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Tue 24 Sep, 2013 10:34 am
I was recently wondering about a statistical mechanic, my course knowledge is way too long ago, google didn't really help, and my friends couldn't pin it down immediately either, so I have turned here for a helping hand. I have two experiments, and need to know for a number of draws N, which one offers the best odds.
Experiment 1:
You have a bowl containing 8 different colored balls. You draw one at a time at random, note what color you got, and place it back into the bowl, so there are always 8. What is the chance you have seen all 8 colors after N draws?
Experiment 2:
The bowl still contains the 8 balls, but for every ball, there are 93 blank balls (such that the odds of getting a colored one is 7%). You repeat the drawing experiment, but you are allowed to draw 10 times as much as in experiment one. (So for example I would compare the odds of getting it done in 10 draws in experiment 1, and getting it done in 100 draws in experiment 2).
All I know is they will both approach a chance of 1 approaching infinity, and my gut feeling tells one will be superior in the lower range, while the other is better at higher ranges, but i can't seem to pin down how to calculate it exactly. If anyone could help, it would be greatly appreciated.
@goesting,
Column 1 is the draw number (N).
Column 2 is the probability of getting the 8th color on the Nth draw. The assumption is that the drawing stops after getting the 8th color. If you want to know the probability of getting all 8 colors on or before the Nth draw, then you will have to add the probabilities from rows 1 through N.
Column 3 is the probability of getting the 8th color (out of 101 balls - 8 colored and 93 plain) on the 10*N - 9th through 10*Nth draws. Again, if you want to know the probability of getting all 8 colors on or before the 10*Nth draw, then you will have to add the probabilities from rows 1 through N.
Do you need data for more than 100 (1000 for experiment 2) draws?
Code:
1 : 0.0000000000000000000000000000 1.545587292454827061089719046E-10
2 : 0.0000000000000000000000000000 2.900139254748128493126858838E-7
3 : 0.0000000000000000000000000000 0.0000088157965029380909877264
4 : 0.0000000000000000000000000000 0.0000722767262922601586356404
5 : 0.0000000000000000000000000000 0.0003077729578654400906899483
6 : 0.0000000000000000000000000000 0.0008900641725022228198718107
7 : 0.0000000000000000000000000000 0.0019988857671924690920934595
8 : 0.0024032592773437500000000000 0.0037610697407550291175081660
9 : 0.0084114074707031250000000000 0.0062142703485091155433218927
10 : 0.0173485279083251953125000000 0.0093000168406940464386474622
11 : 0.0275999307632446289062500000 0.0128802108443802335466862230
12 : 0.0375432986766099929809570313 0.0167661254491430211309236972
13 : 0.0460142595693469047546386719 0.0207497810339128003667665220
14 : 0.0523973302915692329406738281 0.0246308979277541852173400442
15 : 0.0565295456908643245697021485 0.0282361745806252410792897853
16 : 0.0585503220895589038264006376 0.0314303490744408310725781922
17 : 0.0587643296660189662361517549 0.0341201306910628375780944638
18 : 0.0575401273834330595491337590 0.0362528029595690724004906605
19 : 0.0552453036837263766756223049 0.0378113936055510005219824208
20 : 0.0522105980642934008884026298 0.0388080561461989817995211823
21 : 0.0487140817181982987982280519 0.0392769171684711433422790371
22 : 0.0449778261904312456898047401 0.0392672407523556510412695062
23 : 0.0411715257567535873606257013 0.0388374138756542200072908631
24 : 0.0374194010662699207603234392 0.0380499891055799925261534925
25 : 0.0338081380155988063339724328 0.0369678340357150902646217277
26 : 0.0303946050798602022751946695 0.0356513192286605781229172388
27 : 0.0272127305362589468733958847 0.0341564121390435445545400032
28 : 0.0242793067957576298234436480 0.0325335181279002414884618620
29 : 0.0215987058517489416420827503 0.0308269080807189802321540186
30 : 0.0191665997242970371169371904 0.0290745851405769672899117676
31 : 0.0169728255124510484113399379 0.0273084633749332185251465100
32 : 0.0150035442119729137652730172 0.0255547539398996075625483718
33 : 0.0132428332547265848640125035 0.0238344764864243437726732804
34 : 0.0116738351895487632458956842 0.0221640335125789315462999397
35 : 0.0102795649503222872883778997 0.0205558023619724286534128395
36 : 0.0090434588532908308912449848 0.0190187134368664335551755801
37 : 0.0079497312807817378970861994 0.0175587941109148331910518669
38 : 0.0069835904469377679415961075 0.0161796661388431916406226048
39 : 0.0061313527027831716402233588 0.0148829904831534812407011316
40 : 0.0053804852793889902650536896 0.0136688578266785161889731702
41 : 0.0047195998508954268279838779 0.0125361259956178543638329038
42 : 0.0041384134715291855436388022 0.0114827074105717911806707690
43 : 0.0036276889763452961405611487 0.0105058107872529354430470355
44 : 0.0031791635507809065758523929 0.0096021418445414222945462370
45 : 0.0027854716325481521597514442 0.0087680679179406138058448820
46 : 0.0024400664185688961719633010 0.0079997512530164917045860862
47 : 0.0021371428562851842832961396 0.0072932554642305644016081580
48 : 0.0018715639824074337709592014 0.0066446292610597376210113851
49 : 0.0016387917395677395620330494 0.0060499711158955862180658399
50 : 0.0014348228805940344240969824 0.0055054781114483661216295341
51 : 0.0012561302064089762013551092 0.0050074817818601328513612938
52 : 0.0010996091353605829918535201 0.0045524733653622001827557788
53 : 0.0009625294378406801584476839 0.0041371205247789035018231515
54 : 0.0008424918669333516031939707 0.0037582772688372678279813107
55 : 0.0007373893561355752306896509 0.0034129885225322512481645865
56 : 0.0006453724260152796015665440 0.0030984905472869830703844139
57 : 0.0005648184335421422946607176 0.0028122081987779502383749375
58 : 0.0004943043038280293755594040 0.0025517498289081870236871068
59 : 0.0004325823991426378543194910 0.0023149004851108426017653091
60 : 0.0003785592007409735525590218 0.0020996139315558314863000512
61 : 0.0003312765027175608501665291 0.0019040039096705724373007117
62 : 0.0002898948420240457351408332 0.0017263349666695598163887999
63 : 0.0002536789137520965028690498 0.0015650131077994469556265689
64 : 0.0002219847450010155912216896 0.0014185764683298401657290058
65 : 0.0001942484236219239259453681 0.0012856861528352543540135458
66 : 0.0001699761995693893119966528 0.0011651173501818494185273608
67 : 0.0001487357963550447338588049 0.0010557508012734120603332780
68 : 0.0001301487881449576526735510 0.0009565646716772644752815053
69 : 0.0001138839143996966502112346 0.0008666268616015120282073370
70 : 0.0000996512186932152095921138 0.0007850877703700640965847122
71 : 0.0000871969115603216745190610 0.0007111735207373572477980948
72 : 0.0000762988690235051697089373 0.0006441796394311096978780319
73 : 0.0000667626889551128837626508 0.0005834651836535621244658059
74 : 0.0000584182367574944700207093 0.0005284472984511848071963854
75 : 0.0000511166201054552474430533 0.0004785961865029746924410042
76 : 0.0000447275398000837584986254 0.0004334304696697726168680919
77 : 0.0000391369702314465560659108 0.0003925129203400357883911142
78 : 0.0000342451286326178462641884 0.0003554465399970824689470405
79 : 0.0000299646973138185472542877 0.0003218709623540429393946195
80 : 0.0000262192674699255211160446 0.0002914591587229677404356861
81 : 0.0000229419770265142023298482 0.0002639144238976708873034084
82 : 0.0000200743183909961131498950 0.0002389676216515692446575648
83 : 0.0000175650949617494620879799 0.0002163746699154044554015838
84 : 0.0000153695078687708788251653 0.0001959142467531090637536009
85 : 0.0000134483567181165949625866 0.0001773856993566969583140851
86 : 0.0000117673401280660784396046 0.0001606071394018423123447143
87 : 0.0000102964436118480500542125 0.0001454137092212595971686744
88 : 0.0000090094039102126473834086 0.0001316560043456679736983464
89 : 0.0000078832402338221004132877 0.0001191986390193946300305023
90 : 0.0000068978440638850078658144 0.0001079189423107330752784724
91 : 0.0000060356202003680997229708 0.0000977057734002062647346332
92 : 0.0000052811726586740727236648 0.0000884584455394030291099673
93 : 0.0000046210298138540821613676 0.0000800857490273677843223457
94 : 0.0000040434038902581979302000 0.0000725050643503028410311473
95 : 0.0000035379805063279393699795 0.0000656415573742882720679377
96 : 0.0000030957345198010273043990 0.0000594274491712849616332098
97 : 0.0000027087688873990017955222 0.0000538013536978447994873417
98 : 0.0000023701736634039803976009 0.0000487076771360543315969554
99 : 0.0000020739026206758898730332 0.0000440960732498616260571075
100 : 0.0000018146652919894693172805 0.0000399209496097856307200951
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