability of getting

(a) 3 heads,

(b) 2 heads and 1 tail.

Sol. (a) Here, the tossing of each coin is a stage.

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H

T

H

H

H HHH

HTH

THH

TTH

HHT

HTT

THT

TTT

H

H

H

T

T

T

T

T

T

1st Coin 2nd Coin 3rd Coin Outcome

The above diagram is a tree diagram showing the outcomes of tossing three coins. In atree diagram, the result of each stage is shown at the end of a branch for that stage. Byreading along the branches, we get the outcomes of the experiment.

In this case the outcome HHH means the first coin shows a head, the second coin showsa head and the third coin shows a head (as indicated by the branches).

The 8 outcomes are equally likely.

∴ P(3 heads) = P (HHH)

18

=

(b) Let E = {outcomes showing 2 heads and 1 tail}

= {HHT, HTH, THH}

∴ P (2 heads and 1 tail) 38

= .

Mutually Exclusive Events

For any two events, A and B, we denote

� probability that both events A and B will occur as P(A and B);

� probability that either event A or event B will occur, or both will occur as P (A or B).

In a sample space, two events are mutually exclusive if they cannot occur at the sametime.

For example, in rolling a die,

event A = {1, 3, 5} and event B = {2, 4}

are mutually exclusive. This is because when the die shows 1,

3 or 5, it definitely cannot show 2 or 4. Note that A B∩ = φ .

That means, two events, A and B, are mutually exclusive if

A B∩ = φ .

AS

B

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Let us investigate the relationships between P(A), P(B), P(A and B) and P(A or B) for twomutually exclusive events, A and B.

n If A and B are two mutually exclusive events, then the probability of A or Boccurring is

AS

B

P(A or B) = P(A) + P(B).

Independent Events

Let us study another type of combination of events. Intuitively, two events are said to beindependent events if the occurrence or non-occurrence of one event does not affectthe probability of occurrence of the other event.

For example, in tossing a fair coin twice, the event A that the first toss shows a head andthe event B that the second toss shows a head are independent. That is, event A does notaffect how likely event B will occur, and vice versa.

n If A and B are independent events, the probability of both events A and B occurring isthe product of their individual probabilities.

i.e., P(A and B) = P(A) × P(B).

Further Probabilities

Consider drawing two balls at random from a bag containing 3 red balls and 5 greenballs. Suppose the balls are drawn one by one without replacement.

Let A be the event that the first ball drawn is red,

B be the event that the second ball drawn is green.

At first, there are 8 balls in the bag. After drawing the first ball, there will be 7 balls leftin the bag.

1. If event A happens (i.e, the first ball drawn is red), the remaining balls are 2 red onesand 5 green ones.

Hence, P(B) 57

= .

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2. If event A does not happen (i.e., that first ball drawn is not red), the remaining balls are3 red ones and 4 green ones.

Hence,4

P(B)7

= .

That means, the probability of event B depends on whether event A will occur or not.Thus, A and B are NOT independent events. How do we calculate the probability of bothevents A and B occurring if they are not independent?

We can also use a tree diagram to do this. In the tree diagram, the probabilities indicatingthe second stage branches will depend on the results of the first stage.

Hence P the first ball drawn is red and the second ball drawn is green)

= P (RG)

= 3 58 7

×

In general, when the occurrence of the second event depends on the occurrence of thefirst event, we may use a tree diagram to help us compute the probabilities.

Some standard methods of constructing new events in terms of some given eventsassociated with a random expreiment.

Verbal Description of event Equivalent set theoritic notation

Not A A

A or B A∪ B

A and B A∩ B

1st ball 2 ball Outcome Probability

R RR P(RR) = ×3 28 7

G RG P(RG) = ×3 58 7

R GR P(GR) = ×5 38 7

G GG P(GG) = ×5 48 7

R - red ballG - green ball

R

G

38

58

27

57

37

47

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A but not B A B∩

Neither A nor B A B∩ = ( )A B∪

At least one of A, B or C (A ∪ B) ∪ C

Exactly one of A and B ( ) ( )A B A B∩ ∪ ∪

All three of A, B and C A∩ B ∩ C

Exactly two of A, B and C ( ) ( )A B C A B C∩ ∩ ∪ ∩ ∩

( )A B C∪ ∩ ∩

i. Addition theorem for two events

If A and B are two events associated with a random experiment, then

P(A ∪ B) = P(A) + P(B) � P(A ∩ B)

Note:

If A and B are mutually exclusive events, then P(A ∩ B) = 0

∴ P(A ∪ B) = P(A) + P(B)

ii. Addition theorem for three events

If A, B, C are three events associated with a random experiment, then

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) � P(A ∩ B) � P(B ∩ C) � P(A ∩ C)

+P(A ∩ B ∩ C)

Note:

If A, B, C are mutually exclusive events, then

P(A ∩ B) = P(B ∩ C) = P(A ∩ C) = P(A ∩ B ∩ C) = 0

∴ P(A ∪ B ∪ C) = P(A) + P(B) + P(C)

iii. P( A ∩ B) = P(B) � P(A ∩ B)

iv. P(A ∩ B ) = P(A) � P(A ∩ B)

v. If B ⊂ A, then (a) P(A ∩ B ) = P(A) � P(B) (b) P(B) ≤ P(A)

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SOLVED EXAMPLES

Example 1:

Find the probability of obtaining a number greater than 4 on a single toss of a die.

Solution:There are six different possible outcomes and two of these outcomes, {5, 6}, are successful.

Hence, the probability in favour of obtaining a number greater than 4 is 26 .

P (number greater than 4) 2 16 3

= = .

Example 2:

Find the probability of drawing a king (one pick) from a shuffled standard deck of 52cards. (A standard deck of cards is the most common type of deck used in most cardgames containing 52 cards).

Solution:

There are 52 different possible outcomes. Four of these outcomes are successful: king ofspades, king of hearts, king of clubs, king of diamonds. Therefore, the probability in

favour of obtaining a king is 452 .

P(K) 4 152 13

= = .

Example 3:

Find the probability of obtaining 7 on a single toss of one die.

Solution:

There are six different possible outcomes and none of these out comes would produce a7. That is, zero of these outcomes would be successful. The probability in favour of

obtaining a 7 on a single toss of one die is 06 , or 0:

0P(7) 0

6= =

When an event cannot possibly succeed, we say it is an impossible event. The probabilityof an impossible event is zero:

P (impossible event) 0

0 (T 0)T

= = ≠

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Solution:

III

M

11

755

III

IV

H

9

A Venn diagram using the given information is shown in above figure. With it, we cansummarize the information that 11 of the students did not register for either amathematics course or a history course, 9 students registered only for history, and 75students registered only for mathematics.This information can also be used to solve the probability problem. We note that thetotal number of students is 100. Whereas 9 of them are registered only for history.

Hence, the answer is 9100 .

Example 7:

In a certain group of 75 students, it has been determined that 16 students are takingstatistics, chemistry, and psychology; 24 students are taking statistics and chemistry;30 students are taking statistics and psychology; 22 students are taking chemistry andpsychology; 6 students are taking only statistics; 9 students are taking only chemistry;and 5 students are taking only psychology.

Solution:

98

16

S

11

III

6

6

V

IV VI

14

5

VIIVIII

C

III

P

(a) What is the probability that a student is not taking any of the three subjects?

(b) What is the probability that a student is taking chemistry?

Solution:

We first complete the necessary Venn diagram (See above figure). After completing thediagram, we can answer the questions.

(a) The probability that a student is not taking any of the three subjects is 1175 .

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(b) The probability that a student is taking chemistry is 3975 .

The answer to question (b) is obtained by adding the number of students in each partitionof the chemistry circle. Hence, there are 39 students taking chemistry.

Example 8:

Mr. Examination is preparing a quickie quiz for his mathematics class to see if thestudents did their assignment. The quiz is to consist of three true-false questions. Howmany different arrangements of the answers are possible? What are the possibleoutcomes?

Solution:

We have three questions and each questions has two possible outcomes (true or false).Using the counting principle, we compute 2 × 2 × 2 = 8 total possible outcomes.

We can determine the various outcomes by means of a tree diagram. Remember thatthe quickie quiz consists of three questions and the answer to each question is eithertrue or false.

T

Start

F

T T T T

T F T T

T T F T

T F F T

F

F

T

T

F T T F

F F T F

F T F F

F F F F

Firstquestion

SecondQuestion

ThirdQuestion

Firstquestion

Secondquestion

Possible outcomes

Thirdquestion

The sample space is listed beside the tree diagram.

Example 9:

What are the odds is favour of obtaining a sum of 7 when a pair of dice is tossed once?Solution:

Using our definition, we first find probability of obtaining a sum of 7 when a pair of dice

is tossed. It is 636

. Next, we find the probability of not getting a 7. Recall that if P(A) = S/T,

then P(not A) = 1 � ST

. Hence, we have

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Solution:

(a)246246246246

Bag A Bag B Outcome

1

3

5

7

(1, 2)(1, 4)(1, 6)(3, 2)(3, 4)(3, 6)(5, 2)(5, 4)(5, 6)(7, 2)(7, 4)(7, 6)

The above diagram is the required tree diagram. There are 12 equally likely outcomesin the sample space.

(b) (i) The favourable outcomes for a sum of 7 are: (1, 6), (3, 4) and (5, 2)

∴ P (the sum is 7) 3 1

12 4= = .

(ii) Since all the sums of the outcomes are odd,

P(the sum is odd) 12

112

= =

(iii) There are no even sums.

∴ P( the sum is even) 0

012

= =

Example 12:

A club holds an election for the post of chairperson. The probabilities that thecandidates Anjani and Laxmi will be elected are 0.36 and 0.47 respectively. Find theprobability that(a) either Anjani or Laxmi will be elected,(b) neither Anjani nor Laxmi will be elected.

Solution:

(a) Let A be the event that Anjani will be elected,

B be the event the Laxmi will be elected.

Since there is only 1 chairperson, the events, A and B, are mutually exclusive.

P (either Anjani or Laxmi will be elected)

= P(A or B)

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= P(A) + P(B) (addition of probabilities of mutually exclusive events)

= 0.36 + 0.47

= 0.83

(b) P (neither Anjani nor Laxmi will be elected)

= 1 � P (either Monica or Roland will be elected)

= 1 � 0.83

= 0.17

Example 13:

A card is drawn at random from a pack of 52 playing cards. Find the probability that thecard drawn is(a) an ace or a 3,(b) an ace or a red card.

Solution:(a) Let A be event that the card drawn is an ace,

B be event that the card drawn is a 3,C be event that the card drawn is red,

Then, P(A) 4 4 26, P(B) and P(C)

52 52 52= = = .

We cannot draw a card which is both an ace and a 3.

∴ events A and B are mutually exclusive events.

P (an ace or a 3) = P(A or B)

= P(A) + P(B)

4 452 52

= +

852

= 213

=

(b) As there are 2 red aces, the diamond ace and the heart ace, events A and C are notmutually exclusive events. Therefore, we cannot apply the addition of probabilities ofmutually exclusive events to evaluate P(A or C). We have to find P(A or C) here bycounting.

Apart from the two red aces, there are 24 red cards.

P(an ace or a red card) = P (A or C)

4 2452+= 28 7

52 13= =

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Similarly, P (Srinu will miss) 31

4= −

14

=

Hence, we have a tree diagram as shown below.

P (the target will be hit by both) = P(HH)

4 3 35 4 5

= × =

(b) P(the target will be hit by only one of them)

= P(Guru will hit and Srinu will miss)+ P(Guru will miss and Srinu will hit)

= P (HM) + P(MH)

4 1 1 3 4 3 75 4 5 4 20 20 20

= × + × = + =

Example 16:

The probability that a worker with occupational exposure to dust contracts a lung diseaseis 0.2. Three such workers are checked at random. Find the probability that

(a) none of the three workers contracted a lung disease,

(b) at least one of them contracted a lung disease.Solution:

(a) P (a worker does not contract a lung disease)

= 1 � P(a worker contracts a lung disease) = 1 � 0.2 = 0.8

Hence, we have a tree diagram as shown below,

Guru Srinu Outcome

H HH

H

M HM

H MH

M

M MM

H - HitM - Miss

45

15

34

14

34

14

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P (none of them contracted a lung disease)= P(NNN) = 0.8 × 0.8 × 0.8 (multiplication of probabilities) = 0.512

(b) P (at least one of them contracted a lung disease)= 1 � P (none of them contracted a lung disease) = 1 � 0.512 = 0.488

Example 17:

In a library, shelf A has 10 Mathematics books and 6 Science books, while shelf B has 8Mathematics books and 12 Science Books. Ravi goes to one of these shelves and picks upa book at random. Find the probability that the book picked is a Mathematics book.

Solution:

P (picking a Mathematics book)

= P ({shelf A and Mathematics book}or {Shelf B and Mathematics book})

= P(AM) + P(BM) (addition of probabilities)

=1 10 1 8

2 16 2 20× + ×

4180

= .

1st worker 2nd worker 3rd worker Outcome C - Contracts

N - Does not contract

0.2

0.8

C

N

0.2

0.8

C

N

0.2

0.8

C

N

0.2

0.8

C

N

0.2

0.8

C

N

0.2

0.8

C

N

0.2

0.8

C

N

CCC

CCN

CNC

CNNNCC

NCN

NNC

NNN

A

B

M

S

M

S

AM

M - Mathematics

S - Science

12

12

820

1220

1016

616 AS

BM

BS

Shelf Book Outcome

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BASIC PRACTICE

1. In a simultaneous toss of two coins, find the probability of getting(i) 2 heads (ii) exactly one head (iii) exactly 2 tails(iv) exactly one tail (v) no tails.

2. A die is thrown. Find the probability of getting(i) an even number (ii) a prime number(iii) a number greater than or equal to 3 (iv) a number less than or equal to 4(v) a number more than 6 (vi) a number less than or equal to 6.

3. Two dice are thrown simultaneously. Find the probability of getting:(i) an even number as the sum(ii) the sum as a prime number(iii) a total of at least 10(iv) a doublet of even number(v) a multiple of 2 on one dice and a multiple of 3 on the other dice.(vi) same number on both dice(vii) a multiple of 3 as the sum.

4. Find the probability that a leap year, selected at random, will contain 53 sundays.

5. An urn contains 9 red, 7 white and 4 black balls. If two balls are drawn at random, findthe probability that:(i) both the balls are red,(ii) one ball is white(iii) the balls are of the same colour(iv) one is white and other red.

6. Four persons are to be chosen at random from a group of 3 men, 2 women and 4 children.Find the probability of selecting(i) 1 man, 1 woman and 2 children(ii) exactly 2 children(iii) 2 women

7. A five digit number is formed by the digits 1, 2, 3, 4, 5 without repetition. Find theprobability that the number is divisible by 4.

8. There are 4 letters and 4 addressed envelopes. Find the probability that all the lettersare not dispatched in right envelopes.

9. The odds in favour of an event are 3 : 5. Find the probability of occurrence of this event.

10. A and B are two non-mutually exclusive events. If P(A) = 14

, P(B) = 25 and

1P(A B)

2∪ = ,

find the values of P(A B) and P(A B)∩ ∩ .

11. The probability that at least one of the events A and B occurs is 0.6. If A and B occursimultaneously with probability 0.2, then find P(A) P(B)+ .

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12. The probability of two events A and B are 0.25 and 0.50 respectively. The probability oftheir simultaneous occurrence is 0.14. Find the probability that neither A nor B occurs.

13. One number is chosen from numbers 1 to 200. Find the probability that it is divisible by4 or 6?

14. In an essay competition, the odds is favour of competitors P, Q, R, S are 1 : 2, 1: 3, 1 : 4,and 1 : 5 respectively. Find the probability that one of them wins the competition.

15. A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges aredefective. If a person takes out 2 at random what is the probability that either both areapples or both are good?

FURTHER PRACTICE

1. The probability of raining on day 1 is 0.2 and on day 2 is 0.3. What is the probability ofraining on both the days?

(A) 0.2 (B) 0.1 (C) 0.06 (D) 0.25

2. A bag contains 5 red balls and 8 blue balls. It also contains 4 green and 7 black balls. If aball is drawn at random, then find the probability that it is not green.

(A) 5/6 (B) 1/4 (C) 1/6 (D) 7/4

3. A bag contains 2 red, 3 green and 2 blue balls. 2 balls are to be drawn randomly. What isthe probability that the balls drawn contain no blue ball?

(A)57 (B)

1021

(C) 27 (D)

1121

4. If the probability that A will live 15 years is 78 and that B will live 15 years is

910 , then

what is the probability that both will live after 15 years?

(A) 120 (B)

6380 (C)

15 (D) None of these

5. Suppose six coins are flipped. Then the probability of getting at least one tail is

(A) 7172 (B)

5354 (C)

6364 (D)

112

6. The probability that a student is not a swimmer is 1/5. Then the probability that out ofthe five students, four are swimmers, is

(A) 2

54

4 1C

5 5

(B) 44 1

5 5

(C) 4

51

1 4C

5 5

(D) None of these

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7. A set A is containing n elements. A subset P of A is chosen at random. The set isreconstructed by replacing the elements of P. A subset of A is again chosen at random.The probability that P and Q have no common element is:

(A) 5n (B) n3

4

(C) n3

5

(D) 2n

8. If events A and B are independent and P(A) = 0.15, P(A B) 0.45, then∪ = P(B) = ______.

(A) 6

13 (B)6

17 (C) 6

19 (D) 623

9. One hundered identical coins each with probability p of showing up heads are tossed. If0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads on51 coins; then the value of p is:

(A) 12

(B) 49

101 (C) 50

101 (D) 51

10110. The probability that Kumar will hit a target is given as 1/5. Then, his probability of

atleast one hit in 10 shots is:

(A) 10

16 (B)

1041

5 −

(C) 10

11

5− (D) 19

11

5−

11. Two dice are tossed. The probability that the total score is a prime number is:

(A) 16 (B)

512

(C) 12

(D) 79

12. If the probability that A will live 15 years is 78 and that B will live 15 years is

910 , then

what is the probability that both will live after 15 years?

(A) 120 (B)

6380 (C)

15 (D) None of these

13. Four different objects 1, 2, 3, 4 are distributed at random in four places marked 1, 2, 3,4. What is the probability that none of the objects occupy the place corresponding to itsnumber?(A) 17/24 (B) 3/8 (C) 1/2 (D) 5/8

14. Three students try to solve a problem independently with a probability of solving it as1/3, 2/5, 5/12 respectively. What is the probability that the problem is solved?(A) 1/18 (B) 12/30 (C) 23/30 (D) 1/2

15. If the probability of rain on any given day in Pune city is 50%, then what is the probabilitythat it rains on exactly 3 days in a 5-day period?

(A) 8/125 (B) 5/16 (C) 8/25 (D) 2/25

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26. The probability of getting an even number when a die is rolled is:

(A) 16

(B) 1

36(C)

12

(D) none

27. A card is drawn from a packet of 100 cards numbered 1 to 100. The probability of drawinga number which is a square is:

(A) 1

10(B)

9100

(C) 1

100(D)

2100

28. The probability for a randomly selected number out of 1, 2, 3, 4, ....., 25 to be a primenumber is:

(A) 2

25(B)

2325

(C) 1025

(D) 9

2529. In a single throw of two dice, the probability of getting a sum of 10 is:

(A) 1

12(B)

136

(C) 16

(D) none

30. Three letters, to each of which corresponds an addressed envelope are placed in theenvelopes at random. The probability that all letters are placed in the right envelopes is:

(A) 13

(B) 1 (C) 16

(D) 0

BRAIN WORKS

1. A coin is tossed n times: what is the chance that the head will present itself an oddnumber of times?

2. A speaks truth in 75% of the cases and B in 80% of the cases. In what percentage of casesare they likely to contradict each other in stating the same fact?

3. In a multiple choice question there are four alternative answers of which one or morethan one is correct. A candidate will get marks on the question only if he ticks thecorrect answers. The candidate decides to tick answers at random. If he is allowed uptothree chances to answer the question, find the probability that he will get marks on it.

4. An article manufactured by a company consists of two parts X and Y. In the process ofmanufacture of the part X, 9 out of 104 parts may be defective. Similarly 5 out of 100 arelikely to be defective in part Y. Calculate the probability that the assembled product willnot be defective.

5. In a certain city only 2 newspapers A and B are published. It is known that 25% of thecity population reads A and 20% reads B while 8% reads both A and B. It is also knownthat 30% of those who read A but not B look into advertisement and 40% of those whoread B but not A look into advertisements while 50% of those who read both A and Blook into advertisements. What is the percentage of the population who read anadvertisement?

6. A and B throw alternatively with a pair of dice and the score is the sum of the number of

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points on the dice. A wins if he gets a score of 6 before B gets a score of 7 and B wins ifhe gets a score of 7 before A gets a score of 6. If A begins, show that his chance ofwinning is 30/61.

7. Suppose the probability for A to win a game against B is 0.4. If A has an option of playingeither a �best of 3 games� or a �best of five games� match against B, which option shouldA choose so that the probability of his winning the match is higher? No game ends in adraw.

8. Two numbers are selected at random from 1, 2, 3, ..., 100 and multiplied. Find theprobability correct to two places of decimals that the product thus obtained, is divisibleby 3.

9. In a test an examinee either guesses or copies or knows the answer to a multiple choicequestion with four choices. The probability that he makes a guess is 1/3 and the probabilitythat he copies the answer is 1/6. The probability that his answer is correct given that hecopied it is 1/8. Find the probability that he knew the answer to the question, given thathe correctly answered it.

10. A and B are two independent events. The probability that both A and B occur is 1/6 and the

probability the neither of them occurs is 13

. Find the probability of the occurrence of A.

MULTIPLE ANSWER QUESTIONS

1. If A and B are two independent events, the probability that both A and B occur is 18 and

the probability that neither of them occurs in 38 . The probability of the occurrence of A

is:

(A) 12

(B) 13 (C)

14

(D) 15

2. For any two events A and B in a sample space:

(A) A P(A) P(B) 1

P , P(B) 0B P(B)

+ − ≥ ≠ , is always true

(B) P(A B) P(A) P(A B)∩ = − ∩ does not hold

(C) P(A B) 1 P(A)P(B)∪ = − , if A and B are independent

(D) P(A B) 1 P(A)P(B),∪ = − if A and B are disjoint

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3. If A and B are two events that the value of the determinant chosen at random from allthe determinants of order 2 with entries 0 or 1 only is positive or negative respectivelythen ________

(A) P(A) P(B)≥ (B) P(A) P(B)≤ (C) 1

P(A) P(B)2

= = (D) None of these

4. If P(A) = 18 and P(B)

58

= . Which of the following statements is not correct?

(A) 3

P(A B)4

∪ ≤ (B) 1P(A B)

8∩ ≤ (C) 5

P(A B)8

∩ ≤ (D) None of these

5. A bag contains four tickets marked with 112, 121, 211, 222, one ticket is drawn at randomfrom the bag. Let Ei (i = 1, 2, 3) denote the event that ith digit on the ticket is 2. Then:

(A) E1 and E2 are independent

(B) E2 and E3 are independent

(D) E3 and E1 are independent

(D) E1, E2, E3 are independent

PARAGRAPH QUESTIONS

Passage - I

� If A and B are two events associated with a random experiment, then

P(A B) P(A) P(B) P(A B)∪ = + − ∩

� If A, B, C are three events associated with a random experiment, thenP(A ∪ B ∪ C) = P(A) + P(B) + P(C) � P(A ∩ B) � P (B ∩ C) � P (A ∩ C) + P(A ∩ B ∩ C)

(i) Two cards are drawn from a pack of 52 cards. What is the probability that either bothare red or both are kings?

(ii) A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges aredefective. If a person takes out 2 at random what is the probability that either both areapples or both are good?

(iii) The probability that a person will get an electric contract is 25 and the probability that

he will not get plumbing contract is 47 . If the probability of getting at least one contract

is 23 , what is the probability that he will get both?

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(iv) For a post three persons A, B and C appear in the interview. The probability of A beingselected is twice that of B and the probability of B being selected is thrice that of C. Whatare the individual probabilities of A, B, C being selected?

(v) Probability that Chakri passes in mathematics is 23 and the probability that he passes in

English is 49 . If the probability of passing both courses is

14

, what is the probability that

Chakri will pass in at least one of these subjects?

ANSWERS

Basic Practice

1. (i) 14

(ii) 12

(iii) 14

(iv) 12

(v) 14

2. (i) 12

(ii) 12

(iii) 23

(iv) 23

(v) 0 (vi) 1

3. (i)12

(ii) 5

12(iii)

16

(iv) 1

12(v)

1136

(vi) 16

(vii) 13

4.27

5. (i) 1895

(ii) 91

190(iii)

63190

(iv) 63

190

6. (i) 27

(ii) 1021

(iii) 16

7. 15

8. 2324

9. 38

10. 3 1

;20 10

11. 1.2 12. 0.39 13. 67200

14. 114120

15. 316435

Further Practice

1. D 2. A 3. A 4. B 5. C 6. B 7. A 8. B 9. D 10.B11. B 12. B 13. C 14. C 15. B 16. B 17.A 18. D 19.A 20. A

21. C 22. C 23. B 24. A 25.B 26. C 27. A 28.D 29. A 30. C

Brain Works

1.12

2. 35

100 3. 15 4. 361/416 5. 13.9%

6. Proof 7. Best of three games 8. 0.55 9. 24/29 10. 1/3.

Multiple Answer Questions

1. A, C 2. A, C 3. A, B 4. A, B, C 5. A, B, C

Paragraph Questions

(i) 55/221 (ii) 316/435 (iii) 17/105 (iv) 6/10; 3/10; 1/10 (v) 31/36

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CLASS - X

IIT F

oundatio

n &

Olym

pia

d E

xplo

rer - M

ath

em

atic

s Cla

ss - X

FOUNDATION OLYMPIAD&

IntegratedSyllabus

UNIQUE ATTRACTIONS●

● Cross word Puzzles

● Graded Exercise

Basic Practice■

Further Practice■

Brain Works■

● Multiple Answer Questions

● Paragraph Questions

Detailed solutionsfor all problems

of IIT Foundation &Olympiad Explorer

are available in this book

CLASS - X

www.bmatalent.com

� Simple, clear and systematic presentation

� Concept maps provided for every chapter

� Set of objective and subjective questions at the

end of each chapter

� Previous contest questions at the end of each

chapter

� Designed to fulfill the preparation needs for

international/national talent exams, olympiads

and all competitive exams

YOUR

COACH

India’s FIRST scientifically designed portalfor Olympiad preparation• Olympiad & Talent Exams preparation packages

Analysis Reports Previous question papers• •Free Demo Packages Free Android Mobile App• •

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