Statistics

Problem Statement for "CollectingBonuses"

Problem Statement

A juice company is running a promotion where each juice bottle they sell contains a code that you can redeem for a prize. There are n different codes, each corresponding to a different type of prize. The codes are evenly distributed, so the probability of winning a certain type of prize is 1/n for each bottle. The codes are written underneath the bottle caps, so you can't read them until you buy the bottles. Your goal is to collect k different types of prizes. Return the expected number of bottles you must buy to achieve this goal.

Definition

Class:
CollectingBonuses
Method:
expectedBuy
Parameters:
String, String
Returns:
double
Method signature:
double expectedBuy(String n, String k)
(be sure your method is public)

Notes

  • The returned value must be accurate to within a relative or absolute value of 1E-9.

Constraints

  • n and k will contain digits ('0' - '9') only.
  • n and k will represent positive integers without leading zeros.
  • n will represent an integer between 1 and 10^18, inclusive.
  • k will represent an integer between 1 and the integer n represents, inclusive.

Examples

  1. "1"

    "1"

    Returns: 1.0

    With only 1 type of prizes you just need to buy 1 bottle.

  2. "2"

    "1"

    Returns: 1.0

    Now there are 2 types of prizes, but any one will satisfy you, so you again need only 1 bottle.

  3. "2"

    "2"

    Returns: 3.0

    First you buy 1 bottle and collect some type of prizes. After this, you need to collect another type of prizes. Only half of bottles contains it, so in average you must buy 2 more bottles to achieve your goal.

  4. "999999999999999999"

    "999999999"

    Returns: 9.999999995E8

  5. "999999999999999999"

    "999999999999999999"

    Returns: 4.202374733879435E19

  6. "12345"

    "123"

    Returns: 123.61182725183491

  7. "123456789"

    "123456789"

    Returns: 2.3714342296692758E9

  8. "123456789"

    "12345"

    Returns: 12345.617207203177

  9. "1234567890123456"

    "1234567890"

    Returns: 1.2345685072843559E9

  10. "7"

    "3"

    Returns: 3.566666666666667

  11. "218"

    "41"

    Returns: 45.30366038127212

  12. "1692038591"

    "2359193"

    Returns: 2360839.2296567466

  13. "198275398293849"

    "112939489382893"

    Returns: 1.6715837985117944E14

  14. "12397539298439"

    "12397539298000"

    Returns: 2.983205152156711E14

  15. "12345678"

    "12345678"

    Returns: 2.0871643110346735E8

  16. "56781234"

    "56781234"

    Returns: 1.0465878505214562E9

  17. "123456789"

    "123454321"

    Returns: 1.3358068774344294E9

  18. "123456789"

    "123456780"

    Returns: 2.0221788928514185E9

  19. "1234567890123"

    "1234566789000"

    Returns: 1.7197407941607326E13

  20. "5555555512345"

    "12345"

    Returns: 12345.000013714802

  21. "49000000"

    "1"

    Returns: 1.0

  22. "8597705003058896"

    "4263000"

    Returns: 4263000.001056861

  23. "17247006936141401"

    "6006801"

    Returns: 6006801.001046026

  24. "9158096943131503"

    "49999999"

    Returns: 4.999999913649123E7

  25. "4390464402"

    "4389792928"

    Returns: 3.8572291205878876E10

  26. "9158096943131503"

    "4999999"

    Returns: 4999999.001364912

  27. "9158096943131503"

    "499999"

    Returns: 499999.00001364906

  28. "999999999999999999"

    "714095618"

    Returns: 7.140956182549663E8

  29. "999999999999999999"

    "635124699"

    Returns: 6.351246992016917E8

  30. "999999999999999999"

    "525848194"

    Returns: 5.2584819413825816E8

  31. "999999999999999999"

    "133946281"

    Returns: 1.339462810089708E8

  32. "999999999999999999"

    "513446915"

    Returns: 5.134469151318139E8

  33. "999999999999999999"

    "436935328"

    Returns: 4.3693532809545624E8

  34. "999999999999999999"

    "593242844"

    Returns: 5.932428441759685E8

  35. "999999999999999999"

    "132949346"

    Returns: 1.3294934600883776E8

  36. "999999999999999999"

    "859519858"

    Returns: 8.595198583693871E8

  37. "999999999999999999"

    "497127650"

    Returns: 4.9712765012356794E8

  38. "999999999999999999"

    "313332197"

    Returns: 3.1333219704908854E8

  39. "999999999999999999"

    "843442550"

    Returns: 8.434425503556976E8

  40. "999999999999999999"

    "580128563"

    Returns: 5.801285631682745E8

  41. "999999999999999999"

    "915226265"

    Returns: 9.152262654188195E8

  42. "999999999999999999"

    "241876283"

    Returns: 2.4187628302925208E8

  43. "999999999999999999"

    "504259864"

    Returns: 5.0425986412713903E8

  44. "999999999999999999"

    "989599133"

    Returns: 9.895991334896532E8

  45. "999999999999999999"

    "211899745"

    Returns: 2.1189974502245075E8

  46. "999999999999999999"

    "182974866"

    Returns: 1.829748660167399E8

  47. "999999999999999999"

    "723884458"

    Returns: 7.238844582620044E8

  48. "4"

    "3"

    Returns: 4.333333333333333

  49. "60506298085492492"

    "60482287309403175"

    Returns: 4.7388532696195898E17

  50. "4665942929939122"

    "335743874596"

    Returns: 3.35755954614849E11

  51. "5319470233654"

    "5126342963"

    Returns: 5.128814664763474E9

  52. "9813652408233"

    "9813652407749"

    Returns: 2.3289443921309678E14

  53. "573213573970"

    "573213561362"

    Returns: 1.0107129728913447E13

  54. "73129388"

    "73045284"

    Returns: 4.949342515358152E8

  55. "40728449215116"

    "40721158848733"

    Returns: 3.514102861854884E14

  56. "71382128232039"

    "71375745451819"

    Returns: 6.654384071671254E14

  57. "37379737142218905"

    "8974356504341557"

    Returns: 1.0262613697449582E16

  58. "39563965771"

    "39563142909"

    Returns: 4.265248795326505E11

  59. "84628205771417595"

    "84628145200144149"

    Returns: 1.19748550865484467E18

  60. "24536410"

    "23878648"

    Returns: 8.87989733711081E7

  61. "5913563619733"

    "521613038"

    Returns: 5.2163604410755926E8

  62. "37791927064759"

    "37791926967241"

    Returns: 7.473474225307289E14

  63. "37653154115181"

    "66239351664"

    Returns: 6.629768415224618E10

  64. "3820594460558616"

    "475687228"

    Returns: 4.756872576129777E8

  65. "5807435335"

    "1476954194"

    Returns: 1.704252278823898E9

  66. "64619176272"

    "64618114013"

    Returns: 7.118363321577192E11

  67. "38910086"

    "38909426"

    Returns: 4.273793126816553E8

  68. "74796428358507442"

    "118414"

    Returns: 118414.00000009373

  69. "6252130839182"

    "6252130838521"

    Returns: 1.4360790277529075E14

  70. "543357237"

    "543354085"

    Returns: 6.551435604766216E9

  71. "66540041012841108"

    "66539244951890869"

    Returns: 7.5414078334731712E17

  72. "905931919927123003"

    "1"

    Returns: 1.0

  73. "486849920209"

    "573694"

    Returns: 573694.338014321

  74. "87"

    "37"

    Returns: 47.8199468929702

  75. "496739394316243"

    "496739394316204"

    Returns: 1.4983030772153946E16

  76. "535667913269973"

    "215659429719"

    Returns: 2.1570285352247037E11

  77. "94151983529"

    "98551303"

    Returns: 9.860291711175844E7

  78. "379966603576794"

    "374352261569526"

    Returns: 1.6014676495951338E15

  79. "513583686136220168"

    "7469960"

    Returns: 7469960.000054324

  80. "978108330390709159"

    "803040"

    Returns: 803040.0000003297

  81. "429426496696119737"

    "429423507382296035"

    Returns: 5.0995092145787484E18

  82. "210554823153328092"

    "210554823153327331"

    Returns: 7.0016284277728819E18

  83. "853690371508865930"

    "853291399201239146"

    Returns: 6.546466421561686E18

  84. "326405084590734646"

    "326396321135677028"

    Returns: 3.4355128206642739E18

  85. "464036341552130300"

    "791588715360"

    Returns: 7.915893905370829E11

  86. "474531252245507565"

    "474444492582179939"

    Returns: 4.0842624684043228E18

  87. "292525114927782994"

    "292525114924319635"

    Returns: 7.3598082386360238E18

  88. "338187425126862149"

    "21414"

    Returns: 21414.000000000677

  89. "309367936096480831"

    "6019072547196"

    Returns: 6.019131101590341E12

  90. "193386337027428543"

    "193385937127692660"

    Returns: 2.5312311811119585E18

  91. "950469204046516353"

    "950469204046515467"

    Returns: 3.2894266854380958E19

  92. "509705434963486863"

    "30974186471"

    Returns: 3.0974187412132072E10

  93. "627596967838889090"

    "619587611393956203"

    Returns: 2.7371309838323512E18

  94. "844160530410783067"

    "21843926295104"

    Returns: 2.18442089222409E13

  95. "277271702306922347"

    "277921799310"

    Returns: 2.7792193859680334E11

  96. "774354797031945431"

    "738891232580620502"

    Returns: 2.3877418696483615E18

  97. "166060830199166566"

    "166060822608479443"

    Returns: 2.8065845607596708E18

  98. "150619633367322726"

    "2477022091"

    Returns: 2.4770221113679905E9

  99. "196960745448280138"

    "5833569075734"

    Returns: 5.833655466552742E12

  100. "349160207070283774"

    "617527"

    Returns: 617527.0000005461

  101. "839197076132227748"

    "10760876806063487"

    Returns: 1.0830464739248106E16

  102. "137958372494407101"

    "74702440931841191"

    Returns: 1.07574846579969072E17

  103. "939263848532610904"

    "939263848532605400"

    Returns: 3.078019441774887E19

  104. "512676880019313111"

    "512676601271567598"

    Returns: 7.3952866345569577E18

  105. "501610210123375766"

    "6890"

    Returns: 6890.000000000047

  106. "942865292384038211"

    "942856480033102044"

    Returns: 1.0918874569992776E19

  107. "389685216376644418"

    "389684419787389003"

    Returns: 5.1050753277104937E18

  108. "112276843502786937"

    "498"

    Returns: 498.0000000000011

  109. "1000000000000000000"

    "498403686"

    Returns: 4.9840368612420315E8

  110. "1000000000000000000"

    "4971"

    Returns: 4971.000000000013

  111. "1000000000000000000"

    "2366"

    Returns: 2366.0000000000027

  112. "1000000000000000000"

    "999999999185105559"

    Returns: 2.092796253070798E19

  113. "1000000000000000000"

    "119390"

    Returns: 119390.00000000713

  114. "1000000000000000000"

    "788"

    Returns: 788.0000000000003

  115. "1000000000000000000"

    "999999999440641158"

    Returns: 2.130422991223281E19

  116. "1000000000000000000"

    "230046"

    Returns: 230046.00000002646

  117. "1000000000000000000"

    "991917756477153814"

    Returns: 4.8180857812719821E18

  118. "1000000000000000000"

    "7337813760151"

    Returns: 7.337840682038088E12

  119. "1000000000000000000"

    "999999999255054531"

    Returns: 2.101771009552991E19

  120. "1000000000000000000"

    "29148703"

    Returns: 2.9148703000424825E7

  121. "1000000000000000000"

    "999999999999999997"

    Returns: 4.0190414005461025E19

  122. "1000000000000000000"

    "999999999409401374"

    Returns: 2.1249884472222515E19

  123. "1000000000000000000"

    "999999999999477768"

    Returns: 2.8280663503891202E19

  124. "1000000000000000000"

    "999999999999999955"

    Returns: 3.762879922324303E19

  125. "1000000000000000000"

    "999750537587707974"

    Returns: 8.2962023062580398E18

  126. "1000000000000000000"

    "999999999999999938"

    Returns: 3.7311354450961605E19

  127. "1000000000000000000"

    "694730772280"

    Returns: 6.947310136055348E11

  128. "1000000000000000000"

    "999999999999993548"

    Returns: 3.2674308741299286E19

  129. "1000000000000000000"

    "862111303675314"

    Returns: 8.624831353474951E14

  130. "1000000000000000000"

    "999999754920228772"

    Returns: 1.5221682082545816E19

  131. "1000000000000000000"

    "999999928483493014"

    Returns: 1.6453337546948065E19

  132. "1000000000000000000"

    "13214732800"

    Returns: 1.3214732887314583E10

  133. "1000000000000000000"

    "999999999999998070"

    Returns: 3.3880997347008344E19

  134. "1000000000000000000"

    "999999999999999994"

    Returns: 3.957374733879435E19

  135. "1000000000000000000"

    "264295500303380793"

    Returns: 3.0692673585136877E17

  136. "1000000000000000000"

    "1843393311"

    Returns: 1.8433933126990495E9

  137. "1000000000000000000"

    "2634322"

    Returns: 2634322.0000034696

  138. "1000000000000000000"

    "999998486134095005"

    Returns: 1.3400843976887755E19

  139. "1000000000000000000"

    "1"

    Returns: 1.0

  140. "1000000000000000000"

    "1000000000000000000"

    Returns: 4.202374733879435E19

  141. "98675537619876512"

    "986755376198"

    Returns: 9.867603100077731E11

  142. "1000000000000000000"

    "10000000000"

    Returns: 1.000000005E10

  143. "923450342389412399"

    "1234"

    Returns: 1234.000000000001

  144. "1000000000000000000"

    "12342364332"

    Returns: 1.2342364408166979E10

  145. "1235345612332"

    "1235345612332"

    Returns: 3.510801279628695E13

  146. "987605248730638741"

    "590479265483902874"

    Returns: 8.9973751949385613E17

  147. "1000000000000000000"

    "2345983459825"

    Returns: 2.3459862116485005E12

  148. "1000000000000000000"

    "1000000000000000"

    Returns: 1.0005003335835335E15

  149. "41415151223332"

    "3213212315225"

    Returns: 3.3447089157931855E12

  150. "1000000000000000000"

    "100000000000000000"

    Returns: 1.05360515657826304E17

  151. "1000045698467934"

    "400084679847"

    Returns: 4.0016473141660626E11

  152. "100000000000"

    "50000000000"

    Returns: 6.931471805549454E10

  153. "999999999999999999"

    "199999999999999999"

    Returns: 2.2314355131420976E17

  154. "10000000"

    "5000000"

    Returns: 6931471.305599478

  155. "948494948948941651"

    "489489415616516448"

    Returns: 6.8843110151150208E17

  156. "50000000"

    "50000000"

    Returns: 9.152374619146976E8

  157. "10000000000"

    "1"

    Returns: 1.0

  158. "637843264384"

    "2343434"

    Returns: 2343438.3048930094

  159. "10000000000000"

    "1000000000000"

    Returns: 1.0536051565782074E12

  160. "1337328974"

    "666199233"

    Returns: 9.220443660135001E8

  161. "100000000000000000"

    "10000000000000000"

    Returns: 1.053605156578263E16

  162. "34534643655675"

    "4356456643454"

    Returns: 4.65677585104082E12

  163. "1000000000000000000"

    "999999999999999999"

    Returns: 4.102374733879435E19

  164. "1000000000000000000"

    "500000000000000000"

    Returns: 6.9314718055994534E17

  165. "1000000000000000"

    "1000000000000000"

    Returns: 3.511599205981222E16

  166. "1000000000000000"

    "100000000000000"

    Returns: 1.0536051565782625E14

  167. "1000000000000000000"

    "11000000"

    Returns: 1.10000000000605E7

  168. "500000000"

    "50"

    Returns: 50.00000245000016

  169. "1000000000000000000"

    "1000000000"

    Returns: 1.0000000005E9

  170. "300000000"

    "299999999"

    Returns: 5.728952609756602E9

  171. "100000000000000001"

    "100000000000000000"

    Returns: 3.872116224580031E18

  172. "987654321987654321"

    "987654321987654321"

    Returns: 4.149266653071088E19

  173. "100000003"

    "50000001"

    Returns: 6.931471863543609E7

  174. "10000000"

    "10000000"

    Returns: 1.669531136585985E8

  175. "100000000000000000"

    "1000000000"

    Returns: 1.000000005E9

  176. "99999999999999999"

    "123456789"

    Returns: 1.2345678907620789E8

  177. "1000000000000000"

    "98745612382"

    Returns: 9.875048805095222E10

  178. "1000000000000000000"

    "5"

    Returns: 5.0

  179. "100000000000000000"

    "99999999999999999"

    Returns: 3.872116224580031E18

  180. "10000000000"

    "9000000000"

    Returns: 2.3025850925440456E10

  181. "99999999999999999"

    "999999999"

    Returns: 1.000000004E9

  182. "100000000"

    "100000000"

    Returns: 1.8997896413853898E9

  183. "999999999999999999"

    "999999999995999999"

    Returns: 2.6244726629808665E19

  184. "100000000000"

    "1"

    Returns: 1.0

  185. "1000000000000000000"

    "60000000"

    Returns: 6.00000000018E7

  186. "1000000000000000000"

    "4"

    Returns: 4.0

  187. "100000000000000000"

    "100000000000000000"

    Returns: 3.972116224580031E18

  188. "1000000000000000000"

    "30000010"

    Returns: 3.000001000045E7

  189. "10200000"

    "10199999"

    Returns: 1.602941627201915E8

  190. "1000000000000000000"

    "14574857348957574"

    Returns: 1.4682114025218992E16

  191. "123456789123456789"

    "1"

    Returns: 1.0

  192. "1000000000000000000"

    "2"

    Returns: 2.0

  193. "1000000000000000"

    "100000"

    Returns: 100000.00000499995

  194. "1000000000000000000"

    "99999999999998"

    Returns: 1.0000500033335633E14

  195. "1000000000000000000"

    "1000001"

    Returns: 1000001.0000005

  196. "1000000000000000000"

    "10000001"

    Returns: 1.000000100005E7

  197. "2000000"

    "1500000"

    Returns: 2772587.222240406

  198. "50000001"

    "1"

    Returns: 1.0

  199. "92346278348723564"

    "1111111"

    Returns: 1111111.0000066843

  200. "30000001"

    "1"

    Returns: 1.0

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