Coupon CollectING Problem.math
Krapstuff Coupon Collect Problem Calculator
Average number of draws to collect at least 1 distinct coupon from total set of coupons (N).
Max number of coupons: about 40,000,000.
Your mileage will vary.
collecting at least 1 (one) coupon of each kind from a set of coupons. One (1) coupon at a time. For 1 or more coupons at a time see the Baseball Card Collector Problem page.
Coupons can be different types of items.
example: roll a standard die (1d6)
How many rolls on average to get at least 1 of each face?
N = 6 (coupons = 6)
answer 14.7 rolls on average
R code below data from pari gp
pari gp code. formula used: i*sum(k=1,i,1/k)
results: 1 to 100 coupons
\\pari GP function code.
meanCC.tMax=function(tMax=40)=
{
prob=vector(tMax);seqN=Col([1..tMax]);
for(i=1,tMax,prob[i]=i*sum(k=1,i,1/k));
probC=Col(prob);probDec=prob*1.;
probDecC=Col(probDec);
cc=matconcat([seqN,probDecC,probC,seqN]);
return(cc);
}
\p6;
meanCC.tMax(100)
results: coupons, average draws (decimal), average draws (fraction), coupons
[ 1 1.00000 1 1]
[ 2 3.00000 3 2]
[ 3 5.50000 11/2 3]
[ 4 8.33333 25/3 4]
[ 5 11.4167 137/12 5]
[ 6 14.7000 147/10 6]
[ 7 18.1500 363/20 7]
[ 8 21.7429 761/35 8]
[ 9 25.4607 7129/280 9]
[ 10 29.2897 7381/252 10]
[ 11 33.2187 83711/2520 11]
[ 12 37.2385 86021/2310 12]
[ 13 41.3417 1145993/27720 13]
[ 14 45.5219 1171733/25740 14]
[ 15 49.7734 1195757/24024 15]
[ 16 54.0917 2436559/45045 16]
[ 17 58.4724 42142223/720720 17]
[ 18 62.9119 42822903/680680 18]
[ 19 67.4071 275295799/4084080 19]
[ 20 71.9548 279175675/3879876 20]
[ 21 76.5525 56574159/739024 21]
[ 22 81.1979 19093197/235144 22]
[ 23 85.8887 444316699/5173168 23]
[ 24 90.6230 1347822955/14872858 24]
[ 25 95.3990 34052522467/356948592 25]
[ 26 100.215 34395742267/343219800 26]
[ 27 105.069 312536252003/2974571600 27]
[ 28 109.961 315404588903/2868336900 28]
[ 29 114.888 9227046511387/80313433200 29]
[ 30 119.850 9304682830147/77636318760 30]
[ 31 124.845 290774257297357/2329089562800 31]
[ 32 129.872 586061125622639/4512611027925 32]
[ 33 134.930 590436990861839/4375865239200 33]
[ 34 140.019 54062195834749/386105756400 34]
[ 35 145.137 54437269998109/375074163360 35]
[ 36 150.284 54801925434709/364655436600 36]
[ 37 155.459 2040798836801833/13127595717600 37]
[ 38 160.660 2053580969474233/12782132672400 38]
[ 39 165.888 2066035355155033/12454385680800 39]
[ 40 171.142 2078178381193813/12143026038780 40]
[ 41 176.420 85691034670497533/485721041551200 41]
[ 42 181.723 86165190925345133/474156254847600 42]
[ 43 187.050 532145396070491417/2844937529085600 43]
[ 44 192.400 5884182435213075787/30583078437670200 44]
[ 45 197.773 5914085889685464427/29903454472388640 45]
[ 46 203.168 5943339269060627227/29253379375162800 46]
[ 47 208.584 280682601097106968469/1345655451257488800 47]
[ 48 214.022 282000222059796592919/1317620962689624450 48]
[ 49 219.481 13881256687139135026631/63245806209101973600 49]
[ 50 224.960 13943237577224054960759/61980890084919934128 50]
[ 51 230.459 14004003155738682347159/60765578514627386400 51]
[ 52 235.978 14063600165435720745359/59597009697038398200 52]
[ 53 241.516 748469853272339196210427/3099044504245996706400 53]
[ 54 247.073 751511508063543600385227/3041654791204404174800 54]
[ 55 252.649 251499286680120823312889/995450658939623184480 55]
[ 56 258.242 252476961434436524654789/977674754315701341900 56]
[ 57 263.854 253437484000080020709989/960522565643496055200 57]
[ 58 269.483 254381445831833111660789/943961831753090950800 58]
[ 59 275.129 15063255090319832863132951/54749786241679275146400 59]
[ 60 280.792 15117092380124150817026911/53837289804317953893960 60]
[ 61 286.472 925372872575832277072279171/3230237388259077233637600 61]
[ 62 292.168 928551009361054917576341971/3178136785222640504062800 62]
[ 63 297.881 931678699530639103469229171/3127690169584185892887200 63]
[ 64 303.609 623171679694215690971693339/2052546673789621992207225 64]
[ 65 309.353 625192648726870088010174299/2020969032654397038480960 65]
[ 66 315.112 627182997016605479032920699/1990348289735391022746400 66]
[ 67 320.887 14050874595745034300902316411/43787662374178602500420800 67]
[ 68 326.676 14094018321907827923954201611/43143726162793623051885200 68]
[ 69 332.480 42409610330030873613929048033/127555364307389842066443200 69]
[ 70 338.299 42535343474848157886823113473/125733144817284272894065440 70]
[ 71 344.131 3028810706851429109067025637383/8801320137209899102584580800 71]
[ 72 349.978 9112469359293533278712889630349/26037238739245951511812718200 72]
[ 73 355.839 667084944417653637854891458725877/1874681189225708508850515710400 73]
[ 74 361.714 668934292077295215167676426926677/1849347659641577312784968200800 74]
[ 75 367.602 670758981768141571449624262218133/1824689690846356281947835291456 75]
[ 76 373.503 672559662384108370412072783887333/1800680615966798962448521669200 76]
[ 77 379.418 674336957537530146011372623456933/1777295153421775599299839569600 77]
[ 78 385.345 61462860623241058403302042280303/159500847101954220449985602400 78]
[ 79 391.285 4868007055309996043055960217131137/12441066073952429195098876987200 79]
[ 80 397.238 4880292608058024066886120358155997/12285552748028023830160141024860 80]
[ 81 403.204 44031838385838021258243173365847173/109204913315804656268090142443200 81]
[ 82 409.182 44139711531918267321142140457772773/107873146080246062898967091925600 82]
[ 83 415.172 3672441655127796364812512959533039359/8845597978580177157715301537899200 83]
[ 84 421.174 3681181948368536301765969745576439759/8740293240739936953456786043400400 84]
[ 85 427.188 3689819414629973415931738804725211919/8637466261437114165769059148772160 85]
[ 86 433.213 3698356445237207772956045432953649519/8537030607234357024306628228437600 86]
[ 87 439.251 3706795349055853229324900260857622319/8438903818645456368854827903972800 87]
[ 88 445.300 40866521918642154860585199122889549709/91773079027769338011296253455704200 88]
[ 89 451.360 3645196481713595484337076792241271893701/8076030954443701744994070304101969600 89]
[ 90 457.431 3653182778990767589396015372875328285861/7986297277172105058938580634056392160 90]
[ 91 463.514 3661081314759399341652108474601318124261/7898535768631752256093101725989838400 91]
[ 92 469.607 3668893996878372053122809260004199377461/7812682118972711470700785402881253200 92]
[ 93 475.712 3676622671662732154792749821908124918261/7728674784360101669940561903925540800 93]
[ 94 481.827 3684269126502577787295988888472646995861/7646454839845632503239066564522077600 94]
[ 95 487.953 3691835092344109255246562280652279367381/7565965841531467950573392179632371520 95]
[ 96 494.089 3699322246041458103739317199996707235031/7487153697348848492754919344427867650 96]
[ 97 500.236 359553024620966925518018240656745677092407/718766754945489455304472257065075294400 97]
[ 98 506.393 360264457021270114060513483605065190394007/711432400303188542495242948319513301600 98]
[ 99 512.560 360968703235711654233892612988250163157207/704246214441540173379129383184972763200 99]
[100 518.738 361665906988008779005537951077603286192775/697203752297124771645338089353123035568 100]
pari gp code. formula used: i*sum(k=1,i,1/k)
results: 1 to 1000 coupons
\\pari GP function code.
meanCC.tMax=function(tMax=40)=
{
prob=vector(tMax);seqN=Col([1..tMax]);
for(i=1,tMax,prob[i]=i*sum(k=1,i,1/k));
probC=Col(prob);probDec=prob*1.;
probDecC=Col(probDec);
cc=matconcat([seqN,probDecC]);
return(cc);
}
\p6;
meanCC.tMax(1000)
results: coupons, average number of draws
[ 1 1.00000]
[ 2 3.00000]
[ 3 5.50000]
[ 4 8.33333]
[ 5 11.4167]
[ 6 14.7000]
[ 7 18.1500]
[ 8 21.7429]
[ 9 25.4607]
[ 10 29.2897]
[ 11 33.2187]
[ 12 37.2385]
[ 13 41.3417]
[ 14 45.5219]
[ 15 49.7734]
[ 16 54.0917]
[ 17 58.4724]
[ 18 62.9119]
[ 19 67.4071]
[ 20 71.9548]
[ 21 76.5525]
[ 22 81.1979]
[ 23 85.8887]
[ 24 90.6230]
[ 25 95.3990]
[ 26 100.215]
[ 27 105.069]
[ 28 109.961]
[ 29 114.888]
[ 30 119.850]
[ 31 124.845]
[ 32 129.872]
[ 33 134.930]
[ 34 140.019]
[ 35 145.137]
[ 36 150.284]
[ 37 155.459]
[ 38 160.660]
[ 39 165.888]
[ 40 171.142]
[ 41 176.420]
[ 42 181.723]
[ 43 187.050]
[ 44 192.400]
[ 45 197.773]
[ 46 203.168]
[ 47 208.584]
[ 48 214.022]
[ 49 219.481]
[ 50 224.960]
[ 51 230.459]
[ 52 235.978]
[ 53 241.516]
[ 54 247.073]
[ 55 252.649]
[ 56 258.242]
[ 57 263.854]
[ 58 269.483]
[ 59 275.129]
[ 60 280.792]
[ 61 286.472]
[ 62 292.168]
[ 63 297.881]
[ 64 303.609]
[ 65 309.353]
[ 66 315.112]
[ 67 320.887]
[ 68 326.676]
[ 69 332.480]
[ 70 338.299]
[ 71 344.131]
[ 72 349.978]
[ 73 355.839]
[ 74 361.714]
[ 75 367.602]
[ 76 373.503]
[ 77 379.418]
[ 78 385.345]
[ 79 391.285]
[ 80 397.238]
[ 81 403.204]
[ 82 409.182]
[ 83 415.172]
[ 84 421.174]
[ 85 427.188]
[ 86 433.213]
[ 87 439.251]
[ 88 445.300]
[ 89 451.360]
[ 90 457.431]
[ 91 463.514]
[ 92 469.607]
[ 93 475.712]
[ 94 481.827]
[ 95 487.953]
[ 96 494.089]
[ 97 500.236]
[ 98 506.393]
[ 99 512.560]
[ 100 518.738]
[ 101 524.925]
[ 102 531.122]
[ 103 537.329]
[ 104 543.546]
[ 105 549.773]
[ 106 556.009]
[ 107 562.254]
[ 108 568.509]
[ 109 574.773]
[ 110 581.046]
[ 111 587.328]
[ 112 593.619]
[ 113 599.919]
[ 114 606.228]
[ 115 612.546]
[ 116 618.873]
[ 117 625.208]
[ 118 631.552]
[ 119 637.904]
[ 120 644.264]
[ 121 650.633]
[ 122 657.010]
[ 123 663.396]
[ 124 669.789]
[ 125 676.191]
[ 126 682.600]
[ 127 689.017]
[ 128 695.443]
[ 129 701.876]
[ 130 708.317]
[ 131 714.765]
[ 132 721.222]
[ 133 727.685]
[ 134 734.157]
[ 135 740.636]
[ 136 747.122]
[ 137 753.615]
[ 138 760.116]
[ 139 766.624]
[ 140 773.140]
[ 141 779.662]
[ 142 786.191]
[ 143 792.728]
[ 144 799.272]
[ 145 805.822]
[ 146 812.379]
[ 147 818.944]
[ 148 825.515]
[ 149 832.093]
[ 150 838.677]
[ 151 845.268]
[ 152 851.866]
[ 153 858.470]
[ 154 865.081]
[ 155 871.699]
[ 156 878.323]
[ 157 884.953]
[ 158 891.590]
[ 159 898.233]
[ 160 904.882]
[ 161 911.537]
[ 162 918.199]
[ 163 924.867]
[ 164 931.541]
[ 165 938.221]
[ 166 944.907]
[ 167 951.599]
[ 168 958.298]
[ 169 965.002]
[ 170 971.712]
[ 171 978.428]
[ 172 985.150]
[ 173 991.877]
[ 174 998.611]
[ 175 1005.35]
[ 176 1012.09]
[ 177 1018.85]
[ 178 1025.60]
[ 179 1032.36]
[ 180 1039.13]
[ 181 1045.90]
[ 182 1052.68]
[ 183 1059.47]
[ 184 1066.26]
[ 185 1073.05]
[ 186 1079.85]
[ 187 1086.66]
[ 188 1093.47]
[ 189 1100.28]
[ 190 1107.11]
[ 191 1113.93]
[ 192 1120.76]
[ 193 1127.60]
[ 194 1134.44]
[ 195 1141.29]
[ 196 1148.14]
[ 197 1155.00]
[ 198 1161.87]
[ 199 1168.73]
[ 200 1175.61]
[ 201 1182.48]
[ 202 1189.37]
[ 203 1196.26]
[ 204 1203.15]
[ 205 1210.05]
[ 206 1216.95]
[ 207 1223.86]
[ 208 1230.77]
[ 209 1237.69]
[ 210 1244.61]
[ 211 1251.53]
[ 212 1258.47]
[ 213 1265.40]
[ 214 1272.34]
[ 215 1279.29]
[ 216 1286.24]
[ 217 1293.19]
[ 218 1300.15]
[ 219 1307.12]
[ 220 1314.09]
[ 221 1321.06]
[ 222 1328.04]
[ 223 1335.02]
[ 224 1342.00]
[ 225 1349.00]
[ 226 1355.99]
[ 227 1362.99]
[ 228 1370.00]
[ 229 1377.00]
[ 230 1384.02]
[ 231 1391.03]
[ 232 1398.06]
[ 233 1405.08]
[ 234 1412.11]
[ 235 1419.15]
[ 236 1426.19]
[ 237 1433.23]
[ 238 1440.28]
[ 239 1447.33]
[ 240 1454.38]
[ 241 1461.44]
[ 242 1468.51]
[ 243 1475.58]
[ 244 1482.65]
[ 245 1489.73]
[ 246 1496.81]
[ 247 1503.89]
[ 248 1510.98]
[ 249 1518.07]
[ 250 1525.17]
[ 251 1532.27]
[ 252 1539.37]
[ 253 1546.48]
[ 254 1553.60]
[ 255 1560.71]
[ 256 1567.83]
[ 257 1574.96]
[ 258 1582.08]
[ 259 1589.22]
[ 260 1596.35]
[ 261 1603.49]
[ 262 1610.64]
[ 263 1617.78]
[ 264 1624.94]
[ 265 1632.09]
[ 266 1639.25]
[ 267 1646.41]
[ 268 1653.58]
[ 269 1660.75]
[ 270 1667.92]
[ 271 1675.10]
[ 272 1682.28]
[ 273 1689.47]
[ 274 1696.65]
[ 275 1703.85]
[ 276 1711.04]
[ 277 1718.24]
[ 278 1725.44]
[ 279 1732.65]
[ 280 1739.86]
[ 281 1747.07]
[ 282 1754.29]
[ 283 1761.51]
[ 284 1768.74]
[ 285 1775.97]
[ 286 1783.20]
[ 287 1790.43]
[ 288 1797.67]
[ 289 1804.91]
[ 290 1812.16]
[ 291 1819.41]
[ 292 1826.66]
[ 293 1833.91]
[ 294 1841.17]
[ 295 1848.44]
[ 296 1855.70]
[ 297 1862.97]
[ 298 1870.24]
[ 299 1877.52]
[ 300 1884.80]
[ 301 1892.08]
[ 302 1899.37]
[ 303 1906.66]
[ 304 1913.95]
[ 305 1921.25]
[ 306 1928.54]
[ 307 1935.85]
[ 308 1943.15]
[ 309 1950.46]
[ 310 1957.77]
[ 311 1965.09]
[ 312 1972.41]
[ 313 1979.73]
[ 314 1987.05]
[ 315 1994.38]
[ 316 2001.71]
[ 317 2009.05]
[ 318 2016.39]
[ 319 2023.73]
[ 320 2031.07]
[ 321 2038.42]
[ 322 2045.77]
[ 323 2053.12]
[ 324 2060.48]
[ 325 2067.84]
[ 326 2075.20]
[ 327 2082.57]
[ 328 2089.93]
[ 329 2097.31]
[ 330 2104.68]
[ 331 2112.06]
[ 332 2119.44]
[ 333 2126.82]
[ 334 2134.21]
[ 335 2141.60]
[ 336 2148.99]
[ 337 2156.39]
[ 338 2163.79]
[ 339 2171.19]
[ 340 2178.59]
[ 341 2186.00]
[ 342 2193.41]
[ 343 2200.83]
[ 344 2208.24]
[ 345 2215.66]
[ 346 2223.08]
[ 347 2230.51]
[ 348 2237.94]
[ 349 2245.37]
[ 350 2252.80]
[ 351 2260.24]
[ 352 2267.68]
[ 353 2275.12]
[ 354 2282.57]
[ 355 2290.01]
[ 356 2297.46]
[ 357 2304.92]
[ 358 2312.37]
[ 359 2319.83]
[ 360 2327.29]
[ 361 2334.76]
[ 362 2342.23]
[ 363 2349.70]
[ 364 2357.17]
[ 365 2364.65]
[ 366 2372.12]
[ 367 2379.61]
[ 368 2387.09]
[ 369 2394.58]
[ 370 2402.07]
[ 371 2409.56]
[ 372 2417.05]
[ 373 2424.55]
[ 374 2432.05]
[ 375 2439.55]
[ 376 2447.06]
[ 377 2454.57]
[ 378 2462.08]
[ 379 2469.59]
[ 380 2477.11]
[ 381 2484.63]
[ 382 2492.15]
[ 383 2499.67]
[ 384 2507.20]
[ 385 2514.73]
[ 386 2522.26]
[ 387 2529.79]
[ 388 2537.33]
[ 389 2544.87]
[ 390 2552.41]
[ 391 2559.96]
[ 392 2567.50]
[ 393 2575.05]
[ 394 2582.61]
[ 395 2590.16]
[ 396 2597.72]
[ 397 2605.28]
[ 398 2612.84]
[ 399 2620.40]
[ 400 2627.97]
[ 401 2635.54]
[ 402 2643.11]
[ 403 2650.69]
[ 404 2658.27]
[ 405 2665.85]
[ 406 2673.43]
[ 407 2681.01]
[ 408 2688.60]
[ 409 2696.19]
[ 410 2703.78]
[ 411 2711.38]
[ 412 2718.97]
[ 413 2726.57]
[ 414 2734.18]
[ 415 2741.78]
[ 416 2749.39]
[ 417 2757.00]
[ 418 2764.61]
[ 419 2772.22]
[ 420 2779.84]
[ 421 2787.46]
[ 422 2795.08]
[ 423 2802.70]
[ 424 2810.33]
[ 425 2817.95]
[ 426 2825.58]
[ 427 2833.22]
[ 428 2840.85]
[ 429 2848.49]
[ 430 2856.13]
[ 431 2863.77]
[ 432 2871.42]
[ 433 2879.06]
[ 434 2886.71]
[ 435 2894.36]
[ 436 2902.02]
[ 437 2909.67]
[ 438 2917.33]
[ 439 2924.99]
[ 440 2932.66]
[ 441 2940.32]
[ 442 2947.99]
[ 443 2955.66]
[ 444 2963.33]
[ 445 2971.00]
[ 446 2978.68]
[ 447 2986.36]
[ 448 2994.04]
[ 449 3001.72]
[ 450 3009.41]
[ 451 3017.10]
[ 452 3024.79]
[ 453 3032.48]
[ 454 3040.17]
[ 455 3047.87]
[ 456 3055.57]
[ 457 3063.27]
[ 458 3070.97]
[ 459 3078.68]
[ 460 3086.38]
[ 461 3094.09]
[ 462 3101.80]
[ 463 3109.52]
[ 464 3117.23]
[ 465 3124.95]
[ 466 3132.67]
[ 467 3140.40]
[ 468 3148.12]
[ 469 3155.85]
[ 470 3163.58]
[ 471 3171.31]
[ 472 3179.04]
[ 473 3186.77]
[ 474 3194.51]
[ 475 3202.25]
[ 476 3209.99]
[ 477 3217.74]
[ 478 3225.48]
[ 479 3233.23]
[ 480 3240.98]
[ 481 3248.73]
[ 482 3256.49]
[ 483 3264.24]
[ 484 3272.00]
[ 485 3279.76]
[ 486 3287.52]
[ 487 3295.29]
[ 488 3303.05]
[ 489 3310.82]
[ 490 3318.59]
[ 491 3326.37]
[ 492 3334.14]
[ 493 3341.92]
[ 494 3349.70]
[ 495 3357.48]
[ 496 3365.26]
[ 497 3373.05]
[ 498 3380.83]
[ 499 3388.62]
[ 500 3396.41]
[ 501 3404.20]
[ 502 3412.00]
[ 503 3419.80]
[ 504 3427.59]
[ 505 3435.40]
[ 506 3443.20]
[ 507 3451.00]
[ 508 3458.81]
[ 509 3466.62]
[ 510 3474.43]
[ 511 3482.24]
[ 512 3490.06]
[ 513 3497.87]
[ 514 3505.69]
[ 515 3513.51]
[ 516 3521.33]
[ 517 3529.16]
[ 518 3536.98]
[ 519 3544.81]
[ 520 3552.64]
[ 521 3560.47]
[ 522 3568.31]
[ 523 3576.14]
[ 524 3583.98]
[ 525 3591.82]
[ 526 3599.66]
[ 527 3607.51]
[ 528 3615.35]
[ 529 3623.20]
[ 530 3631.05]
[ 531 3638.90]
[ 532 3646.75]
[ 533 3654.61]
[ 534 3662.46]
[ 535 3670.32]
[ 536 3678.18]
[ 537 3686.05]
[ 538 3693.91]
[ 539 3701.78]
[ 540 3709.64]
[ 541 3717.51]
[ 542 3725.38]
[ 543 3733.26]
[ 544 3741.13]
[ 545 3749.01]
[ 546 3756.89]
[ 547 3764.77]
[ 548 3772.65]
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1r. R code for average number of attempts to get at least 1 of each coupon in the set.
call with N.mean(x) or to see the fraction
fractions(N.mean(x))
2r. The actual probabilities of the distribution. R code uses a Markov chain.
Function is called as an example: cc.draws(6, 6, 0)
default function(Number of Coupons=6, draws=15, print raised Matrix=0 for NO)
Print raised Matrix option is for seeing the probabilities of ending in each state. (how many distinct coupons were collected after X number of Draws). For the number of unique coupons collected after X draws, the R code on the Roulette page in this site has a way better result view.
Coupons must be between 2 and 800 for function to run as is.
3r. The actual probabilities of the distribution. R code uses a Markov chain.
returns a Matrix of probabilities for trials up to a maximun number of trials.
Function is called as an example: tMax.dist.cum(6, 100)
default function(Number of Coupons=6, trials Max = 100)
probabilities shown are the mass function (on Draw X) and cumulative (Draw X or less). example: number of draws is 15 so probability is for getting complete set BY and including the 15th draw. Number of coupons must be greater than 1 for function to run.
4r. Plot graph for coupon collecting problem, R code.
shows mean, median and mode as well as distributions.
Function is called as an example: ccplot(x) where x is and integer
a few plot examples. plot returns an image in R.
collect 6 data.
collect 7 data
