Five random positive integers are chosen independently with equal probability from 1 to N inclusively, where N is a positive integer.
Find the probability that:
(i) The number chosen last is equal to the sum of the four numbers that were chosen first.
(ii) The largest number chosen is the sum of the other four.
There are n^5 total outcomes overall.
For n < 4 there are no ways the last number could equal the sum of the first four.
There is 1 way the last could be a 4 and have the first four add up to it.
There are 4 ways the last could be a 5 and have the first four add up to it.
...
There are C(k+3-4,3) ways the last could be k and have the first four add up to it.
i.e., There are C(k-1,3) ways the last could be k and have the first four add up to it.
...
There are C(n-1,3) ways the last could be n and have the first four add up to it.
(i) Sigma{k=4 to n} C(k-1,3) / n^5
(ii) When success (positive result) is found by chance in part (i), it is also necessarily a success in part (ii).
Call the success event of part (i) S, and that of part (ii) T, and use the | symbol to represent "given":
P(S) = P(T)*P(S|T)
My first thought was that P(S|T) was 1/5; my second thought was that this ignores cases where two of the numbers equal the max; but on third thought, no two numbers could equal the max, as one of the maxes would be the total, which must be larger than any of the four positive addends.
So P(S) = P(T)/5
P(T) = 5*P(S) = 5 * Sigma{k=4 to n} C(k-1,3) / n^5
reduced
N num den fraction decimal times 5 for part ii
4 1 1024 1//1024 0.0009765625 0.0048828125
5 5 3125 1//625 0.0016 0.008
6 15 7776 5//2592 0.0019290123456790122 0.0096450617283950616
7 35 16807 5//2401 0.0020824656393169512 0.0104123281965847563
8 70 32768 35//16384 0.00213623046875 0.01068115234375 ** maximum here **
9 126 59049 14//6561 0.002133821063862216 0.0106691053193110805
10 210 100000 21//10000 0.0021 0.0105
11 330 161051 30//14641 0.002049040366095212 0.0102452018304760603
12 495 248832 55//27648 0.0019892939814814814 0.0099464699074074073
13 715 371293 55//28561 0.0019257028815517663 0.0096285144077588319
14 1001 537824 143//76832 0.0018612036651395251 0.0093060183256976259
15 1365 759375 91//50625 0.0017975308641975308 0.0089876543209876542
16 1820 1048576 455//262144 0.001735687255859375 0.008678436279296875
17 2380 1419857 140//83521 0.0016762251409825073 0.0083811257049125369
18 3060 1889568 85//52488 0.0016194177716811461 0.0080970888584057307
19 3876 2476099 204//130321 0.0015653655205224023 0.0078268276026120118
20 4845 3200000 969//640000 0.0015140625 0.0075703125
21 5985 4084101 95//64827 0.0014654387832230397 0.0073271939161151989
22 7315 5153632 665//468512 0.0014193873369305374 0.0070969366846526876
23 8855 6436343 385//279841 0.001375781247208236 0.0068789062360411804
24 10626 7962624 1771//1327104 0.0013344847125771604 0.0066724235628858024
25 12650 9765625 506//390625 0.00129536 0.0064768
26 14950 11881376 575//456976 0.0012582717691957564 0.0062913588459787822
27 17550 14348907 650//531441 0.0012230896750532984 0.0061154483752664924
28 20475 17210368 2925//2458624 0.0011896898427738442 0.0059484492138692211
29 23751 20511149 819//707281 0.0011579556074601183 0.0057897780373005919
30 27405 24300000 203//180000 0.0011277777777777777 0.0056388888888888888
31 31465 28629151 1015//923521 0.0010990545964845412 0.0054952729824227061
32 35960 33554432 4495//4194304 0.0010716915130615234 0.0053584575653076171
33 40920 39135393 1240//1185921 0.0010456008452502316 0.0052280042262511583
34 46376 45435424 341//334084 0.0010207013804911339 0.0051035069024556698
35 52360 52521875 1496//1500625 0.0009969179508538108 0.0049845897542690545
36 58905 60466176 6545//6718464 0.0009741810032769394 0.0048709050163846974
37 66045 69343957 1785//1874161 0.0009524261789675486 0.0047621308948377433
38 73815 79235168 3885//4170272 0.0009315939104211907 0.0046579695521059537
39 82251 90224199 703//771147 0.0009116290408962233 0.004558145204481117
40 91390 102400000 9139//10240000 0.00089248046875 0.00446240234375
41 101270 115856201 2470//2825761 0.0008741008174435134 0.0043705040872175671
42 111930 130691232 2665//3111696 0.0008564461309845177 0.0042822306549225888
43 123410 147008443 2870//3418801 0.0008394755939289826 0.0041973779696449134
44 135751 164916224 12341//14992384 0.000823151274673861 0.0041157563733693053
45 148995 184528125 3311//4100625 0.0008074378905654625 0.0040371894528273128
46 163185 205962976 7095//8954912 0.0007923025932583145 0.0039615129662915726
47 178365 229345007 3795//4879681 0.0007777147727484644 0.0038885738637423224
48 194580 254803968 5405//7077888 0.0007636458785445601 0.0038182293927228009
49 211876 282475249 4324//5764801 0.0007500692565103287 0.0037503462825516439
50 230300 312500000 2303//3125000 0.00073696 0.0036848
51 249900 345025251 4900//6765201 0.0007242948140047871 0.0036214740700239357
52 270725 380204032 20825//29246464 0.0007120518911277615 0.0035602594556388081
53 292825 418195493 5525//7890481 0.0007002107983024101 0.003501053991512051
54 316251 459165024 11713//17006112 0.0006887523732643886 0.0034437618663219435
55 341055 503284375 6201//9150625 0.0006776586298750085 0.0033882931493750426
56 367290 550731776 26235//39337984 0.0006669126714780299 0.0033345633573901498
57 395010 601692057 770//1172889 0.0006564986115480663 0.0032824930577403317
58 424270 656356768 7315//11316496 0.0006464015009593075 0.0032320075047965376
59 455126 714924299 7714//12117361 0.0006366072612675317 0.0031830363063376588
60 487635 777600000 32509//51840000 0.00062710262345679 0.0031355131172839506
61 521855 844596301 8555//13845841 0.0006178750716550911 0.0030893753582754561
62 557845 916132832 17995//29552672 0.0006089127913712844 0.0030445639568564222
63 595665 992436543 9455//15752961 0.0006002046218485527 0.0030010231092427639
64 635376 1073741824 39711//67108864 0.0005917400121688842 0.0029587000608444213
65 677040 1160290625 10416//17850625 0.0005835089807779839 0.0029175449038899197
66 720720 1252332576 455//790614 0.0005755020781316799 0.0028775103906583996
67 766480 1350125107 11440//20151121 0.0005677103521933096 0.0028385517609665487
68 814385 1453933568 47905//85525504 0.0005601253165371582 0.0028006265826857915
69 864501 1564031349 12529//22667121 0.0005527389208360426 0.0027636946041802132
70 916895 1680700000 26197//48020000 0.0005455435235318616 0.0027277176176593085
71 971635 1804229351 13685//25411681 0.000538531866506588 0.0026926593325329402
72 1028790 1934917632 57155//107495424 0.0005316970515879819 0.00265848525793991
73 1088430 2073071593 14910//28398241 0.0005250325187394529 0.0026251625936972645
74 1150626 2219006624 15549//29986576 0.0005185320257971433 0.0025926601289857167
75 1215450 2373046875 5402//10546875 0.0005121896296296296 0.0025609481481481481
76 1282975 2535525376 67525//133448704 0.0005059996686067479 0.0025299983430337397
77 1353275 2706784157 17575//35153041 0.000499956746274099 0.002499783731370495
78 1426425 2887174368 36575//74030112 0.0004940557161388597 0.002470278580694299
79 1502501 3077056399 19019//38950081 0.0004882916674807428 0.0024414583374037142
80 1581580 3276800000 79079//163840000 0.0004826599121093749 0.0024132995605468749
81 1663740 3486784401 20540//43046721 0.0004771559719961016 0.0023857798599805081
82 1749060 3707398432 10665//22606088 0.0004717755677143254 0.0023588778385716272
83 1837620 3939040643 22140//47458321 0.0004665146076280279 0.0023325730381401398
84 1929501 4182119424 3403//7375872 0.00046136917777315 0.0023068458888657503
85 2024785 4437053125 23821//52200625 0.0004563355323810777 0.002281677661905389
86 2123555 4704270176 49385//109401632 0.000451410084997635 0.0022570504249881756
87 2225895 4984209207 25585//57289761 0.0004465894001547676 0.002232947000773838
88 2331890 5277319168 105995//239878144 0.000441870185555546 0.0022093509277777303
89 2441626 5584059449 27434//62742241 0.0004372492847362592 0.0021862464236812962
90 2555190 5904900000 28391//65610000 0.0004327236701722298 0.0021636183508611491
91 2672670 6240321451 29370//68574961 0.0004282904367966027 0.0021414521839830138
92 2794155 6590815232 121485//286557184 0.0004239467959037453 0.0021197339795187266
93 2919735 6956883693 10465//24935067 0.0004196900694110827 0.0020984503470554139
94 3049501 7339040224 64883//156149792 0.0004155176844551928 0.0020775884222759643
95 3183545 7737809375 33511//81450625 0.0004114271682998135 0.0020571358414990676
96 3321960 8153726976 138415//339738624 0.0004074161435350959 0.0020370807176754798
97 3464840 8587340257 35720//88529281 0.0004034823235489735 0.0020174116177448678
98 3612280 9039207968 9215//23059204 0.0003996235082529301 0.0019981175412646507
99 3764376 9509900499 38024//96059601 0.0003958375800457467 0.0019791879002287339
100 3921225 10000000000 156849//400000000 0.0003921225 0.0019606125
4 kill "rndsumrs.txt"
5 open "rndsumrs.txt" for output as #2
10 for N=4 to 100
20 Tot=0
30 for K=4 to N
40 Tot=Tot+combi(K-1,3)
50 next
60 print N,Tot;N^5,Tot//N^5,Tot/N^5,5*Tot/N^5
65 print #2,N,Tot;N^5,Tot//N^5,Tot/N^5,5*Tot/N^5
70 next N
75 close #2
Simulation for n=6 (as in the roll of a 5 dice (part ii), or one die 5 times (part i))
DefDbl A-Z
Dim crlf$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
n = 6
For tr = 1 To 10000
tot = 0: high = 0
For i = 1 To 5
r = Int(Rnd(1) * n + 1)
tot = tot + r
If r > high Then high = r
Next
tot = tot - high
If high = tot Then part2 = part2 + 1
If r = tot Then part1 = part1 + 1
Next
Text1.Text = Text1.Text & part1 / 10000 & Str(part2 / 10000) & crlf
Text1.Text = Text1.Text & crlf & " done"
End Sub
finding
0.0018 .0095
comparing favorably (off by only 1 count in each number) to the expected
0.0019290123456790122 0.0096450617283950616
with Randomize Timer added, and each run changed to 100,000 trials, several runs gave
0.00217 .01022
0.00204 .01009
0.00186 .00976
0.00196 .00934
0.002 .00963
Here are single sets of results for 100,000 trials each of n = 3 through 25:
n i ii
3 0.00000 0.00000
4 0.00094 0.00526
5 0.00158 0.00822
6 0.00183 0.00926
7 0.00229 0.01045
8 0.00229 0.01066
9 0.00214 0.01075
10 0.00194 0.01021
11 0.00199 0.01033
12 0.00191 0.00987
13 0.00190 0.00942
14 0.00179 0.00909
15 0.00186 0.00884
16 0.00175 0.00872
17 0.00166 0.00834
18 0.00167 0.00791
19 0.00141 0.00781
20 0.00170 0.00832
21 0.00153 0.00739
22 0.00154 0.00690
23 0.00129 0.00678
24 0.00131 0.00661
25 0.00130 0.00631
and a second set
3 0.00000 0.00000
4 0.00086 0.00486
5 0.00148 0.00774
6 0.00180 0.00950
7 0.00201 0.01026
8 0.00200 0.01058
9 0.00212 0.01052
10 0.00236 0.01091
11 0.00191 0.00993
12 0.00207 0.00996
13 0.00192 0.00970
14 0.00161 0.00894
15 0.00176 0.00922
16 0.00185 0.00845
17 0.00183 0.00845
18 0.00169 0.00814
19 0.00158 0.00792
20 0.00165 0.00801
21 0.00174 0.00806
22 0.00129 0.00679
23 0.00135 0.00666
24 0.00138 0.00681
25 0.00159 0.00719
|
|
Posted by Charlie
on 2016-06-06 10:51:52 |
solution
