exercise (all)

8/10/2019 Exercise (ALL)

1/39

Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 31

I T P A CE-ians Rg-PB -12I PROBABILITYEdu.Pvt.Ltd

EXERCISE 1 (A)

ONLY ONE OPTION IS CORRECT

1. A committee of three persons is to be randomly selected from a group of three men and two women and thechair person will be randomly selected from the committee. The probability that the committee will have

exactly two women and one man, and that the chair person will be a woman, is/are(A) 1/5 (B) 8/15 (C) 2/3 (D) 3/10

2. 6 married couples are standing in a room. If 4 people are chosen at random, then the chance that exactly onemarried couple is among the 4 is :

(A)33

16(B)

33

8(C)

33

17(D)

33

24

3. A sample space consists of 3 sample points with associated probabilities given as 2p, p2, 4p 1 then(A) p = 311 (B) p = 310 (C) 1/4 < p < 1/2 (D) none

4. A committee of 5 is to be chosen from a group of 9 people. The probability that a certain married couple willeither serve together or not at all is :(A) 1/2 (B) 5/9 (C) 4/9 (D) 2/3

5. There are only two women among 20 persons taking part in a pleasure trip. The 20 persons are divided intotwo groups, each group consisting of 10 persons. Then the probability that the two women will be in the samegroup is :(A) 9/19 (B) 9/38 (C) 9/35 (D) none

6. The probability that a positive two digit number selected at random has its tens digit at least three more thanits unit digit is

(A) 14/45 (B) 7/45 (C) 36/45 (D) 1/6

7. A 5 digit number is formed by using the digits 0, 1, 2, 3, 4 & 5 without repetition. The probability that thenumber is divisible by 6 is :(A) 8 % (B) 17 % (C) 18 % (D) 36 %

8. An experiment results in four possible out comes S1, S2, S3& S4with probabilities p1, p2, p3& p4respectively.Which one of the following probability assignment is possbile.[Assume S1S2S3S4are mutually 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) p

1= 0.20, p

2= 0.30, p

3= 0.40, p

4= 0.10

9 In throwing 3 dice, the probability that atleast 2 of the three numbers obtained are same is(A) 1/2 (B) 1/3 (C) 4/9 (D) none

10. There are 4 defective items in a lot consisting of 10 items. From this lot we select 5 items at random. Theprobability that there will be 2 defective items among them is

(A)2

1(B)

5

2(C)

21

5(D)

21

10


2/39


3/39



21. There are ndifferent gift coupons, each of which can occupy N(N > n) different envelopes, with the sameprobability 1/NP1: The probability that there will be one gift coupon in each of n definite envelopes out of N given envelopesP2: The probability that there will be one gift coupon in each of n arbitrary envelopes out of N given envelopesConsider the following statements

(i) P1= P2 (ii) P1= nN

!n

(iii) P2= !)nN(N

!N

n (iv) P2= !)nN(N

!nn (v) P1= nN

!N

Now, which of the following is true(A) Only (i) (B) (ii) and (iii) (C) (ii) and (iv) (D) (iii) and (v)

22. The probability that an automobile will be stolen and found within one week is 0.0006. The probability that anautomobile will be stolen is 0.0015. The probability that a stolen automobile will be found in one week is(A) 0.3 (B) 0.4 (C) 0.5 (D) 0.6

23 One bag contains 3 white & 2 black balls, and another contains 2 white & 3 black balls. A ball is drawn fromthe second bag & placed in the first, then a ball is drawn from the first bag & placed in the second. When the

pair of the operations is repeated, the probability that the first bag will contain 5 white balls is:

(A) 1/25 (B) 1/125 (C) 1/225 (D) 2/15

24. A child throws 2 fair dice. If the numbers showing are unequal, he adds them together to get his final score.On the other hand, if the numbers showing are equal, he throws 2 more dice & adds all 4 numbers showingto get his final score. The probability that his final score is 6 is:

(A)1296

145(B)

1296

146(C)

1296

147(D)

1296

148

25. Events A and C are independent. If the probabilities relating A, B and C are P (A) = 1/5; P (B) = 1/6;P(A C) = 1/20; P(B C) = 3/8 then(A) events B and C are independent

(B) events B and C are mutually exclusive(C) events B and C are neither independent nor mutually exclusive(D) events B and C are equiprobable

26. Assume that the birth of a boy or girl to a couple to be equally likely, mutually exclusive, exhaustive andindependent of the other children in the family. For a couple having 6 children, the probability that their "threeoldest are boys" is

(A)64

20(B)

64

1(C)

64

2(D)

64

8

27. If a, b and c are three numbers (not necessarily different) chosen randomly and with replacement from the set

{1, 2, 3, 4, 5}, the probability that (ab + c) is even, is

(A)125

50(B)

125

59(C)

125

64(D)

125

75

28. A examination consists of 8 questions in each of which one of the 5 alternatives is the correct one. On theassumption that a candidate who has done no preparatory work chooses for each question any one of thefive alternatives with equal probability, the probability that he gets more than one correct answer is equal to(A) (0.8)8 (B) 3(0.8)8 (C) 1 (0.8)8 (D) 1 3 (0.8)8


4/39



29. A key to room C3 is dropped into a jar with five other keys, and the jar is throughly mixed. If keys arerandomly drawn from the jar without replacement until the key to room C3 is chosen, then what are the oddsthat the key to room C3 will be botained on the 2ndtry?(A) 1:4 (B) 1:5 (C) 1:6 (D) 5:6

30. A bowl has 6 red marbles and 3 green marbles. The probability that a blind folded person will draw a redmarble on the second draw from the bowl without replacing the marble from the first draw, is(A) 2/3 (B) 1/4 (C) 5/12 (D) 5/8

31. 5 out of 6 persons who usually work in an office prefer coffee in the mid morning, the other always drink tea.This morning of the usual 6, only 3 are present. The probability that one of them drinks tea is :(A) 1/2 (B) 1/12 (C) 25/72 (D) 5/72

32. Pals gardner is not dependable, the probability that he will forget to water the rose bush is 2/3. The rosebush is in questionable condition . Any how if watered, the probability of its withering is 1/2 & if notwatered then the probability of its withering is 3/4. Pal went out of station & after returning he finds thatrose bush has withered. What is the probability that the gardner did not water the rose bush.(A) 1/4 (B) 1/2 (C) 3/4 (D) 3/8

33. The probability that a radar will detect an object in one cycle is p. The probability that the object will bedetected in n cycles is :(A) 1 pn (B) 1 (1 p)n (C) pn (D) p(1 p)n1

34. Nine cards are labelled 0, 1, 2, 3, 4, 5, 6, 7, 8. Two cards are drawn at random and put on a table in asuccessive order, and then the resulting number is read, say, 07 (seven), 14 (fourteen) and so on. The

probability that the number is even, is(A) 5/9 (B) 4/9 (C) 1/2 (D) 2/3

35. Two cards are drawn from a well shuffled pack of 52 playing cards one by one. IfA : the event that the second card drawn is an ace and

B : the event that the first card drawn is an ace card.then which of the following is true?

(A) P (A) =17

4; P (B) =

13

1(B) P (A) =

13

1; P (B) =

13

1

(C) P (A) =13

1; P (B) =

17

1(D) P (A) =

221

16; P (B) =

51

4

36. A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds,and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the

probability that the colour changes while she is watching?

(A) 63

1

(B) 21

1

(C) 10

1

(D) 7

1

37. Player A has 2 fair coins and B has one more coin than the player B. Both the players throw all of their coinssimultaneously and observe the number that come up heads. Assuming that all coins are fair, the probabilitythat B obtains more heads than A, is

(A)32

16(B)

32

18(C)

32

13(D)

32

12


5/39



38. A box contains 5 red and 4 white marbles. Two marbles are drawn successively from the box withoutreplacement and the second drawn marble drawn is found to be white. Probability that the first marble is alsowhile is(A) 3/8 (B) 1/2 (C) 1/3 (D) 1/4

39 A and B in order draw a marble from bag containing 5 white and 1 red marbles with the condition thatwhosoever draws the red marble first, wins the game. Marble once drawn by them are not replaced into the

bag. Then their respective chances of winning are(A) 2/3 & 1/3 (B) 3/5 & 2/5 (C) 2/5 & 3/5 (D) 1/2 & 1/2

40. In a maths paper there are 3 sections A, B & C. Section A is compulsory. Out of sections B & C a student hasto attempt any one. Passing in the paper means passing in A & passing in B or C. The probability of thestudent passing in A, B & C are p, q & 1/2 respectively. If the probability that the student is successful is 1/2 then :(A) p = q = 1 (B) p = q = 1/2 (C) p = 1, q = 0 (D) p = 1, q = 1/2


6/39



EXERCISE 1 (B)

MORE THAN ONE OPTIONS MAY BE CORRECT

1 A pair of fair dice having six faces numbered from 1 to 6 are thrown once, suppose two events E and F aredefined as

E : Product of the two numbers appearing is divisible by 5.F : At least one of the dice shows up the face one.

Then the vents E and F are(A) mutually exclusive (B) independent(C) neither independent nor mutually exclusive (D) are equiprobable

2 If A & B are two events such that P(B) 1, BC denotes the event complementry to B, then

(A) P A BC = P A P A BP B

( ) ( )

( )

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

(C) P(A) > < P A B according as P A BC > < P(A)

(D) P A BC + P A BC C = 1

3 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 BA = 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

4 If E1and E

2are two events such that P(E

1) = 1/4, P(E

2/E

1) =1/2 and P(E

1/ E

2) = 1/4

(A) then E1and E2are independent(B) E1and E2are exhaustive(C) E2is twice as likely to occur as E1(D) Probabilities of the events E1E2, E1and E2are in G.P.

5 For the 3 events A, B and C, P (atleast one occurring) =4

3, P (atleast two occurring) =

2

1

and P (exactly two occurring) =5

2. Which of the following relations is / areCORRECT?

(A) P(ABC) =10

1

(B) P(AB) + P(BC) + P(CA) =10

7

(C) P(A) + P(B) + P(C) =20

27

(D) )CBA(P)CBA(P)CBA(P =4

1


7/39



6 Each of 2010 boxes in a line contains one red marble, and for 1 k 2010, the box in the kthpositionalso contains k white marbles. A child begins at the first box and successively draws a single marble atrandom from each box in order. He stops when he first draws a red marble. Let P(n) be the probability that

he stops after drawing exactly n marbles. The possible value(s)of n for which P(n) 0 then roots of the quadratic equation are always real.

MATRIX MATCH TYPE

38 Column I Column II

(A) Six different ball's are kept into 3 different boxes randomly so that (P)2

1

no box being empty. Chance that the balls are evenly distributed in the box, is

(B) Two number x and y are chosen at random without replacement from the set (Q)3

1

of first 15 natural numbers. The probability that (x3+ y3) is divisible by 3, is

(C) You have 5 blue cards and 5 red cards. Every morning you choose a card at (R)4

9

random and throw it down a well. The probability that the first card youthrow and the 3rdcard you throw is one of the same colour, is

(D) There are 10 boxes and each box can hold any number of balls. (S)

1

6A man having 5 balls randomly puts one ball in each of the arbitrarychosen five boxes. Then another man having five balls, again puts one ball

in each of the arbitrary chosen five boxes. The probability that there are (T)1

4ball(s) in atleast 8 boxes, is


13/39



39 Column-I Column-II

(A) Let S be a set consisting of first five prime numbers. Suppose A and B (P) 2%are two matrices of order 2 each with distinct entries S. The chancethat the matrix AB has atleast one odd entry, is

(B) Box A has 3 white & 2 red balls, box B has 2 white & 4 red balls. If two (Q) 25%balls are selected at random (without replacement) from A & two more areselected at random from B, the probability that all the four balls are white is

(C) One percent of the population suffers from a certain disease. There is a (R) 50%blood test for this disease, and it is 99% accurate, in other words, theprobability that it gives the correct answer is 0.99, regardles of whether theperson is sick or healthy. A person takes the blood test, and the result says (S) 96%that he has the disease. The probability that he actually has the disease, is

40 Column I Column II

(A) A gambler has one rupee in his pocket. He tosses an unbiased normal (P)21

coin unless either he is ruined or unless the coin has been tossed for amaximum of five times. If for each head he wins a rupee and for each tailhe looses a rupee, then the probability that the gambler is ruined is (Q)

4

5

(B) The probability at least one of the events A and B occur is 0.6.If A and B occur simultaneously with probability 0.2, then

1/2 P(A) P(B) is

(C) 3 firemen X, Y and Z shoot at a common target. The probabilities (R)3

5that 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 the proficiency of Z to hit the target is (S)

16

11


14/39



EXERCISE 1 (C)

INTEGER TYPE

1. When three cards are drawn from a standard 52-card deck, let p is the probability that they are all the

same rank (e.g. all three are kings). Find the value of1

p

.

2. In each of a set of games it is 2 to 1 in favour of the winner of the previous game. If the chance that the player

who wins the first game shall win three at least, of the next four, bep

q, where p and q are coprimes

then find the value of p + q.

3. A coin is tossed n times, If the chance that the head will present itself an odd number of times isp

q, where p

and q are coprimes then find the value of p + q.

4. Counters marked 1, 2, 3 are placed in a bag, and one is withdrawn and replaced. The operation being

repeated three times. If the chance of obtaining a total of 6 bep

q, where p and q are coprimes


5. A normal coin is continued tossing unless a head is obtained for the first time. If the probability thatnumber of tosses needed are at most 3 is p and the probability that number of tosses are even is q,then find the value of 24pq.

6. A and B each throw simultaneously a pair of dice. If the probability that they obtain the same score be

pq

, where p and q are coprimes then find the value of p + q.

7. A is one of the 6 horses entered for a race, and is to be ridden by one of two jockeys B or C. It is 2 to 1 thatB rides A, in which case all the horses are equally likely to win; if C rides A, his chance is trebled.

If the odds against his winning isp

q, where p and q are coprimes then find the value of p + q.

8. A, B are two inaccurate arithmeticians whose chance of solving a given question correctly are (1/8) and(1/12) respectively. They solve a problem and obtained the same result. It is 1000 to 1 against their making

the same mistake. If the chance that the result is correct is

p

q, where p and q are coprimes then find the valueof p + q.

9. A person flips 4 fair coins and discards those which turn up tails. He again flips the remaining coin and then

discards those which turn up tails. If P =n

m(expressed in lowest form) denotes the probability that he

discards atleast 3 coins, find the value of (m + n).


15/39



10. The probability that a person will get an electric contract is5

2and the probability that he will not get plumbing

contract is7

4. If the probability of getting at least one contract is

3

2, then the probability that he will get both

isp


11. Five horses compete in a race. John picks two horses at random and bets on them. Assuming dead heat,

if the probability that John picked the winner isp

q, where p and q are coprimes,


12. An electrical system has open-closed switchesS1, S2and S3as shown.

The switches operate independently of one another and the current will flow from A to B either if S1is closed

or if both S2and S3are closed. Given P(S1) = P(S2) = P(S3) = 1/2. If the probability that the circuit

will work isp


13. A certain team wins with probability 0.7, loses with probability 0.2 and ties with probability 0.1.The team plays three games. If the probability that the team wins at least two of the games, but lose none is

p

100and that the team wins at least one game is

q

1000, wherep, q N , then find the value of

7q

p

14. If the probability of at most two tails or at least two heads in a toss of three coins is

p

q, where p and q arecoprimes then find the value of p + q.

15. Two cubes have their faces painted either red or blue. The first cube has five red faces and one blue face.When the two cubes are rolled simultaneously, the probability that the two top faces show the same colour is1/2, then find the number of red faces on the second cube.

16. There are 6 red balls and 6 green balls in a bag. Five balls are drawn out at random and placed in a red box.The remaining seven balls are put in a green box. If 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, isp

qwherepand qare relatively

prime, then find the value of (p + q).

17. N fair coins are flipped once. The probability that at most 2 of the coins show up as heads is2

1.

Find the value of N.


16/39



18. If the number of functions f : {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} which satisfy theproperty that f (i) + f (j) = 11 for all values of i and j such that i + j = 11, is (625)k, find k.

19. Purse 'A' contains 9 green and 1 red ball, Purse 'B' contains 10 balls all of them being green. 9 balls aredrawn randomly from the purse A and put into the purse B, and then 9 balls randomly transferred from purse

B to the purse A. Ifq

p (in the lowest form) is the chance that the red ball is still there in the

purse 'A', find the value of (p + q).

20. A best of 9 series is to be played between two teams T1and T2, that is, the first team to win 5 games is the

winner. The team T1has a chance of 3

2of winning any game. Let P =

b

abe the probability (expressed as

lowest rational) that exactly 7 games will need to the played to determine a winner, find (a + b).


17/39



EXERCISE 2 (A)

ONLY ONE OPTION IS CORRECT

1. Two red counters, three green counters and 4 blue counters are placed in a row in random order.

The probability that no two blue counters are adjacent is(A)

99

7(B)

198

7(C)

42

5(D) none

2. South African cricket captain lost the toss of a coin 13 times out of 14. The chance of this happening was

(A) 132

7(B) 132

1(C) 142

13(D) 132

13

3. There are ten prizes, five A's, three B's and two C's, placed in identical sealed envelopes for the top tencontestants in a mathematics contest. The prizes are awarded by allowing winners to select an envelope atrandom from those remaining. When the 8thcontestant goes to select the prize, the probability that theremaining three prizes are one A, one B and one C, is

(A) 1/4 (B) 1/3 (C) 1/12 (D) 1/10

4. An unbaised cubic die marked with 1, 2, 2, 3, 3, 3 is rolled 3 times. The probability of getting a total score of4 or 6 is

(A)216

16(B)

216

50(C)

216

60(D) none

5. A bag contains 3 R & 3 G balls and a person draws out 3 at random. He then drops 3 blue balls into the bag& again draws out 3 at random. The chance that the 3 later balls being all of different colours is(A) 15% (B) 20% (C) 27% (D) 40%

6. A biased coin with probability P, 0 < P < 1, of heads is tossed until a head appears for the first time. If theprobability that the number of tosses required is even is 2/5 then the value of P is(A) 1/4 (B) 1/6 (C) 1/3 (D) 1/2

7. Two numbers a and b are selected from the set of natural number then the probability that a2+ b2is divisibleby 5 is

(A)9

25(B)

7

18(C)

36

11(D)

17

81

8. In an examination, one hundred candidates took paper in Physics and Chemistry. Twenty five candidatesfailed in Physics only. Twenty candidates failed in chemistry only. Fifteen failed in both Physics and Chemistry.

A candidate is selected at random. The probability that he failed either in Physics or in Chemistry but not inboth is

(A)20

9(B)

5

3(C)

5

2(D)

20

11

9. When a missile is fired from a ship, the probability that it is intercepted is 1/3. The probability that the missilehits the target, given that it is not intercepted is 3/4. If three missiles are fired independently from the ship, the

probability that all three hits the target, is(A) 1/12 (B) 1/8 (C) 3/8 (D) 3/4


18/39



10. A fair die is tossed repeatidly. Mr. A wins if it is 1 or 2 on two consecutive tosses and Mr. B wins if it is3, 4, 5 or 6 on two consecutive tosses. The probability that A wins if the die is tossed indefinitely, is

(A)3

1(B)

21

5(C)

4

1(D)

5

2

11. A purse contains 2 six sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives.

A die is picked up and rolled. Because of some secret magnetic attraction of the unfair die, there is 75%chance of picking the unfair die and a 25% chance of picking a fair die. The die is rolled and shows up theface 3. The probability that a fair die was picked up, is

(A)7

1(B)

4

1(C)

6

1(D)

24

1

12. Box A contains 3 red and 2 blue marbles while box B contains 2 red and 8 blue marbles. A fair coin is tossed.If the coin turns up heads, a marble is drawn from A, if it turns up tails, a marble is drawn from bag B.The probability that a red marble is chosen, is(A) 1/5 (B) 2/5 (C) 3/5 (D) 1/2

13. The germination of seeds is estimated by a probability of 0.6. The probability that out of 11 sown seedsexactly 5 or 6 will spring is :

(A)11

55

10

6

5

C .(B)

11

556

11

5

)23(C(C) 11C5

11

6

5

(D) none of these

14. An instrument consists of two units. Each unit must function for the instrument to operate. The reliability of thefirst unit is 0.9 & that of the second unit is 0.8. The instrument is tested & fails. The probability that "only thefirst unit failed & the second unit is sound" is :(A) 1/7 (B) 2/7 (C) 3/7 (D) 4/7

15. Lot A consists of 3G and 2D articles. Lot B consists of 4G and 1D article. A new lot C is formed by taking

3 articles from A and 2 from B. The probability that an article chosen at random from C is defective, is(A) 1/3 (B) 2/5 (C) 8/25 (D) none

16. 'A' and 'B' each have a bag that contains one ball of each of the colours blue, green, orange, red and violet.'A' randomly selects one ball from his bag and puts it into B's bag. 'B' then randomly selects one ball from his

bag and puts it into A's bag. The probability that after this process the contents of the two bags are the same,is(A) 1/6 (B) 1/5 (C) 1/3 (D) 1/2

17 A box has four dice in it. Three of them are fair dice but the fourth one has the number five on all of its faces.A die is chosen at random from the box and is rolled three times and shows up the face five on all the threeoccassions. The chance that the die chosen was a rigged die, is

(A)217

216(B)

219

215(C)

219

216(D) none

18. On a Saturday night 20% of all drivers in U.S.A. are under the influence of alcohol. The probability that adriver under the influence of alcohol will have an accident is 0.001. The probability that a sober driver willhave an accident is 0.0001. If a car on a saturday night smashed into a tree, the probability that the driver wasunder the influence of alcohol, is(A) 3/7 (B) 4/7 (C) 5/7 (D) 6/7


19/39


20/39



26. If at least one child in a family with 3 children is a boy then the probability that 2 of the children are boys, is

(A)7

3(B)

7

4(C)

3

1(D)

8

3

27. The probabilities of events, AB, A, B & A B are respectively in A.P. with probability of second termequal to the common difference. Therefore the events A and B are

(A) compatible (B) independent(C) such that one of them must occur (D) such that one is twice as likely as the other

28. From an urn containing six balls, 3 white and 3 black ones, a person selects at random an even number ofballs (all the different ways of drawing an even number of balls are considered equally probable, irrespectiveof their number). Then the probability that there will be the same number of black and white balls among them

(A)5

4(B)

15

11(C)

30

11(D)

5

2

29. Three numbers are chosen at random without replacement from {1, 2, 3,...... , 10}. The probability that theminimum of the chosen numbers is 3 or their maximum is 7 is

(A) 1/2 (B) 1/3 (C) 1/4 (D) 11/40

30. A box contains 100 tickets numbered 1, 2, 3,.... ,100. Two tickets are chosen at random. It is given that themaximum number on the two chosen tickets is not more than 10. The minimum number on them is 5, with

probability

(A)9

1(B)

11

2(C)

19

3(D) none

31. Sixteen players s1 , s2 ,..... , s16play in a tournament. They are divided into eight pairs at random. From eachpair a winner is decided on the basis of a game played between the two players of the pair. Assume that all theplayers are of equal strength. The probability that "exactly one of the two players s1& s2is among the eightwinners" is

(A)15

4(B)

15

7(C)

15

8(D)

15

9

32. The number 'a' is randomly selected from the set {0, 1, 2, 3, ...... 98, 99}. The number 'b' is selected from thesame set. Probability that the number 3a+ 7bhas a digit equal to 8 at the units place, is

(A)16

1(B)

16

2(C)

16

4(D)

16

3

33. A circle of radius 3 is placed at the centre of a circle of radius r > 3 such that the length of a chord of the largercircle tangent to the smaller circle is 8. The probability that point randomly selected from inside of the largercircle lies inside the annular region of the two circles is

(A)4

3(B)

25

9(C)

25

16(D)

3

7

34. On a normal standard die one of the 21 dots from any one of the six faces is removed at random with eachdot equally likely to be chosen. The die is then rolled. The probability that the top face has an odd number ofdots is

(A)11

5(B)

12

5(C)

21

11(D)

11

6


21/39



35. Two boys A and B find the jumble ofnropes lying on the floor. Each takes hold of one loose end randomly.

If the probability that they are both holding the same rope is101

1then the number of ropes is equal to

(A) 101 (B) 100 (C) 51 (D) 50

36. 2 cards are drawn from a well shuffled pack of 52 cards. The probability that one is a heart card & the otheris a king is :(A) 5/52 (B) 1/34 (C) 20/221 (D) 2/52

37. The entries in a two-by-two determinantdc

baare integers that are chosen randomly and independently,,

and, for each entry, the probability that the entry is odd is p. If the probability that the value of the determinantis even is 1/2, then the value of p, is

(A)3

1(B)

2

1(C)

3

2(D)

2

2

38. An experiment resulting in sample space as S = {a, b, c, d, e, f}

with P(a) =16

1, P(b) =

16

1, P(c) =

16

2, P(d) =

16

3, P(e) =

16

4and P(f) =

16

5.

Let three events A, B and C are defined as A = {a, c, e,}, B = {c, d, e, f} and C = {b, c, f}.

If P(A/B) = p1, P(B/C) = p2, P(C/Ac) = p3and P(A

c/C) = p4, then the correct order sequance is(A) p1< p3< p2 < p4 (B) p1< p4< p3 < p2(C) p1< p3< p4 < p2 (D) p3< p1< p4 < p2

39. A random selector can only select one of the nine integers 1, 2, 3,.... , 7, 8, 9 and it makes these selectionswith equal probability. The probability that after n selections (n > 1), the product of n numbers selected will

be divisible by 10 is

(A) 1

nnn

9

4

9

5

9

8(B) 1

nnn

9

4

9

5

9

8

(C) 1

nnn

9

4

9

5

9

8(D) 1

nnn

9

4

9

5

9

8

40. A die is thrown three times. The chance that the highest number shown on the die is 4 is

(A)19

27(B)

1

216(C)

37

216(D)

19

216

41. Two coins look similar, but have different probabilities of falling "head". One is a fair coin, withP (H) = 1/2, but the other is weighed so that P (H) = 4/5. One of the coin is chosen at random and is tossed10 times, let X is the number of heads to appear, and F is the event that the fair coin was drawn.If X = 7 is observed, the probability that the coin was fair, is

(A) 108

10

85

5

(B) 1010

8

45

5

(C) 710

10

85

5

(D) 810

10

85

5


22/39



42. A fair coin is tossed until a head or five tails occur. If the probability that the coin is tossed for a maximum

number of times can be expressed as a rationalq

p(in the lowest form), then (p + q) equals

(A) 17 (B) 34 (C) 19 (D) 18

43. A coin that comes up head with probability p > 0 and tails with probability 1 p > 0 independently on each

flip, is flipped eight times. Suppose the probability of three heads and five tails is equal to25

1of the probability

of five heads and three tails. Let p =n

m, where m and n are relative prime positive integers.

The value of (m + n) equals(A) 9 (B) 11 (C) 13 (D) 15

44. A test is made up of 5 questions, for each question there are 4 possible answers and only one is correct. Forevery right choice you gain1 mark while for each wrong choice there is a penalty of 1 mark.The probability of getting atleast 2 marks answering to every question in a random way is

(A) 1/16 (B) 1/64 (C) 15/64 (D) 1/102445. In the world series the two teams play best of 7. If the probability of either team winning in a single game is

1/2, then the probability that the series will go a full seven games, is

(A)128

35(B)

64

5(C)

32

5(D)

16

5

46. Consider the following three hands of five cards, dealt from an ordinary deck of 52 cardsH1: All four queens and a two.H2: The ace, king, queen, jack and ten, all in the same suit.H3: The ten of hearts, the ten of clubs, and three kings.(A) H1is more likely than H2or H3. (B) H2is more likely than H1or H3.

(C) H3is more likely than H1or H2. (D) All three hands (H1, H2& H3) are equally likely.

47. In a table tennis singles, two players play, and one of them must win. Probability that A beats B is p,B beats C is q and C beats A is r. If B plays with C and then the winner plays with A, then the probability thatA will be the final winner, is(A) qp + (1 q)r (B) (1 r)q + pq (C) pq + (1 q)(1 r) (D) qp + (1 r)

48. A student has managed to enter in Harward University. It was informed that the probability of getting ascholarship is 40%. In case of getting it, the probability of completing the course is 0.8, while in the case ofnot getting the scholarship, the probability is only 0.4. If after some years from now, you meet the student,completing his course from Harward, the probability that he was given the scholarship, is

(A)17

8(B)

7

4(C)

15

8(D)

9

4

49. Rajdhani Express stops at six intermediate stations between Kota and Bombay. Five passengers board atKota. Each passengers can get down at any station till Bombay. The probability that all five passengers willget down at different stations, is

(A) 55

6

6

P(B) 5

56

6

C(C) 5

57

7

P(D) 5

57

7

C


23/39



50. Three teams participate in a tournament in which each team plays both of the other two teams exactly once.The teams are evenly matched so that in each game, each team has a 50% chance of winning the game. Nogame can end in a tie. At the end of the tournament, if one team has more wins than both of the other twoteams, that team is declared the unique winner of the tournament. Otherwise, the tournament ends in a tie.The probability that the tournament ends in a tie, is(A) 1/8 (B) 1/4 (C) 3/8 (D) 1/2

51. A firing squad is composed of three policemen A, B and C who have probabilities 0.6, 0.7 and 0.8 respectivelyof hitting the victim. Only one of the three bullets is live and is allocated at random.If the victim was found to be hit by live bullet, the probability that it was C who had the live round, is

(A)3

1(B)

21

8(C)

21

6(D)

21

9

PASSAGE - 1

Let Bn denotes the event that n fair dice are rolled once with P(Bn) = n2

1, n N.

e.g. P(B1) = 2

1

, P(B2) = 22

1, P(B3) = 32

1, .......... and P(Bn) = n2

1

Hence B1, B2, B3,...........Bnare pairwise mutually exclusive and exhaustive events as n .The event A occurs with atleast one of the event B1, B2,........, Bnand denotes that the sum of the numbersappearing on the dice is S.

52. If even number of dice has been rolled, the probability that S = 4, is

(A) very closed to2

1(B) very closed to

4

1

(C) very closed to8

1(D) very closed to

16

1

53. Probability that greatest number on the dice is 4 if three dice are known to have been rolled, is

(A)216

37(B)

216

64(C)

216

27(D)

216

31

54. If S = 3, P(B2/S) has the value equal to

(A)169

8(B)

169

24(C)

169

72(D)

169

16

PASSAGE - 2

Three bags A, B and C are given, each containing 6 marbles. The first bag A has 5 black marbles and 1 white.

The second bag B has 4 black marbles and 2 white marbles. The third bag C has 3 black marbles and 3 whitemarbles. Two marbles are drawn randomly one from each of two different bags (we do not know whichbags) and found to be one white and the other black. Let P denote the probability that a marble drawn fromthe remaining bag is white.

55. The probability of drawing one white and one black marble from any two of the selected bags, is

(A)54

25(B)

162

25(C)

54

27(D) None


24/39


25/39



EXERCISE 2 (B)

SUBJECTIVE TYPE

1. If m different cards are placed at random and independently into n boxes lying in a straight line

(n > m), find the probability that the cards go into m adjacent boxes.2. Out of 21 tickets consecutively numbered, three are drawn at random. Find the probability that the numbers

on them are in A.P.

3. A has 3 shares in a lottery containing 3 prizes and 9 blanks. B has 2 shares in a lottery containing 2 prizes and6 blanks. Compare their chances of success.

4. A coin is tossed m + n times (m > n). Show that the probability of at least m consecutive heads come

up is m 2n 2

2

.

5. There are four six faced dice such that each of two dice bears the numbers 0, 1, 2, 3, 4 and 5 and the other

two dice are ordinary dice bearing numbers 1, 2, 3, 4, 5 and 6. If all the four dice are thrown, find theprobability that the total of numbers coming up on all the dice is 10.

6. A die is thrown 7 times. What is the probability that an odd number turns up (i) exactly 4 times (ii) atleast4 times.

7. If m things are distributed among a men and b women, show that the probability that the number of things

received by men is odd, is

m m

m

b a b a12 b a

.

8. An artillery target may be either at point A with probability

8

9 or at point B with probability

1

9 . We have 21shells each of which can be fixed either at point A or B. Each shell may hit the target independently of the other

shell with probability1

2. How many shells must be fired at point A to hit the target with maximum probability?

9. Let p be the probability that a man aged x years will die within a year. Let A1, A

2, . . . , A

nbe n men each aged

x years. Find the probability that out of these n men A1will die with in a year and is first to die.

10. Each of three bags A, B, C contains white balls and black balls. A has a1white & b

1black, B has a

2white &

b2black and C has a

3white & b

3black balls. A ball is drawn from a bag and found to be white. What are the

probabilities that the ball is from bag A, B and C.

11. A bar of unit length is broken into three parts x, y and z. Find the probability that a triangle can be formed fromthe resulting parts.

12. There are n students in a class and probability that exactly out of n pass the examination is directly

proportional to 2 0 n .(i) Find out the probability that a student selected at random was passed the examination.(ii) If a selected student has been found to pass the examination then find out the probability that he is theonly student to have passed the examination.


26/39



13. Let A and B be two independent witnesses in a case. The probability that A will speak the truth is x and theprobability that B will speak the truth is y. A and B agree in a certain statement. Show that the probability that

the statement is true isxy

1 x y 2xy .

14. Find the minimum number of tosses of a pair of dice, so that the probability of getting the sum of the numbers

on the dice equal to 7 on atleast one toss, is greater than 0.95.(Given log

102 = 0.3010, log

103 = 0.4771).

15. Two teams A and B play a tournment. The first one to win (n + 1) games, win the series. The probability thatA wins a game is p and that B wins a game is q (no ties). Find the probability that A wins the series.

Hence or otherwise prove that

nn r

r n rr 0

1C . 1

2

.

16. A natural number is chosen at random from the first one hundred natural numbers. The probability that

x 20 x 400

x 30

is

17. If1 3p 1 p

,3 2

and

1 p

2

are the probabilities of three mutually exclusive events,

then the set of all values of p is

18. For independent events A1, . . ., A

n, P(A

i) =

1,

i 1i = 1, 2, . . ., n. Then the probability that none of the

events will occur is

19. A bag contains a large number of white and black marbles in equal proportions. Two samples of 5 marbles

are selected (with replacement) at random. The probability that the first sample contains exactly 1 blackmarble, and the second sample contains exactly 3 black marbles, is

20. If two events A and B are such that P A = 0.3, P(B) = 0.4 and A B = 0.5, then PB

A B

=

21. A is a set containing n elements. A subset P1of A is chosen at random. The set A is reconstructed by

replacing the elements of P1. A subset P

2is again chosen at random. The probability that 1 2P P contains

exactly one element, is

22. The probability that in a group of N (< 365)people, at least two will have the same birthday is

23. Let E and F be two independent events such that P(E) > P(F).

The probability that both E and F happen is1

12and the probability that neither E nor F happens is

1

2,

then find P(E) & P(F).

24. A draw two cards at random from a pack of 52 cards. After returning them to the pack and shuffling it, Bdraws two cards at random. The probability that there is exactly one common card, is


27/39



25. A company has two plants to manufacture televisions. Plant I manufacture 70% of televisions and plant IImanufacture 30%. At plant I, 80% of the televisions are rated as of standard quality and at plant II, 90%of the televisions are rated as of standard quality. A television is chosen at random and is found to be ofstandard quality. The probability that it has come from plant II is

26. x1, x2, x3, . . . , x50are fifty real numbers such that xr< xr + 1for r = 1, 2, 3, . . ., 49.Five numbers out of these are picked up at random. The probability that the five numbers

have x20

as the middle number is

27. The probability that a man can hit a target is3

4. He tries 5 times. The probability that he will hit the target

at least three times is

28. A die is thrown 7 times. The chance that an odd number turns up at least 4 times, is


28/39



EXERCISE 3

SUBJECTIVE TYPE

1. The probabilities that three men hit a target are, respectively, 0.3, 0.5 and 0.4. Each fires once at the

target. (As usual, assume that the three events that each hits the target are independent)(a) Find the probability that they all : (i) hit the target ; (ii) miss the target(b) Find the probability that the target is hit : (i) at least once, (ii) exactly once.(c) If only one hits the target, what is the probability that it was the first man?

2. Let A & B be two events defined on a sample space . Given P(A) = 0.4 ; P(B) = 0.80

and P B/A = 0.10. Then find ; (i) P BA & P BABA .

3. Three shots are fired independently at a target in succession. The probabilities that the target is hit in thefirst shot is 1/2, in the second 2/3 and in the third shot is 3/4. In case of exactly one hit, the probabilityof destroying the target is 1/3 and in the case of exactly two hits, 7/11 and in the case of three hits is1.0. Find the probability of destroying the target in three shots.

4. In a game of chance each player throws two unbiased dice and scores the difference between the largerand smaller number which arise . Two players compete and one or the other wins if and only if he scoresatleast 4 more than his opponent . Find the probability that neither player wins.

5. A certain drug , manufactured by a Company is tested chemically for its toxic nature. Let the event"THE DRUG IS TOXIC" be denoted by H & the event "THECHEMICALTESTREVEALSTHATTHEDRUGIS

TOXIC" be denoted by S. Let P(H) = a, P H/S =P H/S = 1 a. Then show that the probability thatthe drug is not toxic given that the chemical test reveals that it is toxic, is free from a.

6. A plane is landing. If the weather is favourable, the pilot landing the plane can see the runway. In this casethe probability of a safe landing is p1. If there is a low cloud ceiling, the pilot has to make a blind landing by

instruments. The reliability (the probability of failure free functioning) of the instruments needed for a blindlanding is P. If the blind landing instruments function normally, the plane makes a safe landing with the sameprobability p1as in the case of a visual landing. If the blind landing instruments fail, then the pilot may makea safe landing with probability p2 < p1. Compute the probability of a safe landing if it is known that in K

percent of the cases there is a low cloud ceiling. Also find the probability that the pilot used the blindlanding instrument, if the plane landed safely.

7. In a multiple choice question there are five alternative answers of which one or more than one is correct.A candidate will get marks on the question only if he ticks the correct answers.The candidate ticks the answers at random. If the probability of the candidate getting marks on the questionis to be greater than or equal to 1/3, then find the least number of chances he should be allowed.

8. n people are asked a question successively in a random order & exactly 2 of the n people knowthe answer :

(a) If n >5, find the probability that the first four of those asked do not know the answer.

(b) Show that the probability that the rthperson asked is the first person to know the answer is :

2

1

( )

( )

n r

n n

, if 1 < r < n .


29/39



9. A box contains three coins two of them are fair and one two headed. A coin is selected at random andtossed. If the head appears the coin is tossed again, if a tail appears, then another coin is selected from theremaining coins and tossed.(i) Find the probability that head appears twice.(ii) If the same coin is tossed twice, find the probability that it is two headed coin.(iii) Find the probability that tail appears twice.

10. The ratio of the number of trucks along a highway, on which a petrol pump is located, to the number of carsrunning along the same highway is 3 : 2. It is known that an average of one truck in thirty trucks and two carsin fifty cars stop at the petrol pump to be filled up with the fuel. If a vehicle stops at the petrol pump to be filledup with the fuel, find the probability that it is a car.

11. A batch of fifty radio sets was purchased from three different companies A, B and C. Eighteen of them weremanufactured by A, twenty of them by B and the rest were manufactured by C.The companies A and C produce excellent quality radio sets with probability equal to 0.9 ; B produces thesame with the probability equal to 0.6.What is the probability of the event that the excellent quality radio set chosen at random is manufactured bythe company B?

12. There are 6 red balls & 8 green balls in a bag . 5 balls are drawn out at random & placed in a red box ; theremaining 9 balls are put in a green box . What is the probability that the number of red balls in the green box plusthe number of green balls in the red box is not a prime number?

13. Two cards are randomly drawn from a well shuffled pack of 52 playing cards, without replacement. Let x bethe first number and y be the second number.Suppose that Ace is denoted by the number 1; Jack is denoted by the number 11 ; Queen is denoted by thenumber 12 ; King is denoted by the number 13.Find the probability that x and y satisfy log3(x + y) log3x log3y + 1 = 0.

14.

(a) Two numbers x & y are chosen at random from the set {1,2,3,4,....3n}. Find the probability thatx y is divisible by 3 .

(b) If two whole numbers x and y are randomly selected from the set of natural numbers, then find the probabilitythat x3+ y3is divisible by 8.

15. A hunters chance of shooting an animal at a distance r is 2

2

r

a(r > a) . He fires when r = 2a & if he

misses he reloads & fires when r = 3a, 4a, ..... If he misses at a distance na, the animal escapes. Find theodds against the hunter.

16. An unbiased normal coin is tossed 'n' times. Let :

E1 : event that both Heads and Tails are present in 'n' tosses.E2 : event that the coin shows up Heads atmost once.Find the value of 'n' for which E1& E2are independent.

17. There are two lots of identical articles with different amount of standard and defective articles. There are Narticles in the first lot, n of which are defective and M articles in the second lot, m of which are defective. Karticles are selected from the first lot and L articles from the second and a new lot results. Find the probabilitythat an article selected at random from the new lot is defective.


30/39



18. The probability of the simultaneous occurrence of two events, A and B, isx ; the probability of the

occurrence of exactly one of A, B isy. Evaluate P(AB) and P(A BA B).

19. A fair coin is tossed n > 7 times. Letxdenote the number of occurrences of head. If P(x = 4), P(x = 5), P(x = 6) are in A.P. , find n.

20. The digits 1, 2, ... , 9 are randomly written to form a nine digit number. What is the chance that the

number is divisible by 36 ?

21. An urn contains nblack balls and 2nwhite balls (n 1). A ball is drawn at random. If it is black , then it is put back ; if it is white, then it is thrown away. What is the probability of drawing a white ball in the second draw ?

22. Three persons A , B , and C , throw a die (in that order). Whosoever throws 6 first is declared the winner. What is the proportion of the probabilities of their victories ?

23. The co-efficients, band c, in the quadratic equationx2 +bx +c = 0, are determined by tossing a die twice. What is the probability that the resulting equation has real roots ?

24. Let the events A1 , ... , Anbe mutually independent and let P(Ai )=p, for all i . What is the probability that

(i) atleast one of the events occurs ?(ii) atleast mof the events occur ?(iii) exactlymof the events occur ?

25. An elevator carries two persons and stops at three floors. Find the probability that(i) The two persons get off at different floors ;

(ii) exactly one person gets off at the 1stfloor.

26. S = {1, 2, 3, 4, 5}. Two subsets , A and B , are chosen randomly from the set of all subsets of S

(repetition allowed.) Find the probability that A and B have the same cardinality.

27. A rod is broken into two parts. What is the chance that the smaller part is more thanrd

3

1of the rod ?

28. An integer, n, is chosen at random from the integers 1 through 104.

What is the probability that the digit 8 appears atleast once in n?

29. : whitew : redr

w+ r= n

Select two balls, simultaneously and randomly , from an urn containingwwhite balls and rred balls.

(All the balls are distinguishable.) If the probability that they have different colours is 1/2, show that nis an integer square.

30. The probability that a hunter shoots down a fox satisfies the inverse square law. When the fox is 100 m

from the hunter, the hunter takes aim. If he misses, he reloads his rifle. In the mean while, the fox has

receeded 50m. The hunter again takes aim. If he misses, he reloads his rifle for the last time. Now the

fox is 200 m away, and the hunter makes his final attempt. Given that the probability of the hunter

shooting down the fox at a distance of 100 m is2

1, find the probability of a successful hit.


31/39



31. A person correctly recalls all but the last (units) digit of a telephone number.

?

recalled correctly

Speaking from a public booth, he has money only for two telephone calls. He chooses the

forgotten digit randomly. What is the probability that he calls the right person ?

32. Integers from 1 through 100, written on separate slips of paper, are kept in a box. A draws one slip

randomly and replaces it. B then draws one slip randomly.

What is the probability that B draws a bigger number than A ?

33. mand nare integers chosen randomly from the set {1, 2, ... , 100}, repetition allowed.

What is the probability that 5 divides 3m+ 3n?

34.

: black ballsn

: white ballsn

A person draws balls, one-by-one, randomly and without replacement, until all the 2nballs are

withdrawn.

Let P(n) be the probability that the sequence of balls withdrawn shows colours alternately, starting with

any colour first. (Balls of the same colour are distinguishable.)

Show that P(n) =)!(2)!(2

2

n

n .

35. Two hunters, A and B , have equal skill at shooting down a duck successfully. (The probability of a

successful shot, for either of them , is2

1)

A shoots at 50 ducks and B shoots at 51 ducks.

Find the probability that B bags more ducks than A.

36. A fair die is tossed until the digit on the top face is found to be 1 or 6. Given that 1 did not appear in the

first two throws, find the probability that atleast three throws are necessary.

37. wand bare positive integers and n= w+ b.

Prove thatw

n

wwn

wn

nn

wnwn

n

wn

1)(...1)(

2.1...)(...

2)1)((

1)()(

11

(Hint : use a probability model.)

38. A fair die is tossedntimes. Show that the probability of getting an even number of sixes is

n

3

21

2

1.


32/39



39. The probability of choosing an integer , k, randomly from amongst the integers 1, 2, ... , 2n,

is proportional to log k.

Show that the conditional probability of choosing the integer 2, given that an even integer is chosen, is

)!(loglog 2

log 2

nn .

40. A bag contains 5 white and 7 black , balls. Two balls are randomly drawn. What is the chance that one is white and the other black? (i) the balls are simultaneously drawn; (ii) the balls are drawn successively and without replacement; (iii) the balls are drawn successively and with replacement.

41. npersons sit at a round table randomly. Find the probability that two particular persons, A and B, are

not next to each other.

42. A bag contains 6ntickets marked with the numbers 0, 1, 2, ... , 6n 1. Three tickets are drawn randomly and simultaneously. What is the chance that the sum of the numbers on them is 6n?

43. A permutation of 5 digits from the set {1, 2, 3, 4, 5} where each digit is used exactly once, is chosen

randomly. Letq

pexpressed as rational in lowest form be the probability that the chosen permutation changes

from increasing to decreasing, or decreasing to increasing at most once e.g. the stringslike 1 2 3 4 5, 5 4 3 2 1, 1 2 5 4 3 and 5 3 2 1 4 are acceptable but strings like 1 3 2 4 5 or 5 3 2 4 1 are not,find (p + q).

44. A bag contains N balls, some of which are white, the others are black, white being more in number thanblack. Two balls are drawn at random from the bag, without replacement. It is found that the probability thatthe two balls are of the same colour is the same as the probability that they are of different colour.

It is given that 180 < N < 220.If K denotes the number of white balls, find the exact value of (K + N).

45. There are two packs A and B of 52 playing cards. All the four aces from the pack A are removed whereasfrom the pack B, one ace, one king, one queen and one jack is removed. One of these two packs is slectedrandomly and two cards are drawn simultaneously from it, and found to be a pair (i.e. both have same rank

e.g. two 9's or two king etc). If Q =n

m(expressed in lowest form) denotes the probability that the pack AA

was selected, find (m + n).

46. In a tournament, team X, plays with each of the 6 other teams once. For each match the probabilities of a win,

draw and loss are equal. If the probability that the team X, finishes with more wins than losses, can beexpressed as rational

q

pin their lowest form, then find (p + q).

47. There are two purses, one containing 3 Red balls and 1 Green ball, and the other containing 3 green balls and1 red ball. One ball is taken from one of the purses (it is not known which) and dropped into the other, andthen on drawing a ball from each purse, they were found to be 2 green balls. If P = (n)1,is the probability that two more balls drawn, one from each bag are both green, find n N.


33/39



48. What is the probability that in a group of(i) 2 people, both will have the same date of birth.(ii) 3 people, at least 2 will have the same date of birth.Assume the year to be ordinarry consisting of 365 days.

49. A purse containsncoins of unknown value, a coin drawn at random is found to be a rupee, what is the chancethat it is the only rupee in the purse? Assume all numbers of rupee coins in the purse to be equally likely.

50. The contents of 3 bags w.r.t green and red marbles is as given in the table shown.

31C22B13ARGBagA child randomly selects one of the bags, and draws a marble from it and retains it.

If the marble is green, the child draws the second marbles randomly from one of the tworemaining bags. If the first marble drawn is red the child draws one more marble from the

same bag. The probability that the second drawn marble is green is expressed asn

m

(where m and n are coprime). Find the value of (m + n).


34/39



WIND OW TO I. I.T. - JEE

Q.1 Two cards are drawn at random from a pack of playing cards. Find the probability that one card is a heartand the other is an ace. [REE ' 2001]

Q.2

(a) An urn contains 'm' white and 'n' black balls. A ball is drawn at random and is put back into the urn along withK 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.

(b) An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6 is thrown n times and the list of n numbers showing upis noted. What is the probability that among the numbers 1, 2, 3, 4, 5, 6, only three numbers appear in the list.

[JEE ' 2001]

Q.3 A box contains N coins, m of which are fair and the rest are biased. The probability of getting a head when afair 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 randomand is tossed twice. The first time it shows head and the second time it shows tail. What is the probability thatthe coin drawn is fair? [JEE ' 2002]

Q.4(a) A person takes three tests in succession. The probability of his passing the first test is p, that of his passing

each successive test is p or p/2 according as he passes or fails in the preceding one. He gets selectedprovided he passes at least two tests. Determine the probability that the person is selected.

(b) In a combat, A targets B, and both B and C target A. The probabilities of A, B, C hitting their targets are2/3 , 1/2 and 1/3 respectively. They shoot simultaneously and A is hit. Find the probability that B hits his targetwhereas C does not. [JEE' 2003]

Q.5(a) Three distinct numbers are selected from first 100 natural numbers. The probability that all the three numbers

are divisible by 2 and 3 is

(A)25

4(B)

35

4(C)

55

4(D)

1155

4

(b) If A and B are independent events, prove that P (AB) P (A' B') P (C), where C is an event definedthat exactly one of A or B occurs.

(c) A bag contains 12 red balls and 6 white balls. Six balls are drawn one by one without replacement of whichatleast 4 balls are white. Find the probability that in the next two draws exactly one white ball is drawn (leavethe answer in terms of nCr). [JEE 2004]

Q.6(a) A six faced fair dice is thrown until 1 comes, then the probability that 1 comes in even number of trials is

(A) 5/11 (B) 5/6 (C) 6/11 (D) 1/6[JEE 2005]

(b) A person goes to office either by car, scooter, bus or train the probability of which being7

1and

7

2,

7

3,

7

1

respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is9

1and

9

4,

9

1,

9

2

respectively. Given that he reached office in time, then what is the probability that he travelled by a car.[JEE 2005]


35/39



Q.7 Comprehension (3 questions)There arenurns each containingn+ 1 balls such that the ithurn contains iwhite balls and (n + 1 i) red balls.Let uibe the event of selecting i

thurn, i= 1, 2, 3, ......, nand wdenotes the event of getting a white ball.

(a) If P(ui) i where i = 1, 2, 3,....., n then )w(PLimn

is equal to

(A) 1 (B) 2/3 (C) 3/4 (D) 1/4

(b) If P(ui) = c, where c is a constant then P(un/w) is equal to

(A)1n

2

(B)

1n

1

(C)

1n

n

(D)

2

1

(c) If nis even and E denotes the event of choosing even numbered urn (n

1)u(P i

), then the value of EwP ,

is

(A)1n2

2n

(B) 1n2

2n

(C)

1n

n

(B)

1n

1

[JEE 2006]

Q.8

(a) One Indian and four American men and their wives are to be seated randomly around a circular table. Thenthe conditional probability that the Indian man is seated adjacent to his wife given that each American man isseated adjacent to his wife is(A) 1/2 (B) 1/3 (C) 2/5 (D) 1/5

(b) Let Ecdenote the complement of an event E. Let E, F, G be pairwise independent events with P(G) > 0 andP(E F G) = 0. Then P(EcFc| G) equals(A) P(Ec) + P(Fc) (B) P(Ec) P(Fc) (C) P(Ec) P(F) (D) P(E) P(Fc)

(c) Let H1, H2, ....... , Hnbe mutually exclusive and exhaustive events with P(Hi) > 0, i = 1, 2, ...., n. Let E be anyother event with 0 < P(E) < 1.Statement-1: P(Hi/ E) > P(E / Hi) P(Hi) for i = 1, 2, ....., n.

Statement-2:

n

1ii )H(P = 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.(D) Statement-1 is false, statement-2 is true. [JEE 2007]

Q.9(a) An experiment has 10 equally likely outcomes. Let A and B be two 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(A) 2, 4 or 8 (B) 3, 6, or 9 (C) 4 or 8 (D) 5 or 10

(b) Consider the system of equationsax + 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 is8

3.

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-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 [JEE 2008]


36/39



Q.10 Comprehension (3 questions)A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.

(a) The probability that X = 3 equals

(A) 216

25(B) 36

25(C) 36

5(D) 216

125

(b) The probability that X3 equals

(A)216

125(B)

36

25(C)

36

5(D)

216

25

(c) The conditional probability that X 6 given X > 3 equals

(A)216

125(B)

216

25(C)

36

5(D)

36

25[JEE 2009]

Q.11(a) Let be a complex cube root of unity with 1. A fair die is thrown three times. If r1, r2and r3are the

numbers obtained on the die, then the probability that 0321 rrr is

(A)18

1(B)

9

1(C)

9

2(D)

36

1

(b) A signal which can be green or red with probability5

4and

5

1respectively, is received by station A and then

transmitted to station B. The probability of each station receiving the signal correctly is4

3.

If the signal received at station B is green, then the probability that the original signal was green is

(A)5

3(B)

7

6(C)

23

20(D)

20

9[JEE 2010]


37/39



ANSWER KEY

EXERCISE 1 (A)

1. A 2. A 3. A 4. C 5. A6. A 7. C 8. D 9. C 10. D

11. A 12. A 13. C 14. B 15. C16. D 17. D 18. D 19. B 20. A21. B 22. B 23. C 24. D 25. A26. D 27. B 28. D 29. B 30. A31. A 32. C 33. B 34. A 35. B36. D 37. A 38. A 39. D 40. D

EXERCISE 1 (B)

1. C,D 2. A,B,C,D 3. A,B,C 4. A,C,D5. A,B,C,D 6. B,C,D 7. A,B,D 8. A,B,C,D9. A,C 10. A,D 11. A,B,D 12. A,B,C,D13. A,C 14. A,B,C 15. A,D 16. A,B,C,D17. A,B,C 18. A,C 19. B 20. A21. D 22. B 23. A 24. C25. C 26. B 27. D 28. B29. C 30. B 31. A 32. B33. B 34. A 35. B 36. C37. A 38. (A) S, (B) Q, (C) R; (D) P

39. (A) S; (B) P; (C) R 40. (A) S ; (B) R; (C) P

EXERCISE 1 (C)

1. 425 2. 13 3. 3 4. 34 5. 7

6. 721 7. 18 8. 27 9. 445 10. 12211. 7 12. 13 13. 139 14. 15 15. 316. 37 17. 5 18. 160 19. 29 20. 101

EXERCISE 2 (A)

1. C 2. A 3. A 4. B 5. C6. C 7. A 8. A 9. B 10. B11. A 12. B 13. A 14. B 15. C16. C 17. C 18. C 19. C 20. C21. A 22. A 23. C 24. B 25. A26. A 27. D 28. B 29. D 30. A31. C 32. D 33. C 34. C 35. C36. D 37. D 38. C 39. A 40. C

41. D 42. A 43. B 44. B 45. D


38/39



46. D 47. C 48. B 49. C 50. B51. B 52 D 53 A 54 B 55 A56 B 57 B 58 D 59 B 60 C61 D

EXERCISE 2 (B)

1.

m

m! n m 1

n

2. 10/133 3. 952 : 715 5.

125

1296

6. (i)35

128 (ii)

1

28. 12 9.

n11 1 p

n

10.31 2

1 2 3 1 2 3 1 2 3

pp p, &

p p p p p p p p p 11. 1/4

12. (i)

3 n 1

2 2n 1

(ii) n

14. 17 15. P(A) =n

n r r nn

r 0

C .q .p

16.7

2517 . 18.

1

n 1 19.25

512

20.1

421 . n

3n

422.

365 !1

365 N ! 365 !

23. P(E) =1

3, P(F) =

1

424.

50

66325.

27

83

26.

30 192 250

5

C C

C

27 .

459

51228.

1

2

EXERCISE 3

1. (a) 6% , 21% ; (b) 79% , 44% , (c) 9/44 20.45% 2. (i)0.82,(ii)0.76

3.5

84. 74/81 5. P H S/ = 1/2

6. P E K

p K

P p P p( ) ( ) [ ( ) ] 1100 100

11 1 2 ; P(H 2/A) =

]p)P1(pP[

100

Kp

100

K1

]p)P1(pP[100

K

211

21

7. 11 8. (a)( ) ( )

(( )

n n

n n

4 5

19. 1/2, 1/2, 1/12

10.4

911.

13

412.

213

1001

13.663

1114. (a)

( )

( )

5 3

9 3

n

n

(b) 16

515. (n + 1) : (n 1)


39/39


16. n = 3 17. )LK(NM

NmLMnK

18. 1 x y ,yx

x

19 . 14 20. 9

2 21.

1)(39

818

n

n

22 . 36 : 30 : 25 23. 36

19

24 . (i) 1(1p)n (ii) nCmpm (1p)n m + nC

m + 1pm + 1 (1p)n m 1 + ... + nC

npn ;

(iii) nCmpm(1p)n m.

25 . 3

2,9

4 26.

105

10

2

C 27.3

1

28 . 1

4

10

9

30.

144

95 31.

5

1

32 . 200

99 33.

4

1 35.

2

1

36 . 25

16 40. i]

66

35 ii]

66

35 iii]

72

35 41.

1

3

n

n

42 . 2)(61)(6

3

nn

n

43. 5 44 . 301 45. 35

46 . 341 47. 16 48 . (i)365

1, (ii) 1 2)365(

363364

49 .)1n(n

2

50. 217

WIND OW TO I. I.T. - JEE

1.1

262. (a)

m

m n; (b)

6 3 3 3 2 36

C n n

n

.

3. 4. (a) p2 (2 p); (b) 1/2

5. (a) D, (c) 660125611246212212

11

111

56

112

12

110

46

212

CCCCCCC

CCCCCCCC

6. (a) A, (b) 7

17. (a) B, (b) A, (c) B

8. (a)

C; (b) C; (c) D 9. (a) D, (b) B

10. (a) A, (b) B, (c) D 11. (a) C; (b) C

exercise (all)

Documents