Exercise (ALL)

I I T-ian’s P A C E Edu.Pvt.Ltd

PROBABILITY

Rg-PB -12

EXERCISE 1 (A)

ONLY ONE OPTION IS CORRECT

1.

A comm committe itteee of of thre threee pe perso rsons ns is to be be rando randomly mly select selected ed from from a grou groupp of thre threee men men and two two w wom omen en and and the the chair person will be randomly selected from the committee. The probabilit p robabilityy that the committee will have exactly two women and one man, man, and that tha t the chair person pers on will be a woman, is/are (A) 1/5 (B) 8/15 (C) 2/3 (D) 3/10

2.

6 marrie marriedd coupl couples es are stand standin ingg in a room room.. If 4 pe peopl oplee are cho chosen sen at at random random,, then then the chan chance ce that that exactl exactlyy one one married couple is among the 4 is : 8 17 16 24 (A) (B) (C) (D) 33 33 33 33 A sampl samplee space space con consi sists sts of 3 sample sample poi point ntss with with assoc associa iate tedd probab probabil iliti ities es giv given en as as 2p, p2, 4p – 1 tthen hen (A) p = 11  3 (B) p = 10  3 (C) 1/4 < p < 1/2 (D) none

3.

4.

A comm committee ittee of 5 is is to be cho chosen sen from from a group group of 9 people. people. The probabi probability lity that a certai certainn marrie marriedd couple couple will will either serve together or not at all is : (A) 1/2 (B) 5/9 (C) 4/9 (D) 2/3

5.

There There are only only two two w wom omen en amon amongg 20 per person sonss takin takingg part part in a pleasur pleasuree trip. trip. The The 20 person personss are div divide idedd into into two groups, each group consisting of 10 persons. Then the probability prob ability that the two t wo women women will be in the same group is : (A) 9/19 (B) 9/38 (C) 9/35 (D) none

6.

The probabili probability ty that that a positive positive two digit digit numbe numberr selec selected ted at random random has its its tens tens digit digit at least least three three more more than its unit digit is (A) 14/45 (B) 7/45 (C) 36/45 (D) 1/6

7.

A 5 digi digitt number number is is forme formedd by using using the dig digits its 0, 1, 2, 3, 4 & 5 witho without ut re repetit petition. ion. The probabi probability lity that the the number is divisible by 6 is : (A) 8 % (B) 17 % (C) 18 % (D) 36 %

8.

An expe experi rime ment nt resu result ltss in fo four ur po possi ssibl blee out out come comess S1, S2, S3 & S4 with probabilities p1, p2, p3 & p4 respectively resp ectively.. Which one of the follow following ing probability assignment is possbile. p ossbile. [Assume S1 S2 S3 S4 are mutually exclusive] (A) p 1 = 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 , p2 = 0.30 , p3 = 0.40 , p4 = 0.10

9

In throw throwin ingg 3 dice dice,, the the proba probabil bilit ityy that that atlea atleast st 2 of the the thr three ee num numbe bers rs oobtai btaine nedd are are same same is (A) 1/2 (B) 1/3 (C) 4/9 (D) none

10.

There There are 4 defe defecti ctive ve items items in in a lot consi consisting sting of 10 items. items. From this lot lot we selec selectt 5 items items at random. random. The probabili probability ty that there will will be 2 def defecti ective ve items items among among them is (A)

1 2

Head Office : Andheri

(B)

2 5

(C)

5 21

(D)

10 21

26245 223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # : 26245223

31


PROBABILITY

Rg-PB -12

11.

From a pack of 52 playing playing cards, cards, face cards cards and tens are are rem remove oved d and kept aside then a card card is draw drawn n at random from the ramaining cards. If A : The event that the card drawn drawn is an ace H : The event that the t he card card drawn is a heart S : The event event that the card drawn drawn is a spade then which of the following holds ? (A) 9 P(A) = 4 P(H) (B) P(S) = 4P (A  H) H) (C) 3 P(H) = 4 P(A  S) (D) P(H) = 12 P(A  S)

12.

Wheneve Wheneverr horses horses a, b, c race race together together,, the their ir re respectiv spectivee probabili probabilities ties of winning winning the race race are 0.3, 0.5 0.5 and 0.2 respectively. If If they race three times the probability probabilit y that “the same sa me horse wins all the three races” races” and the probablity that a, b, c each wins one race, are respectively respectively (Assume no dead heat) (A)

8 50

;

9 50

(B)

16 100

,

3 100

(C)

12 50

;

15 50

(D)

10 50

;

8 50

13

Let Let A & B be two two eve evennts. ts. Sup Suppo pose se P(A) P(A) = 0.4 0.4 , P(B) = p & P(A  B) = 0.7. The value of p for which A & B are independent is : (A) 1/3 (B) 1/4 (C) 1/2 (D) 1/5

14 14..

A & B are are tw two inde indepe pend nden entt eve event ntss suc suchh tha thatt P ( A ) = 0.7, P ( B ) = a & P(A  B) = 0.8, then, a = (A) 5/7 (B) 2/7 (C) 1 (D) none

15.

A pair pair of numbe numbers rs is pick picked ed up up random randomly ly (with (withou outt replace replaceme ment) nt) from from the the set {1, 2, 3, 5, 7, 11, 12, 13, 17, 1 7, 19}. The probability that the number 11 was picked given that the sum of the numbers was even, is nearly : (A) 0.1 (B) 0.125 (C) 0.24 (D) 0.18

16.

For a biased biased die the the probab probabili ilitie tiess for the the diff difffe feren rentt faces faces to turn turn up ar aree gi give venn bel below ow : Faces : 1 2 3 4 5 6 Probabilities : 0.10 0.32 0.21 0.15 0.05 0.17 The die is tossed toss ed & you are told that either face one or face two has turned up. Then the probability that it is face one is : (A) 1/6 (B) 1/10 (C) 5/49 (D) 5/21

17.

A determinan determinantt is chosen chosen at random random from from the set of all determi determinants nants of orde orderr 2 with with elements elements 0 or 1 only only.. The probability probabilit y that the determinant chosen has the value non negative negative is : (A) 3/16 (B) 6/16 (C) 10/16 (D) 13/16

18.

15 coupons coupons are are numbere numberedd 1, 2, 3,.... 3,...... , 15 respectiv respectivel elyy. 7 coupons coupons are are selecte selectedd at random random one one at a time time with with replacement. replacement. The probability p robability that the largest number appearing on a selected coupon is is 9 is :

 9  (A)    16 

6

 8  (B)    15 

7

 3  (C)    5 

7

97  87 (D) 157

19.

A card card is drawn drawn & replace replacedd in an an ordina ordinary ry pack pack of of 52 playing playing cards. cards. Minimum Minimum num number ber of of times times must must a card card be drawn so that there is atleast atleast an even even chance chance of drawing drawing a heart, heart, is (A) 2 (B) 3 (C) 4 (D) more than four

20.

A license license plate plate is 3 letters letters (of (of Englis Englishh alphabets) alphabets) follow followed ed by 3 dig digits. its. If If all possible possible lic license ense plates plates are equal equally ly likely, likely, the probability that t hat a plate has either a letter palindrome or a digit palindrome (or both), is (A) 7/52 (B) 9/65 (C) 8/65 (D) none



32


PROBABILITY

Rg-PB -12

11.

From a pack of 52 playing playing cards, cards, face cards cards and tens are are rem remove oved d and kept aside then a card card is draw drawn n at random from the ramaining cards. If A : The event that the card drawn drawn is an ace H : The event that the t he card card drawn is a heart S : The event event that the card drawn drawn is a spade then which of the following holds ? (A) 9 P(A) = 4 P(H) (B) P(S) = 4P (A  H) H) (C) 3 P(H) = 4 P(A  S) (D) P(H) = 12 P(A  S)

12.

Wheneve Wheneverr horses horses a, b, c race race together together,, the their ir re respectiv spectivee probabili probabilities ties of winning winning the race race are 0.3, 0.5 0.5 and 0.2 respectively. If If they race three times the probability probabilit y that “the same sa me horse wins all the three races” races” and the probablity that a, b, c each wins one race, are respectively respectively (Assume no dead heat) (A)

8 50

;

9 50

(B)

16 100

,

3 100

(C)

12 50

;

15 50

(D)

10 50

;

8 50

13

Let Let A & B be two two eve evennts. ts. Sup Suppo pose se P(A) P(A) = 0.4 0.4 , P(B) = p & P(A  B) = 0.7. The value of p for which A & B are independent is : (A) 1/3 (B) 1/4 (C) 1/2 (D) 1/5

14 14..

A & B are are tw two inde indepe pend nden entt eve event ntss suc suchh tha thatt P ( A ) = 0.7, P ( B ) = a & P(A  B) = 0.8, then, a = (A) 5/7 (B) 2/7 (C) 1 (D) none

15.

A pair pair of numbe numbers rs is pick picked ed up up random randomly ly (with (withou outt replace replaceme ment) nt) from from the the set {1, 2, 3, 5, 7, 11, 12, 13, 17, 1 7, 19}. The probability that the number 11 was picked given that the sum of the numbers was even, is nearly : (A) 0.1 (B) 0.125 (C) 0.24 (D) 0.18

16.

For a biased biased die the the probab probabili ilitie tiess for the the diff difffe feren rentt faces faces to turn turn up ar aree gi give venn bel below ow : Faces : 1 2 3 4 5 6 Probabilities : 0.10 0.32 0.21 0.15 0.05 0.17 The die is tossed toss ed & you are told that either face one or face two has turned up. Then the probability that it is face one is : (A) 1/6 (B) 1/10 (C) 5/49 (D) 5/21

17.

A determinan determinantt is chosen chosen at random random from from the set of all determi determinants nants of orde orderr 2 with with elements elements 0 or 1 only only.. The probability probabilit y that the determinant chosen has the value non negative negative is : (A) 3/16 (B) 6/16 (C) 10/16 (D) 13/16

18.

15 coupons coupons are are numbere numberedd 1, 2, 3,.... 3,...... , 15 respectiv respectivel elyy. 7 coupons coupons are are selecte selectedd at random random one one at a time time with with replacement. replacement. The probability p robability that the largest number appearing on a selected coupon is is 9 is :

 9  (A)    16 

6

 8  (B)    15 

7

 3  (C)    5 

7

97  87 (D) 157

19.

A card card is drawn drawn & replace replacedd in an an ordina ordinary ry pack pack of of 52 playing playing cards. cards. Minimum Minimum num number ber of of times times must must a card card be drawn so that there is atleast atleast an even even chance chance of drawing drawing a heart, heart, is (A) 2 (B) 3 (C) 4 (D) more than four

20.

A license license plate plate is 3 letters letters (of (of Englis Englishh alphabets) alphabets) follow followed ed by 3 dig digits. its. If If all possible possible lic license ense plates plates are equal equally ly likely, likely, the probability that t hat a plate has either a letter palindrome or a digit palindrome (or both), is (A) 7/52 (B) 9/65 (C) 8/65 (D) none



32


PROBABILITY

Rg-PB -12

21.

There are n different gift coupons, each of which can occupy N(N N(N > n) different envelopes, with the same probabi pro bability lity 1/N P1: The probability that t hat there will be one gift coupon in each of n definite envelopes out of N given envelopes P2: The probability that tha t there will be one gift gift coupon in each of n arbitrary envelopes out of N given given envelopes Consider the following statements N ! n! N! n! (i) P1 = P2 (ii) P1 = n (iii) P2 = n (iv) P2 = n (v) P1 = n N ( N  n ) ! N N ( N  n ) ! N Now, Now, which which of the follow following ing is true (A) Only (i) (B) (ii) and (iii) (C) (ii) and (iv) (D) (iii) and (v)

22.

The probability probability that that an automobile automobile will will be stolen and foun foundd within within one week week is is 0.0006. The probabili probability ty that an automobile will be stolen is 0.0015. 0.0015 . The probability that a stolen st olen automobile will will be found in one week week is (A) 0.3 (B) 0.4 (C) 0.5 (D) 0.6

23

One bag contai contains ns 3 white white & 2 black black balls, balls, and anothe anotherr contai contains ns 2 white white & 3 black black balls. balls. A ball is draw drawnn from from the second bag & placed in the first, first , then a ball is drawn from the first bag ba g & placed in the second. When the pair of the oper operation ationss is repeated repeated,, the probabili probability ty that the first first bag will will contain contain 5 white white balls balls is: (A) 1/25 (B) 1/125 (C) 1/225 (D) 2/15

24.

A child child throw throwss 2 fair dice. dice. If the num numbers bers show showing ing are are unequal, unequal, he he adds adds them together together to get get his final final score. score. On the other hand, ha nd, if the numbers showing are equal, he throws 2 more dice & adds all 4 numbers showing showing to get his final score. The probability probabilit y that his final score is 6 is: 145 146 147 148 (A) (B) (C) (D) 1296 1296 1296 1296

25.

Events Events A and C are ind indepen ependen dent. t. If the probabil probabilitie itiess relatin relating g A, A, B and and C are P (A) (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 events B and C are equiprobable equiprobable

26.

Assume Assume that the birth of a boy or or gir girll to a couple couple to be equally equally likely likely,, mutually mutually exclusiv exclusive, e, exhausti exhaustive ve and and independent of the other children in the t he family. family. For a couple having 6 children, the probability proba bility that their t heir "three oldest are boys" is 2 1 8 20 (A) (B) (C) (D) 64 64 64 64

27.

If a, b and c are are three three numbers numbers (not (not ne necessa cessarily rily diff differe erent) nt) chose chosenn randoml randomlyy and with with replac replaceme ement nt fro from m the set set {1, 2, 3, 4, 5}, the probability that (ab + c) is even, is (A)

28.

50 125

(B)

59 125

(C)

64 125

(D)

75 125

A examin examination ation con consists sists of 8 questio questions ns in each each of w whic hich h one of of the 5 altern alternativ atives es is the cor correc rectt one. On the assumption assump tion that a candidate ca ndidate who has done no preparatory work chooses for each question any one of the five alternatives alternat ives with equal probability probabilit y, the probability probabilit y that he gets more than one correct answer is equal to 8 8 (A) (0.8) (B) 3 (0.8) (C) 1  (0.8)8 (D) 1  3 (0.8)8



33

I I T-ian’s P A CE Edu.Pvt.Ltd

PROBABILITY

Rg-PB -12

29.

A key to room C3 is dropped into a jar with five other keys, and the jar is throughly mixed. If keys are randomly drawn from the jar without replacement until the key to room C3 is chosen, then what are the odds that the key to room C3 will be botained on the 2nd try? (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 red marble 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.

Pal’s gardner is not dependable, the probability that he will forget to water the rose bush is 2/3. The rose bush is in questionable condition . Any how if watered, the probability of its withering is 1/2 & if not watered then the probability of its withering is 3/4. Pal went out of station & after returning he finds that rose 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 be detected in n cycles is : (A) 1  pn (B) 1  (1  p)n (C) pn (D) p(1 – p) n–1

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 a successive 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. If A : 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?

36.

(A) P (A) =

4 1 ; P (B) = 17 13

(B) P (A) =

1 1 ; P (B) = 13 13

(C) P (A) =

1 1 ; P (B) = 13 17

(D) P (A) =

16 4 ; P (B) = 221 51

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 probabilitythat the colour changes while she is watching? (A)

37.

1 63

(B)

1 21

(C)

1 10

(D)

1 7

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

16 32


(B)

18 32

(C)

13 32

(D)

12 32

: 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW #

34


PROBABILITY

Rg-PB -12

38.

A box contains 5 red and 4 white marbles. Two marbles are drawn successively from the box without replacement and the second drawn marble drawn is found to be white. Probability that the first marble is also while 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 that whosoever 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. SectionA is compulsory. Out of sections B & C a student has to attempt any one. Passing in the paper means passing in A & passing in B or C. The probability of the student 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



35


PROBABILITY

Rg-PB -12

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 are defined 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 P (A)  P (A  B) (A) P  A BC  = 1  P (B) (B) P (A  B)  P(A) + P(B)  1 (C) P(A) > < P A B according as P  A B C  > < P(A) (D) P  A BC  + P  A C BC  = 1

3

For any two events A & B defined on a sample space , (A) P A B 

P (A)

 P (B)  1 P (B)

, P (B)  0 is always true

(B) P AB  = P (A) – P (A  B) (C) P (A  B) = 1 – P (Ac). P (Bc) , if A & B are independent (D) P (A  B) = 1 – P (Ac). P(Bc) , if A & B are disjoint 4

If E1 and E2 are two events such that P(E1) = 1/4, P(E2/E1) =1/2 and P(E1/ E2) = 1/4 (A) then E1 and E2 are independent (B) E1 and E2 are exhaustive (C) E2 is twice as likely to occur as E1 (D) Probabilities of the events E1  E2 , E1 and E2 are in G.P.

5

For the 3 events A, B and C, P (atleast one occurring) = and P (exactly two occurring) = (A) P(ABC) =

3 1 , P (atleast two occurring) = 4 2

2 . Which of the following relations is / are CORRECT? 5

1 10

(B) P(AB) + P(BC) + P(CA) = (C) P(A) + P(B) + P(C) =

7 10

27 20

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


1 4


36

I I T-ian’s P A CE Edu.Pvt.Ltd 6

PROBABILITY

Rg-PB -12

Each of 2010 boxes in a line contains one red marbl e, and for 1  k  2010, the box in the k th position

also contains k white marbles. A child begins at the first box and successively draws a single marble at random 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) < (A) 44 7

(B) 45

(C) 46

1 , is 2010

(D) 47

Which of the following is / are TRUE? (A) A and B are two independent events. If the probability that both A and B occur is

1 and the probability 12

1 1 , then P(A – B) = . (Given: P(A) < P(B) ) 2 6 (B) A fair coin is tossed repeatedly. If the head appears on the first three tosses, then the probability that 1 the tail appearing on the fourth toss equals . 2 (C) If the letters of the word "FREEDOM" are written down at random in a row, the probability that no 6 two E's occur next to each other is . 7 (D) For two given events A and B, P(A) + P(B) – 1  P (A  B)  P(A) + P(B) that neither A nor B occurs is

8

Events A and B satisfy P(A) = 0.4, P(B) = 0.5 and P(A  B) = 0.8. Which of the following statement(s) is/ are CORRECT? (A) PA  B  =

2 5

(B) PB A  =

(C) P(A  B) < P(A) · P(B) 9

(D) P (exactly one of either A or B occurs) =

7 10

A certain coin lands head with probability p. Let Q denote the probability that when the coin is tossed four times the number of heads obtained is even. Then (A) there is no value of p, if Q =

1 4

(C) there are exactly two values of p if Q = 10

2 3

(B) there is exactly one value of p if Q =

3 4

3 4 (D) there are exactly four values of p if Q = 5 5

In an experimental performance of a single throw of a pair of unbiased normal dice, let three events E1, E2 and E3 are defined as follows: E1 : getting prime numbered face on each dice. E2 : getting the same number on each dice. E3 : getting total on two dice equal to 4. Which of the following is/are TRUE? (A) The probabilities P(E1), P(E2), P(E3) are in A.P. (B) The events E1 and E2 are independent. (C) P(E3 / E1) =

2 . 9

(D) P(E1 + E2) + P(E2 – E3) = Head Office : Andheri

17 . 36


37


PROBABILITY

Rg-PB -12

Which of the following statement(s) is/are CORRECT? (A) An event A is known to be independent of the events B, B  C and B  C, then A is also independent of C. (B) Let AC denote the complement of an event A. If A and B are two events, then the probability that atmost one of A, B occurs is P(AC) + P(BC) – P(AC  BC). (C) After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. The probability that exactly nine letters were inserted in the proper envelopes is

1 . 10!

(D) An aircraft is equipped with three engines that operate independently. The probability of an engine failure is 0.01. If only one engine is needed for the successful operation of the aircraft, the probability of a successful flight, is 0.999999. 12

Let all the letters of the word 'MATHEMATICS' are arranged in all possible order. Three events A, B and C are defined as A : Both M are together. B : Both T are together. C : Both A are together. Which of the following hold(s) good? 2 2 (A) P(A) = P(B) = . (B) P(A  B) = P(B  C) = P(C  A) = 55 11 4 58 (C) P(A  B  C) = (D) P( A  B) C  = 495 405

13

A fair coin is thrown large number of times say n and m denotes the number of times it shows up heads. As m n   the fraction is also equal to n (A) the probability of a six digit number N whose six digits are 1, 2, 3, 4, 5, 6 written as random order is divisible by 6. (B) the chance that the product of the outcomes of three rolls of a fair dice is a prime number.

 n n 1 · e   n  . n   (n  1) n  

(C) Lim 

n

1  4i   where [x] denotes the greatest integer less than or equal to x.  n

  n  i 1 n

(D) Lim

14



A normal coin is tossed four times. Two event E and F are defined as E : no two consecutive heads occur in 4 tosses. F : At least 2 consecutive heads occur in 4 tosses. The events E and F are (A) equally likely (B) mutually exclusive (C) exhaustive (D) such that one is twice as likely to occur as other.



38


PROBABILITY

Rg-PB -12

15

A regular tetrahedron has three of its four sides in Red, Green and Yellow colour. Its fourth side is in mixed colour with Red, Green and Yellow simultaneously. Consider a random experiment of tossing this tetrahedron, the side facing down is taken as the outcome. Let R : a side with red colour shows up G : a side with green colour shows up Y : a side with yellow colour shows up the events R, G and Y are (A) pairwise independent (B) mutually independent (C) independent (D) equiprobable

16

A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in the tests I, II and III are p, q and probability that the student is successful is

1 respectively. If the 2

1 , then 2

(A) p = 1, q = 0 (B) p =

2 1 ,q= 3 2

3 2 ,q= 5 3 (D) there are infinitely many values of p and q.

(C) p =

17

Identify the correct statement(s)? (A) Two persons each make a single throw with a die. The probability that they get equal values is P1. Four persons each make a single throw and probability of exactly three being equal values is P2. Then P1 > P2. (B) Each of A and B throw 2 dice. If A throws 9, then B's probability of throwing a higher number is

1 . 6

(C) If P(A  B) = 1  P(A ) P( B ) then A and B are independent events. ( A denotes compliment of event A.) (D) Two even numbers are selected from the given first 20 natural numbers. The probability that their sum is even, is 18

9 . 38

A fair coin is tossed 3 times. Consider the events A : first toss is head B : second toss is head C : exactly two consecutive heads in a row. Then (A) A and B are independent. (B) B and C are independent. (C) C and A are independent. (D) A, B, C are mutually independent.



39


PROBABILITY

Rg-PB -12

COMPREHENSION TYPE PASSAGE - 1

From a pack of 52 playing cards, all the face card are removed. From the remaining pack, cards are dealt one by one until an ace appears. Let P1 denote the probability that exactly 10 cards are dealt before the first ace. If cards continue to be dealt until a second ace appears. Let P2 denotes the probability that exactly 20 cards are dealt before the second ace. In case the first ace appears in one of the 20 cards then the probability is P3. 19

20

The value of P1 equals (A)

(27)(28)(29) (37)(38)(39)

(B)

(C)

(30)(29)(28)(27) ( 40)(39)(38)(37)

(D) None

The value of P2 equals (A)

21

(27)(28)(29) (10)(37)(38)(39)

9 (10)(13)(37)

(B)

9 (13)(37)

The value of P3 equals (A) 2P2 (B) 10P2

(C)

18 (13)(37)

(C) 11P2

(D)

6 (13)(37)

(D) 20P2

PASSAGE - 2

Let S and T are two events defined on a sample space with probabilities P(S) = 0.5, P(T) = 0.69, P(S/T) = 0.5 22

23

24

Events S and T are: (A) mutually exclusive (C) mutually exclusive and independent

(B) independent (D) neither mutually exclusive nor independent

The value of P(S and T) (A) 0.3450 (B) 0.2500

(C) 0.6900

(D) 0.350

The value of P(S or T) (A) 0.6900 (B) 1.19

(C) 0.8450

(D) 0

PASSAGE - 3

A JEE aspirant estimates that she will be successful with an 80 percent chance if she studies 10 hours per day, with a 60 percent chance if she studies 7 hours per day and with a 40 percent chance if she studies 4 hours per day. She further believes that she will study 10 hours, 7 hours and 4 hours per day with probabilities 0.1, 0.2 and 0.7, respectively 25

The chance she will be successful, is (A) 0.28 (B) 0.38


(C) 0.48

(D) 0.58


40


27

PROBABILITY

Given that she is successful, the chance she studied for 4 hours, is 7 6 8 (A) (B) (C) 12 12 12

Rg-PB -12

(D)

9 12

Given that she does not achieve success, the chance she studied for 4 hour, is 20 21 18 19 (A) (B) (C) (D) 26 26 26 26

PASSAGE - 4

Mr. A randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and arranges them in the descending order to form a three digit number. Mr. B randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8} and also arranges them in descending order to form a 3 digit number. 28

The probability that Mr. A's 3 digit number is always greater than Mr. B's 3 digit number, is (A)

29

(B)

1 3

(C)

2 5

(D)

1 2

(D)

1 54

(D)

39 56

The probability that A and B has the same three digit number (A)

30

1 4 1 72

(B)

1 63

(C)

1 56

The probability that Mr. A's number is larger than Mr. B's number, is (A)

47 72

(B)

37 56

(C)

49 72

ASSERTION / REASON TYPE

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. 31

A fair dice is tossed twice. Let E denotes the event that the sum of the numbers appearing on two rolls equals 5 and F denotes the event that an even number comes up in the first roll. Statement-1: Event E and F are independent. Statement-2: For two independent event E and F defined on S P(E  F) = P(E) · P(F)

32

From a well shuffled pack of 52 cards. A card is drawn, outcome is noted and the card is replaced in the pack. 16 such trials were made. Let X denotes a binomial variate denoting the number of times Heart occurs. Statement-1: Mean and variance of X are 4 and 3 respectively. Statement-2: In a binomial probability distribution mean is always greater than variance.

33

Four children A, B, C and D have 1, 3, 5 and 7 identical unbiased dice respectively and roll them with the condition that one who obtains an even score, wins. They keep playing till some one or the other wins. Statement-1: All the four children are equally likely to win provided they roll their dice simultaneously. Statement-2: The child A is most probable to win the game if they roll their dice in order of A, B, C and D respectively.



41


PROBABILITY

Rg-PB -12

Statement-1: Let f (x) = x3 – ax2 + bx + 6 where a, b  {1, 2, 3, 4, 5, 6}. The probability that f (x) is

strictly increasing function is 4 9 . Statement-2: If y = g(x) is differentiable function in (– , ) then g(x) is strictly increasing provided

g(x)  0 and g(x) = 0 does not form an interval. 35

A fair coin is tossed n times. Let pn denotes the probability that no two (or more) consecutive heads occur in n tosses. Statement-1: The probabilities p2, p3, p4 are in arithmetic progression. Statement-2: The probabilities p1, p2, p3 ,..........., pn are in decreasing order.

36.

Statement-1: Two non-negative integers are chosen at random. The probability that

the sum of their squares is divisible by 5 is 9/25. Statement-2: If at the unit place is any number is zero that number is only divisible by 5. 37

Statement - 1: If P is chosen at random in the closed interval [0,5]. then the probability

3 1 that the equation x 2  px  (P  2)  0 has real not is 5 4 Statement -2 : If discriminant > 0 then roots of the quadratic equation are always real.

MATRIX MATCH TYPE

38 (A)

Column I

Six different ball's are kept into 3 different boxes randomly so that

Column II

(P)

1 2

(Q)

1 3

(R)

4 9

(S)

1 6

(T)

1 4

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

(D)

There are 10 boxes and each box can hold any number of balls. A man having 5 balls randomly puts one ball in each of the arbitrary chosen 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 ball(s) in atleast 8 boxes, is



42


PROBABILITY

Column-I

Rg-PB -12 Column-II

(A)

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

(P)

2%

(B)

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

(Q)

25%

(C)

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

(R)

50%

(S)

96%

40 (A)

(B)

Column I

Column II

A gambler has one rupee in his pocket. He tosses an unbiased normal

(P)

1 2

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

(Q)

4 5

3 firemen X, Y and Z shoot at a common target. The probabilities

(R)

3 5

that X and Y can hit the target are 2/3 and 3/4 respectively. If the probability that exactly two bullets are found on the target is 11/24, then the proficiency of Z to hit the target is

(S)

11 16

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)



43


PROBABILITY

Rg-PB -12

EXERCI SE 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 of p 1 .

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, be

p , where p and q are coprimes q

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 is

p , where p q

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 be

p , where p and q are coprimes q

then find the value of p + q. 5.

A normal coin is continued tossing unless a head is obtained for the first time. If the probability that number 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 p , where p and q are coprimes then find the value of p + q. 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 that B 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 is

8.

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

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 , where p and q are coprimes then find the value q

of p + q. 9.

A person flips 4 fair coins and discards those which turn up tails. He again flips the remaining coin and then m discards those which turn up tails. If P = (expressed in lowest form) denotes the probability that he n discards atleast 3 coins, find the value of (m + n).



44

I I T-ian’s P A CE Edu.Pvt.Ltd 10.

Rg-PB -12

2 The probability that a person will get an electric contract is and the probability that he will not get plumbing 5 4 2 contract is . If the probability of getting at least one contract is , then the probability that he will get both 7 3 is

11.

PROBABILITY


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 is

p , where p and q are coprimes, q

then find the value of p + q.

12.

An electrical system has open-closed switches S1, S2 and S3 as shown. The switches operate independently of one another and the current will flow from A to B either if S1 is closed or if both S 2 and S3 are closed. Given P(S1) = P(S2) = P(S3) = 1/2. If the probability that the circuit will work is

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 7q p q  and that the team wins at least one game is , where p, q  N , then find the value of p 100 1000

14.

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

p , where p and q are q

coprimes 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 is 1/2, 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, is

p where p and q are relatively q

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 is

1 . 2

Find the value of N.



45


PROBABILITY

Rg-PB -12

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 the property 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 are drawn randomly from the purse A and put into the purse B, and then 9 balls randomly transferred from purse p (in the lowest form) is the chance that the red ball is still there in the q purse 'A', find the value of (p + q). B to the purse A. If

20.

A best of 9 series is to be played between two teams T1 and T2, that is, the first team to win 5 games is the 2 a of winning any game. Let P = be the probability (expressed as 3 b lowest rational) that exactly 7 games will need to the played to determine a winner, find (a + b). winner. The team T1 has a chance of



46


PROBABILITY

Rg-PB -12

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 7 7 5 (A) (B) (C) (D) none 99 42 198

2.

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

7 213

(B)

1 213

(C)

13 214

(D)

13 213

3.

There are ten prizes, five A's, three B's and two C's, placed in identical sealed envelopes for the top ten contestants in a mathematics contest. The prizes are awarded by allowing winners to select an envelope at random from those remaining. When the 8th contestant goes to select the prize, the probability that the remaining 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 of 4 or 6 is (A)

16 216

(B)

50 216

(C)

60 216

(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 the probability 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 + b2 is divisible by 5 is (A)

8.

25

(B)

7 18

(C)

11 36

(D)

17 81

In an examination, one hundred candidates took paper in Physics and Chemistry. Twenty five candidates failed 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 in both is (A)

9.

9

9 20

(B)

3 5

(C)

2 5

(D)

11 20

When a missile is fired from a ship, the probability that it is intercepted is 1/3. The probabilitythat the missile hits 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



47


PROBABILITY

Rg-PB -12

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 is 3, 4, 5 or 6 on two consecutive tosses. The probability that A wins if the die is tossed indefinitely, is

(A) 11.

1 3

(B)

5 21

(C)

1 4

(D)

2 5

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 the face 3. The probability that a fair die was picked up, is (A)

1 7

(B)

1 4

(C)

1 6

(D)

1 24

12.

Box A contains 3 red and 2 blue marbles while box B contains 2 red and 8 blue marbles. Afair 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 seeds exactly 5 or 6 will spring is : 11

(A)

C 5 . 65 510

11

(B)

C6 (35 25 ) 511

11

(C)

11C 5

 5     6 

(D) none of these

14.

An instrument consists of two units. Each unit must function for the instrument to operate. The reliability of the first unit is 0.9 & that of the second unit is 0.8. The instrument is tested & fails. The probability that "only the first 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. Anew 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 three occassions. The chance that the die chosen was a rigged die, is (A)

18.

216 217

(B)

215 219

(C)

216 219

(D) none

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



48


PROBABILITY

Rg-PB -12

In a horse race there are 18 horses numbered from 1 to 18. The probability that horse 1 would win 1 1 1 , horse 2 is and 3 is . Assuming a tie is impossible, the chance that one of the three horses wins the 6 10 8 race, is

is

(A) 20.

143 420

(B)

119 120

(C)

47 120

(D)

1 5

The contents of three Urns w.r.t. the Red, White and Green balls is as shown in the table given.

Urn R W G I 2 3 1 II 3 2 1 III 3 1 2 A coin when tossed is twice as likely to come heads as compared to tails. Such a coin is tossed two times. If both heads and tails are present then 3 balls are drawn simultaneously from the Urn-I, if head appears on both the occassions then 3 balls are drawn in a similar manner from Urn-II and if no head appears in both the tosses then 3 balls from the Urn-III are drawn in the same manner. The probability that 3 drawn balls are 1 each of different colours, is (A) 10% (B) 15% (C) 30% (D) 90% 21.

An Urn contains 'm' white and 'n' black balls. All the balls except for one ball, are drawn from it. The probability that the last ball remaining in the Urn is white, is (A)

22.

m mn

(B)

n mn

1 (C) (m  n ) !

mn (D) (m  n ) !

A Urn contains 'm' white and 'n' black balls. Balls are drawn one by one till all the balls are drawn. Probability that the second drawn ball is white, is (A)

m mn

(B)

n( m  n  1) (m  n )(m  n  1)

(C)

m( m  1) (m  n )(m  n  1)

(D)

mn (m  n )(m  n  1)

23.

Mr. Dupont is a professional wine taster. When given a French wine, he will identify it with probability 0.9 correctly as French, and will mistake it for a Californian wine with probability 0.1. When given a Californian wine, he will identify it with probability 0.8 correctly as Californian, and will mistake it for a French wine with probability 0.2. Suppose that Mr. Dupont is given ten unlabelled glasses of wine, three with French and seven with Californian wines. He randomly picks a glass, tries the wine, and solemnly says : "French". The probability that the wine he tasted was Californian, is nearly equal to (A) 0.14 (B) 0.24 (C) 0.34 (D) 0.44

24.

7 persons are stopped on the road at random and asked about their birthdays. If the probability that 3 of them K are born on Wednesday, 2 on Thursday and the remaining 2 on Sunday is 6 , then K is equal to 7 (A) 15 (B) 30 (C) 105 (D) 210

25.

Let P be a point chosen at random on the line segment between the points (0, 1) and (3, 4) on the coordinate plane. The probability that the area of the triangle with vertices (0, 0), (3, 0) and P is greater than 2, is (A) 8/9 (B) 3/4 (C) 2/3 (D) 1/2



49


PROBABILITY

Rg-PB -12

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)

3 7

(B)

4 7

(C)

1 3

(D)

3 8

27.

The probabilities of events, A  B, A, B & A  B are respectively in A.P. with probability of second term equal 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 of balls (all the different ways of drawing an even number of balls are considered equally probable, irrespective of their number). Then the probability that there will be the same number of black and white balls among them (A)

4 5

(B)

11 15

(C)

11 30

(D)

2 5

29.

Three numbers are chosen at random without replacement from {1, 2, 3,...... , 10}. The probability that the minimum 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 the maximum number on the two chosen tickets is not more than 10. The minimum number on them is 5, with probability (A)

31.

(C)

3 19

(D) none

4 15

(B)

7 15

(C)

8 15

(D)

9 15

1 16

(B)

2 16

(C)

4 16

(D)

3 16

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 larger circle tangent to the smaller circle is 8. The probability that point randomly selected from inside of the larger circle lies inside the annular region of the two circles is (A)

34.

2 11

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

33.

(B)

Sixteen players s1 , s2 ,..... , s16 play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair. Assume that all the players are of equal strength. The probability that "exactly one of the two players s1 & s2 is among the eight winners" is (A)

32.

1 9

3 4

(B)

9 25

(C)

16 25

(D)

7 3

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

5 11


(B)

5 12

(C)

11 21

(D)

6 11


50


PROBABILITY

Rg-PB -12

Two boys A and B find the jumble of n ropes lying on the floor. Each takes hold of one loose end randomly. If the probability that they are both holding the same rope is (A) 101

(B) 100

(C) 51

1 then the number of ropes is equal to 101 (D) 50

36.

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

37.

The entries in a two-by-two determinant

a b are integers that are chosen randomly and independently,, c d

and, for each entry, the probability that the entry is odd is p. If the probability that the value of the determinant is even is 1/2, then the value of p, is (A) 38.

1 3

(B)

1 2

(C)

2 3

(D)

2 2

An experiment resulting in sample space as S = {a, b, c, d, e, f} 1 1 2 3 4 5 , P(b) = , P(c) = , P(d) = , P(e) = and P(f) = . 16 16 16 16 16 16 Let three events A, B and C are defined as A = {a, c, e,}, B = {c, d, e, f} and C = {b, c, f}. with P(a) =

If P(A/B) = p 1, P(B/C) = p2, P(C/Ac) = p3 and P(Ac/C) = p4, then the correct order sequance is (A) p 1< p3 < p2 < p4 (B) p1< p4 < p3 < p2 (C) p1< p3 < p4 < p2 (D) p3< p1 < p4 < p2 39.

40.

A random selector can only select one of the nine integers 1, 2, 3,.... , 7, 8, 9 and it makes these selections with equal probability. The probability that after n selections (n > 1), the product of n numbers selected will be divisible by 10 is

 8  n  5  n  4 n  (A) 1 –           9   9   9  

 8  n  5  n  4  n  (B) 1 –  9    9    9         

 8  n  5  n  4 n  (C) 1 –  9    9    9         

 8  n  5  n  4  n  (D) 1 –  9    9    9         

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

41.

19 27

(B)

1 216

(C)

37 216

(D)

19 216

Two coins look similar, but have different probabilities of falling "head". One is a fair coin, with P (H) = 1/2, but the other is weighed so that P (H) = 4/5. One of the coin is chosen at random and is tossed 10 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

510 (A) 8 10 5 8 Head Office : Andheri

(B)

58 510  410

(C)

510 510  87

(D)

510 510  88


51


PROBABILITY

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 rational (A) 17

43.

Rg-PB -12

(B) 34

p (in the lowest form), then (p + q) equals q (C) 19 (D) 18

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 to of five heads and three tails. Let p = The value of (m + n) equals (A) 9 (B) 11

1 of the probability 25

m , where m and n are relative prime positive integers. n (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. For every 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) 1/1024

45.

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)

35 128

(B)

5 64

(C)

5 32

(D)

5 16

46.

Consider the following three hands of five cards, dealt from an ordinary deck of 52 cards H1: 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) H1 is more likely than H2 or H3. (B) H2 is more likely than H1 or H3. (C) H3 is more likely than H1 or 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 that A 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 a scholarship is 40%. In case of getting it, the probability of completing the course is 0.8, while in the case of not 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)

49.

8 17

(B)

4 7

(C)

8 15

(D)

4 9

Rajdhani Express stops at six intermediate stations between Kota and Bombay. Five passengers board at Kota. Each passengers can get down at any station till Bombay. The probability that all five passengers will get down at different stations, is 6

(A)

P5

65


6

(B)

C5

65

7

(C)

P5

75

7

(D)

C5

75


52


PROBABILITY

Rg-PB -12

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. No game can end in a tie. At the end of the tournament, if one team has more wins than both of the other two teams, 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 respectively of 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)

1 3

(B)

8 21

(C)

6 21

(D)

9 21

PASSAGE - 1

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

1 2n

, n  N.

1 1 1 1 , P(B2) = 2 , P(B3) = 3 , .......... and P(Bn) = n 2 2 2 2 Hence B1, B2, B3,...........B n are pairwise mutually exclusive and exhaustive events as n  . The event A occurs with atleast one of the event B1, B2,........, Bn and denotes that the sum of the numbers appearing on the dice is S.

e.g. P(B1) =

52.

If even number of dice has been rolled, the probability that S = 4, is (A) very closed to

1 2

(B) very closed to

1 4

1 1 (D) very closed to 8 16 Probability that greatest number on the dice is 4 if three dice are known to have been rolled, is (C) very closed to

53.

(A) 54.

37 216

(B)

64 216

(C)

27 216

(D)

31 216

(C)

72 169

(D)

16 169

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

8 169

(B)

24 169

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 white marbles. Two marbles are drawn randomly one from each of two different bags (we do not know which bags) and found to be one white and the other black. Let P denote the probability that a marble drawn from the remaining bag is white. 55.

The probability of drawing one white and one black marble from any two of the selected bags, is (A)

25 54


(B)

25 162

(C)

27 54

(D) None


53


Rg-PB -12

If one white and one black marble has been drawn, the probability that bags A and B were selected, is (A)

57.

PROBABILITY

6 25

(B)

7 25

(C)

8 25

m (as a reduced fraction), then the value of (m + n) equals n (A) 25 (B) 33 (C) 42

(D)

9 25

If P =

(D) 47

PASSAGE - 3

Read the passage given below carefully before attempting these questions. A standard deck of playing cards has 52 cards. There are four suit (clubs, diamonds, hearts and spades), each of which has thirteen numbered cards (2, ....., 9, 10, Jack, Queen, King, Ace) In a game of card, each card is worth an amount of points. Each numbered card is worth its number (e.g. a 5 is worth 5 points) ; the Jack, Queen and King are each worth 10 points ; and the Ace is worth your choice of "either 1 point or 11 points". The object of the game is to have more points in your set of cards than your opponent without going over 21. Any set of cards with sum greater than 21 automatically loses. Here's how the game is played. You and your opponent are each dealt two cards. Usually the first card for each player is dealt face down, and the second card for each player is dealt face up. After the initial cards are dealt, the first player has the option of asking for another card or not taking any cards. The first player can keep asking for more cards until either he or she goes over 21, in which case the player loses, or stops at some number less than or equal to 21. When the first player stops at some number less than or equal to 21, the second player then can take more cards until matching or exceeding the first player's number without going over 21, in which case the second player wins, or until going over 21, in which case the first player wins. We are going to simplify the game a little and assume that all cards are dealt face up, so that all cards are visible. Assume your opponent is dealt cards and plays first. 58.

The chance that the second card will be a heart and a Jack, is (A)

59.

13 52

(C)

17 52

(D)

1 52

13 52

(B)

16 52

(C)

17 52

(D) none

Given that the first card is a Jack, the chance that it will be the heart, is (A)

61.

(B)

The chance that the first card will be a heart or a Jack, is (A)

60.

4 52

1 13

(B)

4 13

(C)

1 4

(D)

1 3

Suppose your opponent is dealt a King and a 10, and you are dealt a Queen and a 9. Being smart, your opponent does not take any more cards and stays at 20. The chance that you will win if you are allowed to take as many cards as you need, is (A)

97 564


(B)

25 282

(C)

15 188

(D)

1 6


54


PROBABILITY

Rg-PB -12

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 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

n2 2m  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 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 1   b  a    b  a  received by men is odd, is  m 2  b  a  m



8.

m

 . 

8 1 An artillery target may be either at point A with probability or at point B with probability . We have 21 9 9 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 probability

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

9.

Let p be the probability that a man aged x years will die within a year. Let A1, A2, . . . , An be n men each aged x years. Find the probability that out of these n men A1 will 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 a1 white & b1 black, B has a2 white & b2 black and C has a 3 white & b3 black 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 from the 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 the only student to have passed the examination.



55


PROBABILITY

Rg-PB -12

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

xy the statement is true is 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 log102 = 0.3010, log103 = 0.4771).

15.

Two teams A and B play a tournment. The first one to win (n + 1) games, win the series. The probability 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. n

Hence or otherwise prove that 16.

r0

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

 x  20  x  40 x  30

17.

1

 n r Cr . 2nr  1.

If

 0 is

1  3p 1  p 1 p , and are the probabilities of three mutually exclusive events, 3 2 2

then the set of all values of p is 18.

For independent events A1, . . ., An, P(Ai) =

1

i 1

, i = 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 black marble, 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 P 

21.

A is a set containing n elements. A subset P1 of A is chosen at random. The set A is reconstructed by

 B   =  A  B 

replacing the elements of P1. A subset P2 is again chosen at random. The probability that P1  P2 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 is

1 1 and the probability that neither E nor F happens is , 12 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, B draws two cards at random. The probability that there is exactly one common card, is



56


PROBABILITY

Rg-PB -12

25.

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 at plant II, 90% of the televisions are rated as of standard quality. A television is chosen at random and is found to be of standard quality. The probability that it has come from plant II is

26.

x1, x2, x3, . . . , x 50 are fifty real numbers such that xr < xr + 1 for 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 is

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

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



57


PROBABILITY

Rg-PB -12

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 a re 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 A/B  = 0.10. Then find ; (i) P AB  & P AB AB  .

3.

Three shots are fired independently at a target in succession. The probabilities that the target is hit in the first shot is 1/2 , in the second 2/3 and in the third shot is 3/4. In case of exactly one hit, the probability of destroying the target is 1/3 and in the case of exactly two hits, 7/11 and in the case of three hits is 1.0. Find the probability of destroying the target in three shots. In a game of chance each player throws two unbiased dice and scores the difference between the larger and smaller number which arise . Two players compete and one or the other wins if and only if he scores atleast 4 more than his opponent . Find the probability that neither player wins.

4.

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 " THE CHEMICAL TEST REVEALS THAT THE DRUG IS TOXIC" be denoted by S. Let P(H) = a, P S / H = P S / H = 1  a. Then show that the probability that the 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 case the 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 blind landing is P. If the blind landing instruments function normally, the plane makes a safe landing with the same probability p1 as in the case of a visual landing. If the blind landing instruments fail, then the pilot may make a safe landing with probability p 2 < 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 blind landing 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 question is to be greater than or equal to 1/3, then find the least number of chances he should be allowed.

8. (a)

n people are asked a question successively in a random order & exactly 2 of the n people know the answer : 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 r th person asked is the first person to know the answer is :

 2 ( n  r)   n ( n  1)  , if 1 < r < n .  



58


PROBABILITY

Rg-PB -12

9.

A box contains three coins two of them are fair and one two  headed. A coin is selected at random and tossed. If the head appears the coin is tossed again, if a tail appears, then another coin is selected from the remaining 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 cars running along the same highway is 3 : 2. It is known that an average of one truck in thirty trucks and two cars in fifty cars stop at the petrol pump to be filled up with the fuel. If a vehicle stops at the petrol pump to be filled up 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 were manufactured 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 the same 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 by the 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 ; the remaining 9 balls are put in a green box . What is the probability that the number of red balls in the green box plus the number of green balls in the red box is not a prime number?

13.

Two cards are randomly drawn from a well shuffled pack of 52 playing cards, without replacement. Let x be the 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 the number 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)

(b)

15.

Two numbers x & y are chosen at random from the set {1,2,3,4,....3n}. Find the probability that x²  y² is divisible by 3 . If two whole numbers x and y are randomly selected from the set of natural numbers, then find the probability that x3 + y3 is divisible by 8.

a2 A hunter’s chance of shooting an animal at a distance r is 2 (r > a) . He fires when r = 2a & if he r misses he reloads & fires when r = 3a, 4a, ..... If he misses at a distance ‘na’, the animal escapes. Find the odds 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 & E2 are independent.

17.

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



59


PROBABILITY

Rg-PB -12

18.

The probability of the simultaneous occurrence of two events, A and B, is x ; the probability of the occurrence of exactly one of A, B is y. Evaluate P(A  B) and P(A  BA  B).

19.

A fair coin is tossed n > 7 times. Let x denote 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 n black balls and 2n white 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, b and c , in the quadratic equation x2 + 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 , ... , An be mutually independent and let P(Ai )= p , for all i . What is the probability that (i) atleast one of the events occurs ? (ii) atleast m of the events occur ? (iii) exactly m of 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 1st floor.

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 than

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.

: white

w

: red

1 3

rd

of the rod ?

w + r = n

r

Select two balls, simultaneously and randomly , from an urn containing w white balls and r red balls. (All the balls are distinguishable.) If the probability that they have different colours is 1/2, show that n is 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 probabilit y of the hunter shooting down the fox at a distance of 100 m is


1 2

, find the probability of a successful hit.


60


PROBABILITY

Rg-PB -12

A person correctly recalls all but the last (unit’s) 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.

m and n are integers chosen randomly from the set {1, 2, ... , 100}, repetition allowed.

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

34.

: white balls

n

: black balls

n

A person draws balls, one-by-one, randomly and without replacement, until all the 2n balls 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 ) = 35.

2 (n !)2 (2 n )!

.

Two hunters, A and B , have equal skill at shooting down a duck successfully. (The probability of a successful shot, for either of them , is

1 2

)

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.

w and b are positive integers and n = w + b .

 (n  w )(n  w  1) (n  w )...2.1   ...   n  1 (n  1)(n  2) (n  1)...(w  1)w (Hint : use a probability model.) Prove that 1 

n w

n w

  2 n  38. A fair die is tossed n times. Show that the probability of getting an even number of sixes is  1     .  2    3   1



61


PROBABILITY

Rg-PB -12

The probability of choosing an integer , k , randomly from amongst the integers 1, 2, ... , 2 n , is proportional to log k . Show that the conditional probability of choosing the integer 2, given that an even integer is chosen, is log 2 n log 2



log (n !)

.

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.

n persons 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 6 n tickets 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 p expressed as rational in lowest form be the probability that the chosen permutation changes q from increasing to decreasing, or decreasing to increasing at most once e.g. the strings like 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). randomly. Let

44.

A bag contains N balls, some of which are white, the others are black, white being more in number than black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the 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 whereas from the pack B, one ace, one king, one queen and one jack is removed. One of these two packs is slected randomly 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 =

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

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 be p expressed as rational in their lowest form, then find (p + q). q

47.

There are two purses, one containing 3 Red balls and 1 Green ball, and the other containing 3 green balls and 1 red ball. One ball is taken from one of the purses (it is not known which) and dropped into the other, and then on drawing a ball from each purse, th ey 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.



62


PROBABILITY

Rg-PB -12

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 contains n coins of unknown value, a coin drawn at random is found to be a rupee, what is the chance that 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. A 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 two remaining bags. If the first marble drawn is red the child draws one more marble from the m same bag. The probability that the second drawn marble is green is expressed as n (where m and n are coprime). Find the value of (m + n).


Bag G R A 3 1 B 2 2 C 1 3


63


PROBABILITY

Rg-PB -12

WIND OW TO I. I.T. - JEE

Q.1 Q.2 (a)

Two cards are drawn at random from a pack of playing cards. Find the probability that one card is a heart and the other is an ace. [REE ' 2001] An urn contains 'm' white and 'n' black balls.A ball is drawn at random and is put back into the urn along with K additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white.

(b)

An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6 is thrown n times and the list of n numbers showing up is noted. What is the 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 a fair coin is tossed is 1/2, while it is 2/3 when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. What is the probability that the coin drawn is fair? [JEE ' 2002]

Q.4 (a)

(b)

Q.5 (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 selected provided he passes at least two tests. Determine the probability that the person is selected. In a combat, A targets B, and both B and C target A. The probabilities of A, B, C hitting their targets are 2/3 , 1/2 and 1/3 respectively. They shoot simultaneously and A is hit. Find the probability that B hits his target whereas C does not. [JEE' 2003] 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)

4 25

(B)

4 35

(C)

4 55

(D)

4 1155

(b)

If A and B are independent events, prove that P (A  B) · P (A'  B')  P (C), where C is an event defined that 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 which atleast 4 balls are white. Find the probability that in the next two draws exactly one white ball is drawn (leave the answer in terms of nCr ). [JEE 2004]

Q.6 (a)

(b)

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] 1 3 2 1 A person goes to office either by car, scooter, bus or train the probability of which being , , and 7 7 7 7 2 1 4 1 respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is , , and 9 9 9 9 respectively. Given that he reached office in time, then what is the probability that he travelled by a car. [JEE 2005]



64


PROBABILITY

Rg-PB -12

Q.7

Comprehension (3 questions) There are n urns each containing n + 1 balls such that the ith urn contains i white balls and (n + 1 – i) red balls. Let ui be the event of selecting ith urn, i = 1, 2, 3, ......, n and w denotes the event of getting a white ball.

(a)

If P(ui)  i where i = 1, 2, 3,....., n then Lim P(w ) is equal to n 

(A) 1 (b)

(D) 1/4

1 n 1 n 1 n 1 2 1 If n is even and E denotes the event of choosing even numbered urn ( P(u i )  ), then the value of Pw E  , n is (A)

Q.8 (a)

(C) 3/4

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

(c)

(B) 2/3

2

(B)

n2 2n  1

(B)

1

n2 2n  1

(C)

(C)

n

n n 1

(D)

(B)

1

[JEE 2006]

n 1

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

(b)

Let Ec denote the complement of an event E. Let E, F, G be pairwise independent events with P(G) > 0 and P(E  F  G) = 0. Then P(Ec  Fc | G) equals (A) P(Ec) + P(Fc) (B) P(Ec) – P(Fc) (C) P(Ec) – P(F) (D) P(E) – P(F c)

(c)

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

Statement-2:

 P(Hi ) = 1 i 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)

(b)

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 Consider the system of equations ax + by = 0, cx + dy = 0, where a, b, c, d  {0, 1}. Statement-1 : The probability that the system of equations has a unique solution is

3 . 8

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] Head Office : Andheri


65

I I T-ian’s P A CE Edu.Pvt.Ltd Q.10

PROBABILITY

Rg-PB -12

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

(b)

25 25 (B) 216 36 The probability that X  3 equals

(c)

125 25 5 (B) (C) 216 36 36 The conditional probability that X  6 given X > 3 equals

(A)

(C)

5 36

(A)

(A) Q.11 (a)

125 216

(B)

25 216

(C)

5 36

(D)

125 216

(D)

25 216

(D)

25 36

[JEE 2009]

Let  be a complex cube root of unity with   1. A fair die is thrown three times. If r 1, r 2 and r 3 are the numbers obtained on the die, then the probability that r 1  r 2  r 3  0 is (A)

(b)

1 18

(B)

1 9

(C)

A signal which can be green or red with probability

2 9

(D)

1 36

4 1 and respectively, is received by station A and then 5 5

3 . 4 If the signal received at station B is green, then the probability that the original signal was green is

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

(A)

3 5


(B)

6 7

(C)

20 23

(D)

9 20

[JEE 2010]


66


PROBABILITY

Rg-PB -12

ANSWER KEY

EXERCISE 1 (A)

1. 6. 11. 16. 21. 26. 31. 36.

A A A D B D A D

2. 7. 12. 17. 22. 27. 32. 37.

A C A D B B C A

3. 8. 13. 18. 23. 28. 33. 38.

A D C D C D B A

4. 9. 14. 19. 24. 29. 34. 39.

C C B B D B A D

5. 10. 15. 20. 25. 30. 35. 40.

A D C A A A B D

EXERCISE 1 (B)

1. 5. 9. 13. 17. 21. 25. 29. 33. 37. 39.

C,D 2. A,B,C,D A,B,C,D 6. B,C,D A,C 10. A,D A,C 14. A,B,C A,B,C 18. A,C D 22. B C 26. B C 30. B B 34. A A (A) S; (B) P; (C) R

3. 7. 11. 15. 19. 23. 27. 31. 35. 38.

A,B,C 4. A,C,D A,B,D 8. A,B,C,D A,B,D 12. A,B,C,D A,D 16. A,B,C,D B 20. A A 24. C D 28. B A 32. B B 36. C (A) S, (B) Q, (C) R; (D) P 40. (A) S ; (B) R; (C) P

EXERCI SE 1 (C)

1. 6. 11. 16.

425 721 7 37

2. 7. 12. 17.

13 18 13 5

3. 8. 13. 18.

3 27 139 160

4. 9. 14. 19.

34 445 15 29

5. 10. 15. 20.

7 122 3 101

4. 9. 14. 19. 24. 29. 34. 39. 44.

B B B C B D C A B

5. 10. 15. 20. 25. 30. 35. 40. 45.

C B C C A A C C D

EXERCISE 2 (A)

1. 6. 11. 16. 21. 26. 31. 36. 41.

C C A C A A C D D

2. 7. 12. 17. 22. 27. 32. 37. 42.


A A B C A D D D A

3. 8. 13. 18. 23. 28. 33. 38. 43.

A A A C C B C C B


67

I I T-ian’s P A CE Edu.Pvt.Ltd 46. 51. 56 61

D B B D

47. 52 57

PROBABILITY

C D B

48. 53 58

B A D

Rg-PB -12

49. 54 59

C B B

50. 55 60

B A C

EXERCISE 2 (B)

1.

m! n  m  1 nm

35 1 (ii) 128 2

2.

10/133

3.

952 : 715

8.

12

9.

1 n 1 1 p      n 

6.

(i)

10.

p1 p2 p3 , & p1  p 2  p 3 p1  p 2  p 3 p1  p 2  p 3

12.

(i)

16.

7 25

17 .

20.

1 4

21 .

23.

P(E) = 30

26.

3  n  1  (ii) 2  2n  1 n

14.

3n 4

n

1 1 , P(F) = 3 4

C2  19 C2 50

17

459 512

27 .

C5

11.

1/4

15.

P(A) =

5.

 n n r    C n .q r .p n   r 0 

1

18.

 n  1

22.

1

24.

50 663

28.

1 2

125 1296

19.

25 512

25.

27 83

 365  !  365  N ! 365!

EXERCI SE 3 1. 3.

(a) 6% , 21 % ; (b) 79 % , 44 % , (c) 9/44

5

4.

8

 20.45 %

74 /81

2.

(i) 0.82, (ii) 0.76

5.

P H / S = 1/2





6.

K [ P p1  (1  P) p2 ] K K 100 P( E)  (1  ) p1  [P p1  (1  P ) p2 ] ; P (H /A ) = 2  1  K  p  K [ P p  (1  P) p ] 100 100   1 1 2 100  100 

7.

11

10.

13.

4 9

11 663


8.

(a)

11.

4 13

14.

(a)

( n  4 ) (n  5) n ((n  1)

(5n  3) (9n  3)

(b)

5 16

9.

1/2,1/2,1/12

12.

213 1001

15.

(n + 1) :(n

 1)


68

Exercise (ALL)

Recommend Documents