All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Peoples Day (Posted on 2019-08-12) Difficulty: 3 of 5
A company has the policy that any employee's birthday is a holiday for the entire company. How many people should the company employ if the expected value of the total number of "people-days" (the product of the number of employees and the number of days worked) is to be maximized?

Answer the question assuming that there are 365 days in a year, that each day is equally likely to be a birthday, and that the employees have no days off except for the birthday/holidays (i.e. no weekends off).

No Solution Yet Submitted by Danish Ahmed Khan    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer-aided solution | Comment 2 of 3 |
Any given day has 364/365 probability that one given employee does not have it. The probability that none of the n employees has it is (364/365)^n.

That can get to be a small number, so I got the idea of using a Poisson approximation of the number of days in the year that have no employee having that as a birthday. The expected number would by 365 * (364/365)^n. Multiply this by the number of employees, n, and you get the number of person-days worked per year (or peoples days as the puzzle calls them).

Playing around with smaller numbers of days per year resulted in a feeling that it's not just an approximation, but, being an expected value, is an actual result. My initial trepidation was the result of a lack of independence of whether one day or another was not present in the workplace.

The program is:

DefDbl A-Z
Dim crlf$
Function mform$(x, t$)
  f$ = Format$(x, t$)
  If Len(f$) < Len(t$) Then f$ = Space$(Len(t$) - Len(f$)) & f$
  mform$ = f$
End Function

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 
 daysInYear = 365
 
 p0 = (daysInYear - 1) / daysInYear
 For n = 1 To 500
   p = p0 ^ n
   p2 = daysInYear * p
   Text1.Text = Text1.Text & mform(n, "####") & mform(p2, " ##0.00000000") & mform(n * p2, " #####.00000") & crlf
   DoEvents
 Next
  
  Text1.Text = Text1.Text & " done"

End Sub

The list shows a peak of expected person-days worked per year occurs when the number of employees is either 364 or 365, and is 48943.52381, using an expected 134.46023023 or 134.09184604 workday year.

emps work days/yr  person-days/yr
   1 364.00000000   364.00000
   2 363.00273973   726.00548
   3 362.00821167  1086.02464
   4 361.01640835  1444.06563
   5 360.02732230  1800.13661
   6 359.04094608  2154.24568
   7 358.05727225  2506.40091
   8 357.07629342  2856.61035
   9 356.09800221  3204.88202
  10 355.12239124  3551.22391
  11 354.14945319  3895.64399
  12 353.17918071  4238.15017
  13 352.21156652  4578.75036
  14 351.24660332  4917.45245
  15 350.28428386  5254.26426
  16 349.32460089  5589.19361
  17 348.36754719  5922.24830
  18 347.41311555  6253.43608
  19 346.46129880  6582.76468
  20 345.51208976  6910.24180
  21 344.56548130  7235.87511
  22 343.62146628  7559.67226
  23 342.68003760  7881.64086
  24 341.74118819  8201.78852
  25 340.80491096  8520.12277
  26 339.87119887  8836.65117
  27 338.94004490  9151.38121
  28 338.01144204  9464.32038
  29 337.08538330  9775.47612
  30 336.16186170 10084.85585
  31 335.24087030 10392.46698
  32 334.32240216 10698.31687
  33 333.40645037 11002.41286
  34 332.49300804 11304.76227
  35 331.58206829 11605.37239
  36 330.67362427 11904.25047
  37 329.76766914 12201.40376
  38 328.86419607 12496.83945
  39 327.96319827 12790.56473
  40 327.06466896 13082.58676
  41 326.16860138 13372.91266
  42 325.27498877 13661.54953
  43 324.38382442 13948.50445
  44 323.49510161 14233.78447
  45 322.60881366 14517.39661
  46 321.72495390 14799.34788
  47 320.84351567 15079.64524
  48 319.96449234 15358.29563
  49 319.08787729 15635.30599
  50 318.21366393 15910.68320
  51 317.34184567 16184.43413
  52 316.47241595 16456.56563
  53 315.60536824 16727.08452
  54 314.74069600 16995.99758
  55 313.87839272 17263.31160
  56 313.01845192 17529.03331
  57 312.16086712 17793.16943
  58 311.30563187 18055.72665
  59 310.45273972 18316.71164
  60 309.60218427 18576.13106
  61 308.75395911 18833.99151
  62 307.90805785 19090.29959
  63 307.06447413 19345.06187
  64 306.22320160 19598.28490
  65 305.38423393 19849.97521
  66 304.54756479 20100.13928
  67 303.71318790 20348.78359
  68 302.88109698 20595.91459
  69 302.05128575 20841.53872
  70 301.22374798 21085.66236
  71 300.39847744 21328.29190
  72 299.57546791 21569.43369
  73 298.75471321 21809.09406
  74 297.93620714 22047.27933
  75 297.11994356 22283.99577
  76 296.30591632 22519.24964
  77 295.49411929 22753.04719
  78 294.68454636 22985.39462
  79 293.87719144 23216.29812
  80 293.07204845 23445.76388
  81 292.26911133 23673.79802
  82 291.46837404 23900.40667
  83 290.66983055 24125.59594
  84 289.87347485 24349.37189
  85 289.07930094 24571.74058
  86 288.28730286 24792.70805
  87 287.49747463 25012.28029
  88 286.70981032 25230.46331
  89 285.92430399 25447.26305
  90 285.14094973 25662.68548
  91 284.35974165 25876.73649
  92 283.58067386 26089.42200
  93 282.80374051 26300.74787
  94 282.02893574 26510.71996
  95 281.25625372 26719.34410
  96 280.48568865 26926.62611
  97 279.71723470 27132.57177
  98 278.95088612 27337.18684
  99 278.18663711 27540.47707
 100 277.42448194 27742.44819
 101 276.66441487 27943.10590
 102 275.90643017 28142.45588
 103 275.15052214 28340.50378
 104 274.39668510 28537.25525
 105 273.64491336 28732.71590
 106 272.89520126 28926.89133
 107 272.14754318 29119.78712
 108 271.40193347 29311.40881
 109 270.65836653 29501.76195
 110 269.91683676 29690.85204
 111 269.17733858 29878.68458
 112 268.43986642 30065.26504
 113 267.70441473 30250.59886
 114 266.97097797 30434.69149
 115 266.23955064 30617.54832
 116 265.51012721 30799.17476
 117 264.78270220 30979.57616
 118 264.05727014 31158.75788
 119 263.33382557 31336.72524
 120 262.61236303 31513.48356
 121 261.89287711 31689.03813
 122 261.17536237 31863.39421
 123 260.45981344 32036.55705
 124 259.74622491 32208.53189
 125 259.03459141 32379.32393
 126 258.32490760 32548.93836
 127 257.61716813 32717.38035
 128 256.91136767 32884.65506
 129 256.20750091 33050.76762
 130 255.50556255 33215.72313
 131 254.80554731 33379.52670
 132 254.10744992 33542.18339
 133 253.41126512 33703.69826
 134 252.71698769 33864.07635
 135 252.02461238 34023.32267
 136 251.33413399 34181.44222
 137 250.64554732 34338.43998
 138 249.95884719 34494.32091
 139 249.27402843 34649.08995
 140 248.59108589 34802.75202
 141 247.91001442 34955.31203
 142 247.23080890 35106.77486
 143 246.55346422 35257.14538
 144 245.87797527 35406.42844
 145 245.20433699 35554.62886
 146 244.53254428 35701.75147
 147 243.86259211 35847.80104
 148 243.19447541 35992.78236
 149 242.52818918 36136.70019
 150 241.86372839 36279.55926
 151 241.20108804 36421.36429
 152 240.54026314 36562.12000
 153 239.88124872 36701.83105
 154 239.22403982 36840.50213
 155 238.56863149 36978.13788
 156 237.91501880 37114.74293
 157 237.26319683 37250.32190
 158 236.61316068 37384.87939
 159 235.96490544 37518.41997
 160 235.31842625 37650.94820
 161 234.67371823 37782.46864
 162 234.03077654 37912.98580
 163 233.38959633 38042.50420
 164 232.75017278 38171.02834
 165 232.11250107 38298.56268
 166 231.47657641 38425.11168
 167 230.84239401 38550.67980
 168 230.20994909 38675.27145
 169 229.57923690 38798.89104
 170 228.95025269 38921.54296
 171 228.32299173 39043.23159
 172 227.69744928 39163.96128
 173 227.07362066 39283.73637
 174 226.45150115 39402.56120
 175 225.83108607 39520.44006
 176 225.21237077 39637.37726
 177 224.59535058 39753.37705
 178 223.98002085 39868.44371
 179 223.36637696 39982.58148
 180 222.75441428 40095.79457
 181 222.14412821 40208.08721
 182 221.53551416 40319.46358
 183 220.92856755 40429.92786
 184 220.32328380 40539.48422
 185 219.71965837 40648.13680
 186 219.11768670 40755.88973
 187 218.51736427 40862.74712
 188 217.91868656 40968.71307
 189 217.32164906 41073.79167
 190 216.72624729 41177.98698
 191 216.13247674 41281.30306
 192 215.54033297 41383.74393
 193 214.94981151 41485.31362
 194 214.36090792 41586.01614
 195 213.77361776 41685.85546
 196 213.18793662 41784.83558
 197 212.60386008 41882.96044
 198 212.02138375 41980.23398
 199 211.44050325 42076.66015
 200 210.86121420 42172.24284
 201 210.28351224 42266.98596
 202 209.70739303 42360.89339
 203 209.13285222 42453.96900
 204 208.55988551 42546.21664
 205 207.98848856 42637.64015
 206 207.41865708 42728.24336
 207 206.85038679 42818.03007
 208 206.28367340 42907.00407
 209 205.71851265 42995.16914
 210 205.15490029 43082.52906
 211 204.59283207 43169.08757
 212 204.03230376 43254.84840
 213 203.47331115 43339.81527
 214 202.91585002 43423.99190
 215 202.35991619 43507.38198
 216 201.80550546 43589.98918
 217 201.25261366 43671.81716
 218 200.70123664 43752.86959
 219 200.15137024 43833.15008
 220 199.60301032 43912.66227
 221 199.05615276 43991.40976
 222 198.51079343 44069.39614
 223 197.96692825 44146.62500
 224 197.42455310 44223.09989
 225 196.88366391 44298.82438
 226 196.34425661 44373.80199
 227 195.80632714 44448.03626
 228 195.26987145 44521.53069
 229 194.73488550 44594.28878
 230 194.20136527 44666.31401
 231 193.66930674 44737.60986
 232 193.13870589 44808.17977
 233 192.60955876 44878.02719
 234 192.08186133 44947.15555
 235 191.55560966 45015.56827
 236 191.03079977 45083.26875
 237 190.50742772 45150.26037
 238 189.98548956 45216.54651
 239 189.46498137 45282.13055
 240 188.94589923 45347.01581
 241 188.42823923 45411.20565
 242 187.91199748 45474.70339
 243 187.39717009 45537.51233
 244 186.88375318 45599.63578
 245 186.37174290 45661.07701
 246 185.86113538 45721.83930
 247 185.35192679 45781.92592
 248 184.84411330 45841.34010
 249 184.33769107 45900.08508
 250 183.83265630 45958.16407
 251 183.32900519 46015.58030
 252 182.82673394 46072.33695
 253 182.32583878 46128.43721
 254 181.82631593 46183.88425
 255 181.32816164 46238.68122
 256 180.83137216 46292.83127
 257 180.33594374 46346.33754
 258 179.84187266 46399.20315
 259 179.34915520 46451.43120
 260 178.85778765 46503.02479
 261 178.36776632 46553.98701
 262 177.87908751 46604.32093
 263 177.39174754 46654.02960
 264 176.90574275 46703.11609
 265 176.42106948 46751.58341
 266 175.93772409 46799.43461
 267 175.45570293 46846.67268
 268 174.97500237 46893.30064
 269 174.49561880 46939.32146
 270 174.01754861 46984.73813
 271 173.54078821 47029.55360
 272 173.06533399 47073.77085
 273 172.59118239 47117.39279
 274 172.11832984 47160.42238
 275 171.64677277 47202.86251
 276 171.17650764 47244.71611
 277 170.70753091 47285.98606
 278 170.23983904 47326.67525
 279 169.77342852 47366.78656
 280 169.30829584 47406.32284
 281 168.84443750 47445.28694
 282 168.38185000 47483.68170
 283 167.92052986 47521.50995
 284 167.46047361 47558.77451
 285 167.00167780 47595.47817
 286 166.54413895 47631.62374
 287 166.08785364 47667.21399
 288 165.63281842 47702.25171
 289 165.17902988 47736.73964
 290 164.72648459 47770.68053
 291 164.27517916 47804.07713
 292 163.82511017 47836.93217
 293 163.37627425 47869.24836
 294 162.92866802 47901.02840
 295 162.48228811 47932.27499
 296 162.03713116 47962.99082
 297 161.59319381 47993.17856
 298 161.15047273 48022.84087
 299 160.70896459 48051.98041
 300 160.26866605 48080.59982
 301 159.82957382 48108.70172
 302 159.39168458 48136.28874
 303 158.95499503 48163.36349
 304 158.51950189 48189.92858
 305 158.08520189 48215.98658
 306 157.65209174 48241.54007
 307 157.22016821 48266.59164
 308 156.78942802 48291.14383
 309 156.35986794 48315.19919
 310 155.93148474 48338.76027
 311 155.50427519 48361.82959
 312 155.07823608 48384.40966
 313 154.65336420 48406.50300
 314 154.22965636 48428.11210
 315 153.80710935 48449.23945
 316 153.38572001 48469.88752
 317 152.96548516 48490.05880
 318 152.54640164 48509.75572
 319 152.12846630 48528.98075
 320 151.71167598 48547.73631
 321 151.29602755 48566.02484
 322 150.88151789 48583.84876
 323 150.46814386 48601.21047
 324 150.05590237 48618.11237
 325 149.64479031 48634.55685
 326 149.23480459 48650.54629
 327 148.82594211 48666.08307
 328 148.41819980 48681.16953
 329 148.01157460 48695.80804
 330 147.60606343 48710.00093
 331 147.20166326 48723.75054
 332 146.79837103 48737.05918
 333 146.39618371 48749.92918
 334 145.99509828 48762.36282
 335 145.59511171 48774.36242
 336 145.19622099 48785.93025
 337 144.79842312 48797.06859
 338 144.40171512 48807.77971
 339 144.00609398 48818.06586
 340 143.61155673 48827.92929
 341 143.21810041 48837.37224
 342 142.82572206 48846.39694
 343 142.43441871 48855.00562
 344 142.04418742 48863.20047
 345 141.65502527 48870.98372
 346 141.26692931 48878.35754
 347 140.87989662 48885.32413
 348 140.49392431 48891.88566
 349 140.10900944 48898.04430
 350 139.72514914 48903.80220
 351 139.34234052 48909.16152
 352 138.96058068 48914.12440
 353 138.57986676 48918.69297
 354 138.20019589 48922.86935
 355 137.82156522 48926.65565
 356 137.44397189 48930.05399
 357 137.06741306 48933.06646
 358 136.69188590 48935.69515
 359 136.31738759 48937.94214
 360 135.94391529 48939.80950
 361 135.57146621 48941.29930
 362 135.20003753 48942.41359
 363 134.82962647 48943.15441
 364 134.46023023 48943.52381
 365 134.09184604 48943.52381
 366 133.72447112 48943.15643
 367 133.35810271 48942.42369
 368 132.99273804 48941.32760
 369 132.62837438 48939.87014
 370 132.26500897 48938.05332
 371 131.90263908 48935.87910
 372 131.54126199 48933.34946
 373 131.18087497 48930.46636
 374 130.82147531 48927.23177
 375 130.46306031 48923.64762
 376 130.10562727 48919.71585
 377 129.74917349 48915.43841
 378 129.39369631 48910.81720
 379 129.03919303 48905.85416
 380 128.68566099 48900.55118
 381 128.33309754 48894.91016
 382 127.98150001 48888.93300
 383 127.63086576 48882.62159
 384 127.28119216 48875.97779
 385 126.93247656 48869.00348
 386 126.58471635 48861.70051
 387 126.23790891 48854.07075
 388 125.89205163 48846.11603
 389 125.54714190 48837.83820
 390 125.20317712 48829.23908
 391 124.86015472 48820.32050
 392 124.51807211 48811.08427
 393 124.17692670 48801.53219
 394 123.83671594 48791.66608
 395 123.49743727 48781.48772
 396 123.15908813 48770.99890
 397 122.82166597 48760.20139
 398 122.48516825 48749.09696
 399 122.14959245 48737.68739
 400 121.81493603 48725.97441
 401 121.48119648 48713.95979
 402 121.14837129 48701.64526
 403 120.81645794 48689.03255
 404 120.48545395 48676.12339
 405 120.15535681 48662.91951
 406 119.82616405 48649.42261
 407 119.49787319 48635.63439
 408 119.17048176 48621.55656
 409 118.84398729 48607.19080
 410 118.51838732 48592.53880
 411 118.19367941 48577.60224
 412 117.86986111 48562.38278
 413 117.54692999 48546.88208
 414 117.22488360 48531.10181
 415 116.90371954 48515.04361
 416 116.58343538 48498.70912
 417 116.26402870 48482.09997
 418 115.94549712 48465.21780
 419 115.62783822 48448.06421
 420 115.31104962 48430.64084
 421 114.99512894 48412.94928
 422 114.68007379 48394.99114
 423 114.36588181 48376.76801
 424 114.05255063 48358.28147
 425 113.74007788 48339.53310
 426 113.42846123 48320.52449
 427 113.11769833 48301.25718
 428 112.80778682 48281.73276
 429 112.49872439 48261.95276
 430 112.19050871 48241.91875
 431 111.88313745 48221.63224
 432 111.57660831 48201.09479
 433 111.27091897 48180.30791
 434 110.96606714 48159.27314
 435 110.66205052 48137.99197
 436 110.35886682 48116.46593
 437 110.05651376 48094.69651
 438 109.75498906 48072.68521
 439 109.45429046 48050.43351
 440 109.15441569 48027.94291
 441 108.85536250 48005.21486
 442 108.55712863 47982.25085
 443 108.25971184 47959.05234
 444 107.96310989 47935.62079
 445 107.66732055 47911.95764
 446 107.37234159 47888.06435
 447 107.07817079 47863.94234
 448 106.78480594 47839.59306
 449 106.49224482 47815.01793
 450 106.20048525 47790.21836
 451 105.90952502 47765.19578
 452 105.61936193 47739.95159
 453 105.32999382 47714.48720
 454 105.04141849 47688.80400
 455 104.75363378 47662.90337
 456 104.46663753 47636.78671
 457 104.18042756 47610.45540
 458 103.89500173 47583.91079
 459 103.61035789 47557.15427
 460 103.32649390 47530.18719
 461 103.04340761 47503.01091
 462 102.76109691 47475.62677
 463 102.47955966 47448.03612
 464 102.19879374 47420.24029
 465 101.91879704 47392.24063
 466 101.63956746 47364.03844
 467 101.36110289 47335.63505
 468 101.08340124 47307.03178
 469 100.80646042 47278.22994
 470 100.53027833 47249.23082
 471 100.25485291 47220.03572
 472  99.98018208 47190.64594
 473  99.70626378 47161.06277
 474  99.43309593 47131.28747
 475  99.16067649 47101.32133
 476  98.88900340 47071.16562
 477  98.61807463 47040.82160
 478  98.34788812 47010.29052
 479  98.07844185 46979.57365
 480  97.80973379 46948.67222
 481  97.54176192 46917.58748
 482  97.27452422 46886.32067
 483  97.00801867 46854.87302
 484  96.74224328 46823.24575
 485  96.47719603 46791.44008
 486  96.21287495 46759.45723
 487  95.94927803 46727.29840
 488  95.68640330 46694.96481
 489  95.42424877 46662.45765
 490  95.16281247 46629.77811
 491  94.90209244 46596.92739
 492  94.64208670 46563.90666
 493  94.38279331 46530.71710
 494  94.12421032 46497.35990
 495  93.86633577 46463.83621
 496  93.60916773 46430.14719
 497  93.35270425 46396.29401
 498  93.09694342 46362.27782
 499  92.84188330 46328.09977
 500  92.58752198 46293.76099


  Posted by Charlie on 2019-08-12 22:46:02
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (3)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information