Probability Jee Adv

7/24/2019 Probability Jee Adv

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MATHEMATICS PROBABILITYM.R.S FIITJEE COIMBATORE

M.R.S FIITJEE COIMBATORE

CONCEPTUAL QUESTIONS

Multip le choice Questions wi th One Correct Answer

1. You are given a box with 20 cards in it. 10 of these cards have the letter Iprinted on them. The

other ten have the letter T printed on them. if you pick up 3 cards at random and keep them in thesame order, the probability of making the word IIT is

(A)9

80(B)

1

8

(C)4

27(D

5

38

2. In a box containing 100 bulbs, 10 are defective. If 5 bulbs are drawn one after another withreplacement, then the probability that none is defective, is

(A) 510 (B)

51

2

(C)

59

10

(D9

10

3. In suffling a pack of cards 3 are accidentally dropped, then the chance that missing card shouldbe of different suits is

(A)169

425(B)

261

425

(C) 104425

(D) 103425

.

4. A five digit number is formed by the digits 1, 2, 3, 4, 5, 6 and 8. The probability that the number haseven digit at both ends is

(A)2

7(B)

3

7

(C)4

7(D

5

7

5. Two bags contain 3 white, 2 black and 2 white, 4 black balls respectively. A ball is chosen atrandom then the probability of its being black is

(A)8

15(B)

2

3

(C)6

4(D)

7

15.


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PROBABILITY MATHEMATICSM.R.S FIITJEE COIMBATORE


6. Two events A and B have probabilities 0.25 and 0.50 respectively. The probability that bothA and B occur simultaneously is 0.14. Then the probability that neither A nor B occurs is(A) 0.39 (B) 0.25(C) 0.11 (D) 0.27

7. Two fair dice are tossed. Let x be the event that the first die shows an even number and y be the

event that the second die shows an odd number. The two events x and y are :(A) mutually exclusive (B) independent and mutually exclusive(C) dependent (D) none of these

8. The probability that an event A happens in one trial of an experiments is 0.4. Three independenttrials of the experiment are performed. The probability that the event A happens at least once is(A) 0.936 (B) 0.784(C) 0.904 (D) 0.90

9. A team of 8 couples, (husband and wife) attend a lucky draw in which 4 persons picked up for aprize. The probability that there is at least one couple is(A) 11/39 (B) 15/39(C) 14/39 (D) 12/39

10. Five boys and three girls are seated at random in a row. The probability that no boy sits betweentwo girls is

(A)1

56(B)

1

8

(C)3

28(D)

3

56.

11. In a convex hexagon two diagonals are drawn at random. The probability that the diagonals intersectat an interior point of the hexagon is

(A)

5

12 (B)

7

12

(C)2

5(D)

1

12.

12. 4 five-rupee coins, 3 two-rupee coins and 2 one-rupee coins are stacked together in column atrandom. The probability that the coins of the same denomination are consecutive is

(A)13

9!(B)

1

210

(C)1

35(D)

1

5.

13. Two cards are drawn at random from a pack of 52 cards. The probability of getting at least a spade

and an ace is

(A)1

34(B)

8

221

(C)1

26(D)

2

51.


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SINGLE CORRECT ANSWER TYPE

LEVEL - I

Axi omati c Defini ti on o f Probab il it y

1. The probability that a card drawn from a pack of 52 cards will be a diamond or king is

(A)4

52(B)

4

13

(C)1

52(D

2

13


2. Three mangoes and three apples are in box. If two fruits are chosen at random, the probability thatone is a mango and the other is an apple is

(A)2

3(B)

3

5

(C)1

3(D)

1

5.


3. From 4 childreen, 2 women and 4 men, 4 are selected. The probability that there are exactly 2children among the selected is

(A)11

21(B)

9

21

(C)10

21(D)

8

25.

Conditional Probability

4. If A and B are two events such that P(A) > 0, and 1)B(P , then )B/A(P is equal to

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

(C))B(P

)A(P(D)

1 P(A B)

P(B)

Set Theoretic Princip les

5. If A and B are two events, then which is not correct-(A) P(AB

c) = P(A) P(AB)

(B) P(ABc) + P(A

cB) = P(A B) P (AB)

(C) P(AB) = P(A) + P(B) P (A B)(D) If A and B are independent events then P(AB) = 0

Binomial Distribution

6. India and Pakistan play a 5 match test series of hockey, the probability that India wins at leastthree matches is-

(A)2

1(B)

5

3

(C)5

4(D)

1

3


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7. A bag contains 3 red and 3 white balls. Two balls are drawn one by one. The probability that theyare of different colours is

(A)3

10(B)

2

5

(C) 35

(D) 15

.


8. A bag contains 5 brown and 4 white socks. A man pulls out 2 sockes. The probability that they areof the same colour is

(A)5

108(B)

1

6

(C)5

18(D)

4

9


9. Two dice are thrown together. The probability of getting the sum of digits as a multiple of 4 is

(A)1

9(B)

1

3

(C)1

4(D)

5

9

Independent Event

10. If P(A B) =3

2, P(A B) =

6

1and P(A) =

3

1then-

(A) A and B are independent events (B) A and B are disjoint events(C) A and B are dependent events (D) none of the above


11. If a person throws 3 dice, the probability of getting sum of digit exactly 15 is-

(A)72

5(B)

108

5

(C)36

5(D)

1

72.


12. If the letters of INTERMEDIATE are arranged, then the probability no two Es occur together is-

(A)11

6(B)

11

5

(C)11

2(D)

3

11


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Set Theoretic Princip les

13. Two dice are thrown together. The probability that atleast one will show its digit greater than 3 is

(A)1

4(B)

3

4

(C)1

2(D)

1

8


14. One number is selected at random from first two hundred positive integers. The probability that it isdivisible by 6 or 8 is

(A)1

3(B)

2

3

(C)3

4(D)

1

4Axi omati c Defini ti on o f Probab il it y

15. Four persons are selected at random out of 3 men, 2 women and 4 children. The probability that

there are exactly 2 children in the selection is

(A)11

21(B)

9

21

(C)10

21(D)

12

21.

Set Theoretic Principle

16. The probability that an anti aircraft gun can hit an enemy plane at the first, second and third shotare 0.6, 0.7 and 0.1 respectively. The probability that the gun hits the plane is(A) 0.108 (B) 0.892(C) 0.14 (D) 0.91


17. The probability that atleast one of the events A and B happens is 0.6. If probability of their simultaneous

happening is 0.2, then P A P B is(A) 0.4 (B) 0.8(C) 1.2 (D) 1.4


18. A bag contains 5 white and 3 black balls. Two balls are drawn at random. The probability that oneball is white and other is black will be

(A)15

28(B)

2

7

(C)8

28(D)

1

7

Total Probability Theorem

19. A purse contains 4 copper and 3 silver coins and another purse contains 6 copper and 2 silvercoins. One coin is drawn from any one of these two purses. The probability that it is a copper coinis

(A)4

7(B)

3

4

(C)2

7(D)

37

56


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20. Three letters are written to different persons, and addresses on three envelopes are also written.Without looking at the addresses, the probability that the letters go into right envelopes is

(A)1

27(B)

1

6

(C)1

9(D)

1

3.


21. A number is chosen at random among the first 120 natural numbers. The probability of the numberchosen being a multiple of 5 or 15 is

(A)1

5(B)

1

8

(C)1

6(D)

1

4.


22. One hundred identical coins, each with probability, p, of showing up heads are tossed once. If0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51coins, then the value of p is

(A)2

1(B)

101

49

(C)101

50(D)

101

51


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

that

x 20 x 40

0x 30

is

(A)1

50(B)

3

50

(C)3

25(D)

7

25Set Theoretic Principle

24. If A and B are two events such that5

P (A B)6

, P 1 3

A , P(B)3 4

, then A and B are

(A) mutually exclusive (B) dependent

(C) independent (D) none of theseSet Theoretic Principle

25. If1 3p 1 p

,3 2

and

1 2p

2

are the probabilit ies of three mutually exclusive events, then the set

of all values of p is

(A) (B)1 1

,2 3

(C) [0, 1] (D)1 2

,3 3

.


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LEVEL - II


26. 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

(A) n/(n + 1) (B) (n 1)/(n + 1)(C) 1/(n + 1) (D) n + ( 1/(n+1) )

Total Probability Theorem

27. A bag contains a large number of white and black marbles in equal proportions. Two samples of 5marbles are selected (with replacement) at random. The probability that the first sample containsexactly 1 black marble, and the second sample contains exactly 3 black marbles, is

(A)25

512(B)

15

32

(C)15

1024(D)

35

256


28. 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

=

(A)1

4(B)

1

5

(C)3

5(D)

2

5


29. A is a set containing n elements. A subset P1of A is chosen at random. The set A is reconstructedby replacing the elements of P

1. A subset P

2is again chosen at random. The probability that

1 2P P contains exactly one element, is

(A) n3n

4(B)

n

n

3

4

(C)3

4(D) n

3

4.


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

(A)

365 !

1365 N ! 365 !

(B)

N365 365 !

365 N ! 1

(C) 1

N365 365 !

365 N !(D)

N

365 !1

365 N ! 365


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31. 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

(A) P(E) =

1

3 , P(F) =1

4 (B) P(E) =1

2 , P(F) =1

6

(C) P(E) = 1, P(F) =1

12(D)

1 1P E , P F

3 2 .


32. A drawn two cards at random from a pack of 52 cards. After returning them to the pack andshuffling it, B draws two cards at random. The probability that there is exactly one common card,is

(A)5

546

(B)

50

663

(C)25

663(D)

25

273.

Bayes Theorem33. A company has two plants to manufacture televisions. Plant I manufacture 70% of televisions and

plant II manufacture 30%. At plant I, 80% of the televisions are rated as of standard quality and atplant II, 90% of the televisions are rated as of standard quality. A television is chosen at randomand is found to be of standard quality. The probability that it has come from plant II is

(A)17

50(B)

27

83

(C)3

5(D)

9

83.


34. x1, x

2, x

3, . . . , x

50are fifty real numbers such that x

r< x

r + 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 middlenumber is

(A)

20 302 2

505

C C

C

(B)

30 192 250

5

C C

C

(C)

19 312 3

505

C C

C

(D)

0 302 2

495

C C

C

.

Axi omati c Defini ti on o f Probab il it y35. If the integers m and n are chosen at random from 1 to 100, then the probability that a number of

the form 7n+ 7mis divisible by 5 equals

(A)1

4(B)

1

2

(C)1

8(D)

1

3.


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36. 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

(A)

291

364 (B)371

461

(C)471

502(D)

459

512Binomial Distribution

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

(A)1

4(B)

1

2

(C)1

8(D)

3

4.


38. Fifteen coupons are numbered 1,2,........15, respectively. Seven coupons are selected at randomone at a time with replacement. The probability that the largest number appearing on a selectedcoupon is 9, is

(A)

6

16

9

(B)

7

15

8

(C)

7

5

3

(D)

77

15

.


39. India plays two matches each with West Indies and Australia. In any match the probabilities ofIndia getting points 0, 1 and 2 are 0. 45, 0.05 and 0.50 respectively. Assuming that the outcomesare independent, the probability of India getting at least 7 points is(A) 0.8750 (B) 0.0875(C) 0.0625 (D) 0.0250


40. Two numbers are selected at random from 40 consecutive natural numbers. The probability that thesum of the selected numbers is odd will be

(A)14

29(B)

20

39

(C)1

2

(D)19

39

.


41. If all letters of the word MISSISSIPPI are rearranged then the probability that all S come togetherwill be

(A)1

165(B)

4

165

(C)8

165(D)

2

165.


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42. If the letter of the word SUCCESS are arranged, then the probability that similar letters occurs

together is

(A)35

4(B)

35

3

(C)351 (D)

352

Axi omati c Defini ti on o f Probab il it y43. If two persons p1and p2throw a dice, the probability that p1throws a number higher than p2throws

is

(A)2

1(B)

36

15

(C)36

1(D)

11

36.


44. The probability of having at least one tail in 4 throws with a coin is-

(A)16

15(B)

16

1

(C)4

1(D) 1

Bayes theorem45. The chances of defective screws in three boxes A, B, C are 1/5, 1/6, 1/7, respectively. A box is

selected at random and a screw drawn from it at random is found to be defective. Then the probabilitythat it came from box A is(A) 16/29 (B) 1/15(C) 27/59 (D) 42/107.


46. If the letters of the word ATTEMPT are written down at random, the chance that all Ts are consecutiveis-

(A)42

1(B)

7

6

(C)7

1(D)

41

42.

Axi omati c Defini ti on o f Probab il it y47. 4 gentlemen and 4 ladies take seats at random round a table. The probability that they are sitting

alternately is-

(A)35

4(B)

70

1

(C) 352 (D) 351

Axi omati c Defini ti on o f Probab il it y48. From a group of 10 persons consisting of 5 lawyers, 3 doctors and 2 engineers, four persons are

selected at random. The probability that the selection contains at least one of each category is-

(A)2

1(B)

3

1

(C)3

2(D)

1

4.


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Axi omati c Defini ti on o f Probab il it y49. A and B draw two cards each, one after another, from a pack of well shuffled pack of 52 cards. The

probability that all the four cards drawn are of the same suit is-

(A)4985

44

(B)

4985

11

(C) 492517

2413

(D)

11

17 49 .

Axi omati c Defini ti on o f Probab il it y50. Three numbers are chosen at random without replacement from the set A = {x | 1 x 10, x

N}. The probability that the minimum of the chosen numbers is 3 and maximum is 7, is

(A)1

12(B)

1

15

(C)1

40(D)

39

40.

LEVEL - III

51. Three natural numbers are taken at random from the set A ={ x| 1 x 100, x N}. The probabilitythat the AM of the numbers taken is 75, is

(A)

772

1003

C

C(B)

252

1003

C

C

(C)

7472

10097

C

C(D)

752

1003

C

C.

52. Let S be the universal set and n(X) = k. The probability of selecting two subsets A and B of the set

X such that B A is

(A)

1

2 (B) k

1

2 1

(C) k1

2(D) k

1

3

53. If ten objects are distributed at random among ten persons, the probability that at least one ofthem will not get anything is

(A)10

10

10 10

10

(B)

10

10

10 10!

10

(C)10

10

10 1

10

(D) 10

10!

10.

54. 10 different books and 2 different pens are given to 3 boys so that each gets equal number of

things. The probability that the same boy does not receive both the pens is

(A)5

11(B)

7

11

(C)2

3(D)

6

11.


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55. Two distinct numbers are selected at random from the first twelve natural numbers. The probabilitythat the sum will be divisible by 3 is

(A)1

3(B)

23

66

(C)1

2(D)

2

3.

56. The probability of a number n showing in a throw of a dice marked 1 to 6 is proportional to n. Thenthe probability of the number 3 showing in a throw is

(A)1

2(B)

1

6

(C)1

7(D)

1

21.

57. The probability that out of 10 persons, all born in April, at least two have the same birthday is

(A)

3010

10

C

30 (B)

3010C1

30!

(C)

10 30

10

10

30 C

30

(D)

10

30

30!

58. If one ball is drawn at random from each of the three boxes containing 3 white and 1 black, 2 whiteand 2 black, 1 white and 3 black balls then the probability that 2 white and 1 black ball will bedrawn is

(A)13

32(B)

1

4

(C)1

32(D)

3

16

59. Four numbers are nultiplied together. Then, the probability that the product will be divisible by 5 or10 is

(A)369

625(B)

399

625

(C)123

625(D)

133

625.

60. In a multiple choice question there are four 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 answer. The

candidate decides to tick answers at random. If he is allowed up to three chances to answer the

answer the question, then the probability that he will get marks on it is(A) 1/3 (B) 2/3

(C) 1/5 (D) 2/15.


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MULTIPLE CORRECT ANSWER TYPE

LEVEL - I

1. Cards are drawn one by one without replacement untill two aces are drawn. Let P(m) be theprobability that the event occurs in exactly m trials, then P(m) must be zero at

(A) m = 2 (B) m = 50(C) m = 51 (D) m = 52

2. A number is chosen at random from the set of integer 1, 2, 3, . . . n. Let A and B be the events thatthe number drawn is divisible by 2 and 3 respectively. Then(A) A and B are always independent (B) A and B are independent if n = 6k(C) A and B are dependent if n = 10 (D) A and B are independent if n= 6k + 2

3. Let p be the probability the in a pack of playing cards two kings are adjacent and q be theprobability that no two kings are together, then(A) p = q (B) p < q

(C) p + q = 1 (D)48 47 46

q52 51 50

4. Which of the following statements are true?

(A) The probability that birthday of twelve people will fall in 12 calender months =12!

612

(B) The probability that birthday of six people will fall in exactly two calender months is

6

122 12

2 2C

6

(C) The probability that birthday of six people will fall is exactly two calender months is

612

2 6

2 2C

12

(D) The probability that birthday of n (n 365) people are different is

365n

n

P

365

5. A bag contains N tickets numbered 1, 2, 3, . . ., N. If r tickets are drawn one by one withreplacement then the probability that all dif ferent numbers are drawn is

(A)

Nr

Nr

C

P (B)

r

N N 1 N 2 . . . N r 1

N

(C)r

r

P

N

(D)1

r!


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6. Which of the following statements are true for two events A and B of the same sample space?

(A) P A B 0 , if A and B are independent

(B) P A B 0 , if A and B are mutually exclusive.

(C) P A B P A P B 1

(D) P A B P B / A 7. A die is thrown twice. Let X

1and X

2be the outcomes of these trials respectively. Consider the

following eventsA

1= {X

1is divisible by 2, X

2is divisible by 3}

A2= {X

1is divisible by 3, X

2is divisible by 2}

A3= {X

1is divisible by X

2}

A4= {X

2is divisible by X

1}

A5= {X

1+ X

2is divisible by 2}

A6= {X

1+ X

2is divisible by 3}. Then

(A) A1and A

2are independent (B) A

1and A

5are independent

(C) A3and A

4are independent (D) A

3and A

6are independent

8. If M and N are any two events, the probablility that exactly one of them occur is

(A) P M P N 2P M N (B) P M P N P M N

(C) P M P N P M N (D) P M N P M N

9. Let 0 P A 1, 0 P B 1 and P A B P A P B P A P B , then

(A) P B / A P B P A (B) P A B P A P B

(C) P A B P A P B (D) P A / B P A

10. Let P(n) be the probability of getting n heads when a coins is tossed m times, if P(4), P(5), P(6)are in A.P., then the possible values of m could be(A) 10 (B) 11(C) 7 (D) 14

11. A five-digit number is written down at random. The probability that the number is divisible by 5 andno two consecutive digits are identical, is

(A)1

5(B)

31 9

.5 10

(C)

43

5

(D)3

5.

12. Each of the n bags contains a white and b black balls. One ball is transferred from 1st bag to thesecond bag then one ball is transferred from second bag to the third bag and so on. Let p

nbe the

probability that ball trnasferred from nth bag is white, then

(A) 1a

pa b

(B) 2a

pa b

(C) 3a

pa b

(D) 4a

pa b

.


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13. In a single cast with two dice the odds against drawing 7 is

(A)1

6(B)

1

12

(C) 5 : 1 (D) 1 : 5.

14. 7 white balls and 3 black balls are placed in a row at random. The probability that no two blackballs are adjacent is

(A)1

2(B)

7

15

(C)2

15(D)

1

3.

15. Given that x 0, 1 and y 0, 1 . Let A be the event of (x, y) satisfying 2y x and B be the event

of (x, y) satisfying 2x y . Then

(A)

1P A B

3

(B) A, B are exhaustive(C) A, B are mutually exclusive (D) A, B are independent.

16. 10 apples are distributed at random among 6 persons. The probability that at least one of them willreceive none is

(A)6

143(B)

144

155

C

C

(C)137

143(D)

135

143.

17. 4 gentlemen and 4 ladies take seats at random round a table. The probability that they are sittingalternately is

(A)4

35(B)

1

70

(C)2

35(D)

1

35

18. Let x = 33n. The index n is given a positive integral value at random. The probability that the valueof x will have 3 in the units place is

(A)1

4(B)

1

2

(C)1

3(D)

2

3.

19. Three dice are thrown simultaneously. The probability of getting a sum of 15 is

(A)1

72(B)

5

36

(C)5

72(D)

5

108.


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20. If E and F are the complementary events of the events E and F respectively then

(A) P E / F P E / F 1 (B) P E / F P E / F 1

(C) P E / F P E / F 1 (D) P E / F P E / F 1

LEVEL - II21. Three dice are thrown. The probability of getting a sum which is a perfect square is

(A)2

5(B)

9

20

(C)1

4(D)

17

108.

22. The probability of getting a sum of 12 in four throws of an ordinary dice is

(A)

31 5

6 6

(B)

45

6

(C)

2

1 536 6

(D)

3

56

.

23. Three different numbers are selected at random from the set A = {1, 2, 3, , 10}. The probabilitythat the product of two of the numbers is equal to the third is

(A)3

4(B)

1

40

(C)1

8(D)

1

4.

24. If A and B are two events such that 3

P A B4

and 1 3

P A B8 8

then

(A) 11

P A P B8

(B) 3

P A P B8

(C) 7

P A P B8

(D) 1

P A P B8

.

25.. There are 7 seats in a row. Three persons take seats at random. The probability that the middleseat is always occupied and no two persons are consecutive is

(A)9

70(B)

9

35

(C)

4

35 (D)

2

35 .

26. A second-order determinant is written down at random using the numbers 1, 1 as elements. Theprobability that the value of the determinant is nonzero is

(A)1

2(B)

3

8

(C)5

8(D)

1

3.


17/31



27. If E and F are two events with P(E) P(F) > 0 then(A) occurrence of E occurrence of F(B) occurrence of F occurrence of E(C) nonoccurrence of E nonoccurrence of F(D) none of the above implications hold.

28. I f two events A and B are such that

cP A 0.3, P(B) = 0.4 and

cP AB 0.5, then

CP B / A B is(A) less than 0.3 (B) 1/4(C) 1/6 (D) 1/7.

29. Numbers 1, 2, 3, , 100 are written down on each of the cards A, B and C. One number isselected at random from each of the cards. The probability that the numbers so selected can bethe measures (in cm) of three sides of right-angled triangles no two of which are similar, is

(A) 34

100(B) 3

3

50

(C)3

3!

100(D)

21 3

100 50

.

30. The probabilities that a student passes in mathematics, physics and chemistry are m, p and crespectively. Of these subjects, a student has a 75% chance of passing in at least one, a 50%chance of passing in at least two, and a 40% chance of passing in exactly two subjects. Whichof the following relations are true?

(A)19

p m c20

(B)27

p m c20

(C)1

pmc10

(D)1

pmc4

LEVEL - III

31. Let A and B be two events such that 1 5P A B , P A B3 6 and 1P A 2 . Then(A) A, B are independent (B) A, B are mutually exclusive

(C) P(A) = P(B) (D) P B P A .

32. The probability that exactly one of the independent events A and B occurs is equal to

(A) P A P B 2P A B (B) P A P B P A B

(C) P A P B 2P A B (D) P A P B P A B .33. If A and B are independent events such that 0 < P(A) < 1, 0 < P(B) < 1 then

(A) A, B are mutually exclusive (B) A and B are independent

(C) A, B are independent (D) P A / B P A / B 1 .34. For any two events A and B

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

(C) P A B P A P B P A B (D) P A B P A P B P A B

35. A coin is tossed repeatedly. A and B call alternately for winning a prize of Rs. 30 One who callscorrectly f irst wins the prize. A starts the cell. Then the expectation of(A) A is Rs 10 (B) B is Rs 10(C) A is Rs 20 (D) B is Rs 20.


18/31



COMPREHENSION TYPE

Passage - I:

The probability of happening of an event in one trial being known, then the probability of its happening

exactly x times in n trials is given by nCxqn-x. px where

p = probability of happening the event

q = probability of not happening the event = 1- p.

Now nCxqn-xpxis (x + 1)th term in the expansion of (q + p)nwhose expansion gives the happening

of the event 0, 1, 2, .... n times respectively.

1. In four throws with a pair of dice, the chance of throwing doublets atleast twice is

(A)19

144(B)

125

144

(C)17

144(D)

18

144.

2. A man takes a forward step with probability (.8) and backward step with probability (.2). What isthe probability that at the end of 9 steps he is exactly three steps away from starting point

(A) 869888

5(B) 8

5377

5

(C) 85378

5(D) 8

5376

5

3. Unbiassed coin is tossed 6 times. The probability of getting utmost 4 heads is

(A)7

64

(B)57

64

(C)21

32(D)

11

32

Passage - II:

A commander of an army battalion is punishing two of his soldiers X and Y. He arranged a duel

between them. The rules of the duel are that they are to pick up their guns and shoot at each other

simultaneously.

If one or both hit, then the duel is over. If both shot miss then they repeat the process. Suppose

that the results of the shots are independent and that each shot of X will hit Y with probability 0.4

and each shot of Y will hit X with probability 0.2. Now answer the following questions.4. The probability that the duel ends after first round is

(A) 11/25 (B) 12/25

(C) 13/25 (D) 2/5

5. The probability that X is not hit, is

(A) 3/25 (B) 7/25

(C) 5/13 (D) 8/13


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6. The probability that both the soldiers are hit, is

(A) 5/13 (B) 2/13

(C) 8/13 (D) 1/13

Passage - III:

. Let E1, E

2, E

3, .. E

nbe a set of mutually exclusive and exhaustive events and A be an event,

then P(A) =

n

i

i 1 i

AP(E ).P

E

andj

jj

AP(E ).P

EEP

A P(A)

for j = 1, 2, n, where

FP

G

,

denotes the probability of occurring the event F given that G has already occurred. There are two

bags of red and yellow colours. Red bag contains 4 fair coins and 3 biased coins and yellow bag

contains 5 fair coins and 7 biased coins. Biased coin has tail on both sides. Two coins are

transferred from red bag to yellow bag and then a coin is taken from yellow bag and tossed.

7. Probability that both coins, transferred from red bag to yellow bag, were fair, is

(A) 1/7 (B) 3/7

(C) 4/7 (D) 2/7

8. Probability that both coins, transferred from red bag to yellow bag, were of mixed type, is

(A) 3/7 (B) 4/7

(C) 2/7 (D) 1/7

9. If both coins transferred from red bag to yellow bag were biased, then the probability that tossing

of coin results in head, is

(A) 23/28 (B) 9/28

(C) 5/28 (D) 19/28


20/31



MATRIX MATCH TYPE

1. A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed

by replacing the elements of the subset P. A subset Q of A is again chosen at random. The

probability that

List - I List-II

(A) P Q (p) n(3n 1)/4n

(B) P Q is a singleton (q) (3/4)n

(C) P Q contains 2 elements (r) 2nCn/4n

(D) |P| = |Q| (s) n 2 n3 n n 1 / 2 4 where |X| = number of elements in X

2. A player tosses a coin and is to score one point for every head and two points for every tail turned

up. He is to play unitl his score reaches or passes n. Pnis the chance of obtaining exactly a score

of n, then

Column I Column II

(A) P1equals to (p) 5/8

(B) P2equals to (q) 1/2

(C) P3equals to (r) 3/4

(D) P4equals to (s) 11/16

3. Two dice are thrown. Let A be the event that sum of the points on the two dice is odd and B be the

event that atleast one 3 is there, then match the following

Column I Column II

(A) P(AB) (p) 12/36

(B) P(AB) (q) 6/36

(C) P(A B) (r) 23/36

(D) P(B) (s) 11/36


21/31



ASSERTION - REASONING TYPE

1. Let A and B be two independent events of a random experiment.

Statement1 : P A B = P A P(B) .

Statement2 : P A B = 1 P(A) P(B).2. A fair die is rolled once.

Statement1 : The probability of getting a composite number is1

3.

Statement2 : There are three possibilities for the obtained number (i) the number is a primenumber (ii) the number is a composite number (iii) the number is 1, and hence probability of

each possibility is1

3.

3. Let A and B are two events such that P(A) =3

5and P(B) =

2

3, then

Statement1 :4

15P

3A B

5 .

Statement2 :2 A 9

P5 B 10

.

4. Consider the system of equations ax + by = 0, cx + dy = 0, where a, b, c, d {0, 1}

Statement1: The probability that the system of equations has a unique solution is 3/8 and

Statement2: The probability that the system of equations has a solution is 1.

5. Statement1 : If two events E1and E

2are independent then

1E and 2E are also independent.

Statement2 :1 2 1 2P(E E ) P(E E ) = 1 P(E1E2) = 1 P(E1).P(E2)


22/31



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 thenumbers 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 containing2 prizes and 6 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 1n 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 andthe other two dice are ordinary dice bearing numbers 1, 2, 3, 4, 5 and 6. If all the four dice are

thrown, find the probability 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)atleast 4 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 a1

2 b a

.

8. An artillery target may be either at point A with probability 89

or at point B with probability 19

. Wee

have 21 shells 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

a2white & b

2black 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 f rom bag A, B and C.

11. The probability that at least one of A and B occurs is 0.6. If A and B occur simultaneously with

probability 0.3, then find the value of )B(P)A(P .

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


23/31



directly proportional to2 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 the only student to have passed the examination.

13. Let A and B be two independent witnesses in a case. The probability that A will speak the truth isx and the probability that B wil l 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 ofthe numbers on the dice equal to 7 on atleast one toss, is greater than 0.95.

(Given log10

2 = 0.3010, log10

3= 0.4771).

15. Two teams A and B play a tournment. The first one to win (n + 1) games, win the series. Theprobability that A 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. Suppose the probability for A to win a game against B is 0.4. If A has an option of playing either a" best of 3 games " or a " best of 5 games " match against B, which option should he choose sothat the probability of his winning the match is higher ? (No game ends in a draw).

17. Suppose that it is 9 to 7 against a person A who is now 35 years of age living till he is 65 and3 to 2 against a person B now 45 living till he is 75, then find the chance that one atleast of

these persons will be alive 30 years hence.

18. If a pair of fair dice is rolled 5 times, then find out the probability that 3 times we get sum more than9.

19. In a certain town, 40% of the people have brown hair, 25% have brown eyes and 15% have bothbrown hair and brown eyes. If a person selected at random from the town, having brown hair, thenfind the probability that he also has brown eyes.

20. Four cards are chosen at random one by one, without replacement from a well shuffled pack of 52

playing cards. Show that the probability that all the chosen cards are aces, is

1

270725 .

21. Cards are drawn one by one (without replacement) at random from a well shuffled full pack of 52playing cards until two aces are obtained for the first time. If N is the number of cards required tobe drawn, then find P(N = n).

22. A bag contains 10 fair coins and 25 coins having heads on both sides. A coin is selected atrandom and tossed. If i t gives head, then find out the probability that it was a fair coin.


24/31



PREVIOUS YEARS IIT QUESTIONS

1. Let 0 < P(A) < 1, 0 < P(B) < 1 and )B(P).A(P)B(P)A(P)BA(P , then

(A) P(B/A) = P(B) - P(A) (B) )B(P)A(P)BA(P cccc

(C) )B(P)A(P))BA((P ccc

(D) P(A/B) = P(A) [IIT - 95]2. The probability of India wining a test match against West Indies is 1/2. Assuming independence

from match the probability that in a 5 match series India's second win occurs at the third test is(A) 1/8 (B) 1/4(C) 1/2 (D) 1/3 [IIT - 95]

3. For the three events A, B and C, P(exactly one of the events A or B occurs) = P(exactlyone of the events B or C occurs) = P(exactly one of the events C or A occurs) = p andP(all the three events occur simultaneously ) = p2, where 0 < p < 1/2. Then the probabilityof at least one of three events A, B and C occurring is

(A)

2

p2p3 2

(B)

4

p3p 2

(C)2

p3p 2

(D)4

p2p3 2

[IIT - 97]

4. If P(B) =4

3,

3

1)CBA(P and

3

1)CBA(P , then )CB(P is

(A) 1/12 (B) 1/6(C) 1/15 (D) 1/9 [IIT - 2003]

5. Two number is selected randomly from the set S = {1, 2, 3, 4, 5, 6}without replacement one by one.The probability that minimum of the two numbers is less than 4 is

(A)151 (B)

1514

(C)5

1(D)

5

4[IIT - 2003]

6. If three distinct numbers are chosen randomly from the first 100 natural numbers, then theprobability that all three are divisible by both 2 and 3 is(A) 4/25 (B) 4/35(C) 4/33 (D) 4/1155 [IIT - 2004]

7. A six faced fair die is thrown until 1 comes, then the probability that 1 comes in even number of

trials is [IIT - 2005](A) 5/11 (B) 5/6(C) 6/11 (D) 1/6

8. An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of theexperiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A andB are independent, is(A) 2, 4 or 8 (B) 3, 6 or 9(C) 4 or 8 (D) 5 or 10 [IIT 2008]


25/31



9. Let be a complex cube root of unity with 1. A fair die is thrown three times. If r1, r

2and r

3are

the numbers obtained on the die, then the probability that 31 2 rr r = 0 is [IIT 2010]

(A)1

18(B)

1

9

(C) 29

(D) 136

10. A signal which can be green or red with probability4 1

and5 5

respectively, is received by station AA

and then transmitted to station B. The probability of each station receiving the signal correctly is

3

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

green is [IIT 2010]

(A)3

5(B)

6

7

(C)20

23(D)

9

20

SUBJECTIVE TYPE QUESTIONS

11. In a test an examinee either guesses or copies or knows the answer to a multiple choice question

with four choices. The probability that he make a guess is1

3and the probability that he copies the

answer is6

1. The probability that his answer is correct given that he copied it, is

8

1. Find the

probability that he knew the answer to the question given that he correctly answered it.[IIT - 91]

12. Numbers are selected at random, one at a t ime, from the two digi t numbers00, 01, 02, ............ ,99 with replacement. An event E occurs if and only if the product of the twodigits of a selected number is 18. If four numbers are selected, find the probability that theevent E occurs at least 3 times. [IIT - 93]

13. An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and thenumber obtained by adding the numbers on the two faces is noted. If the result is a tail, a cardfrom a well shuffled pack of eleven cards numbered 2, 3, 4,.........,12 is picked and the numberon the card is noted. What is the probability that the noted number is either 7 or 8. [IIT - 94]

14. In how many ways 3 girls and 9 boys can be seated in two vans, each having numbered seats, 3 inthe front and 4 at the back ? How many seating arrangements are possible if 3 girls should

sit together in a back row on adjacent seats ? Now, if all the seating arrangements are equallylikely, what is the probability of 3 girls sitting together in a back row on adjacent seats ? [IIT-96]

15. 3 players A, B and C toss a coin cyclically in that order (that is A, B, C, A, B, C, A, B,........) till a

head shows. Let p be the probability that the coin shows a head. Let , and be respectively

the probabilities that A, B and C gets the first head. Prove that )p1( . Determine , and (in terms of p). [IIT - 98]


26/31



16. A coin has probability 'p' of showing head when tossed. It is tossed 'n' times. Let Pndenote the

probability that no two (or more) consecutive heads occur. Prove that p1= 1, p

2= 1 - p2and

pn= (1 - p)p

n - 1+ p(1 - p)p

n - 2, for all 3n . [IIT - 2000]

17. An urn contains 'm' white and 'n' black balls. A ball is drawn at random and is put back into the urnalong with K additional balls of the same colour as that of the ball drawn. A ball is again drawn atrandom. What is the probability that the ball drawn now is white? [IIT - 2001]

18. An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6 is thrown n times and the list of n numbersshowing up is noted. What is the probability that among the numbers 1, 2, 3, 4, 5, 6 only threenumbers appear in the list. [IIT - 2001]

19. A box contains N coins, m of which are fair and the rest are biased. The probability of getting ahead when a fair coin is tossed is 1/2, while i t is 2/3 when a biased coin is tossed. A coin is drawnfrom the box at random and is tossed twice. The first time it shows head and the second time itshows tail. What is the probability that the coin drawn is fair ? [IIT - 2002]

20. For a student to qualify, he must pass at least two out of three exams. The probability that he willpass the 1stexam is p. It he fails in one of the exams then the probability of his passing in the next

exam is2

potherwise it remains the same. Find the probability that he will qualify.. [IIT - 2003]

21. A is targeting to B, B and C are targeting to A. Probability of hitting the target by A, B and C are

2

1,

3

2and

3

1respectively. If A is hit then find the probability that B hits the target and C does not.

[IIT - 2003]

22. There are 18 balls, 12 red and 6 white. Six balls are drawn one by one without replacement. If atleast 4 are white, find the probability that next two draw will result in one red and one white ball.

[IIT - 2004]

23. A person goes to office either by car, scooter, bus or train, the probability of which being7

2,

7

3,

7

1

and71 respectively. Probability that he reach office late, if he takes car, scooter, bus or train is

92 ,

91 ,

9

4and

9

1respectively. Given that he reached office in time, then what is the probability that he

travelled by a car? [IIT - 2005]

COMPREHENSIVE TYPEI. There are n urns numbered1, 2,, ....., n each containing (n + 1) balls. Urn i contains i white balls

and (n + 1 i) red balls, i = 1,2, .....n. An urn is selected and a ball is drawn at random from it. LetU

idenote the event that urn numbered i is selected and let W denote the event that a white ball is

drawn from the selected urn. Further, suppose that E denotes the event that an even numbered urnis selected

[IIT 2006]

24. If P(Ui) i, i = 1,2,....., n, then n

lim P(W)

equals

(A)2

3(B)

1

2

(C)1

3(D) 1


27/31



25. If P(Ui) = c, i = 1, 2,.....,n, where c is a constant, then P(U

n/W) equals

(A)i

n 1(B)

2

n 1

(C)n

n 1(D) 1

26. If n is even and E denotes the event of choosing even numbered urn i1

P U , i 1, 2,...., nn

then the value of P W / E is

(A)n 2

2n 1

(B)

n 2

2 n 1

(C)

n 2

2 n 1

(D)

1

n 1

II. A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.27. The probability that X = 3 equals [IIT 2009]

(A)25

216(B)

25

36

(C)5

36(D)

125

216

28. The probabili ty that X 3 equals

(A)125

216(B)

25

36

(C) 536

(D) 25216

29. The conditional probability that X6 given X > 3 equals

(A)125

216(B)

25

216

(C)5

36(D)

25

36

ASSERTION REASON TYPE

30. Let H1, H

2, .... , H

nbe mutually exclusive and exhaustive events with P(H

i) > 0, i = 1, 2, ...., n. Let

E be any other event with 0 < P(E) < 1. [IIT 2007]

STATEMENT- 1 : P(Hi|E) > P(E | H

i) . P(H

i) for i = 1, 2, ... , n.

STATEMENT 2 :n

ii 1

P(H ) 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.


28/31



ANSWERS & KEYS

CONCEPTUAL QUESTIONS

1. (D) 2. (C)

3. (A) 4. (A)

5. (A) 6. (A)

7. (D) 8. (B)

9. (B) 10. (C)

11. (A) 12. (B)

13. (C)

SINGLE CORRECT ANSWER TYPE

1. (B) 2. (B)

3. (B) 4. (D)

5. (D) 6. (A)

7. (C) 8. (D)

9. (C) 10. (A)11. (B) 12. (A)

13. (B) 14. (D)

15. (C) 16. (B)

17. (C) 18. (A)

19. (D) 20. (B)

21. (A) 22. (D)

23. (D) 24. (C)

25. (A) 26. (C)

27. (A) 28. (A)

29. (A) 30. (D)

31 (B) 32 (B)

33. (B) 34. (B)35. (A) 36. (D)

37. (B) 38. (C)

39. (B) 40. (B)

41. (B) 42. (D)

43. (B) 44. (A)

45. (D) 46. (C)

47. (D) 48. (A)

49. (A) 50. (C)

51. (C) 52. (B)

53. (B) 54. (A)

55. (A) 56. (C)

57. (C) 58. (A)59. (A) 60. (C)

MULTIPLE CORRECT ANSWER TYPE

1. (C, D) 2. (B, C, D)

3. (B, D) 4. (A, C, D)

5. (B, C) 6. (B, D)


29/31



7. (A) 8. (A, D)

9. (C, D) 10. (C, D)

11. (C) 12. (A, B, C, D)

13. (C) 14. (B)

15. (A) 16. (C)

17. (D) 18. (A)

19. (D) 20. (A, D)

21. (D) 22. (A)

23. (B) 24. (A, C)

25. (C) 26. (A)

27. (D) 28. (A, B)

29. (D) 30. (B, C)

31. (A) 32. (A, C)

33. (B, C, D) 34. (A, C)

35. (B, C)

COMPREHENSION TYPE

1. (A) 2. (A)3. (B) 4. (C)

5. (D) 6. (B)

7. (D) 8. (B)

9. (C)

MATRIX MATCH TYPE

1. (A-q), (B-p), (C-s), (D-r) 2. (A-q), (B-r), (C-p), (D-s)

3. (A-r), (B-q), (C-p), (D-s)

ASSERTION - REASONING TYPE

1. (C) 2. (C)

3. (A) 4. (B)

5. (C)

SUBJECTIVE TYPE

1.

m

n m 1 m!

n

2.

10

133

3. 952 :715 5.125

1296

6. 12

8 12

9. n1

1 1 pn


30/31



10. From Bag A =1

1 2 3

p

p p p , From Bag B =2

1 2 3

p

p p p , From Bag C =3

1 2 3

p

p p p where

k

k

k k

aP

a b

, where k = 1,2,3

11. 1.1

12. (i)

3 n 1

2 2n 1

; (ii)

2

2

n n 1

14. 17

15. n

n r r n

n

r 0

C .q .p .p where p q 1

16. A must choose the first offer i.e., best of three games.

17. 53/80

18. 5250

6

19. 3

8

21. 1 52 51

50 49 17 13

n n n

22.1

6

PREVIOUS YEARS IIT QUESTIONS

1. (D) 2. (B)

3. (A) 4. (A)

5. (D) 6. (D)

7. (A) 8. (D)

9. (C) 10. (C)

11.

29

2412.

4

97

25

13.193

792= 0.2436

14. (i) 7(13!); (ii) 12!; (iii) 12!/7.13! = 1/91

15.

2

3 3 3

1 p p 1 p pp, ,

1 1 p 1 1 p 1 1 p


31/31


16. 24/29 17.m

m n

18. n n3

n

6C 3 3 2 3

6

19.9m

m 8N

20. 2 32p p 21.1

2

22.

12 610 2 12 6 11 11 51 1 2 4 1 1

12 18 12 182 6 2 6

C CC C C C C C

C C C C

= 12 6 12 6 12 6

2 4 1 5 0 6

1

C C C C C C

23. 7/49 . 24. (A)

25. (B) 26. (B)

27. (A) 28. (B)

29. (D) 30. (B)

HINTS & SOLUTIONS

Probability Jee Adv

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