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guido111
guido111
Joined: Sep 16, 2010
  • Threads: 10
  • Posts: 707
October 15th, 2010 at 9:01:31 PM permalink
From the Wizard of Odds site:
"So the answer is the product of the expected number of spins at each step: (38/38)*(38/37)*(38/36)*Ö*(38/1)=160.66."

I only get the 160.66 when I ADD all the steps instead of multiply.
1
1.027027027
1.055555556
1.085714286
1.117647059
1.151515152
1.1875
1.225806452
1.266666667
1.310344828
1.357142857
1.407407407
1.461538462
1.52
1.583333333
1.652173913
1.727272727
1.80952381
1.9
2
2.111111111
2.235294118
2.375
2.533333333
2.714285714
2.923076923
3.166666667
3.454545455
3.8
4.222222222
4.75
5.428571429
6.333333333
7.6
9.5
12.66666667
19
38
___________
160.6602765
guido111
guido111
Joined: Sep 16, 2010
  • Threads: 10
  • Posts: 707
October 15th, 2010 at 9:12:48 PM permalink
Quote: DorothyGale


For example, with N = 2, the answer is 2*(1/2) + 3*(1/4) + 4*(1/8) + 5*(1/16) + ... = 3. So, the expected number of flips of a coin to get both heads and tails to appear is 3.


So then (2/2)+(2/1)=3 solves for the coin toss example.
Another Wizard Gem!
guido111
guido111
Joined: Sep 16, 2010
  • Threads: 10
  • Posts: 707
October 15th, 2010 at 9:24:40 PM permalink
Quote: Wizard

160.6602765.

I'll refrain from giving a solution to let others enjoy the problem.


A quick Excel simulation shows only a 59.4% chance that in 161 spins, all 38 numbers would hit.


65377 groups of 161 spins so the sample size is small and the error about 1.5%

Nope27 has a nice table showing the probability of NOT hitting "at least 1" number in x spins
http://wizardofvegas.com/member/nope27/blog/#post153
Ace
Ace
Joined: Aug 15, 2013
  • Threads: 6
  • Posts: 43
August 17th, 2016 at 9:44:04 AM permalink
I just found this old thread while searching for something else.

In case you didn't know the answer is closely approximated by 38 x (ln 38 + γ) = 160.16, where γ is the Euler constant .57721...

This is a version of the coupon collectors problem.
mustangsally
mustangsally
Joined: Mar 29, 2011
  • Threads: 25
  • Posts: 2463
May 26th, 2018 at 1:31:31 PM permalink
Quote: Ace

This is a version of the coupon collectors problem.

yes, the basic version
as you know
there are many many versions to the ccp.

one can run R code for this here online
https://sites.google.com/view/krapstuff/coupon-collecting


BruceZ R code provided above

data
       draw          Prob on X      cumulative    
[1,] 37 0 0
[2,] 38 4.86120346e-16 4.86120346e-16
[3,] 39 8.993226401e-15 9.479346747e-15
[4,] 40 8.614564237e-14 9.562498912e-14
[5,] 41 5.691765657e-13 6.648015548e-13
[6,] 42 2.915675884e-12 3.580477438e-12
[7,] 43 1.234202623e-11 1.592250366e-11
[8,] 44 4.493466909e-11 6.085717275e-11
[9,] 45 1.446233166e-10 2.054804894e-10
[10,] 46 4.197640261e-10 6.252445154e-10
[11,] 47 1.115394694e-09 1.740639209e-09
[12,] 48 2.745508363e-09 4.486147572e-09
[13,] 49 6.319513238e-09 1.080566081e-08
[14,] 50 1.370746777e-08 2.451312858e-08
[15,] 51 2.819825894e-08 5.271138752e-08
[16,] 52 5.531184799e-08 1.080232355e-07
[17,] 53 1.039291494e-07 2.119523849e-07
[18,] 54 1.877993422e-07 3.997517271e-07
[19,] 55 3.274732435e-07 7.272249706e-07
[20,] 56 5.526923563e-07 1.279917327e-06
[21,] 57 9.052368025e-07 2.185154129e-06
[22,] 58 1.442202938e-06 3.627357068e-06
[23,] 59 2.239645227e-06 5.867002295e-06
[24,] 60 3.39647971e-06 9.263482005e-06
[25,] 61 5.038511251e-06 1.430199326e-05
[26,] 62 7.322416712e-06 2.162440997e-05
[27,] 63 1.043949409e-05 3.206390406e-05
[28,] 64 1.461897549e-05 4.668287956e-05
[29,] 65 2.013070094e-05 6.68135805e-05
[30,] 66 2.728696108e-05 9.410054158e-05
[31,] 67 3.644333883e-05 0.0001305438804
[32,] 68 4.799841269e-05 0.0001785422931
[33,] 69 6.239222416e-05 0.0002409345173
[34,] 70 8.010345867e-05 0.0003210379759
[35,] 71 0.0001016453376 0.0004226833135
[36,] 72 0.000127560269 0.0005502435825
[37,] 73 0.0001584133524 0.0007086569349
[38,] 74 0.0001947848733 0.0009034418082
[39,] 75 0.000237261963 0.001140703771
[40,] 76 0.0002864296247 0.001427133396
[41,] 77 0.0003428613473 0.001769994743
[42,] 78 0.000407109541 0.002177104284
[43,] 79 0.0004796960279 0.002656800312
[44,] 80 0.0005611028174 0.003217903129
[45,] 81 0.0006517633789 0.003869666508
[46,] 82 0.0007520546081 0.004621721117
[47,] 83 0.0008622896541 0.005484010771
[48,] 84 0.0009827117486 0.006466722519
[49,] 85 0.001113489145 0.007580211664
[50,] 86 0.001254711243 0.008834922907
[51,] 87 0.001406385947 0.01024130885
[52,] 88 0.001568438264 0.01180974712
[53,] 89 0.001740710129 0.01355045725
[54,] 90 0.001922961422 0.01547341867
[55,] 91 0.002114872092 0.01758829076
[56,] 92 0.002316045332 0.01990433609
[57,] 93 0.002526011677 0.02243034777
[58,] 94 0.002744233928 0.0251745817
[59,] 95 0.002970112782 0.02814469448
[60,] 96 0.003202993039 0.03134768752
[61,] 97 0.003442170257 0.03478985777
[62,] 98 0.00368689774 0.03847675551
[63,] 99 0.003936393734 0.04241314925
[64,] 100 0.004189848706 0.04660299796
[65,] 101 0.004446432623 0.05104943058
[66,] 102 0.004705302106 0.05575473268
[67,] 103 0.004965607402 0.06072034009
[68,] 104 0.005226499062 0.06594683915
[69,] 105 0.005487134293 0.07143397344
[70,] 106 0.00574668291 0.07718065635
[71,] 107 0.006004332849 0.0831849892
[72,] 108 0.006259295209 0.08944428441
[73,] 109 0.006510808794 0.0959550932
[74,] 110 0.006758144154 0.1027132374
[75,] 111 0.007000607105 0.1097138445
[76,] 112 0.00723754173 0.1169513862
[77,] 113 0.007468332886 0.1244197191
[78,] 114 0.007692408205 0.1321121273
[79,] 115 0.007909239634 0.1400213669
[80,] 116 0.008118344515 0.1481397114
[81,] 117 0.008319286249 0.1564589977
[82,] 118 0.008511674556 0.1649706722
[83,] 119 0.008695165377 0.1736658376
[84,] 120 0.008869460438 0.1825352981
[85,] 121 0.009034306517 0.1915696046
[86,] 122 0.009189494436 0.200759099
[87,] 123 0.00933485782 0.2100939568
[88,] 124 0.009470271652 0.2195642285
[89,] 125 0.009595650648 0.2291598791
[90,] 126 0.009710947485 0.2388708266
[91,] 127 0.009816150919 0.2486869775
[92,] 128 0.009911283796 0.2585982613
[93,] 129 0.009996401006 0.2685946623
[94,] 130 0.01007158738 0.2786662497
[95,] 131 0.01013695557 0.2888032053
[96,] 132 0.01019264393 0.2989958492
[97,] 133 0.01023881434 0.3092346635
[98,] 134 0.01027565017 0.3195103137
[99,] 135 0.0103033542 0.3298136679
[100,] 136 0.01032214655 0.3401358145
[101,] 137 0.0103322628 0.3504680773
[102,] 138 0.01033395204 0.3608020293
[103,] 139 0.01032747508 0.3711295044
[104,] 140 0.01031310269 0.3814426071
[105,] 141 0.01029111397 0.391733721
[106,] 142 0.01026179471 0.4019955157
[107,] 143 0.01022543601 0.4122209518
[108,] 144 0.01018233277 0.4224032845
[109,] 145 0.01013278246 0.432536067
[110,] 146 0.01007708389 0.4426131509
[111,] 147 0.01001553604 0.4526286869
[112,] 148 0.009948437068 0.462577124
[113,] 149 0.009876083334 0.4724532073
[114,] 150 0.009798768516 0.4822519758
[115,] 151 0.00971678283 0.4919687587
[116,] 152 0.009630412305 0.501599171
[117,] 153 0.009539938145 0.5111391091
[118,] 154 0.009445636154 0.5205847453
[119,] 155 0.009347776235 0.5299325215
[120,] 156 0.009246621943 0.5391791434
[121,] 157 0.00914243011 0.5483215735
[122,] 158 0.009035450517 0.5573570241
[123,] 159 0.008925925627 0.5662829497
[124,] 160 0.008814090359 0.5750970401
[125,] 161 0.008700171916 0.583797212
[126,] 162 0.008584389651 0.5923816016
[127,] 163 0.008466954977 0.6008485566
[128,] 164 0.008348071305 0.6091966279
[129,] 165 0.008227934027 0.6174245619
[130,] 166 0.008106730519 0.6255312924
[131,] 167 0.007984640178 0.6335159326
[132,] 168 0.007861834482 0.6413777671
[133,] 169 0.00773847707 0.6491162442
[134,] 170 0.007614723847 0.656730968
[135,] 171 0.007490723104 0.6642216911
[136,] 172 0.007366615652 0.6715883068
[137,] 173 0.007242534973 0.6788308418
[138,] 174 0.007118607382 0.6859494491
[139,] 175 0.006994952199 0.6929444013
[140,] 176 0.006871681928 0.6998160833
[141,] 177 0.006748902444 0.7065649857
[142,] 178 0.006626713189 0.7131916989
[143,] 179 0.006505207365 0.7196969063
[144,] 180 0.006384472142 0.7260813784
[145,] 181 0.006264588852 0.7323459673
[146,] 182 0.006145633203 0.7384916005
[147,] 183 0.006027675475 0.7445192759
[148,] 184 0.005910780734 0.7504300567
[149,] 185 0.005795009025 0.7562250657
[150,] 186 0.005680415583 0.7619054813
[151,] 187 0.00556705103 0.7674725323
[152,] 188 0.005454961567 0.7729274939
[153,] 189 0.005344189176 0.778271683
[154,] 190 0.005234771804 0.7835064549
[155,] 191 0.005126743552 0.7886331984
[156,] 192 0.005020134854 0.7936533333
[157,] 193 0.004914972662 0.7985683059
[158,] 194 0.004811280611 0.8033795865
[159,] 195 0.004709079192 0.8080886657
[160,] 196 0.004608385915 0.8126970516
[161,] 197 0.004509215465 0.8172062671
[162,] 198 0.004411579859 0.821617847
[163,] 199 0.004315488595 0.8259333356
[164,] 200 0.00422094879 0.8301542843
[165,] 201 0.004127965323 0.8342822497
[166,] 202 0.004036540969 0.8383187906
[167,] 203 0.003946676521 0.8422654672
[168,] 204 0.003858370922 0.8461238381
[169,] 205 0.003771621375 0.8498954595
[170,] 206 0.003686423462 0.8535818829
[171,] 207 0.00360277125 0.8571846542
[172,] 208 0.003520657396 0.8607053116
[173,] 209 0.003440073247 0.8641453848
[174,] 210 0.003361008934 0.8675063937
[175,] 211 0.003283453461 0.8707898472
[176,] 212 0.003207394797 0.873997242
[177,] 213 0.003132819954 0.877130062
[178,] 214 0.003059715063 0.880189777
[179,] 215 0.002988065457 0.8831778425
[180,] 216 0.002917855732 0.8860956982
[181,] 217 0.002849069822 0.888944768
[182,] 218 0.002781691059 0.8917264591
[183,] 219 0.002715702232 0.8944421613
[184,] 220 0.002651085649 0.897093247
[185,] 221 0.002587823185 0.8996810702
[186,] 222 0.002525896339 0.9022069665
[187,] 223 0.002465286276 0.9046722528
[188,] 224 0.002405973878 0.9070782267
[189,] 225 0.002347939785 0.9094261664
[190,] 226 0.00229116443 0.9117173309
[191,] 227 0.002235628087 0.913952959
[192,] 228 0.002181310896 0.9161342699
[193,] 229 0.002128192904 0.9182624628
[194,] 230 0.002076254091 0.9203387168
[195,] 231 0.002025474402 0.9223641912
[196,] 232 0.001975833774 0.924340025
[197,] 233 0.001927312158 0.9262673372
[198,] 234 0.001879889547 0.9281472267
[199,] 235 0.001833545996 0.9299807727
[200,] 236 0.001788261643 0.9317690344
[201,] 237 0.001744016725 0.9335130511
[202,] 238 0.0017007916 0.9352138427
[203,] 239 0.001658566758 0.9368724095
[204,] 240 0.001617322841 0.9384897323
[205,] 241 0.001577040652 0.9400667729
[206,] 242 0.00153770117 0.9416044741
[207,] 243 0.001499285562 0.9431037597
[208,] 244 0.00146177519 0.9445655349
[209,] 245 0.001425151623 0.9459906865
[210,] 246 0.001389396645 0.9473800831
[211,] 247 0.001354492261 0.9487345754
[212,] 248 0.001320420707 0.9500549961
[213,] 249 0.001287164452 0.9513421606
[214,] 250 0.001254706207 0.9525968668
[215,] 251 0.001223028926 0.9538198957
[216,] 252 0.001192115813 0.9550120115
[217,] 253 0.001161950322 0.9561739618
[218,] 254 0.001132516164 0.957306478
[219,] 255 0.001103797306 0.9584102753
[220,] 256 0.001075777975 0.9594860533
[221,] 257 0.001048442655 0.9605344959
[222,] 258 0.001021776095 0.961556272
[223,] 259 0.0009957633018 0.9625520353
[224,] 260 0.000970389547 0.9635224249
[225,] 261 0.0009456403615 0.9644680652
[226,] 262 0.0009215015376 0.9653895668
[227,] 263 0.0008979591278 0.9662875259
[228,] 264 0.0008749994431 0.9671625253
[229,] 265 0.0008526090524 0.9680151344
[230,] 266 0.0008307747802 0.9688459092
[231,] 267 0.0008094837052 0.9696553929
[232,] 268 0.0007887231584 0.970444116
[233,] 269 0.0007684807205 0.9712125968
[234,] 270 0.0007487442199 0.971961341
[235,] 271 0.0007295017299 0.9726908427
[236,] 272 0.0007107415665 0.9734015843
[237,] 273 0.0006924522853 0.9740940366
[238,] 274 0.0006746226787 0.9747686592
[239,] 275 0.0006572417731 0.975425901
[240,] 276 0.0006402988256 0.9760661998
[241,] 277 0.0006237833211 0.9766899832
[242,] 278 0.0006076849692 0.9772976681
[243,] 279 0.0005919937005 0.9778896618
[244,] 280 0.0005766996639 0.9784663615
[245,] 281 0.0005617932228 0.9790281547
[246,] 282 0.0005472649521 0.9795754197
[247,] 283 0.0005331056343 0.9801085253
[248,] 284 0.0005193062567 0.9806278316
[249,] 285 0.0005058580076 0.9811336896
[250,] 286 0.000492752273 0.9816264418
[251,] 287 0.000479980633 0.9821064225
[252,] 288 0.0004675348586 0.9825739573
[253,] 289 0.0004554069079 0.9830293642
[254,] 290 0.0004435889232 0.9834729532
[255,] 291 0.0004320732272 0.9839050264
[256,] 292 0.0004208523196 0.9843258787
[257,] 293 0.0004099188739 0.9847357976
[258,] 294 0.000399265734 0.9851350633
[259,] 295 0.000388885911 0.9855239492
[260,] 296 0.0003787725794 0.9859027218
[261,] 297 0.0003689190746 0.9862716409
[262,] 298 0.0003593188889 0.9866309598
[263,] 299 0.0003499656689 0.9869809254
[264,] 300 0.0003408532121 0.9873217786
[265,] 301 0.0003319754636 0.9876537541
[266,] 302 0.0003233265134 0.9879770806
[267,] 303 0.0003149005929 0.9882919812
[268,] 304 0.0003066920722 0.9885986733
[269,] 305 0.0002986954573 0.9888973687
[270,] 306 0.0002909053867 0.9891882741
[271,] 307 0.0002833166287 0.9894715908
[272,] 308 0.0002759240789 0.9897475148
[273,] 309 0.000268722757 0.9900162376
[274,] 310 0.0002617078041 0.9902779454
[275,] 311 0.0002548744803 0.9905328199
[276,] 312 0.0002482181617 0.990781038
[277,] 313 0.0002417343379 0.9910227724
[278,] 314 0.0002354186097 0.991258191
[279,] 315 0.0002292666861 0.9914874577
[280,] 316 0.0002232743822 0.9917107321
[281,] 317 0.0002174376167 0.9919281697
[282,] 318 0.0002117524093 0.9921399221
[283,] 319 0.0002062148787 0.992346137
[284,] 320 0.00020082124 0.9925469582
[285,] 321 0.0001955678028 0.992742526
[286,] 322 0.0001904509686 0.992932977
[287,] 323 0.0001854672289 0.9931184442
[288,] 324 0.000180613163 0.9932990574
[289,] 325 0.0001758854358 0.9934749428
[290,] 326 0.0001712807963 0.9936462236
[291,] 327 0.0001667960747 0.9938130197
[292,] 328 0.0001624281814 0.9939754479
[293,] 329 0.0001581741043 0.994133622
[294,] 330 0.0001540309076 0.9942876529
[295,] 331 0.0001499957294 0.9944376486
[296,] 332 0.0001460657803 0.9945837144
[297,] 333 0.0001422383416 0.9947259527
[298,] 334 0.0001385107633 0.9948644635
[299,] 335 0.0001348804628 0.9949993439
[300,] 336 0.0001313449229 0.9951306889
[301,] 337 0.0001279016907 0.9952585906
[302,] 338 0.0001245483754 0.9953831389
[303,] 339 0.0001212826472 0.9955044216
[304,] 340 0.0001181022359 0.9956225238
[305,] 341 0.000115004929 0.9957375287
[306,] 342 0.0001119885705 0.9958495173
[307,] 343 0.0001090510597 0.9959585684
[308,] 344 0.0001061903493 0.9960647587
[309,] 345 0.0001034044449 0.9961681632
[310,] 346 0.0001006914027 0.9962688546
[311,] 347 9.804932913e-05 0.9963669039
[312,] 348 9.547637895e-05 0.9964623803
[313,] 349 9.297075443e-05 0.996555351
[314,] 350 9.053070399e-05 0.9966458817
[315,] 351 8.815452109e-05 0.9967340363
[316,] 352 8.584054311e-05 0.9968198768
[317,] 353 8.358715022e-05 0.996903464
[318,] 354 8.139276432e-05 0.9969848567
[319,] 355 7.925584798e-05 0.9970641126
[320,] 356 7.717490343e-05 0.9971412875
[321,] 357 7.514847154e-05 0.9972164359
[322,] 358 7.317513086e-05 0.9972896111
[323,] 359 7.125349666e-05 0.9973608646
[324,] 360 6.938221999e-05 0.9974302468
[325,] 361 6.755998678e-05 0.9974978068
[326,] 362 6.578551697e-05 0.9975635923
[327,] 363 6.405756363e-05 0.9976276499
[328,] 364 6.237491209e-05 0.9976900248
[329,] 365 6.073637919e-05 0.9977507611
[330,] 366 5.91408124e-05 0.997809902
[331,] 367 5.75870891e-05 0.997867489
[332,] 368 5.607411576e-05 0.9979235632
[333,] 369 5.460082725e-05 0.997978164
[334,] 370 5.316618609e-05 0.9980313302
[335,] 371 5.176918172e-05 0.9980830994
[336,] 372 5.040882985e-05 0.9981335082
[337,] 373 4.908417178e-05 0.9981825924
[338,] 374 4.77942737e-05 0.9982303866
[339,] 375 4.653822611e-05 0.9982769249
[340,] 376 4.531514316e-05 0.99832224
[341,] 377 4.412416205e-05 0.9983663642
[342,] 378 4.296444245e-05 0.9984093286
[343,] 379 4.18351659e-05 0.9984511638
[344,] 380 4.073553525e-05 0.9984918993
[345,] 381 3.966477413e-05 0.9985315641
[346,] 382 3.862212641e-05 0.9985701862
[347,] 383 3.760685564e-05 0.9986077931
[348,] 384 3.66182446e-05 0.9986444113
[349,] 385 3.565559477e-05 0.9986800669
[350,] 386 3.471822583e-05 0.9987147851
[351,] 387 3.380547524e-05 0.9987485906
[352,] 388 3.291669774e-05 0.9987815073
[353,] 389 3.205126492e-05 0.9988135586
[354,] 390 3.120856475e-05 0.9988447671
[355,] 391 3.038800123e-05 0.9988751551
[356,] 392 2.958899389e-05 0.9989047441
[357,] 393 2.881097744e-05 0.9989335551
[358,] 394 2.805340137e-05 0.9989616085
[359,] 395 2.731572954e-05 0.9989889242
[360,] 396 2.659743985e-05 0.9990155217
[361,] 397 2.589802382e-05 0.9990414197
[362,] 398 2.52169863e-05 0.9990666367
[363,] 399 2.455384506e-05 0.9990911905
[364,] 400 2.390813051e-05 0.9991150987
from avg draws
0 160.66028
1 159.66028
2 158.63325
3 157.57769
4 156.49198
5 155.37433
6 154.22282
7 153.03532
8 151.80951
9 150.54284
10 149.23250
11 147.87536
12 146.46795
13 145.00641
14 143.48641
15 141.90308
16 140.25090
17 138.52363
18 136.71411
19 134.81411
20 132.81411
21 130.70300
22 128.46770
23 126.09270
24 123.55937
25 120.84508
26 117.92201
27 114.75534
28 111.30079
29 107.50079
30 103.27857
31 98.52857
32 93.10000
33 86.76667
34 79.16667
35 69.66667
36 57.00000
37 38.00000

"A quick Excel simulation shows only a 59.4% chance that in 161 spins, all 38 numbers would hit."

that I calculated using a Markov chain in R (in the spoiler results)
161 spins: all 38 numbers prob: 0.583797212

Sally
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Ace2
Ace2
Joined: Oct 2, 2017
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May 26th, 2018 at 1:44:36 PM permalink
58 % sounds reasonable. We know itís between 50 % and 63% (1 - 1/e).
Itís all about making that GTA
Ace2
Ace2
Joined: Oct 2, 2017
  • Threads: 23
  • Posts: 973
June 28th, 2018 at 9:19:30 PM permalink
Has anyone ever worked the coupon collector problem, but with 2 hits required? So, for instance, what is the expected number of roulette spins for every number to have appeared at least twice ? I thought about it today but no solution so far.
Itís all about making that GTA
mustangsally
mustangsally
Joined: Mar 29, 2011
  • Threads: 25
  • Posts: 2463
June 28th, 2018 at 10:34:32 PM permalink
Quote: Ace2

Has anyone ever worked the coupon collector problem, but with 2 hits required?

yes, I actually have but it recently got off of my A-list.

I was reading this interesting pdf
(Handsome man, sweet family)

https://www.cmc3.org/conference/Tahoe15/Tahoe15_JLee.pdf

"Two Questions I Started To Ask Myself
1) How Many Happy Meals Do I Have To Buy
Before I Collect All 8 Toys?
2) How Much Money Will I Spend Until I Have
Collected All 8 Toys?"
What If We Want To Collect Two Complete Sets?

in the pdf he continues
"Another Method To Calculate The Waiting Time
Is To Use The Dirichlet Type-II C-Integral
This Is Used To Calculate The Lower Tail Of
The Multinomial Distribution"

ok
learn some more stuff

well,
I started with a Markov chain approach (states get large)
but found myself completing other projects first (as I usually do)

not much easy stuff out there that I found

I am more into the actual distribution (the mean is ok too)
as one can create fun games to play from them (imo, of course)
Sally
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mustangsally
mustangsally
Joined: Mar 29, 2011
  • Threads: 25
  • Posts: 2463
June 29th, 2018 at 9:05:57 AM permalink
ok, I have this close to being done using the Markov chain solution
yes, there are other methods too. I like this method and start in Excel


*****
regular coupon collect problem of at least 1 of each:
will start with coupons=3
(coupons=38 asked for will be later as only in Excel right now and NOT online yet. the number of states is (N+1)*(N+2) / 2 = 780 so I have to get it right the 1st time)

mean wait time for at least 1 of each = 11/2 = 5.5
the distribution
(from here: https://sites.google.com/view/krapstuff/coupon-collecting
section 3r.)
Draw xDraw x Probcumulative: x or less
200
30.2222222220.222222222
40.2222222220.444444444
50.1728395060.617283951
60.123456790.740740741
70.0850480110.825788752
80.0576131690.88340192
90.0387136110.922115531
100.0259106840.948026216
110.017307660.965333875
120.011549730.976883605
130.0077035830.984587188
140.0051369770.989724165
150.0034250690.993149234
160.0022835190.995432753
170.0015223920.996955146
180.0010149440.997970089
190.0006766340.998646724
200.0004510910.999097815


*****
coupon collect problem of at least 2 of each:
with coupons=3
mean wait time for at least 1 of each = 347/36 (about 9.638888889)
the distribution
draw xdraw x probcumulative: (x or less)
100
200
300
400
500
60.123456790.12345679
70.1646090530.288065844
80.160036580.448102423
90.1365645480.584666971
100.1088248740.693491846
110.0833206320.776812478
120.0622195130.83903199
130.0457021570.884734147
140.03318650.917920647
150.0238964540.941817101
160.0170960760.958913177
170.0121676170.971080794
180.0086226780.979703471
190.0060880830.985791554
200.0042847620.990076316


what they both look like
1st = at least 1
2nd = at least 2


1st to see
Sally
Last edited by: mustangsally on Jun 29, 2018
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Gabes22
Gabes22
Joined: Jul 19, 2011
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June 29th, 2018 at 1:26:48 PM permalink
The only problem I see in the McDonald's problem is don't they "release" the different toys at different dates. i.e. releasing toy A on June 1st and toy B on June 8th and so on and so forth, They might do it in waves to like release A, B and C on one date then D, E and F on another and so on and so forth. Knowing that information may drastically change the odds.
A flute with no holes is not a flute, a donut with no holes is a danish

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