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).
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 |
computer-aided solution
