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After 1st Lecture Q.1 A quadratic equation is chosen from the set of all the quadratic equations which are unchanged by squaring their roots. The chance that the chosen equation has equal roots is: (A) 1/2 (B) 1/3 (C) 1/4 (D) 2/3 Q.2 If the letters of the word "MISSISSIPPI" are written down at random in a row, the probability that no two S's occur together is : (A) 1/3 (B) 7/33 (C) 6/13 (D) 5/7 Q.3 A sample space consists of 3 sample points with associated probabilities given as 2p, p2, 4p – 1 then (A) p = 11 3 (B) 3 (C) 1/4 < p < 1/2 (D) none Q.4 A committee of 5 is to be chosen from a group of 9 people. The probability that a certain married couple will either serve together or not at all is : (A) 1/2 (B) 5/9 (C) 4/9 (D) 2/3 Q.5 There are only two women among 20 persons taking part in a pleasure trip. The 20 persons are divided into two groups, each group consisting of 10 persons. Then the probability that the two women will be in the same group is: (A) 9/19 (B) 9/38 (C) 9/35 (D) none Q.6 A bag contain 5 white, 7 black, and 4 red balls, find the chance that three balls drawn at random are all white. Q.7 If four coins are tossed, find the chance that there should be two heads and two tails. Q.8 Thirteen persons take their places at a round table, show that it is five to one against two particular persons sitting together. Q.9 In shuffling a pack of cards, four are accidentally dropped, find the chance that the missing cards should be one from each suit. Q.10 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. Q.11 There are three works, one consisting of 3 volumes, one of 4 and the other of one volume. Theyare placed on a shelf at random, prove that the chance that volumes of the same works are all together is 3/140. Q.12 The letter forming the word Clifton are placed at random in a row. What is the chance that the two vowels come together? Q.13 Consider a function f (x) that has zeroes 4 and 9. Given that Mr. A randomly selects a number from the set {– 10, – 9, – 8, 8, 9, 10}, what is the probability that Mr. A chooses a zero of f (x2)? Q.14 There are 'm' rupees and 'n' ten nP's, placed at random in a line. Find the chance of the extreme coins being both ten nP's. Q.15 A fair die is tossed. If the number is odd, find the probability that it is prime. Q.16 Three fair coins are tossed. If both heads and tails appear, determine the probability that exactly one head appears. Q.17 3 boys and 3 girls sit in a row. Find the probability that (i) the 3 girls sit together. (ii) the boys are girls sit in alternative seats. Q.18 A coin is biased so that heads is three times as likely to appear as tails. Find P (H) and P (T). Q.19 In a hand at "whist" what is the chance that the 4 kings are held by a specified player? Q.20 The probability that a positive two digit number selected at random has its tens digit at least three more then

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- MATHEMATICS Daily Practice Problems After 1st Lecture Q.1 A quadratic equation is chosen from the set of all the quadratic equations which are unchanged by squaringtheirroots. The chance thatthe chosen equation has equal roots is: (A) 1/2 (B) 1/3 (C) 1/4 (D) 2/3 Q.2 If the letters of the word "MISSISSIPPI" are written down at random in a row, the probability that no two S's occur together is : (A) 1/3 (B) 7/33 (C) 6/13 (D) 5/7 Q.3 A sample space consists of 3 samplepoints with associated probabilities given as 2p, p2, 4p – 1 then (A) p = 3 11 (B) 3 10 (C) 1/4 < p < 1/2 (D) none Q.4 Acommitteeof5is tobechosenfrom agroupof9 people.Theprobabilitythat acertain married couple will either servetogetheror not atall is : (A) 1/2 (B) 5/9 (C) 4/9 (D) 2/3 Q.5 There are onlytwo women among20persons taking part inapleasure trip. The 20 persons aredivided into twogroups,each group consistingof10 persons. Then theprobabilitythat the twowomenwill be inthesamegroupis: (A) 9/19 (B) 9/38 (C) 9/35 (D) none Q.6 Abagcontain5white,7black,and4redballs,findthechancethatthreeballsdrawnatrandomareallwhite. Q.7 If four coins are tossed, find the chance that there should be two heads and two tails. Q.8 Thirteen persons take their places at a round table, show that it is five to one against two particular persons sittingtogether. Q.9 Inshufflingapackofcards, fourareaccidentallydropped, findthechancethat themissingcards should be one from each suit. Q.10 Ahas3shares inalotterycontaining3prizesand9blanks,Bhas2sharesinalotterycontaining2 prizes and 6 blanks. Compare their chances of success. Q.11 Therearethreeworks,oneconsistingof3volumes,oneof4andtheotherofone volume.Theyareplaced onashelfat random,provethat thechancethatvolumes ofthesameworks are alltogetheris 3/140. Q.12 The letter forming the word Clifton are placed at random in a row. What is the chance that the two vowels cometogether? Q.13 Consider a function f (x)that has zeroes 4 and 9. Given that Mr.A randomlyselects anumberfrom the set {– 10, – 9, – 8, ..... 8, 9, 10}, what is the probability that Mr.Achooses a zero of f (x2)? Q.14 There are 'm' rupees and 'n' ten nP's, placed at random in a line. Find the chance of the extreme coins beingbothten nP's. Q.15 A fairdie is tossed. Ifthe numberis odd,find the probabilitythat it is prime. Q.16 Three fair coins are tossed. If both heads and tails appear, determine the probabilitythat exactly one head appears. Q.17 3 boysand 3girls sitin arow. Findtheprobabilitythat (i)the3 girls sit together. (ii)theboysare girls sit inalternativeseats. Q.18 A coin is biased so that heads is three times as likelyto appear as tails. Find P (H) and P (T). Q.19 In a handat "whist" what is the chance that the 4 kings are held bya specified player? Q.20 The probabilitythat apositivetwodigit numberselectedat random hasits tens digit at least three more thenitsunitdigitis (A) 14/45 (B) 7/45 (C) 36/45 (D) 1/6 CLASS : XIII DPP ON PROBABILITY DPP. NO.- 1
- MATHEMATICS Daily Practice Problems CLASS : XIII (XYZ) DPP ON PROBABILITY DPP. NO.- 2 After 2nd Lecture Q.1 Given two independent eventsA, B such that P(A) = 0.3, P(B) = 0.6. Determine (i) P (A and B) (ii) P (A and not B) (iii) P(notAand B) (iv) P(neitherAnor B) (v) P (A or B) Q.2 Acardisdrawnat randomfromawellshuffleddeck ofcards. Findtheprobabilitythatthecardis a (i) king or a red card (ii)cluboradiamond (iii)kingor aqueen (iv) kingor an ace (v) spade or a club (vi) neither aheart nor aking. Q.3 A coin is tossed and a die is thrown. Find the probabilitythat the outcome will be a head or a number greater than 4. Q.4 LetAand B be events such that ) A ( P ~ = 4/5, P(B) = 1/3, P(A/B) = 1/6, then (a) P(A B) ; (b) P(A B) ; (c) P(B/A) ; (d) Are A and B independent? Q.5 IfAand B are two events such that P (A) = 4 1 , P (B) = 2 1 and P (Aand B) = 8 1 , find (i) P (A or B), (ii) P(notAand not B) Q.6 A5digitnumberisformedbyusingthedigits0,1,2,3,4 &5withoutrepetition.Theprobabilitythatthe numberisdivisibleby6is: (A) 8 % (B) 17 % (C) 18 % (D) 36 % Q.7 An experiment results in four possible out comes S1, S2, S3 & S4 with probabilities p1, p2, p3 & p4 respectively.Whichoneofthefollowingprobabilityassignment ispossbile. [Assume S1 S2 S3 S4 aremutuallyexclusive] (A) p1 = 0.25 , p2 = 0.35 , p3 = 0.10 , p4 = 0.05 (B) p1 = 0.40 , p2 = 0.20 , p3 = 0.60 , p4 = 0.20 (C) p1 = 0.30 , p2 = 0.60 , p3 = 0.10 , p4 = 0.10 (D) p1 = 0.20 , p2 = 0.30 , p3 = 0.40 , p4 = 0.10 Q.8 In throwing3 dice, the probabilitythat atleast 2 ofthe three numbers obtained are same is (A) 1/2 (B) 1/3 (C) 4/9 (D) none Q.9 Thereare4defectiveitemsinalot consistingof10items.Fromthis lotweselect 5itemsat random. The probabilitythat therewillbe2defectiveitemsamongthem is (A) 2 1 (B) 5 2 (C) 21 5 (D) 21 10 Q.10 From a pack of 52 playingcards, face cards and tens are removed and kept aside then a card is drawn at random fromthe ramainingcards.If A : The event that the card drawn is an ace H : The event that the card drawn is a heart S : The event that the card drawn is a spade thenwhichofthefollowingholds? (A) 9 P(A) = 4 P(H) (B) P(S) = 4P (A H) (C) 3 P(H) = 4 P(A S) (D) P(H) = 12 P(A S)
- Q.11 6marriedcouples arestandinginaroom. If4peoplearechosenatrandom, then thechancethat exactly one marriedcouple is amongthe4 is : (A) 33 16 (B) 33 8 (C) 33 17 (D) 33 24 Q.12 The chance that a 13 card combinationfrom a pack of52 playing cards is dealt to a player in a game of bridge, in which 9 cards are of the same suit, is (A) 13 52 4 39 9 13 C C · C · 4 (B) 13 52 4 39 9 13 C C · C · ! 4 (C) 13 52 4 39 9 13 C C · C (D) none Q.13 If two of the 64 squares are chosen at random ona chess board, the probabilitythat theyhave a side in commonis: (A) 1/9 (B) 1/18 (C) 2/7 (D) none Q.14 Two red counters, three green counters and 4 blue counters are placed in a row in random order.The probabilitythat no twoblue counters are adjacent is (A) 99 7 (B) 198 7 (C) 42 5 (D) none Q.15 Theprobabilitiesthat astudentwillreceiveA,B,CorDgradeare0.40, 0.35,0.15 and0.10respectively. Findtheprobabilitythat astudentwill receive (i) not anAgrade (ii) B or C grade (iii) at most C grade Q.16 Inasinglethrowofthreedice, determinethe probabilityofgetting (i) a total of 5 (ii) a total of at most 5 (iii) a total of at least 5. Q.17 A dieis thrownonce. IfE is the event "the numberappearing is a multiple of 3" and F is theevent "the numberappearingiseven",findtheprobabilityoftheevent"EandF".AretheeventsEandFindependent? Q.18 Inthetwodiceexperiment, ifEistheeventofgettingthe sumofnumberondiceas11 and Fis theevent of gettinga number other than 5 on the first die, find P(E and F).AreE and F independent events? Q.19 Anatural numberx is randomlyselectedfrom theset offirst 100 naturalnumbers.Find the probability thatitsatisfiestheinequality. x + 100 x > 50. Q.20 3studentsAandBandCareinaswimmingrace.AandBhavethesameprobabilityofwinningandeach is twice as likelyto win as C. Find the probabilitythat B or C wins.Assume no two reach the winning pointsimultaneously. Q.21 A box contains 7 tickets, numbered from 1 to 7 inclusive. If 3 tickets aredrawn from the box without replacement, oneata time, determine theprobabilitythattheyarealternativelyeither odd-even-odd or even-odd-even. Q.22 5 differentmarbles areplacedin5differentboxes randomly.Find theprobabilitythatexactlytwo boxes remain empty. Giveneachbox can hold anynumberof marbles. Q.23 SouthAfricancricketcaptainlost thetossofacoin13timesoutof14.Thechanceofthishappeningwas (A) 13 2 7 (B) 13 2 1 (C) 14 2 13 (D) 13 2 13 Q.24 There areten prizes, fiveA's, three B's and two C's, placed in identical sealed envelopes for thetop ten contestants inamathematics contest. Theprizes areawardedbyallowingwinners to selectan envelope atrandomfromthoseremaining.When the8th contestantgoestoselecttheprize, theprobabilitythat the remainingthree prizes are oneA, one B and one C, is (A) 1/4 (B) 1/3 (C) 1/12 (D) 1/10
- MATHEMATICS Daily Practice Problems CLASS : XIII (XYZ) DPP ON PROBABILITY DPP. NO.- 3 After 3rd Lecture Q.1 Wheneverhorses a,b, cracetogether, their respectiveprobabilities ofwinning the race are 0.3, 0.5 and 0.2respectively.Iftheyracethreetimestheprobabilitythat“thesamehorsewinsallthethreeraces”and the probablitythat a, b, c each wins one race, are respectively (Assume dead heat) (A) 8 50 ; 9 50 (B) 16 100 , 3 100 (C) 12 50 ; 15 50 (D) 10 50 ; 8 50 Q.2 LetA& B be two events. Suppose P(A) = 0.4, P(B) = p & P(A B) = 0.7.The value of p for which A & Bare independent is : (A) 1/3 (B) 1/4 (C) 1/2 (D) 1/5 Q.3 A & B are two independent events such that P ) A ( = 0.7, P ) B ( = a & P(A B) = 0.8, then, a = (A) 5/7 (B) 2/7 (C) 1 (D) none Q.4 A pair of numbers is picked up randomly (without replacement) from the set {1, 2, 3, 5, 7, 11, 12,13, 17, 19}.The probabilitythat the number 11 was picked given that the sum of the numbers waseven, is nearly: (A) 0.1 (B) 0.125 (C) 0.24 (D) 0.18 Q.5 For abiased die the probabilities for the diffferent faces to turn up are given below : Faces : 1 2 3 4 5 6 Probabilities: 0.10 0.32 0.21 0.15 0.05 0.17 Thedie istossed& youaretold that either faceone orface twohasturned up. Thenthe probabilitythat it is faceone is : (A) 1/6 (B) 1/10 (C) 5/49 (D) 5/21 Q.6 Adeterminantis chosenat randomfrom theset ofall determinants oforder2 with elements 0or1 only. Theprobabilitythat thedeterminant chosenhas thevaluenonnegativeis : (A) 3/16 (B) 6/16 (C) 10/16 (D) 13/16 Q.7 15 coupons are numbered 1, 2, 3,..... , 15 respectively. 7 coupons are selected at random one at a time with replacement.The probabilitythat thelargest numberappearingon aselected coupon is 9 is : (A) 6 16 9 (B) 7 15 8 (C) 7 5 3 (D) 7 7 7 15 8 9 Q.8 A cardis drawn& replacedin an ordinarypack of52 playing cards.Minimum numberoftimesmust a card be drawn so that there is atleast an even chance of drawing a heart, is (A) 2 (B) 3 (C) 4 (D) morethan four Q.9 An electrical systemhas open-closedswitches S1, S2 and S3 as shown. The switches operate independentlyof one another and thecurrent will flow fromAto B either if S1 is closedorif both S2 and S3 areclosed. If P(S1)=P(S2) =P(S3) =1/2, find the probabilitythat thecircuit willwork.
- Q.10 A certain team wins with probability0.7, loses with probability0.2 and ties withprobability0.1. The team plays threegames. Findtheprobability (i) that theteam wins at least twoof the games, but lose none. (ii) that theteam wins at least one game. Q.11 Anintegerischosenatrandomfromthefirst200positiveintegers.Findtheprobabilitythattheintegeris divisible by6 or 8. Q.12 Aclerkwasaskedtomailfourreportcardstofourstudents.Headdressesfourenvelopsthatunfortunately paidno attentiontowhichreport cardbeput in whichenvelope.What is theprobabilitythat exactlyone of the students received his (or her) own card? Q.13 Find the probabilityof at most two tails or at least two heads in a toss of three coins. Q.14 What is the probabilitythat ina group of (i) 2 people, both will have the samedate of birth. (ii) 3 people, at least 2 will have the same date of birth. Assume the year to be ordinarryconsisting of 365 days. Q.15 The probabilitythat a person will get an electric contract is 5 2 and the probabilitythat he will not get plumbingcontractis 7 4 .Iftheprobabilityofgettingatleastonecontractis 3 2 ,whatis theprobabilitythat hewill get both? Q.16 Five horses competein a race. Johnpicks two horses at random and bets on them.Find the probability that Johnpicked the winner.Assumedead heat. Q.17 Twocubeshavetheirfacespaintedeitherred orblue.Thefirstcubehas fivered faces andoneblueface. When the two cubes are rolled simultaneously, the probability that the two top faces show the same colour is 1/2. Number of red faces on the second cube, is (A) 1 (B) 2 (C) 3 (D) 4 Q.18 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 remainingseven balls areput in a green box.If the probabilitythat thenumber ofred balls in the greenbox plus the number of green balls in the red box is not a prime number, is q p where p and q arerelativelyprime,thenfindthe value of (p+ q) Q.19 A bag contains 6R, 4W and 8B balls. If 3 balls are drawn simultaneously at random determine the probabilityoftheevent (a) all 3 are red ; (b) all 3 are black ; (c) 2 are white and 1 is red ; (d) at least 1 is red ; (e) 1 of each colour are drawn (f) the balls are drawn in the order of red, white, black. Q.20 The odds that a book will be favourablyreviewed bythree independent critics are 5 to 2, 4 to 3, and 3 to 4respectively.What is theprobabilitythat of thethree reviewsa majoritywill be favourable? Q.21 When three cardsare drawn from astandard 52-card deck, what isthe probabilitytheyare all the same rank? (e.g. all three are kings). Q.22 A, B, C in order cut a pack of cards, replacing them after each cut, on condition that the first who cuts a spade shall win a prize. Findtheir respective chances. Q.23 AandBinorderdrawfrom apursecontaining3 rupees and4nP's, find theirrespectivechances of first drawing a rupee, the coins once drawn not being replaced.
- MATHEMATICS Daily Practice Problems CLASS : XIII (XYZ) DPP ON PROBABILITY DPP. NO.- 4 After 4th Lecture Q.1 There are n different gift coupons, each of which can occupyN(N > n) different envelopes, with the same probability1/N P1:TheprobabilitythattherewillbeonegiftcouponineachofndefiniteenvelopesoutofNgivenenvelopes P2:Theprobabilitythattherewillbeonegiftcouponineachofnarbitraryenvelopesoutof Ngivenenvelopes Considerthefollowingstatements (i) P1 = P2 (ii) P1 = n N ! n (iii) P2 = ! ) n N ( N ! N n (iv) P2 = ! ) n N ( N ! n n (v) P1 = n N ! N Now,whichofthefollowingistrue (A)Only(i) (B)(ii)and(iii) (C)(ii)and (iv) (D)(iii)and(v) Q.2 Theprobabilitythatanautomobilewillbestolenandfoundwithingoneweekis0.0006.Theprobabilitythat anautomobilewillbestolenis0.0015.Theprobabilitythatastolenautomobilewillbefoundinoneweekis (A) 0.3 (B) 0.4 (C) 0.5 (D) 0.6 Q.3 Onebagcontains 3white& 2black balls, andanothercontains 2 white &3 black balls.Aball is drawn from the secondbag& placedinthe first, thenaball is drawn from thefirst bag & placed in thesecond. Whenthepairoftheoperationsis repeated, theprobabilitythat thefirst bagwill contain5 whiteballsis: (A) 1/25 (B) 1/125 (C) 1/225 (D) 2/15 Q.4 A child throws 2 fair dice. If the numbers showing are unequal, he adds them together to get his final score.Ontheotherhand,ifthenumbers showingareequal,hethrows2moredice&addsall 4numbers showingtogethisfinal score. Theprobabilitythat hisfinal scoreis 6 is: (A) 1296 145 (B) 1296 146 (C) 1296 147 (D) 1296 148 Q.5 A persondraws a card from a packof52cards, replaces it &shuffles the pack. Hecontinues doing this till he drawsaspade. Theprobabilitythat hewill failexactlythefirst twotimes is : (A) 1/64 (B) 9/64 (C) 36/64 (D) 60/64 Q.6 Indicate the correct order sequence in respect of the following : I. Iftheprobabilitythatacomputerwillfailduringthefirsthourofoperationis0.01,thenifweturn on100 computers, exactlyonewillfail in the firsthourofoperation. II. Amanhastenkeysonlyoneofwhichfitsthelock. Hetriestheminadooronebyonediscarding the one hehas tried. The probabilitythat fifth keyfits thelock is 1/10. III. Given the eventsAand B in a sample space. If P(A) = 1, thenAand B are independent. IV. Whenafairsixsideddieistossedonatabletop,thebottomfacecannotbeseen.Theprobability that the product of the numbers on the five faces that can be seen is divisible by6is one. (A) FTFT (B)FTTT (C) TFTF (D) TFFF Q.7 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 Q.8 A bagcontains 3 R &3 G balls anda person draws out 3 at random. Hethen drops 3 blueballs into the bag &again draws out 3 at random. The chancethat the 3 laterballs being all of different colours is (A) 15% (B) 20% (C) 27% (D) 40% Q.9 Abiasedcoin withprobabilityP,0<P<1, ofheads istosseduntil aheadappears forthefirsttime.Ifthe probabilitythat the number of tosses required is even is 2/5 then the valueof P is (A) 1/4 (B) 1/6 (C) 1/3 (D) 1/2 Q.10 If a, b N then the probabilitythat a2 + b2 is divisible by5 is (A) 9/25 (B) 7/18 (C) 11/36 (D) 17/81
- Q.11 Inanexamination,onehundredcandidatestookpaperinPhysicsandChemistry.Twentyfivecandidates failed in Physics only. Twentycandidates failed in chemistryonly. Fifteen failed in both Physics and Chemistry.Acandidate is selected at random. The probability that he failed either in Physics or in Chemistrybut notinbothis (A) 9/20 (B) 3/5 (C) 2/5 (D) 11/20 Q.12 When a missile is fired from a ship, the probabilitythat it is intercepted is 1/3. The probabilitythat the missile hitsthetarget, giventhatitis notinterceptedis3/4.Ifthreemissilesarefiredindependentlyfrom theship,theprobabilitythat all three hitstheirtargets, is (A) 1/12 (B) 1/8 (C) 3/8 (D) 3/4 Q.13 Ineachofaset ofgamesit is 2to1infavourofthewinneroftheprevious game. What isthechancethat the playerwho wins the firstgameshall wins threeat least of the next four? Q.14 A coinis tossed n times, what is thechancethat the head will present itself an odd number of times? Q.15 Afairdieistossedrepeatidly.Awinsifitis 1or2ontwoconsecutivetossesandBwinsifitis3, 4, 5 or 6 ontwoconsecutive tosses.TheprobabilitythatAwins ifthedie is tossedindefinitely, is (A) 1/3 (B) 5/21 (C) 1/4 (D) 2/5 Q.16 Counters marked 1, 2, 3 are placed in a bag, and one is withdrawn and replaced. The operation being repeatedthree times, what is thechance of obtaininga total of 6? Q.17 Anormal coinis continued tossingunless ahead is obtainedforthefirst time. Findtheprobabilitythat (a) number of tosses needed are at most 3. (b) number of tosses are even. Q.18 A purse contains 2 six sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives.Adieis pickedupandrolled. Becauseofsomesecret magneticattraction oftheunfairdie,thereis 75%chanceofpickingtheunfairdieanda25%chanceofpickingafairdie. Thedieisrolledand shows up the face 3. The probabilitythat a fair die was picked up, is (A) 1/7 (B) 1/4 (C) 1/6 (D) 1/24 Q.19 Before a race the chance of three runners,A, B, C were estimated to be proportional to 5, 3, 2, but during the race Ameets with an accident which reduces his chance to 1/3. What are the respective chance of B and C now? Q.20 Afair coin is tossed a large number of times.Assuming the tosses are independent which one of the followingstatement,isTrue? (A)Oncethenumberofflipsislargeenough,thenumberofheadswillalwaysbeexactlyhalfofthetotal number of tosses. For example, after 10,000 tosses one should have exactly5,000 heads. (B) The proportion of heads will be about 1/2 and this proportion will tend to get closer to 1/2 as the number oftosses inreases (C)Asthenumberof tossesincreases, anylongrunofheads will bebalanced byacorrespondingrun of tails so that the overall proportion of heads is exactly1/2 (D)All of the above Q.21 A andBeach throw simultaneouslyapair of dice. Find theprobabilitythat theyobtain the same score. Q.22 A is one of the 6 horses entered for a race, and is to be ridden byone of two jockeys B or C. It is 2 to 1thatB ridesA,inwhichcaseall thehorses areequallylikelyto win; if C ridesA, hischance is trebled, whataretheodds against his winning? Direction for Q.23 to Q.25 LetSandTaretwoeventsdefinedonasamplespacewithprobabilities,P(S)=0.5,P(T)=0.69,P(S/T)=0.5 Q.23 Events S and T are: (A)mutuallyexclusive (B)independent (C)mutuallyexclusiveandindependent (D)neithermutuallyexclusivenorindependent Q.24 The value of P(S and T) (A) 0.3450 (B) 0.2500 (C) 0.6900 (D) 0.350 Q.25 The value of P(S or T) (A) 0.6900 (B) 1.19 (C) 0.8450 (D) 0
- MATHEMATICS Daily Practice Problems CLASS : XIII (XYZ) DPP ON PROBABILITY DPP. NO.- 5 After 5th Lecture Q.1 EventsAand C are independent. If the probabilities relatingA, B and C are P(A) = 1/5; P(B) = 1/6; P(A C) = 1/20; P(B C) = 3/8 then (A) events B and C are independent (B)events Band C aremutuallyexclusive (C)events Band C areneitherindependentnormutuallyexclusive (D) events B and C are equiprobable Q.2 Assumethatthebirthofaboyorgirltoacoupletobeequallylikely,mutuallyexclusive,exhaustiveand independent oftheotherchildrenin thefamily. Foracouplehaving6 children, theprobabilitythat their "three oldest are boys" is (A) 20/64 (B) 1/64 (C) 2/64 (D) 8/64 Q.3 Aand B playa game.Ais to throw a die first, and is to win if he throws 6, If he fails B is to throw, and to win ifhe throws 6 or 5. If he fails,Ais to throw again and to win with 6 or 5 or 4, and so on, find the chance of each player. Q.4 BoxAcontains 3 red and 2 blue marbles while box B contains 2 red and 8 blue marbles.Afair coin is tossed. If thecoin turns up heads, a marble is drawnfromA, if it turns up tails, a marbleis drawn from bag B.The probabilitythat a red marble is chosen, is (A) 1/5 (B) 2/5 (C) 3/5 (D) 1/2 Q.5 Aexaminationconsistsof8questionsineachofwhichoneofthe5alternativesisthecorrectone.Onthe assumptionthat acandidatewhohas donenopreparatoryworkchoosesforeachquestionanyoneofthe fivealternativeswithequalprobability,theprobabilitythathegetsmorethanonecorrectanswerisequalto (A) (0.8)8 (B) 3 (0.8)8 (C) 1 (0.8)8 (D) 1 3 (0.8)8 Q.6 The germinationofseeds is estimatedbya probabilityof0.6.Theprobabilitythat out of11 sown seeds exactly5or6will springis: (A) 11 5 5 10 6 5 C . (B) 11 5 5 6 11 5 ) 2 3 ( C (C) 11C5 11 6 5 (D) none of these Q.7 Theprobabilityofobtainingmore tails than heads in6 tosses of afaircoins is : (A) 2/64 (B) 22/64 (C) 21/64 (D) none Q.8 Aninstrumentconsistsoftwounits.Eachunitmustfunctionfortheinstrumenttooperate.Thereliability ofthefirstunitis0.9&thatofthesecondunitis0.8.Theinstrumentistested&fails.Theprobabilitythat "onlythefirstunitfailed&thesecondunit issound"is: (A) 1/7 (B) 2/7 (C) 3/7 (D) 4/7 Q.9 LotAconsists of3Gand2Darticles.LotBconsistsof4Gand1Darticle.AnewlotCisformedbytaking 3articlesfromAand2 fromB.Theprobabilitythatanarticlechosenatrandom fromCisdefective,is (A) 1/3 (B) 2/5 (C) 8/25 (D) none Q.10 'A' and 'B' each have a bag that contains one ball of each of the colours blue, green, orange, red and violet. 'A'randomlyselects one ball from his bagand puts it into B's bag. 'B' then randomlyselects one ball from his bagand puts it intoA's bag.The probabilitythat afterthis process the containts of thetwo bags are the same, is (A) 1/6 (B) 1/5 (C) 1/3 (D) 1/2 Q.11 If mn coins havebeen distributed into m purses, ninto each find (1) the chance that two specified coins will be found in thesame purse, and (2) what thechancebecomes whenrpurses havebeen examined and found not tocontain eitherof thespecifiedcoins.
- Q.12 A box has four dice in it. Three of them are fair dice but the fourth one has the number five onall of its faces.Adieis chosen at random from thebox andis rolled three times andshows up thefacefiveon all the three occassions. The chance that the die chosen was a rigged die, is (A) 217 216 (B) 219 215 (C) 219 216 (D) none Q.13 OnaSaturdaynight 20%of alldriversinU.S.A.areundertheinfluenceofalcohol.Theprobabilitythat adriver undertheinfluenceofalcoholwill havean accident is0.001. Theprobabilitythatasoberdriver will havean accident is 0.0001. If a car on a saturdaynight smashed into a tree, theprobabilitythat the driverwasundertheinfluenceofalcohol, is (A) 3/7 (B) 4/7 (C) 5/7 (D) 6/7 Q.14 The contents of three Urns w.r.t. the Red, White and Green balls is as shown in the table given. 2 1 3 III 1 2 3 II 1 3 2 I G W R Urn A coin when tossed is twice as likely to come heads as compared to tails. Such a coin is tossed two times. If bothheads and tails arepresent then 3 balls are drawn simultaneouslyfrom theUrn-I, if head appears on both the occassions then 3 balls are drawn in a similar manner from Urn-IIand if no head appears in both the tosses then 3 balls from the Urn-IIIare drawn in the same manner. The probabilitythat 3 drawn balls are 1 each ofdifferent colours, is (A) 10% (B) 15% (C) 30% (D) 90% Direction for Q.15 to Q.17 (3 Questions) AJEEaspirantestimatesthatshewillbesuccessfulwithan80percentchanceifshestudies10hoursper day, with a60 percent chance ifshe studies 7 hours per dayand with a 40 percent chanceif 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 Q.15 Thechanceshewill besuccessful, is (A) 0.28 (B) 0.38 (C) 0.48 (D) 0.58 Q.16 Given that she is successful, thechance she studied for4 hours, is (A) 6/12 (B) 7/12 (C) 8/12 (D) 9/12 Q.17 Given that she does not achieve success, the chance she studied for 4 hour, is (A) 18/26 (B) 19/26 (C) 20/26 (D) 21/26 Q.18 Aletterisknowntohavecomeeitherfrom LondonorClifton.Onthepostmarkonlythetwoconsecutive letters ONare legible. What is the chance that it came from London? Q.19 A purse contains n coins of unknown value, acoin drawnat random is found to be a rupee,what is the chancethatisittheonlyrupeeinthepurse?Assumeallnumbersofrupeecoinsinthepurseisequallylikely. Q.20 One of a pack of 52 cards has been lost, from the remainder of the pack two cards are drawn and are found to be spades, find the chance that the missing card is a spade. Q.21 A,Baretwoinaccuratearithmeticianswhosechanceofsolvingagiven questioncorrectlyare (1/8)and (1/12) respectively.Theysolve a problem and obtained the same result. If it is 1000 to 1 against their making thesame mistake, find thechancethat the result is correct. Q.22 Weconductanexperimentwherewerolladie5times.Theorderinwhichthenumberreadoutisimportant. (a) What is thetotal numberof possibleoutcomesof this experiment? (b) What is the probabilitythat exactly3 times a "2"appears in the sequence (sayevent E)? (c) What is the probabilitythat the face 2 appears at least twice (sayevent F)? (d) Which of the following are true : E F, F E? (e) Compute the probabilities : P(E F), P(E/F), P(F/E) (f) Are the events E and F independent?
- MATHEMATICS Daily Practice Problems CLASS : XIII (XYZ) DPP ON PROBABILITY DPP. NO.- 6 After 6th Lecture Q.1 Abowlhas6redmarblesand3greenmarbles.Theprobabilitythat ablind foldedpersonwilldrawared marble onthesecond drawfrom the bowl without replacingthe marble from thefirst draw, is (A) 2/3 (B) 1/4 (C) 5/12 (D) 5/8 Q.2 5outof6personswhousuallyworkinanofficeprefercoffeeinthemid morning,theotheralwaysdrink tea. This morningof the usual 6, only3 arepresent. The probabilitythat one of them drinks tea is : (A) 1/2 (B) 1/12 (C) 25/72 (D) 5/72 Q.3 Pal’s gardner is not dependable, the probability that he will forget to water therosebush is 2/3.The rose bushis in questionablecondition. Any how if watered, the probabilityofits withering is 1/2&if notwateredthen theprobabilityof itswitheringis 3/4.Pal wentoutof station&afterreturninghe finds that rose bush has withered. What is the probability that the gardner did not water the rose bush. Q.4 Theprobabilitythat aradarwilldetect an object inone cycle is p.Theprobabilitythat theobject will be detectedin ncycles is : (A) 1 pn (B) 1 (1 p)n (C) pn (D) p(1 – p)n–1 Q.5 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 probabilitythat thenumberis even, is (A) 5/9 (B) 4/9 (C) 1/2 (D) 2/3 Q.6 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. thenwhichofthefollowingistrue? (A) P (A) = 4/17 ; P (B) = 1/13 (B) P (A) = 1/13 ; P (B) = 1/13 (C) P (A) = 1/13 ; P (B) = 1/17 (D) P (A) = 16/221 ; P (B) = 4/51 Q.7 If 2 p) 2 1 ( & 4 p) 1 ( , 3 p) 3 1 ( aretheprobabilities ofthreepairwisemutuallyexclusiveevents defined on a sample space S, then the true set of all values of p is (A) 2 1 , 3 1 (B) 1 , 3 1 (C) 3 1 , 4 1 (D) 2 1 , 4 1 Q.8 Alot contains 50defective & 50 non defective bulbs .Two bulbs aredrawn at random, oneat a time, with replacement . The events A, B, C are defined as : A ={ the first bulbis defective}; B = { the second bulb is non defective} C = { the two bulbs are both defective or both non defective} Determinewhether (i)A,B,C arepair wise independent (ii)A,B,C are independent Q.9 An Urn contains 'm' white and 'n' black balls.All the balls except for one ball, are drawn from it. The probabilitythatthelast ball remainingintheUrniswhite,is (A) n m m (B) n m n (C) ! ) n m ( 1 (D) ! ) n m ( n m Q.10 A Urn contains 'm' white and 'n' black balls. Balls are drawn one by one till all the balls are drawn. Probabilitythatthesecond drawn ball is white, is (A) n m m (B) ) 1 n m )( n m ( ) 1 n m ( n (C) ) 1 n m )( n m ( ) 1 m ( m (D) ) 1 n m )( n m ( mn
- Q.11 Mr. Dupont isaprofessional wine taster.Whengiven aFrenchwine, he will identifyit with probability 0.9 correctlyas French, and will mistake it for a Californian wine with probability0.1. When given a Californianwine,hewill identifyitwithprobability0.8correctlyasCalifornian, andwill mistakeitfora Frenchwinewithprobability0.2.SupposethatMr.Dupontisgiventenunlabelledglassesofwine,three withFrenchandseven withCalifornian wines. He randomlypicksaglass, tries thewine, and solemnly says : "French".Theprobabilitythatthewinehetastedwas Californian, is nearlyequal to (A) 0.14 (B) 0.24 (C) 0.34 (D) 0.44 Q.12 LetA, B & C be 3 arbitraryevents defined on a sample space 'S' and if, P(A) + P(B) + P(C) = p1 , P(A B) + P(B C) + P(C A) = p2 & P(A B C) = p3, then the probabilitythatexactlyone of thethree events occurs is given by: (A) p1 p2 + p3 (B) p1 p2 + 2p3 (C) p1 2p2 + p3 (D) p1 2p2 + 3p3 Q.13 Threenumbersarechosen at random without replacement from {1, 2, 3,...... , 10}.The probabilitythat theminimumofthechosennumbers is3ortheirmaximumis7is (A) 1/2 (B) 1/3 (C) 1/4 (D) 11/40 Q.14 A biased coin which comes up heads three times as often as tails is tossed. If it shows heads, a chip is drawn from urn–I which contains 2 white chips and 5 red chips. If the coin comes up tails, a chip is drawnfrom urn–IIwhichcontains 7 whiteand4 redchips.Given that ared chip was drawn, what is the probabilitythat thecoin came up heads? Q.15 In a college, four percent of the men and one percent of the women are taller than 6 feet. Further 60 percent of the students are women. If a randomly selected person is taller than 6 feet, find the probabilitythat thestudent is awomen. Q.16 Ifatleastonechildinafamilywith3childrenisaboythentheprobabilitythat2ofthechildrenareboys,is (A) 3/7 (B) 1/4 (C) 1/3 (D) 3/8 Q.17 The probabilities of events,A B,A, B &A B are respectively inA.P. with probabilityof second term equal to the common difference.Therefore the eventsAand B are (A)compatible (B)independent (C) suchthat one of them must occur (D) such that one is twice as likelyas the other Q.18 Fromanurncontainingsixballs,3whiteand3blackones,apersonselectsatrandomanevennumberofballs (allthedifferentwaysofdrawinganevennumberofballsareconsideredequallyprobable,irrespectiveoftheir number).Thentheprobabilitythattherewillbethesamenumberofblackandwhiteballsamongthem (A) 4/5 (B) 11/15 (C) 11/30 (D) 2/5 Q.19 Onepursecontains 6coppercoins and1 silvercoin; asecondpursecontains 4 coppercoins. Five coins are drawnfrom the first purseand put into thesecond, and then 2 coins are drawn from the second and put intothe first.The probabilitythatthesilver coinis in the secondpurse is (A) 1/2 (B) 4/9 (C) 5/9 (D) 2/3 Q.20 7 persons are stopped ontheroad at random andaskedabout theirbirthdays. Ifthe probabilitythat 3 of them arebornonWednesday, 2onThursdayand theremaining2 onSundayis 6 7 K , thenK isequal to (A) 15 (B) 30 (C) 105 (D) 210 Q.21 TwobusesAandBarescheduledtoarrive atatowncentral busstation atnoon.Theprobabilitythat bus A willbe late is 1/5.Theprobabilitythat bus Bwill be lateis 7/25. The probabilitythat the bus Bis late given that busAis late is 9/10.Then the probabilities (i) neither bus will be late on aparticularday and (ii) busAis late given that bus B is late, are respectively (A) 2/25 and 12/28 (B) 18/25 and 22/28 (C) 7/10 and 18/28 (D) 12/25 and 2/28
- MATHEMATICS Daily Practice Problems Q.1 Aboxcontains anormalcoinandadoublyheadedcoin.Acoinselected atrandomandtossedtwice,fell headwise onboth the occasions. The probabilitythat the drawncoin is a doublyheaded coin is (A) 2 3 (B) 5 8 (C) 3 4 (D) 4 5 Q.2 Abox contains 5 red and 4 white marbles.Two marbles are drawn successivelyfrom the box without replacementandtheseconddrawnmarbledrawnisfoundtobewhite.Probabilitythatthefirstmarble is alsowhileis (A) 8 3 (B) 2 1 (C) 3 1 (D) 4 1 Q.3 A and B in order drawa marble from bagcontaining 5 white and 1 red marbles with the condition that whosoeverdraws theredmarblefirst, winsthegame. Marbleoncedrawn bythem arenot replaced into the bag.Thentheir respectivechancesof winning are (A) 3 2 & 3 1 (B) 5 3 & 5 2 (C) 5 2 & 5 3 (D) 2 1 & 2 1 Q.4 In a maths paper there are 3 sections A, B & C. Section A 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 probabilityof the student passing inA, B & C are p, q & 1/2 respectively. If the probability that the student is successfulis 1/2then: (A) p = q = 1 (B) p = q = 2 1 (C) p = 1, q = 0 (D) p = 1, q = 2 1 Q.5 Abox contains 100 tickets numbered1, 2, 3,.... ,100.Two tickets are chosen at random. It is given that themaximumnumberonthetwochosenticketsisnotmorethan10.Theminimumnumberonthemis5, with probability (A) 9 1 (B) 11 2 (C) 19 3 (D) none Q.6 Sixteenplayers s1 ,s2 ,..... , s16 playin a tournament. Theyaredivided into eight pairsat random.From eachpairawinneris decidedonthe basis ofagameplayed betweenthetwo players ofthe pair.Assume that all the players are of equal strength. Theprobabilitythat "exactlyone of thetwo players s1 & s2 is amongtheeightwinners"is (A) 15 4 (B) 15 7 (C) 15 8 (D) 15 9 Q.7 The number 'a' is randomly selected from the set {0, 1, 2, 3, ...... 98, 99}. The number 'b' is selected from the same set. Probabilitythat the number 3a + 7b has a digit equal to 8 at the units place, is (A) 16 1 (B) 16 2 (C) 16 4 (D) 16 3 CLASS : XIII (XYZ) DPP ON PROBABILITY DPP. NO.- 7
- Q.8 A circleofradius 3 is placed at the centre ofa circle of radius r> 3 such that the length of achord of the largercircletangentto thesmallercircleis8.Theprobabilitythat pointrandomlyselectedfrom insideof thelargercirclelies insidetheannularregionofthetwo circles is (A) 4 3 (B) 25 9 (C) 25 16 (D) 3 7 Q.9 We aregiven two urns as follows : UrnAcontains 5 red marbles, 3 white marbles and 8 blue marbles. Urn B contains 3 red marbles and 5 white marbles Afairdiceistossedif3or6appears,amarbleischosenfromB,otherwiseamarbleischosenfromA.Find theprobabilitythat(i)aredmarbleis chosen, (ii)awhitemarbleischosen,(iii)abluemarbleischosen. (UseTree Diagram) Q.10 Two boys A and B find the jumble of n ropes lying on the floor. Each takes hold of one loose end randomly.Iftheprobabilitythat theyare bothholdingthesamerope is 101 1 thenthenumberofropes is equal to (A) 101 (B) 100 (C) 51 (D) 50 Direction for Q.11 to Q.14 (4 Questions) Read thepassagegivenbelow carefullybeforeattemptingthesequestions. Astandarddeckofplayingcards has52cards.Therearefoursuit (clubs, diamonds,hearts andspades), 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.a5isworth5points);theJack,QueenandKingareeachworth10points;andtheAceisworthyour choice of "either1pointor11points".Theobject ofthegameis to havemorepoints in yoursetofcards thanyouropponentwithoutgoingover21.Anyset ofcardswithsumgreaterthan21automaticallyloses. Here's how the game is played.Youand your opponent are each dealt two cards. Usuallythe first card for each player is dealt face down, and the second card for eachplayeris dealt face up.After the initial cards aredealt,thefirst playerhas theoption ofaskingforanothercardornot takinganycards. Thefirst playercankeepaskingfor morecards until eitherheorshegoes over21, in which casetheplayerloses, or stops at some number less than or equal to 21. When the first player stops at some number less than or equalto21, the second playerthen can take morecards until matchingorexceeding the first player's numberwithout goingover 21, in which case the second player wins, or until goingover 21, in which casethefirst player wins. Weare goingtosimplifythegamea littleandassumethat all cards aredealtfaceup, so thatall cards are visible.Assumeyouropponent is dealt cards and plays first. Q.11 The chance that the second card will be a heart and a Jack, is (A) 52 4 (B) 52 13 (C) 52 17 (D) 52 1 Q.12 The chance that the first card will be a heart or a Jack, is (A) 52 13 (B) 52 16 (C) 52 17 (D) none Q.13 Given that the first card is a Jack, the chancethat it will bethe heart, is (A) 13 1 (B) 13 4 (C) 4 1 (D) 3 1
- Q.14 Suppose your opponent is dealt a Kingand a 10, and you are dealt a Queen and a 9. Being smart, your opponent does not take anymorecards and stays at20. Thechance thatyou will win ifyou areallowed to take as many cards as you need, is (A) 564 97 (B) 282 25 (C) 188 15 (D) 6 1 More than one alternative are correct: Q.15 If A& B are two events such that P(B) 1, BC denotes the event complementryto B, then (A) P A BC = P A P A B P B ( ) ( ) ( ) 1 (B) P (A B) P(A) + P(B) 1 (C) P(A) > < P A B according as P A BC > < P(A) (D) P A BC + P A B C C = 1 Q.16 Two real numbers, x & yare selected at random. Given that 0 x 1 ; 0 y 1. LetAbe the event that y2 x ; B be the event that x2 y, then : (A) P (A B) = 1 3 (B) A& B are exhaustive events (C) A& B aremutuallyexclusive (D) A& B are independent events. Q.17 For any two events A& B defined on a sample space, (A) P A B P A P B P B ( ) ( ) ( ) 1 , P (B) 0 is always true (B) P B A = 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 Q.18 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 likelyto occur as E1 (D) Probabilities of the events E1 E2, E1 and E2 are in G.P. Q.19 Let 0 < P(A) < 1 , 0 < P(B) < 1 & P(A B) = P(A) + P(B) P(A). P(B), then : (A) P(B/A) = P(B) P(A) (B) P(AC BC) = P(AC) + P(BC) (C) P((A B)C) = P(AC). P(BC) (D) P(A/B) = P(A) Q.20 If M & N are independent events such that 0 < P(M) < 1 & 0 < P(N) < 1, then : (A) M & N aremutuallyexclusive (B) M & N are independent (C) M & N are independent (D) P M N + P M N = 1
- ANSWER KEY DPP-1 Q.1 A Q.2 B Q.3 A Q.4 C Q.5 A Q.6 56 1 Q.7 8 3 Q.9 20825 2197 Q.10 952 to 715 Q.12 7 2 Q.13 21 4 Q.14 ) 1 n m )( n m ( ) 1 n ( n Q.15 3 2 Q.16 2 1 Q.17 5 1 , 10 1 Q.18 4 3 , 4 1 Q.19 13 52 9 48 4 4 C C · C Q.20 A DPP-2 Q.1 (i) 0.18, (ii) 0.12, (iii) 0.42, (iv) 0.28, (v) 0.72 Q.2 (i) 13 7 , (ii) 2 1 , (iii) 13 2 , (iv) 13 2 , (v) 2 1 , (vi) 13 9 Q.3 3 2 Q.4 (a) 1/18, (b) 43/90, (c) 5/18, (d) NO Q.5 (i) 8 5 , (ii) 8 3 Q.6 C Q.7 D Q.8 C Q.9 D Q.10 A Q.11 A Q.12 A Q.13 B Q.14 C Q.15 (i) 0.6, (ii) 0.5, (iii) 0.25 Q.16 (i) 36 1 , (ii) 108 5 ,(iii) 54 53 Q.17 P(E) = 3 1 , P(F)= 2 1 , P(E and F) = 6 1 ;Yes Q.18 P(E)= 36 2 , P(F)= 36 30 , P(E F) = 36 1 ; Not independent Q.19 11 20 Q.20 5 3 Q.21 7 2 Q.22 12 25 Q.23 A Q.24 A DPP-3 Q.1 A Q.2 C Q.3 B Q.4 C Q.5 D Q.6 D Q.7 D Q.8 B Q.9 8 5 Q.10 (i) 0.49 ; (ii) 0.973 Q.11 4 1 Q.12 3 1 Q.13 8 7 Q.14 (i) 365 1 , (ii) 1 – 2 ) 365 ( 363 364 Q.15 105 17 Q.16 10 4 Q.17 C Q.18 37 Q.19 (a) 204 5 , (b) 102 7 , (c) 68 3 , (d) 204 149 , (e) 17 4 , 51 2 Q.20 343 209 Q.21 22100 52 Q.22 37 16 , 37 12 , 37 9 Q.23 35 22 , 35 13
- DPP-4 Q.1 B Q.2 B Q.3 C Q.4 D Q.5 B Q.6 B Q.7 B Q.8 C Q.9 C Q.10 A Q.11 A Q.12 B Q.13 9 4 Q.14 2 1 Q.15 B Q.16 27 7 Q.17 (a) 8 7 , (b) 3 1 Q.18 A Q.19 B = 5 2 ; C = 15 4 Q.20 B Q.21 648 73 Q.22 13 to 5 Q.23 B Q.24 A Q.25 C DPP-5 Q.1 A Q.2 D Q.3 A : 324 169 ; B : 324 155 Q.4 B Q.5 D Q.6 A Q.7 B Q.8 B Q.9 C Q.10 C Q.11 (1) 1 mn 1 n , (2) 1 rn mn 1 n Q.12 C Q.13 C Q.14 C Q.15 C Q.16 B Q.17 D Q.18 17 12 Q.19 ) 1 n ( n 2 Q.20 50 11 Q.21 14 13 Q.22 (a) 65; (b) 5 2 6 5 · 10 (c) 1 – 2 · 5 6 5 (d) E F (e) P(E F) =P(E) ; P(E/F) = ) F ( P ) E ( P ; P(F/E) = 1 (f) No DPP-6 Q.1 A Q.2 A Q.3 4 3 Q.4 B Q.5 A Q.6 B Q.7 A Q.8 (i) A,B,C are pairwise independent (ii)A,B,C are not independent. Q.9 A Q.10 A Q.11 C Q.12 D Q.13 D Q.14 193 165 Q.15 11 3 Q.16 A Q.17 D Q.18 B Q.19 C Q.20 B Q.21 C DPP-7 Q.1 D Q.2 A Q.3 D Q.4 D Q.5 A Q.6 C Q.7 D Q.8 C Q.9 (i) 3 1 ; (ii) 3 1 ; (iii) 3 1 Q.10 C Q.11 D Q.12 B Q.13 C Q.14 D Q.15 A, B, C, D Q.16 A, B Q.17 A, B, C Q.18 A, C, D Q.19 C, D Q.20 B, C, D

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