2. There are n letter and n addressed envelopes. The probability that all the letters are not kept in the right envelope, is. (c)

Transcription

1 PAGE # CHAPTER EXERCISE I. A sigle letter is selected at radom from the word PROBABILITY. The probability that the selected letter is a vowel is / / / 0. There are letter ad addressed evelopes. The probability that all the letters are ot kept i the right evelope, is!! Noe of these. From a book cotaiig 00 page oe page is selected radomly. The probability that the sum of the digits of the page umber of the selected page is, is / 9/00 /00 Noe of these. Three letters are to be set to differet persos ad addresses o the three evelopes are also writte. Without lookig at the addresses, the probability that the letters go ito the right evelope is equal to / /9 / /6. A box cotais 0 red balls ad gree balls. If two balls are draw i successio the the probability that oe is red ad other is gree, is / / / Noe of these 6. Six boys ad six girls sit i a row. What is the probability that the boys ad girls sit alteratively /6 /9 / Noe of these. A card is draw at radom from a pack of cards. What is the probability that the draw card is either a heart or a kig / 9/ / /6 8. A ubiased die with faces marked,,,, ad 6 is rolled four times. Out of four face values obtaied the probability that the miimum face value is ot less tha ad the maximum face value is ot greater tha, is Two dice are throw. The probability that the sum of the poits o two dice will be, is The letter of the word ASSASSIN are writte dow at radom i a row. The probability that o two S occur together is Noe of these. The chace of throwig at least 9 i a sigle throw with two dice, is The umbers are selected at radom from the umbers,,. The probability that the differece betwee the first ad secod is ot less tha m (where 0 < m < ), is THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

2 PAGE # m m m m m m m m. Fiftee persos amog whom are A ad B, sit dow at radom at a roud table. The probability that there are persos betwee A ad B, is. Out of tickets marked with umbers from to, three are draw at radom. The chace that the umbers o them are i A.P., is Noe of these. There are apples & orages placed at radom i a lie. The the chace of the extreme fruits beig both orages is: (A) / (B) / (C) / (D) oe 6. A drawer cotais brow socks & blue socks well mixed up. A ma reaches the drawer & pulls out two socks at radom. The probability that they match is : (A) /9 (B) /8 (C) /9 (D) /. I the Frech lottery there were 90 tickets bearig umbers to 90. Suppose five tickets are draw at radom. The the probability that two of the tickets draw bear umbers ad 89 is : (A) /80 (B) /6 (C) /6 (D) oe 8. 0 studets are seated at radom i a row. The probability that two particular studets are ot seated side by side is : (A) / (B) / (C) / (D) oe 9. If two umbers p ad q are chose at radom from the set {,,,,, 6,, 8, 9, 0} with replacemet, determie the probability that the roots of the equatio x + px + q = 0 are real 0. Two squares are chose at radom from the small squares draw o a chessboard. What is the chace that the two squares chose have exactly oe corer i commo.. Five ordiary dice are rolled at radom ad the sum of the umbers show o them is 6. What is the probability that the umbers show o each is ay oe from,, or?. Out of ( + ) tickets umbered cosecutively, three are draw at radom. Fid the chace that the umbers o them are i A.P.. A ma parks his car amog cars stadig i a row, his car ot beig parked at a ed. O his retur he fids that exactly m of the cars are still there. What is the probability that both the cars parked o two sides of his car, have left?. From a ordiary pack of cards a eve umber of cards are draw at radom. Fid the probability of gettig equal umber of black ad red cards.. A ur cotais white ad black balls. Oe ball is draw. What is the probability that it is black? 6. From a pack of cards, four cards are draw. Fid the chace that they will be the four hoours of the same suit. THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

3 PAGE #. Six dice are throw simultaeously. Fid the probability that all dice show differet faces. 8. Six boys ad six girls sit i a row radomly. Fid the probability that (i) (ii) The six girls sit together The boys ad girls sit alterately. 9. What is the probability that i a group of N people, at least two of them will have the same birthday? 0. A group cotais 0 me ad wome. Three member committee is formed which must cotai at least oe woma. Fid the probability that the committee so formed has more wome that me. THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

4 PAGE # CHAPTER EXERCISE II. The probabilities of three mutually exclusive evets are /, / ad /6. The statemet is True Wrog Could be either Do ot kow. A ad B are two evets such that ad P(A) = 0., P(A + B) = 0. ad P (AB) = 0., the P (B) = Noe of these. If the probability of X to fail i the examiatio is 0. ad that for Y is 0., the the probability that either X or Y fail i the examiatio is Noe of these. The probability of happeig a evet A is 0. ad that of B is 0.. If A ad B are mutually exclusive evets, the the probability of happeig either A or B is Noe of these. If A ad B are ay two evets, the the true relatio is P (A B) is ot less tha P (A) + P (B) P (A B) is ot greater tha P (A) + P (B) P (A B) = P (A) + P (B) P (A B) P (A B) = P (A) + P (B) + P (A B) A 6. If A ad B are two evets such that P (A) 0 ad P (B), the P = B A A P P B B P A B P B P P A B. The probability of happeig a eve A i oe trial is 0.. The probability that the evet A happes at least oce i three idepedet trails is Two cois are tossed. Let A be the evet that the first coi shows head ad B be the evet that the secod coi shows a tail. Two evets A ad B are Mutually exclusive Depedet Idepedet ad mutually exclusive Noe of these 9. The probability of happeig at least oe of the evets A ad B is 0.6. If the evets A ad B happes simultaeously with the probability 0., the P A P B = The chaces to fail i physics are 0% ad the chaces to fail i mathematics are 0%. What are the chaces to fail i at least oe subject 8% 8% % 8%. A problem of mathematics is give to three studets whose chaces of solvig the problem are /, / ad / respectively. The probability that the questio will be solved is / / / /. If P (A A ) = P c A P c A, where c stads for complemet, the the evets A ad A are Mutually exclusive Idepedet Equally likely Noe of these THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

5 PAGE #. If A ad B are to evets such that P A, P B ad B p A B, the P = A /0 / /0 Noe of these B. If A ad B are two evets such that A B, the P = A 0 / /. If the probability of a horse A wiig a race is / ad the probability of a horse B wiig the same race is /, the the probability that either of them will wi the race is /0 9/0 /0 9/0 6. Let E ad F be two idepedet evets. The probability that both E ad F happes is ad the probability that either E or F happes is, the P E, P F P E, P F P E, P F Noe of these 6. The probabilities that A ad B will die withi a year are p ad q respectively, the the probability that oly oe of them will be alive at the ed of the year is p + q p + q qp p + q pq p + q + pq 6 8. A ad B are two idepedet evets. The probability that both A ad B occur is 6 ad the probability that either of them occurs is. The the probability of the two evets are respectively. ad ad 6 ad 6 ad 9. If A ad B are two evets, the the probability of the evet that at most oe of A, B occurs, is P A' B P A B' P A' B' P A B P A' P B' P A B All of the above 0. Let A ad B be two evets such that P (A) = 0. ad P (A B) = 0.8. If A ad B are idepedet evets, the P (B) = 6. If E ad F are idepedet evets such that 0 < P (E) < ad 0 < P (F) <, the E ad F c (the complemet of the evet F) are idepedet E c ad F c are idepedet c E E P P All the above c F F THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

6 PAGE # 6. For two give evets A ad B, P (A B) = Not less tha P (A) + P (B) Not greater tha P (A) + P (B) Equal to P (A) + P (B) P (A B) All of the above. P (A B) = P (A B) if a oly if the relatio betwee P (A) ad P (B) is P (A) = P A P (A B) = P (A B) P (A) = P (B) Noe of these. The two evets A ad B have probabilities 0. ad 0.0 respectively. The probability that both A ad B occur simultaeously is 0.. The the probability that either A or B occurs is Noe of these p p. If, ad set of all values of p is p 6 [0, ] are the probabilities of three mutually exclusive ad exhaustive evets, the the, 0, (0, ) 6. Three groups A, B, C are competig for positios o the Board of Directors of a compay. The probabilities of their wiig are 0., 0., 0. respectively. If the group A wis, the probability of itroducig a ew product is 0. ad the correspodig probabilities for group B ad C are 0.6 ad 0. respectively. The probability that the ew product will be itroduced, is If P (A) = 0., P (B) = 0., P (C) = 0.8, P (AB) = 0.08, P (AC) = 0.8, P (ABC) = 0.09, P (A + B + C) 0., ad P (BC) = x, the 0. x x x 0. Noe of these 8. Let 0 < P (A) <, 0 < P (B) < ad P (A B) = P (A) + P (B) P (A) P (B). The P (B / A) = P (B) P (A) P (A c B c ) = P(A c ) + P (B c ) P (A B) c = P (A c ) P (B c ) P (A / B) = P (A) 9. The probabilities that a studet passes i Mathematics, Physics ad Chemistry are m, p, ad c respectively. O these subjects, the studet has a % chace of passig i at least oe, a 0% chace of passig i at least two ad a 0% chace of passig i exactly two. Which of the followig relatios are true 9 p m c 0 pmc 0 p m c pmc 0 0. Two evets A & B have probability 0. & 0. respectively. The probability that both A ad B occur simultaeously is 0.. The the probability that either A or B occurs is (A) 0.9 (B) 0. (C) 0. (D) oe. If M & N are ay two evets, the which oe of the followig represets the probability of the occurrece of exactly oe of them? (A) P (M) + P (N) - P (M N) (B) P (M) + P (N) - P (M N) (C) P M P N P M N (D) P M N P M N THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

7 . If A ad B are two idepedet evets such that 8 P A B' ad 8 P AB ', the P A = PAGE #. If two evets A ad B are such that P(A c ) = 0., P (B) = 0. ad P (AB c ) = 0., the P[B / (A B c )] is equal to Noe of these. For the three idepedet evets A, B ad C, the probability of exactly oe of the evets A or B occurrig = the probability of exactly oe of the evets B or C occurrig = the probability of exactly oe of the vets C or A occurrig = p. If the probability of all the evets occurrig simultaeously be p where 0 < p < 0. the fid the probability of at least oe of the evets A, B ad C occurrig.. Three critics review a book. Odds i favour of the book are :, : ad : respectively for the three critics. Fid the probability that majority are i favour of the book. THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

8 PAGE # 8 CHAPTER EXERCISE III. A purse cotais copper cois ad silver cois, the secod purse cotais 6 copper cois ad silver cois. If a coi is draw out of ay purse, the the probability that it is a copper coi is / / /6 Noe of these. A speaks truth i % cases ad B i 80% cases. I what percetage of cases are they likely to cotradict each other i statig the same fact. % % % %. A bag cotais while ad black balls ad aother bag cotais white ad black balls. A ball is picked up radomly. The probability of its beig black is / 8/ 6/ /. A ati-craft gu take a maximum of four shots at a eemy plae movig away from it. The probability of hittig the plae at the first, secod, third ad fourth shot are 0., 0., 0. ad 0. respectively. The probability that the gu hits the plae is A coi is tossed util a head appears or util the coi has bee tossed five times. If a head does ot occur o the first two tosses, the the probability that the coi will be tossed times is 6. A bag A cotais white ad red balls ad bag B cotais white ad red balls. Oe ball is draw at radom from a radomly chose bag ad is foud to be red. The probability that it was draw from bag B was 6. Ur A cotais 6 red ad black balls ad ur B cotais red ad 6 black balls. Oe ball is draw at radom from ur A ad placed i ur B. The oe ball is draw at radom from ur B ad placed i ur A. If oe ball is ow draw at radom from ur A, the probability that it is foud to be red, is 8 9 Noe of these 8. A box cotais 00 tickets umbered, 00. Two tickets are chose at radom. It is give that the maximum umber o the two chose tickets is ot more tha 0. The miimum umber o them is with probability 8 Noe of these 9. A bag x cotais white balls ad black balls ad aother bag y cotais white balls ad black balls. A bag ad a ball out of it are picked at radom. The probability that the ball is white, is Noe of these 0. A ubiased coi is tossed. If the result is a head, a pair of ubiased dice is rolled ad the umber obtaied by addig the umbers o the two faces is oted. If the result is a tail, a card from a well shuffled pack of eleve cards umbered,,,, is picked ad the umber o the card is oted. The probability that the oted umber is either or 8, is Noe of these. The cotets of ur I ad II are as follows, Ur I : white ad black balls THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

9 PAGE # 9 Ur II : white ad 6 black balls Oe ur is chose at radom ad a ball is draw ad its colour is oted ad replaced back to the ur. Agai a ball is draw from the same ur, colour is oted ad replaced. The process is repeated times ad as a result oe ball of white colour ad of black colour are oted. Fid the probability the chose ur was I. /8 0/0 /0 oe. There are two urs. There are m white & black balls i the first ur & p white & q black balls i the secod ur. Oe ball is take from the first ur & placed ito the secod. Now, the probability of drawig a white ball from the secod ur is : (A) (C) pm m qm m p p q q p q (B) (D) p m p m p q q m q m p q. Oe bag cotais white & black balls, ad aother cotais white & black balls. A ball is draw from the secod bag & placed i the first, the a ball is draw from the first bag & placed i the secod. Whe the pair of the operatios is repeated, the probability that the first bag will cotai white balls is : (A) / (B) / (C) / (D) oe. Two studets A ad B attempt to solve the same questio. Their chaces of solvig the questio are ad 8 respectively. If the odds agaist makig the same mistake by them be 000 :, fid the probability of their results beig correct if they obtai the same result.. Three factories A, B ad C produce the same product. The factory A produces twice as may as B produces while the factories B ad C produce i the same quatity. It is kow that % of the products of A as well as C are defective while % of the products of B are defective. All the products of the three factories are stocked together. If a product is select at radom from the stock, what is the probability that the product is defective? 6. A card from a pack of cards is lost. From the remaiig cards of the pack two cards are draw ad are foud to be spades. Fid the probability of the missig card to be a spade.. I a test, a examiee either guesses or copies or kows the aswer to a multiple-choice questio with four choices, oly oe aswer beig correct. The probability that he makes a guess is ad the probability that he copies the aswer is 6. The probability that his aswer is correct, give that he copies it, is. Fid the 8 probability that he kew the aswer to the questio, give that he correctly aswers it. 8. I a bag there are six balls of ukow colours ; three balls are draw at radom ad foud to be all black. Fid the probability that o black ball is left i the bag (Or Fid the probability that the bag cotaied exactly black balls.) 9. A ur cotais 6 black balls ad ukow umber ( 6) of white balls. Three balls are draw successively ad ot replaced ad are all foud to be white. Prove that the chace that a black ball will be draw i the ext draw is There are two bags, oe of which cotais three black ad four white balls while the other cotais four black ad three white balls. A dice is cast. If the face or turs up, a ball is take out from the first bag. But if ay other face turs up, a ball is take from the secod bag. Fid the probability of gettig a black ball. THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

10 PAGE # 0. Three urs cotai white ad black balls, white ad black balls ; ad white ad black ball respectively. A ball is draw from a ur chose at radom. What is the probability that a white ball is draw if the choices of urs are equiprobable?. A ur cotais two balls each of which is either white or black. A white ball is added to the ur. What is the probability of drawig a white ball from the ur ow?. The probability that a certai electroic compoet whe first use is 0.0. If it does ot fair immediately the probability that it lasts for oe year is What is the probability that a ew compoet will last oe year?. A factory A produces 0% defective values ad aother factory B produces 0% defective valves. A bag cotais valves of factory A ad valves of factory B. If two valves are draw at radom from the bag, fid the probability that at least oe valve is defective.. A ubiased coi is tossed. If the result is a head, a pair of ubiased dice is rolled ad the umber obtaied by addig the umbers show o them is oted. If the result is a tail, a card from a well-shuffled pack of eleve cards umbered,,,, is picked ad the umber of the card is oted. What is the chage that the oted umber is either or 8? 6. A bolt factory has three machies A, B ad C maufacturig %, % ad 0% of the total productio. Of these the machies produce %, % ad % defective bolts respectively. A bolt is selected at radom ad it is foud to be defective. Fid the probability that it was maufactured by the machie A B C.. A ca hit a target times i shots ; B ca hit times i shots ad C twice i shots. They firs oce each. If two of them hit, what is the chace that C has missed it? 8. A bag cotais balls of ukow colours. A ball is draw ad replaced twice. O each occasio it is foud to be red. Agai, two balls are draw at a time. What is the probability of both the balls beig red? 9. A letter is kow to have come from either MAHARASTRA or MADRAS. O the postmark oly cosecutive letters RA ca be read clearly. What is the chace that the letter came from MAHARASTRA? 0. A bag cotais 0 cois of which at least are oe-rupee cois. Two cois are draw ad both are foud to be ot oe-rupee cois. What is the probability of the bag to cotai exactly oe-rupee cois? A ma has three cois A, B, C. The coi A is ubiased. The probability that a head will show whe B is tossed is while it is i case of the coi C. A coi is chose at radom ad tossed times givig heads ad tail. Fid the probability that the coi A was chose THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

11 PAGE # CHAPTER EXERCISE IV. Two card are draw successively with replacemet from a pack of cards. The probability of drawig tow aces is 69. I a box cotaiig 00 eggs, 0 eggs are rotte. The probability that out of a sample of eggs oe is rotte if the samplig is with replacemet is 6 (/0) (/) (9/) (9/0). If the probability that a studet is ot a swimmer is /, the the probability that out of studets oe is swimmer is 66 C C Noe of these. A coi is tossed m + times, where m. The probability of gettig at least m cosecutive heads is m m m Noe of these. A fair coi is tossed times. If the probability that head occurs 6 times is equal to the probability that head occurs 8 times, the is equal to 6. A speaks truth i % cases ad B i 80% cases. I what percetage of cases are they likely to cotradict each other i statig the same fact. % % % %. If a die is throw times, the the probability of obtaiig exactly times is C 6 6 C Cards are draw oe by oe at radom from a well shuffled full pack of cards util two aces are obtaied for the first time. If N is the umber of cards required to be draw, the P r {N = }, where 0, is Oe hudred idetical cois with probability p of showig up heads are tossed oce. If 0 < p < ad the probability of heads showig o 0 cois, the the value of p is The probability that a evet will fail to happe is 0.0. The probability that the evet will take place o cosecutive occasios is THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

12 PAGE #. A fair coi is tossed times. Let X be the umber of times head is observed. If P (X = ), P (X = ) ad P (X = 6) are i H. P., the is equal to 0 Noe of these. A pair of fair dice is rolled together till a sum of either or is obtaied. The the probability that comes before is Noe of these. A locker ca be opeed by dialig a fixed three digit code (betwee 000 ad 999). A strager who does ot kow the code tries to ope the locker by dialig three digits at radom. The probability that the strager succeeds at the k th trial is k 999 k 000 k 000 Noe of these. The items produced by a firm are supposed to cotai % defective items. The probability that a sample of 8 items will cotai less tha defective items, is The probability of Idia wiig a test match agaist West Idies is. Assumig idepedece from match to match, the probability that i a match series Idia s secod wi occurs at the third test, is 6. A ma alteratively tosses a coi ad throws a dice begiig with the coi. The probability that he gets a head i the coi before he gets a or 6 i the dice is 8 Noe of these. A ubiased die is tossed util a umber greater tha appears. The probability that a eve umber of tosses is eeded is / / / / 8. If 8 cois are tossed, the the chace that oe & oly oe will tur up head is : (A) /6 (B) /8 (C) / (D) oe 9. A coi is tossed util a head appears or util the coi has bee tossed five times. If a head does ot occur o the first two tosses, the probability that the coi will be tossed times is: (A) / (B) / (C) / (D) / 0. Two dice are throw util a 6 appears o atleast oe of them. The the probability that for the first time, a 6 appears i the secod throw is : (A) /96 (B) /96 (C) /96 (D) oe. You are give a box with 0 cards i it. 0 of these cards have the letter 'I' prited o them, the other te have the letter 'T' prited o them. If you pick up three cards at radom & keep them i the same order, the probability of makig the word "IIT" is : (A) 9/80 (B) /8 (C) / (D) /8 THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

13 PAGE #. I a series of idepedet trials the probability of exactly success is times as large as the probability of successes. The probability of a success i each trial is : (A) / (B) / (C) / (D) /. A & B havig equal skill, are playig a game of best of poits. After A has wo two poits & B has wo oe poit the probability that A will wi the game is : (A) / (B) / (C) / (D) oe. Two people take turs tossig a coi. The first perso to obtai head is the wier. The probability that the first player wis the game is : (A) / (B) / (C) / (D) oe. Three groups of childre cotai respectively girls ad boy, girls ad boys, oe girl ad boys. Oe child is selected at radom from each group. The chace that three selected cosistig of girl ad boys, is 9 Noe of these 6. A coi i tossed (m + ) times, m >. Show that the probability of gettig (at least) m cosecutive heads is ( + )/ m +.. I a multiple choice questio, there are four alterative aswer of which oe or more aswers are correct. A cadidate gets marks if he ticks all the correct aswers. The cadidate, beig igorat about the aswers, decides to tick at radom. How may attempts at least should he be allowed so that the probability of his gettig marks i the questio may exceed? 8. I a game A throws two ordiary dice. If he throws or he wis. If he throws, or he loses. If he throws ay other umber, he throws agai ad cotiues to throw util either the umber he threw first or turs up. I the first case he wis ad i the secod he loses. Show that the odds agaist his wiig is :. 9. A coi is tossed 0 times. Fid the probability of gettig (i) exactly six heads (ii) at least six heads (iii) at most six heads. 0. A ma takes a step forward with probability 0. ad backward with probability 0.6. Fid the probability that at the ed of eleve steps he is oe step away from the startig poit. The probability of a ma hittig a target i oe fire is. How may times at least must he fire at the target i order that his chace of hittig the target at least oce will exceed? (e) I a sequece of idepedet trials, the probability of success i oe trial is. Fid the probability that the secod success takes place o or after the fourth trial. A lot cotais 0 articles. The probability that the lot cotais exactly defective articles is 0. ad the probability that the lot cotais exactly defective articles is 0.6. Articles are draw from the lot at radom oe by oe without replacemet ad test till all the defective articles are foud. What is the probability that the testig procedure eds at the th testig.? Two persos A ad B toss a coi 0 times each together. Fid the probability that both of them get tails at the same time. THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

14 PAGE # CHAPTER EXERCISE V. If four persos are chose at radom from a group of me, wome ad childre. The the probability that exactly two of them are childre, is 0/ 8/6 / 9/. The probability that a leap year selected radomly will have Sudays is / / / /9. Idia plays two matches each with West Idies ad Australia. Ia ay match the probabilities of Idia gettig poit 0, ad are 0., 0.0 ad 0.0 respectively. Assumig that the outcomes are idepedets, the probability of Idia gettig at least poits is Let p deotes the probability that a ma aged x years will die i a year. The probability that out of me A, A, A A each aged x. A will die i a year ad will be the first to die, is p p p Noe of these. There are differet objects,,, distributed at radom i places marked,,,. The probability that at least three of the objects occupy places correspodig to their umber is /6 /6 / Noe of these 6. If positive itegers are take at radom ad multiplied together, the probability that the last digit of the product is,, 6 or 8, is. For a biased die the probabilities for differet faces to tur up are give below Face: 6 Probability: Noe of these The die is tossed ad you are told that either face or has tured up. The the probability that it is face, is Noe of these 8. A biased die is tossed ad the respective probabilities for various faces to tur up are give below Face: 6 Probability: If a eve face has tured up, the the probability that it is face or face, is A determiat is chose at radom. The set of all determiats of order with elemets 0 or oly. The probability that value of the determiat chose is positive, is Noe of these If the itegers m ad are chose at radom betwee ad 00, the the probability that a umber of the form m + is divisible by equals. The odds agaist a certai evet are : ad the odds i favour of aother idepedet evet are 6 :. The the chace that oe atleast of the evet will happe is : 8 9 THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

15 PAGE # (A) / (B) 8/ (C) / (D) oe. If the homework of each oe of studets ca be checked by oe of the teachers, the the probability of all the papers beig checked by differet teachers is : (A) / (B) 60/ (C) 0/ (D) oe. Sixtee players S, S, S., S 6 play i a touramet. They are dived ito eight pairs at radom. From each pair a wier is decided o the basis of a game played betwee the two players of the pair. Assume that all the players are of equal stregth. Fid the probability that the players S is amog the eight wiers. Fid the probability that exactly oe of the two players S ad S is amog the eight wiers.. A ur cotais white ad black balls. A ball is draw at radom. If it is white, it is ot replaced ito the ur, otherwise it is replaced alog with aother ball of the same colour. The process is repeated. Fid the probability that the third ball draw is black.. A ma ad a woma appear i a iterview for, two vacacies i the same post. The probability of ma s selectio is / ad that of the woma s selectio is /, what is the probability that both of them will be selected oly oe of them will be selected oe of them will be selected 6. If m thigs are distributed amog a me ad b wome, show that the chace that the umber of thigs received by me is odd is m b a b a. m b a m.. If o a average I vessel i every 0 is wrecked, fid the chace that out of vessels expected at least will arrive safely. 8. I a certai experimet the probability of success is twice the probability of failure. Fid the probability of at least four successes i six trials. 9. Suppose the probability for A to wi a game agaist B is 0.. If A has a optio of playig either a best of games or a best of games match agaist B ; which optio should A choose so that the probability of his wiig the match is higher? (No game eds i a draw). 0. A, B, c i order cut a pack of cards, replacig them after each cut, o the coditio that the first who cuts a spade shall wi a prize ; fid their respective chaces. Three players, A, B ad C, toss a coi cyclically i that order (that is A, B, C, A, B, C, A, B,.. ) till a head shows. Let p be the probability that the coi shows a head. Let, ad be, respectively, the probabilities that A, B ad C gets the first head. Prove that = ( p). Determie, ad (i terms of p). A employer seds a letter to his employee but he does ot receive the reply (It is certai that employee would have replied if he did receive that letter). It is kow that oe out of letters does ot reach its destiatio. Fid the probability that employee does ot receive the letter. A set A has elemets. A subset P of A is selected at radom. Returig the elemets of P, the set Q is formed agai ad the a subset Q is elected from it. Fid the probability that P ad Q have o commo elemets. THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

16 PAGE # 6 MISCELLANEOUS EXERCISE I. A lot cotais 0 articles. The probability that the lot cotais exactly defective articles is 0. ad the probability that the lot cotais exactly defective articles is 0.6. Articles are draw from the lot at radom oe day by oe without replacemet ad are tested till all defective articles are foud. What is the probability that the testig procedure eds at the twelth testig? (IIT 986). A ma takes a step forward with probability 0. ad backward with probability 0.6. Fid the probability that at the ed of eleve steps he is oe step away from the startig poit. (IIT 98). A ur cotais white ad black balls. A ball is draw at radom. If it is ot replaced ito the ur. Otherwise it is replaced alog with aother all of the same colour. The process is replaced. Fid the probability that the third ball draw is black. (IIT 98). A box cotais fifty paise cois, twety five paise cois ad a certai fixed umber N of te ad five paise cois. Five cois are take out of the box at radom. Fid the probability that the total value of these cois is less tha oe rupee ad fifty paise. (IIT 988). Suppose the probability for A to wi a game agaist B is 0.. If A has a optio of plyig either a best of games or a best of games match agaist B, which optio should choose so that the probability of his wiig the match is higher? (No game eds i a draw). (IIT 989) 6. A is a set cotaiig elemets. A subset P of A is chose at radom. The set A is recostructed by replacig the elemets of P. A subset Q of A is agai chose at radom. Fid the probability that P ad Q have o commo elemets. (IIT 99). I a test a examiee either guesses or copies of kows the aswer to a multiple choice questio with four choices. The probability that he make a guess is ad the probability that he copies the aswer is 6. The probability that his aswer is correct give that he copied it, is. Fid the probability that he kew the 8 aswer to the questio give that he correctly aswered it. (IIT 99) 8. A lot cotais 0 defective ad 0 o-defective bulbs. Two bulbs are draw at radom, oe at a time, with replacemet. The evets A, B, C are defied as : A = (the first bulb is defective) B = (the secod bulb is o-defective) C = (the two bulbs are both defective or both o-defective). Determie whether (i) A, B, C are pair wise idepedet, (ii) A, B, C are idepedet. (IIT 99) 9. Numbers are selected at radom, oe at a time, from the two-digit umbers 00, 0, 0,...,99 with replacemet. A evet E occurs if ad oly if the product at the two digits of a selected umber is 8. If four umbers are selected, fid probability that the evet E occurs at least times. (IIT 99) 0. A ubiased coi is tossed. If the result i a head, a pair of ubiased dice is rolled ad the umber obtaied by addig the umbers o the two faces is oted. If the result is a tail, a card from a well shuffled pack of eleve cards umbered,,..., is picked ad the umber o the card is oted. What is the probability that the oted umber is either or 8? (IIT 99). I how may ways three girls ad ie boys ca be seated i two vas, each havig umbered seats, i the frot ad at the back? How may seatig arragemets are possible if girls should sit together i a back row o adjacet seats? Now, if all the seatig arragemets are equally likely, what is the probability of girls sittig together i a back row o adjacet seats? (IIT 996) THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

17 PAGE #. Sixtee players S,S,...,S6 play i a touramet. They are divided ito at pairs at radom from each pair a wier is decided o the basis of a game played betwee the two players of the pair. Assume that all the players are of equal stregth. Fid the probability that the players S is amog the eight wiers. Fid the probability that exactly oe of the two players S ad S is amog the eight wiers. (IIT 99). If p ad q are chose radomly from the set {,,,,, 6,, 8, 9, 0 }, with replacemet, determie the probability that the roots of the equatio x px q 0 are real. (IIT 99). Three players A, B. ad C, toss a coi cyclically i that order (that is A, B, C, A, B, C, A, B,...) till a head shows. Let p be the probability that the coi shows a head. Let, ad be, respectively that the probability, the probabilities that A, B, ad C gets the first head. Prove that p. Determie, ad (i terms of p). (IIT 998). Eight players P, P,..., P 8 play a kock-out touramet. It is kow that wheever the players Pi ad Pj play, the player P will wi if i < j. Assumig that the players are paired at radom i each roud, what is the probability that the player P reaches the fial? (IIT 999) 6. A coi has probability p of showig head whe tossed. It is tossed times. Let P deote the probability that o two (or more) cosecutive heads occur. Prove that p p p ad p p. p p p p for all (IIT 000). A ur cotais m white ad black balls. A ball is draw at radom ad is put back ito the ur alog with k additioal balls of the same colour as that of the ball draw. A ball is agai draw at radom. What is the probability that the ball draw ow is white? (IIT 00) 8. A ubiased die, with faces umbered,,,,, 6, is throw times ad the list of umbers showig up is oted. What is the probability that amog the umbers,,,,, 6 oly three umbers appear i this list? (IIT 00) 9. A box cotais N cois, m of which are fair ad the rest are biased. The probability of gettig a head whe a fair coi is tossed is ½, while it is / whe a biased coi is tossed. A coi is draw from the box at radom ad is tossed twice. The first time it shows head ad the secod time it shows tail. What is the probability that the coi draw is fair? (IIT 00) 0. For a studet to qualify, he must pass at least two out of three exams. The probability that the will pass the st exam is p. If he fails i oe of the exams the the probability of his passig i the ext exam is p otherwise it remais the same. Fid the probability that he will qualify. (IIT 00). A is targetig to B, B ad C are targetig to A. probability of hittig the target by A, B ad C are, ad respectively. If A is hit, the fid the probability that B hits the target ad C does ot. (IIT 00). If A ad B are two idepedet evets, prove that P A B.P A B P C, where C is a evet defied that exactly oe of A ad B occurs. (IIT 00) THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,

18 PAGE # 8. (e). A bag cotais red balls ad 6 white balls. Six balls are drows oe by oe without replacemet of which at least balls are white. Fid the probability that i the ext two drows exactly oe white ball is draw. (Leave the aswer i C ). (IIT 00) r A perso goes to office either by car, scooter, bus or trai probability of which beig,, ad respectively. Probability that he reaches offices late, if he takes car, scooter bus or trai is,, ad respectively. Give that he reached office i time, the what is the probability that he traveled by a car? (IIT 00) THE SCHOOL OF MATHEMATICS, 6-A Talwadi, KOTA (Near D.A.V. School) 0-69,