50 Chapter2 Axiomsof Probability which can be generalizedto give "(!^') : ,7 \- /2- P t\ r' 'tI, t' - If r6;a1 + III P(AiA1A1,) i.j.k t<J + . . . +( - 1 ) ' + r P ( A t . - . A n \ If S is finite and ezrchone point set is assumedto have equal probability,then p@):ry tsl where lEl denotesthe number of outcomesin the event E. P(A) canbe interpretecleither as a long-run relative frequencyor as a measureof one's desree of beliei. PROBLEMS 1. A box contains 3 marblos: 1 red. I green, and 1 blue. Consider an experiment that consistsof taking I marble from the box and then replacing it in the box and drawing a secondmarble from the box. I)escribe the sarnplespace.Repeat when the secondmarble is drawn without replacing thrl lirst marble. 2. In an experiment. die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample spaceof this experiment? Let E,, denote the event that n rolls are necessaryto complete the experiment. What points of the sam- plespace arecontained in E,'i Whatis lU f " 1 't \r / 'Iwo /o 3. \' dice are thrown. Let E be the event that the sum of the dice is odd, let ,F be the event that at least one of the dice lands on 1, and let G be lhe event that the surn is 5. Describe the evonts EF, E U F, FG, EF', ANdEFC, 4. A, B, and C take turns llipping a coin. The lirst one to get a head wins. The sample spaceof this experiment can be delined by "" - J 0l ,0o0t .0o. .o. l , o o o l . . . . , I (a) Interpret the sample space. (b) Define the following eventsin terms of S: (i) zl wins :,,4. (ii) B rvins:11. (iii) t,,l i) B)". Assume that,,1flips first, then B, then C, then A. and so on. 5. A system is comprised of -5components.each of which is either rvorking or failed, Consider an experiment that consistsoI observing the statusof eiich component,and let the outcome of the experiment be given by the vector (r1.,r2,x3,x.1,-r5). where -v; is equal to I if cr:rnponent I is working and is equal to 0 if component i is failed. (a) Floiv many outcomes are in tl.resample space of this experiment? (b) Supposethat the system will work if components 1 ancl 2 arrl both rvorking, or if components 3 and 4 are both working. or if cornponents1.3. and 5 are all working.Let l{ be the event that the systemwill rvork. Specifyall the outcclmesin 1,V. (c) Let zl be the event that components4 and .5 are troth failed. How many outcomr:sare containedin the event A? (d) Write out all the outcomesin the eventAW. 6. A hospital administratorcodesirrcomingpatients sutfering gunshot wounds according to whether they have insurzrnce(coding 1 if they clo and 0 iI they clo not) and according to their condition, which is ratecl as goocl (g), fair (f), or serious (s). Consider an experinrentthat consistsnf the cocling ol such a patient. (a) Give the sample spaceof this experirnent. (b) Let,4 be the event.that the patientis in scrious condition.Specifythe outcomesin A. (c) Let B be the event that the patient is uninsured.Specifythe outcomesin B. (d) Give all the outcomesin the eventBc U A. 7. Consider an experirnent that consistsof detcrrnining the type o1 job-either blue-collar or whitecc;llar-and the political affiliation-Republican. Democratic.or Independent-of the 15 mombcrs of an adult soccerteam. Ilolv many outcomesare (a) in the samplespace? (b) in the event that at least one of the team members is a blue-collar worker? ', *9o 3 3 3 / : 3 s L J " Z f : 1*75' Problems 51 (c) in the event that none of the team members considershimself or herself an Independent? 8. Suppose that /4 and B are mtttually exclusive events for which P(A) : .3 and P(B) : .5. What is the probability that (a) either A or B occurs'/ (b) r1 occurs but B does not'? (c) both A t'nd B occur? 9. A retail establishrnentacceptseither the Arncrican Expressor the VISA credit card.A total ol24 percent of its customers carry an American Express card,6l percent carry a VISA card, and 11 percent carry both cards. What percentageof its customers carry a credit card that the establishment will accept? 10. Sixty percent of the students at a certain school wear neither a ring nor a necklace. Twenty percsnt wear a ring and 30 percent wear a necklace. If one of the students is chosenrandomly. wlrat is the probability that this student is wearing (a) a ring or a necklace? (b) a ring and a necklace'? 11. A total of 28 percent of American males smoke cigarettes.7 percent smoke cigars,and 5 percent smokc bo{.hcigarsanc{ciā¬larettes. (a) What percentageof males smokes neither cigarsnor cigarettes? (b) What percentage smokes cigars but not cigarettes? An elementary school is offering 3 language classes:one in Spanish,one in French.and one in German. The classesare open to any of the 10t) studentsin the school.There are 28 studentsin the Spanishclass,26irr the F'renchclass,and I6 in the German class.There ttre 12 studentsthat arc in both Sparrishand French,4 that are in both Spanish and Gerrnan, and 6 that are in both French and German. lrr addition, there are 2 studentstaking all 3 classes. (a) If a student is chosenranclomly.what is the probability that ht: or she is not in any oi the languageclasses? (tr) if a student is chosen randotrly. what is the probabilitythat he or sheis takingexactlyone languageclass} (c) lf 2 studentsare chosenrandomly,what is the probabilitythat at least 1 is taking a language class? 13. A certain town with a population c1f100,000has 3 nervspapers:I, Il, and lll. The proportions of townspeoplewho read these papers are as follows: [: 10 percent I and lI: 8 percent I l : 3 0 p e r c e r r tI a n d I l l : 2 p e r c e n t III: 5 percent II andIII:4 perccnt I and II and III: 1 percent (Thc list tells us. for instance, that 8000 people r e a dn e r v s p a p e rI sa n d I l . ) (a) Find tht: nunber of people lvho read only one Ilewspaper. (b) How many people read at least lwo nervslrapcrs? (c) lf I and III are morning papers and ll is an evcningpaper,how many peopleread at least one morning paper:plus an eveningpaper? (d) How many people clo not rcacl any newspapers'? (e) Horv many people read onlv one morning paper and one evening paper'l 14, The following clatawert: given in a study of a group of I000 subscribersto a certain magazine:[n referenceto job. marital status,and education.there were 312 professionals,470 married persons,525 college graduatcs, 42 professional collcge graduates, 147 married college graduates, {J6 married prolessionals,and 25 tn:rrricd prot.essionalcollcgc graduates.Show that the nunrbersreported in the stuclymust be incorrect. the Hint:LeL M.W, and G den<ltc.respec{.ively. set of professionals,manied persorls,and college graduates.Assume tha( <sneo{ the I(l(Il persons is chosen at raudom. and use Proposition4.4 to show that if the given numbcrs are corrr:ct, then P ( M u W u G ) > t . poker hands are "? ) ecluallylikely, what is the pr<lbabilitvoI beingdealt (a) a flush'?(A hand is said to be a flush if all 5 carclsare o1 the samesuit.) (b) one pair'? ('I'ltis occurs when the cards have dcnominationsa. ri. b. c, d. r.vheren. b. c. and r/ trreall distinct.) (c) trvo pairs? (This occurs rvhen thc cards have denominationsa, a, b. b, c, rvherea, b, and c are all distinct.) (d) three of a kincl? (This occurswhen the carcls have denominations a, (t, e, b, <:.where a, b, and c are all distincl.) (e) four of a kincl'l ('I'his occurs when the cards have denominationsa. a. a, a, b.) Poker dice is playcd bv sirnultaneouslyrolling -s clice. Show that (a) P{no two alike} : .0926: (tt) P{one pair} : .zffljg' (c) P{two pair} : .2315; (d) P{three alike} : .1J43; (e) f' {full house} : .0386: (f) P{four alike} : .0193' (g) P{fivealike} : .0008. 17. If 8 rooks (castles) arc randomly placed on it chessboar:d. compute the prnbabilit;rthat nclneof the rooks can capturc any of the others.That is. 15. If it is assumeclthat all ( 52 C h a p t e r2 Axioms of Probability compute the probability that no row or file contains more than one rook. 18. Two cards are randomly selcctedfront an ordinary playing deck. What is the probability that they form a blackjack? That is, what is the probability that one of the cards is an ace and the other one is either a ten. a jack, a queen,or a king? 19. Two syrnmetric dice have both had two of their sides painted red, two painted black. one painted yellow. and the other painted white. When this pair of dice is rolled, what is the probability that both dice land with the same color face up? 2{}. Supposethat you are playing blackjack against a dealer. Irr a freshly shuffled deck, what is the probability that neither you nor the clealer is clealt a blackjack? 21. A small community organization consists o1 20 families,of which 4 have one child,8 havetwo children, -5have three children, 2 have four children, and t has five children. (a) If one of these families is chosen at random, what is the probability it has i children, I : r.2.3,4.51 (b) If one oI the children is randomly chosen, what is the probability that child corneslt'om a f a m i l yh a v i n gj c h i l d r e n i, : 7 , 2 . 3 , 4 . 5 ' l 22. Consider the following technique lbr shullling a deck of n cards: For any initial orc'leringo1 the cards,go through the deck one card at a time and at each carcl, Ilip a lair coin. If the coin comes utrrheads, ttren leave the card where it is; if the coin comes up tails, then movc that card to the end of the deck. After the coin has been flipped n times, say that one round has been completed.For instance,if n : 4 and the initial orderingis 1. 2, 3, 4. then if the successivellips result in the outcome h. t, t, h. then the ordering at the end of the round is 1,4,2,3. Assumingthat all possibleoutcomesof the sequenceof n coin fiips are equally likely, what is the probability that the ordering after one round is the same as the initial orclering? 23. A pair of fair dice is rolled. What is the probatrility that the seconddie lands on a higher value than doesthe first? 24. If two dice are rolled, what is the probability that the sum of the upturned facesequals l'?Find it lbr i :2.3, ...,11,12 A pair of dice is rolled until a sum of either 5 or 7 appears.Find the probability that a 5 occurs first. Hint: Let E,, denote the event that a 5 occurs on the nth roli and no 5 or 7 occurson thclirst n - 1 rolls. CcrmputeP(Er) and argue that I P(8,') is thll desiredprobability. 26. The game of craps is played as follows: A player rolls two dice. lf the sum of the dice is eithcr a 2, 3, or'12, the player loses;if the sutn is either a 7 or an 11, the player wins. lf the outcome is anything else,the player continuesto roll the diceuntil she rolls either the initial outcorne or a 7. If the 7 comes first. the player loses.whereas if the initial outcome reoccurs before the 7 appears.the player wins. Cornpute the prolrability of a player winning at craps. flint: I-et f; denote the event that the initial outcome is I and the player wins. The clesiredprobability is I ptgil. To compute P(Ei), dsline the i-a events E;,,1to be the event that the initial sum is I and theplayer wins on the nth roll. Argue thal P(Ei): ,,). ,LrP(Ei 27, An urn contains 3 red and 7 black balls. PlayersI and B withdraw balls ffom the urn consecutively until a red ball is selected.Find the probabilitythat ,4 selectsthe red ball. (,,1draws the first ball. then B, anclso on. There is no replacementol the balls drarvn.) 28. An urn contains 5 r:ed,6 blue, and 8 greerrballs. If a set of 3 balls is randomly selected,what is the probabilitythat each of the balls will be (a) of the same color'l (b) of diflerent colors'?Repeat uuder that whenever a ball is selectecl.its the assumptior:r color is noted and it is then replaced in the urn before the next selection. This is known as satn' p lin g with repIacentetfi. 29. An urn contains ruwhite and m blitck balls, where n anclm are positive numbers. (a) If two balls are randonly withdrawn, what is the probability that they are the samecolor? (tr) If a ball is randornly withdrawn and then replacedbefore the secondone is drawn, what is the probability that the withdrawn balls are the same color? (c) Show that the probability in part (b) is always larger than the one in part (a). 30. The chessclubs of trvo schoolsconsist o[, respectively. 8 and 9 players. Four members from each club are rtrndomly chosen to participate in zrcontest between the two schools. T'he chosen players lrorn one team arc then randomly paired with thosefrom the other team, and eachpairing plays a gamc oI chess.Supposethat Rebeccaand her sister Elise are on the chcssclubsitt dilferent schools. What is the probability that (a) Rebeccaarrd Elise will be pairecl? (b) Rebeccaand Elise u'ill be chosento represent their schoolsbut will not play eachother? (c) either Rebecca or Rlise will be chosen to representher school? Problems 53 31. A 3-person basketball team consistsof a guard. a forward. and a center. (a) If a person is chosen at rand<lmfrom each of three different such teams,what is the probability of selectinga complete team? (b) What is the probability that all 3 players selectedplay the same position? 32. A group of irrdividuals containing b boys and g girls is lined up in random orden that is, each cll the (b + g) ! per:rnutationsis assumedto be equally likely. What is the probability that the person in t h e l t h p o s i t i o n ,1 = i < b + g , i s a g i r l ? 33. A forest contains 20 elk, of which 5 are captured, tagged,and then released.A certain time later, 4 of the 20 elk are captured.What is the probability that 2 clf these 4 have been taggcd? What assumptions are you making? 34. The second Earl of Yarborough is reported to have bet at odds of 1000to 1 that a bridge hand of 13 cardswould contain at leastone card that is ten or higher. (By ten or higher we mean thaf a card is either a ten, a jack, a queen.a king, or an ace.)Nowadays,we call a lrand that has no carcls higher than 9 a Yttrborctul4lr.What is the probability that a randomly selected bridge hand is a Yarborough? 35. Seven balls are randomly withdrawn from an urn that contains12 rcd. 16 blue, and lu green balls. Find the probability that (a) 3 red, 2 blue, and 2 green balls are rvithdrawn; (b) at least 2 red balls are withdrawr; (c) all withdrawn balls are the same color; (d) either exactly 3 red balls or exactly 3 blue balls are withdrawn. 36. Two cards are chosenat random front a deck of -52 playing cards.What is the probability that they (a) are both aces? (b) have the same value? 37. An instructor gives her classa set of 10 problems with the inlbrmation that the final exam will consist of a random selectionof 5 of them. If a student hasligured out how to do 7 of the problems,what is the probability that he or she will answercorrectly (a) all,5 problems? (b) at least 4 of the problems'/ 38. There are n socks.3 of which are red, in a drawer. What is the value of n if, lvhen 2 of the socks are chosenrandomly, the probability that they are both red is |'? 39. Thcre are 5 hotels in a certain town. If 3 people check into hotels in a day, what is the probatrility that they each check into a diff'erent hotel? What assumptionsare you making? 40. A town contains 4 people who repair televisions. lf 4 sets break dorvn, what is the probability that exactly I of the repairers aro called? Solve the problem for I : 1,2,3,4. What assumptionsare you making? 41. If a die is rolled 4 times,what is the probabilitythat 6 comesup at leastonce? 'l"wo 42. dice are thrown n times in succession.Compute the probability that double 6 appoarsat lcast once. How large need n be to make this probability a t l e a s t; ? 43. (a) If N people, including A and B, are randomly arrangedin a line . what is the probabiiity that A and Il are Ilext to each other? (b) What would the probability be if the people 'l rvcre randoml-varranged in a circle 44. Irive people. designated as A. I], C, D, E. are arrangedin lincar order. Assumingthat eachpossible order is ecluall;,likely, rvhat is the probability that (a) there is exactly one person betweenA and B'! (b) there are exactly two people between A and ,B? (c) there are three people betweenA and B'! 45. A wornan has n keys, of which one will open her door. (a) If slre tries the ke5rsat random, discarding those that do not work, rvhatis thc probability that she will open the door on her ftth tr:y? (b) What if she does not discard previously tried kcys? 46. How many people have to be in a room in order that the probability that at least two of thern celebrate their birthdav in the same mnnth is at least j? Assumethat all possiblcmonthly outcomcsare equally likely. 47. lf there are 12 strangers in a room, what is the probability that no two c;f them celebratetheir birthday in the samemonth? 48. Given 20 people, what is the probability that. among the 12 months in thr: year. therc are 4 months containing exactly 2 birthctaysancl 4 containing exactly 3 birthdays'/ 49. A gror-rpof 6 men ancl 6 women is ranclomly divided into 2 groups of size 6 each. What is the probability that both groups will have the same number of rnen? 50. In a hand of briclge, fincl the probability that you have 5 spadesand your partner has the remaitring 8. 51. Supposethat n balls are randomly distributed irrto N compartments.Find the probability that n balls will fall into the first compartment. Assume that all N'l arrangementsare equally likely. 54 Chapter 2 Axioms of Probability 52. A closet contains 10 pairs of shoes. If 8 shoes are randomly selected,what is the probability that there will be (a) no complete pair? (b) exactly 1 complete pair? 53. If 4 married couplesare arrangedin a row, find the probability that no husband sits next to his wife. 54. Cornpute the probability that a bridge hand is void in at least one suit. Note that the answer is not (Why not?) I:ILnt: U se Proposition 4.4. Compute the probability that a hand of 13 cards contains (a) the ace and king of at least one suit; (b) all 4 of at least 1 of the 13 denominations. Two players play the following game: Player ,,{ chooses one of the three spinners pictured in Figure 2.6, and then plaver B choosesone of the remaining two spinners. Both players then spin their spinner, and the one that lands on the higher number is declared the winner. Assuming that each spinner is equally likely to land in any of its 3 regions,would you rather be player z1 or player B? Explain your answer! 2.6: Spinners FIGURE EXERCISES THEORETICAL Prove the following relations: . (V n,)n :[ r;r *no l . E F C E C E U F . 2. ltE C .1. then F'' C E'. 3. F:FEU F E ca n d E U F : E U E'F. (l'') UF':frru''"r Answersto SelectedProblems CHAPTER 1 1. 67,600,000; 19,656,000 2. 1296 4. 24:4 5. 144;18 6. 2401 7. 720;72.: I44: 72 8. 120;1260:34,650 9.27,720 10. 40,320;10,080; 1.152;2880l.384l1..720l. 72: 144 12,24,300,000:1.7 ,1,00,720L3. 190 14.2,598,960 16.42:94 17.604,800 18.600 19.896;1000;91020.36:26 21.35 22.18 23.48 25. 521/(Bl)4 27.27,720 28. 65,536:2520 29. 12,600:945 30. 564,480 31. 165;35 32. 1287;L4,lI2 33. 220;572 CHAPTER 2 9.74 10. .4;.1 11.70:2 12..5:.32;149119813.20,000; 12,000; 11,000; 68,000; 10,000 14. 1.057 15..0020;.4226:.0475:.0211:-.A{J024 x 10-o 17.9.1.0947 18. .048 19. 5/18 20. .9052 22. (n + D lzn 23.5tl2 25. .4 26. .492929 27. .0888;.2477;.1243;.2099 30. 1/18;1.16; L12 3L. 219:Il9 33.701323 36. .0045;.0588 37. .0833;.5 38.4 39. .48 40.1164;21t64;36164:6164 41..5177 44..3;.2;.1 46.5 48. 1.0604x 10-3 49..4329 50.2.6084x 10-6 52..091,45:.426853.12135 54. .0511 55. .2198;.0343 CHAPTER 3 l. u3 2 . 1 . 1 6 ; 7 1 5 ; 1 1 4 : 1 1 3 : 13".1. 23 :3l 9 5 , 6 1 9 1 6 . 1 1 2 7 . 2 1 3 8.1,12 9.711,1"I0. .22 ll.ll17:1133 12..504;.3629 14.351768;2101768 15. .4848 16..9835 17..0792;.264 18. .331;.383:.286:48.6219.44.29: 41.18 20..4:1126 21..496;311.4;9162 22.519;116;5154 23.419;Il2 24.lt3: 112 26. 20121;4014128. 31128;2911,536 29. .0893 30.711.2;31533..76, 49176 34.2713L 35..62,101L936.112 37.113:1.15:1. 38.'1213739.461185 40.3113:5113:5152:15152 41. 431459 42.34.48 43.4le 45.1111. 48.213 52..1.1:16/89; 50.17.5;381L65;17133 51..65;56165;8165:1165:14135;72135;9135 12127: 315;9125 55. 9 57.(c) 213 60.213;113;314 6L 116:3120 65.9lI3; ll2 69.9;9:18:L10;4; 4: B:1.20alloverL28 70. ll9;1lLB 7L 38164;13164;1,3164 73.1,116;1,132;5116;114;31132 74.9119 75.314,7112lt. p211t- 2p + 2p2) 79..5550 81..9530 83..5;.6;.B 84.911.9;6119;4119;7115;531L65;7133 89.97l l42: t5 126:3311,02 4 CHAPTER l. p(4) : 6/91:p(2) : 8/9L:p(1) : 32/91;p(0): 1/91;p\-1) : 16/91,; p(*2) :28/91 4. (a)1,12;5118:'5136:5184:51252;11252:0;0;0;0 5. n - 2i; 457 A l. A classhas 10 boys and 6 girls. 3 studentsare selectedat randomfrom the class.What is the probability that 2 of them are girls and 1 is a boy? 2.Dice arethrown until a six occurs.Computethe probabilityof the events 4 : {"Six" obtainedin first two throws} B : {the numberof throws is odd} 3.. 4 fair coinsaretossed"Find the probabilitythat they are all headsif at leastone of coin is head. 4. In telephonenumber5 last numbersare cleaned.Find the probability of eventthat different figuresarecleanedbesides1,3,5. 5. There are4 balls with numbers1,2,3,4in the urn. Chooseat randomone ball after anotherfrom the urn. Find the probability that at least 1 of balls orderednumbercan be the samewith its own. 6. Coin, with the probability of "head" which is equalto l13 thrown until a "tail" occurs.Find the probability of eventsthat: A: {the numberof throwsis odd}; 3 : {"tail" is obtainedin the first threethrows} 7.Dice thrown 10 times.Find probability f eventsthat: A:{at leastoneof them"5"}; B:{ exactlytreeof them"5"} 8. Given is an urn containing 9w and2b balls.Find the probability of eventthat third takenball is white if color of fist 2 takenballs without replacementunknown. g.Thereare 30 ticket, 6 of them are "good". Two studentsone after another aretaking ticket. Find the probabilityof eventsthat 4:{both havegot "good" ticket};B:{secondhavegot "good" ticket}. 10. 90 of 100 itemsare non-defective.Find the probabilityof eventthat at least2 of 10 taking itemsare defective. 11. 80 of 100itemsare non-defective.Find the probabilityof eventthat 3 of 5 taking itemsare defective. "1"}, if it is known that the 12. 3 dice arethrown. Computethe probabilityof eventA:{occur two sumoffaces equal4. 13. Find the probabilityof event,that the 5 digit numberof automobiletakenat randomhasonly two identicallyfigures. Find 14. Given is an um containing 4w and 3b balls. 1 ball took out of urn without replacement. the probabilityof eventthat taking 2 balls with replacementarewhite. 15. Dice arethrown until a six occurs.Computethe probability of the following events:A : {A six is obtainedin the first two throws), B: {the numberof throwsis even}. 3 16. Threecoinsthrown 10 times.Find probabilityof eventthat at leastin one trial appearances "heads". ...20) . Find the probabilityof eventthat 3 of 17 Chooseat random 4 numberfrom (1 2 .... them even,1 odd and exactly 1 divided by 10. 30. 3 of 15 books in Russian.Calculateprobability of event thatat leastone among3 choosing booksin Russian. 18. There are N studentsin a class.Each student'sbirthday is one of 365 days .What is the probabilityP that therearc atleast2 studentswhosebirthdaycoincide? 19.ThereareM tickets numbered 1,2,...,M,of which n, numbered 1,2,...nwin prizes(M>:2n). You buy n tickets. Find the probability of winning at leastone prize 20. Considera family with two children. Ask the probability that both children are boys, if: a) that the older child is a boy; b) ) that at leastone of the childrenis a boy. 2l.Letwe havethe sequences {a,,e2,...,a,},whereo,.{0,r,2,3\, i =ln.Find the probabilitythat :1) the endsare0; 2) exactlym of them are"2". 22.Letwe havethe sequences{o,,ort...ten}, wherea, e{0,1,2\,t=tn. Find the probabilitythat 1) first threeelementsare"l"; 2) exactlym+2 of them are "0" , suchthat2 of them in the end. 23. Chooseat random 10 numbersfrom (1 2 ...30) . Find the probability of eventthatT of them evenand exactly2 divided by 10. 24 .5 of 20 bookson "Probability Theory" .Calculateprobability of eventthat two of 4 choosing bookson "ProbabilityTheory" . 25. Dice thrown twice. Computethe probabilityof the following events:A: {A "3" obtainedat lastonetime); B : {A sumof facesis not lessthan9}.
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