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ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
  • Threads: 101
  • Posts: 4828
July 5th, 2018 at 6:18:03 PM permalink
After some fancy number crunching with big integers to get around the significant digits problem, here is an approximation to the 2-coupon problem for up to 1000 different coupons, along with a ratio of the 2-coupon to 1-coupon values.

Coupons2 of Each1 of EachRatio
1046.22129.2901.578
20108.66771.9551.510
30177.162119.8501.478
40249.529171.1421.458
50324.765224.9601.444
60402.279280.7921.433
70481.672338.2991.424
80562.691397.2381.417
90645.102457.4311.410
100728.775518.7381.405
110813.565581.0461.400
120899.351644.2641.396
130986.093708.3171.392
1401073.683773.1401.389
1501162.080838.6771.386
1601251.196904.8821.383
1701341.026971.7121.380
1801431.5021039.1311.378
1901522.6051107.1051.375
2001614.2831175.6061.373
2101706.5201244.6071.371
2201799.2741314.0851.369
2301892.5391384.0171.367
2401986.2911454.3851.366
2502080.5131525.1691.364
2602175.1481596.3531.363
2702270.2241667.9221.361
2802365.7131739.8611.360
2902461.5931812.1581.358
3002557.8561884.7991.357
3102654.4811957.7741.356
3202751.4552031.0711.355
3302848.7972104.6811.354
3402946.4632178.5951.352
3503044.4512252.8021.351
3603142.7442327.2951.350
3703241.3652402.0661.349
3803340.2562477.1071.348
3903439.4612552.4111.348
4003538.9392627.9721.347
4103638.6952703.7831.346
4203738.6962779.8371.345
4303838.9952856.1301.344
4403939.5522932.6561.343
4504040.3243009.4081.343
4604141.3583086.3831.342
4704242.6293163.5761.341
4804344.1263240.9811.340
4904445.8823318.5941.340
5004547.8283396.4121.339
5104650.0203474.4291.338
5204752.4043552.6431.338
5304855.0033631.0491.337
5404957.8363709.6441.336
5505060.8403788.4241.336
5605164.0623867.3851.335
5705267.4663946.5261.335
5805371.0564025.8411.334
5905474.8684105.3291.334
6005578.8354184.9871.333
6105683.0014264.8111.333
6205787.3524344.8001.332
6305891.8644424.9491.332
6405996.5484505.2581.331
6506101.4024585.7221.331
6606206.4524666.3411.330
6706311.6344747.1101.330
6806417.0054828.0301.329
6906522.5134909.0961.329
7006628.2244990.3071.328
7106734.0435071.6611.328
7206840.0175153.1561.327
7306946.1715234.7901.327
7407052.4775316.5611.327
7507158.9095398.4671.326
7607265.4955480.5061.326
7707372.2235562.6771.325
7807479.0805644.9771.325
7907586.1105727.4061.325
8007693.2665809.9621.324
8107800.5485892.6421.324
8207907.9605975.4461.323
8308015.5376058.3721.323
8408123.2096141.4191.323
8508231.0266224.5841.322
8608338.9616307.8671.322
8708447.0486391.2671.322
8808555.2366474.7811.321
8908663.5866558.4091.321
9008772.0026642.1491.321
9108880.6056726.0011.320
9208989.2796809.9621.320
9309098.0806894.0321.320
9409207.0026978.2101.319
9509316.0467062.4941.319
9609425.1877146.8831.319
9709534.4647231.3761.318
9809643.8647315.9731.318
9909753.3407400.6711.318
10009862.9427485.4711.318


Ace2
Ace2 
Joined: Oct 2, 2017
  • Threads: 23
  • Posts: 965
July 5th, 2018 at 8:42:27 PM permalink
Quote: ThatDonGuy

As a matter of fact, I wouldn't be surprised if the limit of the ratio is 1. If you have, say, Googolplex (that's 1 followed by googol zeroes) different coupons, there's a very, very, very good chance you will be very close to having chosen all of the coupons at least twice by the time you choose every coupon once.

Yes it does appear that way.

These are some great answers from you and Mustang Sally. I found a fairly slick way to simulate this in excel but my computer freezes for anything over about 500 coupons.

Itís amazing how this problem is very simple for 1 of each coupon, but gets quite tedious for any other number, even with a formula.

From what I can tell, the formula is basically an enormous inclusion-exclusion exercise which explodes before the number of coupons gets very large.
Itís all about making that GTA
mustangsally
mustangsally
Joined: Mar 29, 2011
  • Threads: 25
  • Posts: 2463
July 6th, 2018 at 10:53:25 AM permalink
Quote: Ace2

Itís amazing how this problem is very simple for 1 of each coupon, but gets quite tedious for any other number, even with a formula.

yes it is.
of course, if it was real simple, every one could do it.
Quote: Ace2

From what I can tell, the formula is basically an enormous inclusion-exclusion exercise which explodes before the number of coupons gets very large.

agree.
even the approximation formula given (online and in papers) is way off on most results.
I am not wasting my time figuring out how far off and what ranges
others have done that before

I can only think of fun dice games to play with the distribution data
other than that, hard to imagine what good knowing the mean for at least 2 is.
from my R code not published yet
Time difference of 2.952016 mins <<<< takes time even for my machine
> print(sprintf("for %g items at least 2 each, mean:%g ",N,meanTrials))
[1] "for 38 items at least 2 each, mean: 234.833 "
> print(formatC(data, digits=10),quote=FALSE)
Draw X Draw X Prob cumulative: (X or less)
[1,] 75 0 0
[2,] 76 5.925597361e-21 5.925597361e-21
[3,] 77 1.461647349e-19 1.520903323e-19
[4,] 78 1.851099439e-18 2.003189771e-18
[5,] 79 1.603948436e-17 1.804267413e-17
[6,] 80 1.069168303e-16 1.249595045e-16
[7,] 81 5.845270243e-16 7.094865288e-16
[8,] 82 2.728872441e-15 3.43835897e-15
[9,] 83 1.118442664e-14 1.462278561e-14
[10,] 84 4.106353755e-14 5.568632316e-14
[11,] 85 1.371398506e-13 1.928261737e-13
[12,] 86 4.216488998e-13 6.144750736e-13
[13,] 87 1.205063898e-12 1.819538971e-12
[14,] 88 3.22684756e-12 5.046386531e-12
[15,] 89 8.149328924e-12 1.319571546e-11
[16,] 90 1.951922687e-11 3.271494232e-11
[17,] 91 4.455253589e-11 7.726747822e-11
[18,] 92 9.730631824e-11 1.745737965e-10
[19,] 93 2.040930538e-10 3.786668503e-10
[20,] 94 4.123859188e-10 7.910527691e-10
[21,] 95 8.049689046e-10 1.596021674e-09
[22,] 96 1.521708889e-09 3.117730563e-09
[23,] 97 2.792061784e-09 5.909792347e-09
[24,] 98 4.982252249e-09 1.08920446e-08
[25,] 99 8.661951043e-09 1.955399564e-08
[26,] 100 1.469616449e-08 3.425016013e-08
[27,] 101 2.436887984e-08 5.861903997e-08
[28,] 102 3.95456994e-08 9.816473937e-08
[29,] 103 6.288315778e-08 1.610478971e-07
[30,] 104 9.809256193e-08 2.591404591e-07
[31,] 105 1.502659394e-07 4.094063985e-07
[32,] 106 2.262709568e-07 6.356773553e-07
[33,] 107 3.352204269e-07 9.708977822e-07
[34,] 108 4.89020241e-07 1.459918023e-06
[35,] 109 7.029972444e-07 2.162915268e-06
[36,] 110 9.966057708e-07 3.159521038e-06
[37,] 111 1.394208334e-06 4.553729373e-06
[38,] 112 1.925922459e-06 6.479651831e-06
[39,] 113 2.62852195e-06 9.108173782e-06
[40,] 114 3.546377208e-06 1.265455099e-05
[41,] 115 4.73241567e-06 1.738696666e-05
[42,] 116 6.249080252e-06 2.363604691e-05
[43,] 117 8.169260964e-06 3.180530787e-05
[44,] 118 1.057717281e-05 4.238248068e-05
[45,] 119 1.356915178e-05 5.595163246e-05
[46,] 120 1.725434028e-05 7.320597274e-05
[47,] 121 2.175523392e-05 9.496120666e-05
[48,] 122 2.720806277e-05 0.0001221692694
[49,] 123 3.376298279e-05 0.0001559322522
[50,] 124 4.158405603e-05 0.0001975163083
[51,] 125 5.08490024e-05 0.0002483653107
[52,] 126 6.174871013e-05 0.0003101140208
[53,] 127 7.448649723e-05 0.000384600518
[54,] 128 8.927712172e-05 0.0004738776397
[55,] 129 0.0001063455439 0.0005802231836
[56,] 130 0.0001259254492 0.0007061486328
[57,] 131 0.0001482575467 0.0008544061794
[58,] 132 0.0001735876611 0.001027993841
[59,] 133 0.000202164644 0.001230158484
[60,] 134 0.0002342381305 0.001464396615
[61,] 135 0.0002700561739 0.001734452789
[62,] 136 0.0003098627913 0.00204431558
[63,] 137 0.0003538954567 0.002398211037
[64,] 138 0.0004023825774 0.002800593614
[65,] 139 0.0004555409911 0.003256134605
[66,] 140 0.0005135735197 0.003769708125
[67,] 141 0.0005766666157 0.004346374741
[68,] 142 0.0006449881318 0.004991362873
[69,] 143 0.0007186852474 0.00571004812
[70,] 144 0.0007978825768 0.006507930697
[71,] 145 0.0008826804836 0.00739061118
[72,] 146 0.0009731536228 0.008363764803
[73,] 147 0.001069349725 0.009433114528
[74,] 148 0.001171288633 0.01060440316
[75,] 149 0.00127896161 0.01188336477
[76,] 150 0.001392330902 0.01327569567
[77,] 151 0.001511329574 0.01478702525
[78,] 152 0.001635861612 0.01642288686
[79,] 153 0.001765802276 0.01818868913
[80,] 154 0.0019009987 0.02008968783
[81,] 155 0.002041270724 0.02213095856
[82,] 156 0.002186411947 0.02431737051
[83,] 157 0.002336190974 0.02665356148
[84,] 158 0.002490352849 0.02914391433
[85,] 159 0.002648620644 0.03179253497
[86,] 160 0.002810697192 0.03460323217
[87,] 161 0.002976266928 0.03757949909
[88,] 162 0.003144997836 0.04072449693
[89,] 163 0.003316543459 0.04404104039
[90,] 164 0.00349054497 0.04753158536
[91,] 165 0.003666633263 0.05119821862
[92,] 166 0.003844431065 0.05504264969
[93,] 167 0.004023555033 0.05906620472
[94,] 168 0.004203617829 0.06326982255
[95,] 169 0.004384230146 0.06765405269
[96,] 170 0.004565002691 0.07221905538
[97,] 171 0.00474554808 0.07696460346
[98,] 172 0.004925482663 0.08189008613
[99,] 173 0.005104428255 0.08699451438
[100,] 174 0.005282013762 0.09227652814
[101,] 175 0.005457876702 0.09773440485
[102,] 176 0.005631664612 0.1033660695
[103,] 177 0.005803036336 0.1091691058
[104,] 178 0.005971663192 0.115140769
[105,] 179 0.00613723002 0.121277999
[106,] 180 0.006299436105 0.1275774351
[107,] 181 0.006457995976 0.1340354311
[108,] 182 0.006612640088 0.1406480712
[109,] 183 0.006763115387 0.1474111866
[110,] 184 0.006909185754 0.1543203723
[111,] 185 0.007050632352 0.1613710047
[112,] 186 0.007187253849 0.1685582585
[113,] 187 0.007318866558 0.1758771251
[114,] 188 0.007445304472 0.1833224295
[115,] 189 0.00756641921 0.1908888488
[116,] 190 0.007682079887 0.1985709286
[117,] 191 0.007792172895 0.2063631015
[118,] 192 0.007896601627 0.2142597032
[119,] 193 0.007995286127 0.2222549893
[120,] 194 0.008088162683 0.230343152
[121,] 195 0.008175183375 0.2385183353
[122,] 196 0.008256315567 0.2467746509
[123,] 197 0.008331541368 0.2551061923
[124,] 198 0.008400857051 0.2635070493
[125,] 199 0.008464272454 0.2719713218
[126,] 200 0.008521810344 0.2804931321
[127,] 201 0.008573505773 0.2890666379
[128,] 202 0.008619405417 0.2976860433
[129,] 203 0.008659566903 0.3063456102
[130,] 204 0.008694058136 0.3150396684
[131,] 205 0.008722956616 0.323762625
[132,] 206 0.008746348769 0.3325089737
[133,] 207 0.00876432927 0.341273303
[134,] 208 0.008777000386 0.3500503034
[135,] 209 0.00878447132 0.3588347747
[136,] 210 0.008786857574 0.3676216323
[137,] 211 0.008784280324 0.3764059126
[138,] 212 0.008776865814 0.3851827784
[139,] 213 0.008764744769 0.3939475232
[140,] 214 0.008748051821 0.402695575
[141,] 215 0.008726924967 0.4114225
[142,] 216 0.008701505041 0.420124005
[143,] 217 0.008671935207 0.4287959402
[144,] 218 0.008638360485 0.4374343007
[145,] 219 0.008600927288 0.446035228
[146,] 220 0.008559782991 0.454595011
[147,] 221 0.008515075522 0.4631100865
[148,] 222 0.008466952974 0.4715770395
[149,] 223 0.008415563246 0.4799926027
[150,] 224 0.008361053697 0.4883536564
[151,] 225 0.008303570837 0.4966572273
[152,] 226 0.008243260029 0.5049004873
[153,] 227 0.008180265217 0.5130807525
[154,] 228 0.008114728674 0.5211954812
[155,] 229 0.008046790778 0.529242272
[156,] 230 0.007976589795 0.5372188618
[157,] 231 0.007904261691 0.5451231235
[158,] 232 0.007829939962 0.5529530634
[159,] 233 0.007753755478 0.5607068189
[160,] 234 0.007675836346 0.5683826553
[161,] 235 0.007596307791 0.575978963
[162,] 236 0.00751529205 0.5834942551
[163,] 237 0.007432908282 0.5909271634
[164,] 238 0.00734927249 0.5982764359
[165,] 239 0.007264497462 0.6055409333
[166,] 240 0.007178692717 0.612719626
[167,] 241 0.00709196447 0.6198115905
[168,] 242 0.007004415597 0.6268160061
[169,] 243 0.006916145628 0.6337321517
[170,] 244 0.006827250727 0.6405594025
[171,] 245 0.006737823704 0.6472972262
[172,] 246 0.006647954016 0.6539451802
[173,] 247 0.006557727789 0.660502908
[174,] 248 0.006467227837 0.6669701358
[175,] 249 0.006376533697 0.6733466695
[176,] 250 0.006285721659 0.6796323912
[177,] 251 0.006194864813 0.685827256
[178,] 252 0.006104033092 0.6919312891
[179,] 253 0.006013293322 0.6979445824
[180,] 254 0.005922709276 0.7038672917
[181,] 255 0.005832341735 0.7096996334
[182,] 256 0.005742248541 0.7154418819
[183,] 257 0.005652484669 0.7210943666
[184,] 258 0.005563102286 0.7266574689
[185,] 259 0.00547415082 0.7321316197
[186,] 260 0.00538567703 0.7375172968
[187,] 261 0.005297725075 0.7428150218
[188,] 262 0.005210336583 0.7480253584
[189,] 263 0.005123550728 0.7531489091
[190,] 264 0.005037404297 0.7581863134
[191,] 265 0.004951931765 0.7631382452
[192,] 266 0.004867165368 0.7680054106
[193,] 267 0.004783135175 0.7727885457
[194,] 268 0.004699869157 0.7774884149
[195,] 269 0.004617393267 0.7821058082
[196,] 270 0.0045357315 0.7866415397
[197,] 271 0.004454905973 0.7910964456
[198,] 272 0.00437493699 0.7954713826
[199,] 273 0.00429584311 0.7997672257
[200,] 274 0.004217641216 0.803984867
[201,] 275 0.004140346583 0.8081252135
[202,] 276 0.004063972939 0.8121891865
[203,] 277 0.003988532533 0.816177719
[204,] 278 0.003914036196 0.8200917552
[205,] 279 0.003840493399 0.8239322486
[206,] 280 0.003767912319 0.8277001609
[207,] 281 0.003696299895 0.8313964608
[208,] 282 0.003625661879 0.8350221227
[209,] 283 0.003556002902 0.8385781256
[210,] 284 0.003487326519 0.8420654521
[211,] 285 0.003419635266 0.8454850874
[212,] 286 0.003352930707 0.8488380181
[213,] 287 0.003287213488 0.8521252316
[214,] 288 0.003222483383 0.855347715
[215,] 289 0.003158739337 0.8585064543
[216,] 290 0.003095979514 0.8616024338
[217,] 291 0.003034201342 0.8646366352
[218,] 292 0.002973401549 0.8676100367
[219,] 293 0.002913576209 0.8705236129
[220,] 294 0.002854720778 0.8733783337
[221,] 295 0.002796830131 0.8761751638
[222,] 296 0.002739898601 0.8789150624
[223,] 297 0.002683920011 0.8815989824
[224,] 298 0.002628887706 0.8842278701
[225,] 299 0.002574794592 0.8868026647
[226,] 300 0.002521633158 0.8893242979
[227,] 301 0.002469395511 0.8917936934
[228,] 302 0.002418073403 0.8942117668
[229,] 303 0.002367658259 0.8965794251
[230,] 304 0.002318141201 0.8988975663
[231,] 305 0.002269513074 0.9011670793
[232,] 306 0.002221764471 0.9033888438
[233,] 307 0.002174885753 0.9055637296
[234,] 308 0.002128867072 0.9076925966
[235,] 309 0.002083698394 0.909776295
[236,] 310 0.002039369515 0.9118156645
[237,] 311 0.001995870081 0.9138115346
[238,] 312 0.001953189607 0.9157647242
[239,] 313 0.001911317494 0.9176760417
[240,] 314 0.001870243043 0.9195462848
[241,] 315 0.001829955471 0.9213762402
[242,] 316 0.00179044393 0.9231666842
[243,] 317 0.001751697513 0.9249183817
[244,] 318 0.001713705273 0.926632087
[245,] 319 0.001676456233 0.9283085432
[246,] 320 0.0016399394 0.9299484826
[247,] 321 0.001604143771 0.9315526264
[248,] 322 0.001569058349 0.9331216847
[249,] 323 0.001534672151 0.9346563569
[250,] 324 0.001500974214 0.9361573311
[251,] 325 0.001467953607 0.9376252847
[252,] 326 0.001435599441 0.9390608841
[253,] 327 0.001403900872 0.940464785
[254,] 328 0.001372847109 0.9418376321
[255,] 329 0.001342427424 0.9431800595
[256,] 330 0.001312631155 0.9444926907
[257,] 331 0.001283447714 0.9457761384
[258,] 332 0.00125486659 0.947031005
[259,] 333 0.001226877356 0.9482578823
[260,] 334 0.001199469671 0.949457352
[261,] 335 0.00117263329 0.9506299853
[262,] 336 0.00114635806 0.9517763434
[263,] 337 0.001120633931 0.9528969773
[264,] 338 0.001095450952 0.9539924282
[265,] 339 0.001070799283 0.9550632275
[266,] 340 0.001046669188 0.9561098967
[267,] 341 0.001023051045 0.9571329478
[268,] 342 0.0009999353442 0.9581328831
[269,] 343 0.0009773126913 0.9591101958
[270,] 344 0.0009551738092 0.9600653696
[271,] 345 0.0009335095392 0.9609988791
[272,] 346 0.0009123108424 0.96191119
[273,] 347 0.0008915688013 0.9628027588
[274,] 348 0.0008712746199 0.9636740334
[275,] 349 0.0008514196251 0.964525453
[276,] 350 0.0008319952673 0.9653574483
[277,] 351 0.0008129931206 0.9661704414
[278,] 352 0.000794404883 0.9669648463
[279,] 353 0.0007762223771 0.9677410687
[280,] 354 0.0007584375496 0.9684995062
[281,] 355 0.0007410424716 0.9692405487
[282,] 356 0.0007240293383 0.969964578
[283,] 357 0.0007073904684 0.9706719685
[284,] 358 0.0006911183043 0.9713630868
[285,] 359 0.0006752054111 0.9720382922
[286,] 360 0.0006596444764 0.9726979367
[287,] 361 0.0006444283092 0.973342365
[288,] 362 0.0006295498398 0.9739719149
[289,] 363 0.0006150021182 0.974586917
[290,] 364 0.0006007783141 0.9751876953
[291,] 365 0.0005868717153 0.975774567
[292,] 366 0.0005732757271 0.9763478427
[293,] 367 0.0005599838709 0.9769078266
[294,] 368 0.0005469897836 0.9774548164
[295,] 369 0.0005342872162 0.9779891036
[296,] 370 0.0005218700326 0.9785109736
[297,] 371 0.0005097322084 0.9790207058
[298,] 372 0.00049786783 0.9795185737
[299,] 373 0.0004862710928 0.9800048448
[300,] 374 0.0004749363003 0.9804797811
[301,] 375 0.0004638578628 0.9809436389
[302,] 376 0.0004530302958 0.9813966692
[303,] 377 0.0004424482188 0.9818391174
[304,] 378 0.0004321063542 0.9822712238
[305,] 379 0.0004219995253 0.9826932233
[306,] 380 0.0004121226555 0.983105346
[307,] 381 0.0004024707666 0.9835078167
[308,] 382 0.0003930389775 0.9839008557
[309,] 383 0.0003838225028 0.9842846782
[310,] 384 0.0003748166514 0.9846594949
[311,] 385 0.0003660168248 0.9850255117
[312,] 386 0.0003574185161 0.9853829302
[313,] 387 0.0003490173084 0.9857319475
[314,] 388 0.0003408088733 0.9860727564
[315,] 389 0.0003327889696 0.9864055454
[316,] 390 0.000324953442 0.9867304988
[317,] 391 0.0003172982194 0.987047797
[318,] 392 0.0003098193137 0.9873576163
[319,] 393 0.0003025128184 0.9876601292
[320,] 394 0.0002953749073 0.9879555041
[321,] 395 0.000288401833 0.9882439059
[322,] 396 0.0002815899255 0.9885254958
[323,] 397 0.000274935591 0.9888004314
[324,] 398 0.0002684353105 0.9890688667
[325,] 399 0.0002620856384 0.9893309524
[326,] 400 0.0002558832014 0.9895868356

at least I have a few photos to share
at least 1 of 38


at least 2 of 38


will update
https://sites.google.com/view/krapstuff/coupon-collecting/coupon-collecting-at-least-2
for those that want it

Sally
Last edited by: mustangsally on Jul 6, 2018
I Heart Vi Hart
Ace
Ace
Joined: Aug 15, 2013
  • Threads: 6
  • Posts: 43
July 6th, 2018 at 11:01:58 AM permalink
Quote: mustangsally

I can only think of fun dice games to play with the distribution data
other than that, hard to imagine what good knowing the mean for at least 2 is.

True, there probably isn't much of a practical use for knowing the mean of 2 or more. However, these problems are still useful since you sometimes learn things that are applicable for other scenarios.
Ace2
Ace2 
Joined: Oct 2, 2017
  • Threads: 23
  • Posts: 965
September 23rd, 2018 at 7:49:33 PM permalink
Quote: Ace2

Can someone run the number for 1000 coupons, getting at least 2 of each ? Iíd be interested to know if thereís a limit. .

Sure Ace, it would be my pleasure.

The answer of 9,862.97 is given by the integral from zero to infinity of :

(1 - ((x/1000) + 1) / e^(x/1000))^999 * (x/1000) / e^(x/1000) * x dx

I could probably give a few more digits but Iím very confident up to those (and canít imagine any need for more). I had to do this in excel, and couldnít really space more than 0.1, which requires about 300,000 lines. Unfortunately when I plug the equation into an online integral site it times out.

This is very close to the number given by ThatDonGuy.
Last edited by: Ace2 on Sep 23, 2018
Itís all about making that GTA
Ace2
Ace2 
Joined: Oct 2, 2017
  • Threads: 23
  • Posts: 965
February 8th, 2020 at 9:16:53 AM permalink
Quote: Ace

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.

I just learned that the formula for the Nth harmonic number can be further refined by adding the term 1/2n - 1/12n^2 + 1/120n^4.

So the formula for the expected number of roulette spins to get all 38 numbers becomes:

38 * ( Ln(38) + γ + 1/(2*38) - 1/(12*38^2) + 1/(120*38^4) ) = 160.6602765.

This is accurate to at least 10 digits which is as far as my iPhone calculator goes.
Itís all about making that GTA
Ace2
Ace2 
Joined: Oct 2, 2017
  • Threads: 23
  • Posts: 965
March 23rd, 2020 at 8:38:22 AM permalink
The exact answer for expected spins to get all 38 roulette numbers is the integral from zero to infinity of:

1-(1-1/e^(x/38))^38 dx

2053580969474233 / 12782132672400 = 160.66
Itís all about making that GTA

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