Thread Rating:

AlexDinNYC
AlexDinNYC
  • Threads: 1
  • Posts: 5
Joined: Mar 29, 2018
March 29th, 2018 at 6:15:54 PM permalink
First time poster, long time learner here. The Wizard of Odds site has been my go-to for probability and math problems for slot machines and lottery style games for many years. Thanks for being such a great resource! I have a keno question about the number of draws needed to get all 80 numbers.
I know how to calculate the odds for matches per draw using a COMBIN function for keno but how do I figure out how many draws I would need to conduct to draw all 80 numbers, assuming the numbers go back in the hopper and subsequent drawings could (likely will) have duplicates?

I assume its a 1/80 for each number but i can't wrap my head around the duplicates in different drawings and how many drawings I would need to draw every # at least once.

thanks!
CrystalMath
CrystalMath
  • Threads: 8
  • Posts: 1909
Joined: May 10, 2011
Thanked by
AlexDinNYC
March 29th, 2018 at 8:48:01 PM permalink
Assuming you draw 20 numbers each game, then they go back in the hopper, I calculate it will take 17.8604 games on average to hit all 80 numbers.

It's not an easy problem until you've done many similar ones, and I calculated it using an absorbing Markov matrix.

Also, for calculating keno probabilities, I recommend using the hypergeometric distribution function. It's much easier than using Combin once you know how to use it.
I heart Crystal Math.
AlexDinNYC
AlexDinNYC
  • Threads: 1
  • Posts: 5
Joined: Mar 29, 2018
Thanked by
CrystalMath
March 29th, 2018 at 8:53:55 PM permalink
Going to have to look up the Markov so i can tweak the number of #s per drawing and how many #s to choose from.

For the odds i'm using =1/(COMBIN(E4,C4)*COMBIN(A4-E4,E4-C4)/COMBIN(A4,E4)) where E4 is the number of balls drawn, C4 is the number of correct picks and A4 is the number of balls in the hopper.


I'll take a gander at the Hypergeo as well - thanks again for the help!
CrystalMath
CrystalMath
  • Threads: 8
  • Posts: 1909
Joined: May 10, 2011
March 29th, 2018 at 9:02:59 PM permalink
I uploaded my Excel sheet to google drive:

here
I heart Crystal Math.
AlexDinNYC
AlexDinNYC
  • Threads: 1
  • Posts: 5
Joined: Mar 29, 2018
March 29th, 2018 at 9:07:32 PM permalink
Thanks says I need permission.... can you make it public for a min so i can check it out?
RS
RS
  • Threads: 62
  • Posts: 8626
Joined: Feb 11, 2014
March 29th, 2018 at 11:29:35 PM permalink
Quote: CrystalMath

Assuming you draw 20 numbers each game, then they go back in the hopper, I calculate it will take 17.8604 games on average to hit all 80 numbers.

It's not an easy problem until you've done many similar ones, and I calculated it using an absorbing Markov matrix.

Also, for calculating keno probabilities, I recommend using the hypergeometric distribution function. It's much easier than using Combin once you know how to use it.


Hypgeomdist? Or something like that. Yup, once I figured how it worked, it was like holy sh** bbq 1337ness. Ah, I haven’t messed around with Excel in a while, good times.
mustangsally
mustangsally
  • Threads: 25
  • Posts: 2463
Joined: Mar 29, 2011
Thanked by
AlexDinNYC
March 30th, 2018 at 10:45:06 AM permalink
Quote: CrystalMath

Assuming you draw 20 numbers each game, then they go back in the hopper, I calculate it will take 17.8604 games on average to hit all 80 numbers.

I agree
I set this up in Excell too
used just combin() as I wanted to do it different from you (your method is easier to populate the transition matrix)

Quote: CrystalMath

It's not an easy problem until you've done many similar ones, and I calculated it using an absorbing Markov matrix.

me too. using a Markov chain approach and making an absorbing transition matrix

here is my Excel in Google
https://goo.gl/HDYz1j

BruceZ did some R code for these type of coupon collector problems
where you choose more than 1 coupon at a time
It has some small errors in it as choose(a,b) returns very small numbers without using a more accurate method.
the sumproduct still works for the average too.

average
17.86042226
median
17
mode
16
here is the distribution also
draw Xby Xon X1 in
100.
200.
300.
41.31E-291.30772E-2976,468,711,070,730,500,000,000,000,000.00
51.25E-151.2547E-15797,000,608,031,493.00
63.57E-103.56594E-102,804,312,648.77
74.37E-074.36381E-072,291,577.58
83.79E-053.74718E-0526,686.70
90.0007299210.0006920131,445.06
100.0055401740.004810253207.89
110.0229383713050.01739819757.48
120.0631139464010.04017557524.89
130.1309142085730.06780026214.75
140.2225548931480.09164068510.91
150.3282321438880.1056772519.46
160.4369064933730.1086743499.20
170.5397410746970.1028345819.72
180.6313124070240.09157133210.92
190.7092983265460.0779859212.82
200.7735691671790.06427084115.56
210.8252680598490.05169889319.34
220.8661144122360.04084635224.48
230.8979596237780.03184521231.40
240.9225429773230.02458335440.68
250.9413814270610.0188384553.08
260.9557387236430.01435729769.65
270.9666363026880.01089757991.76
280.9748827733030.008246471121.26
290.9811089655290.006226192160.61
300.9858018862270.004692921213.09
310.9893346652850.003532779283.06
320.9919916020100.002656937376.37
330.9939884316940.00199683500.79
340.9954883693680.001499938666.69
350.9966146210860.001126252887.90
360.9974600378200.0008454171,182.85
370.9980945083800.0006344711,576.12
380.9985705899480.0004760822,100.48
390.9989277792420.0003571892,799.64
400.9991957429940.0002679643,731.85
410.9993967560240.0002010134,974.80
420.9995475383250.0001507826,632.08
430.9996606376710.0001130998,841.78
440.9997454692508.48316E-0511,788.06
450.9998090968946.36276E-0515,716.44
460.9998568198464.7723E-0520,954.28
470.9998926133023.57935E-0527,938.07
480.9999194590902.68458E-0537,249.79
490.9999395938212.01347E-0549,665.43
500.9999546950881.51013E-0566,219.61
510.9999660211601.13261E-0588,291.86
520.9999745157838.49462E-06117,721.53
530.9999808867886.37101E-06156,961.11
540.9999856650644.77828E-06209,280.50
550.9999892487833.58372E-06279,039.73
560.9999919365782.6878E-06372,052.18
570.9999939524292.01585E-06496,068.41
580.9999954643191.51189E-06661,423.78
590.9999965982381.13392E-06881,897.21
600.9999974486778.50439E-071,175,863.29
610.9999980865086.37831E-071,567,813.42
620.9999985648804.78372E-072,090,423.35
630.9999989236603.5878E-072,787,223.37
640.9999991927452.69085E-073,716,297.82
650.9999993945592.01814E-074,955,057.63
660.9999995459191.5136E-076,606,765.33
670.9999996594391.1352E-078,809,020.43
680.9999997445798.514E-0811,745,360.59
690.9999998084356.3856E-0815,660,235.52
700.9999998563264.7891E-0820,880,750.06
710.9999998922443.5918E-0827,841,193.82
720.9999999191832.6939E-0837,120,902.74
730.9999999393872.0204E-0849,495,149.41
740.9999999545411.5154E-0865,989,178.02
750.9999999659051.1364E-0887,997,184.37
760.9999999744298.524E-09117,315,813.33
770.9999999808226.393E-09156,421,085.12
780.9999999856164.794E-09208,594,077.03
790.9999999892123.596E-09278,086,767.54
800.9999999919092.697E-09370,782,341.45
810.9999999939322.023E-09494,315,375.71
820.9999999954491.517E-09659,195,794.15
830.9999999965871.138E-09878,734,600.87
840.9999999974408.53E-101,172,332,937.11
850.9999999980806.4E-101,562,500,141.77
860.9999999985604.8E-102,083,333,160.96
870.9999999989203.6E-102,777,777,547.94
880.9999999991902.7E-103,703,703,397.26
890.9999999993932.03E-104,926,109,583.27
900.9999999995441.51E-106,622,516,982.18
910.9999999996581.14E-108,771,927,390.21
920.9999999997448.6E-1111,627,909,016.87
930.9999999998086.4E-1115,624,993,286.17
940.9999999998564.79999E-1120,833,360,521.67
950.9999999998923.6E-1127,777,792,612.51
960.9999999999192.70001E-1137,036,942,596.44
970.9999999999392E-1149,999,995,862.98
980.9999999999541.5E-1166,666,661,150.64
990.9999999999661.2E-1183,333,634,834.68
1000.9999999999771.1E-1190,909,266,895.52


R code used (I corrected the -values in Excel)
Tmax = 100
Ncoupons = 80
r = 20
require(Rmpfr)
prob = rep(0,Tmax)

for (T in ceiling(Ncoupons/r):Tmax) {
k = 0:Ncoupons
prob[T] = as.double(sum((-1)^k*choose(Ncoupons,k)*(chooseZ(Ncoupons - k,r)/chooseZ(Ncoupons,r))^T))
}

#prob
prob = as.matrix(prob)
#rownames(prob) = paste("t=",T," ",sep="")
colnames(prob) = "Probability"
cat("\n")
print(formatC(prob, digits=12),quote=FALSE)


this would be a fun project to convert the transition matrix in Excel into R

Sally
I Heart Vi Hart
AlexDinNYC
AlexDinNYC
  • Threads: 1
  • Posts: 5
Joined: Mar 29, 2018
March 30th, 2018 at 1:34:10 PM permalink
Sally thank you so much for the excel and the explanation. Most of this is way over my head lol

I was looking to see how many times I would have to pick 10 numbers from 99 before i would get all 99 (assuming the 10 go back into the pool after each set of picks).

The more interesting and equally difficult issue is if I have a card with all 99 numbers and i pick 10 at a time - how many draws would I expect to conduct before I got a match for all the numbers on my card - with duplication?

Is it the same as just drawing the numbers or is matching them per drawing another layer?

Doing the hypergeo on the odds (99 choices with 10 chosen) it is a 1 in 2 that i would match 1 number out of ten drawn from 99. So does that mean I could expect to match all 99 after 198 drawings? I'm thinking I have to account for the duplicates and I am not....


Thanks!
mustangsally
mustangsally
  • Threads: 25
  • Posts: 2463
Joined: Mar 29, 2011
Thanked by
AlexDinNYC
March 30th, 2018 at 2:12:00 PM permalink
Quote: AlexDinNYC

Sally thank you so much for the excel and the explanation. Most of this is way over my head lol

Markov chains are difficult at first to understand.
The R code is easier to many as it is just the sum of many calculations (called inclusion exclusion)
of course, when a computer calculates for you(or an online program)

the R code that has small -values where the values should be small positive numbers.
I have to look into that

Quote: AlexDinNYC

I was looking to see how many times I would have to pick 10 numbers from 99 before i would get all 99 (assuming the 10 go back into the pool after each set of picks).

this is the coupon collecting problem with 99 coupons and select 10 at a time without replacement for each selection.
Then all 99 are available for next draw.

Quote: AlexDinNYC

The more interesting and equally difficult issue is if I have a card with all 99 numbers and i pick 10 at a time - how many draws would I expect to conduct before I got a match for all the numbers on my card - with duplication?

on average I get using the R code
about 49.3 draws on average

here is the distribution
10 thru 16 are very small values (I made at 0) just for the average calculation (sumproduct)
99 draw 10 each time
draw Xby Xon X
100
200
300
400
500
600
700
800
900
100.00E+000
110.00E+000
120.00E+000
130.00E+000
140.00E+000
150.00E+000
160.00E+000
178.99E-118.99187E-11
182.07E-091.9811E-09
192.71E-082.5013E-08
202.41E-072.13718E-07
211.56E-061.32202E-06
227.82E-066.25549E-06
233.15E-052.36523E-05
240.0001054877.40166E-05
250.0003027470.00019726
260.0007611090.000458362
270.0017079860.000946876
280.0034754910.001767505
290.0064978930.003022402
300.0112874410.004789548
310.0183907650.007103324
320.0283334960.009942731
330.0415635180.013230022
340.0584028840.016839366
350.079015580.020612696
360.1033943340.024378754
370.1313658460.027971512
380.1626110230.031245177
390.1966953230.0340843
400.233104060.036408737
410.2712781450.038174085
420.3106468420.039368697
430.3506553580.040008515
440.3907862090.040130851
450.430574230.039788021
460.4696156830.039041454
470.5075723460.037956662
480.5441715690.036599223
490.5792033580.03503179
500.6125154050.033312046
510.6440068750.03149147
520.6736216110.029614736
530.7013412280.027719617
540.7271784550.025837227
550.751170960.023992504
560.7733757810.022204821
570.7938644490.020488669
580.8127187910.018854341
590.8300273960.017308605
600.845882710.015855314
610.8603786770.014495967
620.8736088730.013230196
630.8856650670.012056193
640.8966361320.010971066
650.9066072690.009971136
660.9156594610.009052193
670.9238691450.008209683
680.9313080240.007438879
690.938043020.006734996
700.9441363130.006093294
710.9496454580.005509144
720.9546235450.004978088
730.9591194110.004495866
740.9631778580.004058447
750.9668398980.00366204
760.9701429990.0033031
770.9731213290.00297833
780.9758060010.002684673
790.9782253080.002419307
800.9804049420.002179635
810.9823682150.001963273
820.9841362520.001768037
830.9857281870.001591934
840.9871613290.001433143
850.9884513340.001290005
860.9896123490.001161014
870.9906571490.001044801
880.9915972720.000940122
890.9924431250.000845853
900.9932040980.000760973
910.9938886560.000684559
920.9945044320.000615776
930.9950583040.000553871
940.9955564660.000498163
950.9960045020.000448036
960.9964074370.000402935
970.9967697960.00036236
980.9970956550.000325859
990.9973886810.000293026
1000.9976521740.000263493
1010.9978891040.000236931
1020.9981021460.000213041
1030.9982937030.000191557
1040.9984659380.000172236
1050.9986207990.000154861
1060.9987600360.000139237
1070.9988852230.000125187
1080.9989977770.000112554
1090.9990989710.000101194
1100.9991899529.09805E-05
1110.9992717488.17968E-05
1120.9993452887.35396E-05
1130.9994114036.61154E-05
1140.9994708445.94404E-05
1150.9995242835.3439E-05
1160.9995723264.80432E-05
1170.9996155184.31921E-05
1180.9996543493.88307E-05
1190.9996892583.49095E-05
1200.9997206433.13842E-05
1210.9997488572.82148E-05
1220.9997742232.53655E-05
1230.9997970272.28038E-05
1240.9998175272.05007E-05
1250.9998359581.84303E-05
1260.9998525261.65689E-05
1270.9998674221.48955E-05
1280.9998808131.3391E-05
1290.9998928521.20385E-05
1300.9999036741.08226E-05
1310.9999134049.72952E-06
1320.999922158.74681E-06
1330.9999300147.86335E-06
1340.9999370837.06912E-06
1350.9999434386.35511E-06
1360.9999491515.7132E-06
1370.9999542875.13614E-06
1380.9999589054.61735E-06
1390.9999630564.15097E-06
1400.9999667873.73169E-06
1410.9999701423.35477E-06
1420.9999731583.01591E-06
1430.9999758692.71128E-06
1440.9999783072.43742E-06
1450.9999804982.19122E-06
1460.9999824681.96989E-06
1470.9999842391.77091E-06
1480.9999858311.59203E-06
1490.9999872621.43122E-06
1500.9999885491.28666E-06
1510.9999897051.15669E-06
1520.9999907451.03985E-06
1530.999991689.34821E-07
1540.999992528.40395E-07
1550.9999932767.55507E-07
1560.9999939556.79193E-07
1570.9999945666.10589E-07
1580.9999951155.48913E-07
1590.9999956084.93468E-07
1600.9999960524.43622E-07
1610.9999964513.98812E-07
1620.9999968093.58529E-07
1630.9999971313.22313E-07
1640.9999974212.89757E-07
1650.9999976822.60488E-07
1660.9999979162.34177E-07
1670.9999981262.10522E-07
1680.9999983161.89257E-07
1690.9999984861.70141E-07
1700.9999986391.52955E-07
1710.9999987761.37504E-07
1720.99999891.23616E-07
1730.9999990111.11129E-07
1740.9999991119.9904E-08
1750.9999992018.9812E-08
1760.9999992818.0741E-08
1770.9999993547.2585E-08
1780.9999994196.5253E-08
1790.9999994785.8662E-08
1800.9999995315.2736E-08
1810.9999995784.741E-08
1820.9999996214.262E-08
1830.9999996593.8316E-08
1840.9999996933.4445E-08
1850.9999997243.0966E-08
1860.9999997522.7839E-08
1870.9999997772.5026E-08
1880.99999982.2498E-08
1890.999999822.0226E-08
1900.9999998381.8183E-08
1910.9999998551.6346E-08
1920.9999998691.4695E-08
1930.9999998821.321E-08
1940.9999998941.1877E-08
1950.9999999051.0676E-08
1960.9999999159.598E-09
1970.9999999238.629E-09
1980.9999999317.757E-09
1990.9999999386.974E-09
2000.9999999446.269E-09

Quote: AlexDinNYC

Is it the same as just drawing the numbers or is matching them per drawing another layer?

should be the same as each of the 10 draws is without replacement

Quote: AlexDinNYC

Doing the hypergeo on the odds (99 choices with 10 chosen) it is a 1 in 2 that i would match 1 number out of ten drawn from 99. So does that mean I could expect to match all 99 after 198 drawings? I'm thinking I have to account for the duplicates and I am not....

time for a break now.
I was coding in R and javascript with a 4 day headache.

women NCAAB tonight
UConn vs Notre Dame!
Sally
I Heart Vi Hart
AlexDinNYC
AlexDinNYC
  • Threads: 1
  • Posts: 5
Joined: Mar 29, 2018
March 30th, 2018 at 2:14:45 PM permalink
Once again a very big thank you :) Hope the head feels better and have a good watching the game!
mustangsally
mustangsally
  • Threads: 25
  • Posts: 2463
Joined: Mar 29, 2011
May 27th, 2018 at 3:59:39 PM permalink
Quote: AlexDinNYC

Once again a very big thank you :)

you are welcome.
I finished some R code so one can run these type of calculations online.
it can be found here
https://sites.google.com/view/krapstuff/coupon-collecting/baseball-card-collector-problem

It is a Markov chain solution and it uses a simple hypergeometric formula to populate the transition matrix. (I think I said that correctly). actually I used choose(), but that is for extra credit.

(I have yet to add anything to stop large values or negative numbers one may input.
so try to be nice as it is your machine)

results
Keno
avg draws
17.860422

your 99, 10 at a time
avg draws
49.33094

also included at no extra charge
the probability distribution matrix and mean wait time matrix too

well, the code did it!
> coupons.r(80,20,60) #Keno numbers 20 at a time (r=20)
Time difference of 0.1574485 secs
draw Prob on X cumulative
[1,] 3 0 0
[2,] 4 1.307724409e-29 1.307724409e-29
[3,] 5 1.254704187e-15 1.254704187e-15
[4,] 6 3.565936203e-10 3.56594875e-10
[5,] 7 4.363806011e-07 4.367371959e-07
[6,] 8 3.747184596e-05 3.790858316e-05
[7,] 9 0.0006920127152 0.0007299212983
[8,] 10 0.004810252733 0.005540174032
[9,] 11 0.01739819727 0.02293837131
[10,] 12 0.0401755751 0.0631139464
[11,] 13 0.06780026217 0.1309142086
[12,] 14 0.09164068458 0.2225548931
[13,] 15 0.1056772507 0.3282321439
[14,] 16 0.1086743495 0.4369064934
[15,] 17 0.1028345813 0.5397410747
[16,] 18 0.09157133233 0.631312407
[17,] 19 0.07798591952 0.7092983265
[18,] 20 0.06427084063 0.7735691672
[19,] 21 0.05169889267 0.8252680598
[20,] 22 0.04084635239 0.8661144122
[21,] 23 0.03184521154 0.8979596238
[22,] 24 0.02458335354 0.9225429773
[23,] 25 0.01883844974 0.9413814271
[24,] 26 0.01435729658 0.9557387236
[25,] 27 0.01089757905 0.9666363027
[26,] 28 0.008246470615 0.9748827733
[27,] 29 0.006226192226 0.9811089655
[28,] 30 0.004692920698 0.9858018862
[29,] 31 0.003532779059 0.9893346653
[30,] 32 0.002656936725 0.991991602
[31,] 33 0.001996829683 0.9939884317
[32,] 34 0.001499937674 0.9954883694
[33,] 35 0.001126251719 0.9966146211
[34,] 36 0.0008454167342 0.9974600378
[35,] 37 0.0006344705596 0.9980945084
[36,] 38 0.0004760815678 0.9985705899
[37,] 39 0.0003571892942 0.9989277792
[38,] 40 0.0002679637525 0.999195743
[39,] 41 0.0002010130293 0.999396756
[40,] 42 0.0001507823008 0.9995475383
[41,] 43 0.000113099346 0.9996606377
[42,] 44 8.48315791e-05 0.9997454692
[43,] 45 6.362764444e-05 0.9998090969
[44,] 46 4.77229516e-05 0.9998568198
[45,] 47 3.579345625e-05 0.9998926133
[46,] 48 2.684578819e-05 0.9999194591
[47,] 49 2.0134731e-05 0.9999395938
[48,] 50 1.510126662e-05 0.9999546951
[49,] 51 1.132607229e-05 0.9999660212
[50,] 52 8.494622728e-06 0.9999745158
[51,] 53 6.371005423e-06 0.9999808868
[52,] 54 4.778275563e-06 0.9999856651
[53,] 55 3.583718713e-06 0.9999892488
[54,] 56 2.687795779e-06 0.9999919366
[55,] 57 2.015850612e-06 0.9999939524
[56,] 58 1.511890075e-06 0.9999954643
[57,] 59 1.133918741e-06 0.9999965982
[58,] 60 8.504397197e-07 0.9999974487
avg draws
0 17.860422
1 17.816656
2 17.772336
3 17.727448
4 17.681976
5 17.635907
6 17.589223
7 17.541908
8 17.493945
9 17.445316
10 17.396002
11 17.345984
12 17.295240
13 17.243751
14 17.191493
15 17.138443
16 17.084577
17 17.029869
18 16.974293
19 16.917820
20 16.860422
21 16.802067
22 16.742723
23 16.682356
24 16.620930
25 16.558407
26 16.494747
27 16.429909
28 16.363846
29 16.296514
30 16.227861
31 16.157835
32 16.086380
33 16.013437
34 15.938941
35 15.862826
36 15.785020
37 15.705445
38 15.624019
39 15.540655
40 15.455258
41 15.367726
42 15.277949
43 15.185810
44 15.091180
45 14.993922
46 14.893885
47 14.790906
48 14.684806
49 14.575391
50 14.462446
51 14.345736
52 14.225002
53 14.099956
54 13.970278
55 13.835613
56 13.695562
57 13.549675
58 13.397444
59 13.238295
60 13.071567
61 12.896502
62 12.712224
63 12.517707
64 12.311749
65 12.092918
66 11.859499
67 11.609406
68 11.340076
69 11.048302
70 10.730003
71 10.379874
72 9.990841
73 9.553179
74 9.052995
75 8.469455
76 7.769190
77 6.893665
78 5.726619
79 4.000000
> coupons.r(99,10,200) #10 at a time (r=10)
Time difference of 1.206498 secs
draw Prob on X cumulative
[1,] 8 0 0
[2,] 9 0 0
[3,] 10 1.259251356e-38 1.259251356e-38
[4,] 11 4.004930954e-29 4.004930956e-29
[5,] 12 1.409397915e-23 1.40940192e-23
[6,] 13 1.252485574e-19 1.252626514e-19
[7,] 14 1.272579592e-16 1.273832218e-16
[8,] 15 3.021991432e-14 3.034729754e-14
[9,] 16 2.522336965e-12 2.552684263e-12
[10,] 17 9.581957937e-11 9.837226364e-11
[11,] 18 1.974426878e-09 2.072799142e-09
[12,] 19 2.501159242e-08 2.708439156e-08
[13,] 20 2.137172209e-07 2.408016125e-07
[14,] 21 1.322022544e-06 1.562824156e-06
[15,] 22 6.25548924e-06 7.818313397e-06
[16,] 23 2.365231098e-05 3.147062437e-05
[17,] 24 7.401656432e-05 0.0001054871887
[18,] 25 0.0001972603028 0.0003027474914
[19,] 26 0.0004583616628 0.0007611091542
[20,] 27 0.0009468764099 0.001707985564
[21,] 28 0.001767505032 0.003475490597
[22,] 29 0.003022402466 0.006497893063
[23,] 30 0.004789547857 0.01128744092
[24,] 31 0.007103324016 0.01839076494
[25,] 32 0.009942731259 0.02833349619
[26,] 33 0.01323002183 0.04156351802
[27,] 34 0.01683936576 0.05840288378
[28,] 35 0.02061269637 0.07901558015
[29,] 36 0.02437875422 0.1033943344
[30,] 37 0.0279715118 0.1313658462
[31,] 38 0.03124517665 0.1626110228
[32,] 39 0.03408430001 0.1966953228
[33,] 40 0.03640873734 0.2331040602
[34,] 41 0.03817408521 0.2712781454
[35,] 42 0.0393686971 0.3106468425
[36,] 43 0.04000851513 0.3506553576
[37,] 44 0.0401308514 0.390786209
[38,] 45 0.03978802072 0.4305742297
[39,] 46 0.03904145359 0.4696156833
[40,] 47 0.03795666217 0.5075723455
[41,] 48 0.03659922315 0.5441715686
[42,] 49 0.03503178959 0.5792033582
[43,] 50 0.03331204629 0.6125154045
[44,] 51 0.03149147004 0.6440068746
[45,] 52 0.02961473643 0.673621611
[46,] 53 0.02771961716 0.7013412282
[47,] 54 0.02583722733 0.7271784555
[48,] 55 0.02399250405 0.7511709595
[49,] 56 0.02220482128 0.7733757808
[50,] 57 0.02048866862 0.7938644494
[51,] 58 0.01885434126 0.8127187907
[52,] 59 0.01730860512 0.8300273958
[53,] 60 0.01585531415 0.84588271
[54,] 61 0.01449596686 0.8603786768
[55,] 62 0.01323019634 0.8736088732
[56,] 63 0.0120561934 0.8856650666
[57,] 64 0.0109710655 0.8966361321
[58,] 65 0.009971136433 0.9066072685
[59,] 66 0.009052192754 0.9156594613
[60,] 67 0.008209683422 0.9238691447
[61,] 68 0.007438878928 0.9313080236
[62,] 69 0.006734996023 0.9380430196
[63,] 70 0.006093293568 0.9441363132
[64,] 71 0.005509144441 0.9496454576
[65,] 72 0.004978087851 0.9546235455
[66,] 73 0.004495865799 0.9591194113
[67,] 74 0.00405844686 0.9631778581
[68,] 75 0.003662039975 0.9668398981
[69,] 76 0.00330310046 0.9701429986
[70,] 77 0.002978330082 0.9731213287
[71,] 78 0.002684672679 0.9758060013
[72,] 79 0.002419306538 0.9782253079
[73,] 80 0.002179634502 0.9804049424
[74,] 81 0.001963272556 0.9823682149
[75,] 82 0.001768037497 0.9841362524
[76,] 83 0.001591934163 0.9857281866
[77,] 84 0.001433142546 0.9871613291
[78,] 85 0.001290005082 0.9884513342
[79,] 86 0.001161014283 0.9896123485
[80,] 87 0.001044800872 0.9906571494
[81,] 88 0.0009401224854 0.9915972719
[82,] 89 0.0008458530258 0.9924431249
[83,] 90 0.0007609726711 0.9932040976
[84,] 91 0.0006845585652 0.9938886561
[85,] 92 0.0006157761773 0.9945044323
[86,] 93 0.000553871315 0.9950583036
[87,] 94 0.0004981627662 0.9955564664
[88,] 95 0.0004480355398 0.9960045019
[89,] 96 0.0004029346726 0.9964074366
[90,] 97 0.0003623595667 0.9967697962
[91,] 98 0.0003258588228 0.997095655
[92,] 99 0.0002930255324 0.9973886805
[93,] 100 0.0002634929963 0.9976521735
[94,] 101 0.0002369308345 0.9978891044
[95,] 102 0.0002130414558 0.9981021458
[96,] 103 0.000191556858 0.9982937027
[97,] 104 0.0001722357298 0.9984659384
[98,] 105 0.0001548608275 0.9986207992
[99,] 106 0.0001392366032 0.9987600358
[100,] 107 0.000125187061 0.9988852229
[101,] 108 0.0001125538205 0.9989977767
[102,] 109 0.0001011943688 0.9990989711
[103,] 110 9.098048268e-05 0.9991899516
[104,] 111 8.179680559e-05 0.9992717484
[105,] 112 7.353956434e-05 0.9993452879
[106,] 113 6.611541237e-05 0.9994114033
[107,] 114 5.944038719e-05 0.9994708437
[108,] 115 5.343897119e-05 0.9995242827
[109,] 116 4.804324553e-05 0.9995723259
[110,] 117 4.319212829e-05 0.9996155181
[111,] 118 3.883068837e-05 0.9996543488
[112,] 119 3.490952787e-05 0.9996892583
[113,] 120 3.138422618e-05 0.9997206425
[114,] 121 2.821483956e-05 0.9997488574
[115,] 122 2.536545092e-05 0.9997742228
[116,] 123 2.280376461e-05 0.9997970266
[117,] 124 2.050074189e-05 0.9998175273
[118,] 125 1.843027298e-05 0.9998359576
[119,] 126 1.6568882e-05 0.9998525265
[120,] 127 1.48954616e-05 0.9998674219
[121,] 128 1.339103424e-05 0.999880813
[122,] 129 1.203853748e-05 0.9998928515
[123,] 130 1.082263094e-05 0.9999036741
[124,] 131 9.729522546e-06 0.9999134037
[125,] 132 8.746812442e-06 0.9999221505
[126,] 133 7.863352503e-06 0.9999300138
[127,] 134 7.069120057e-06 0.9999370829
[128,] 135 6.355104326e-06 0.999943438
[129,] 136 5.713204302e-06 0.9999491512
[130,] 137 5.136136939e-06 0.9999542874
[131,] 138 4.617354587e-06 0.9999589047
[132,] 139 4.150970761e-06 0.9999630557
[133,] 140 3.731693401e-06 0.9999667874
[134,] 141 3.354764858e-06 0.9999701422
[135,] 142 3.015907941e-06 0.9999731581
[136,] 143 2.711277412e-06 0.9999758694
[137,] 144 2.437416364e-06 0.9999783068
[138,] 145 2.191217016e-06 0.999980498
[139,] 146 1.969885455e-06 0.9999824679
[140,] 147 1.770909945e-06 0.9999842388
[141,] 148 1.592032427e-06 0.9999858308
[142,] 149 1.431222908e-06 0.999987262
[143,] 150 1.28665642e-06 0.9999885487
[144,] 151 1.156692322e-06 0.9999897054
[145,] 152 1.03985568e-06 0.9999907452
[146,] 153 9.348205323e-07 0.9999916801
[147,] 154 8.403948455e-07 0.9999925205
[148,] 155 7.555069855e-07 0.999993276
[149,] 156 6.791935589e-07 0.9999939552
[150,] 157 6.105884801e-07 0.9999945657
[151,] 158 5.489131434e-07 0.9999951147
[152,] 159 4.934675872e-07 0.9999956081
[153,] 160 4.436225506e-07 0.9999960518
[154,] 161 3.988123326e-07 0.9999964506
[155,] 162 3.585283722e-07 0.9999968091
[156,] 163 3.223134767e-07 0.9999971314
[157,] 164 2.897566339e-07 0.9999974212
[158,] 165 2.604883468e-07 0.9999976817
[159,] 166 2.341764408e-07 0.9999979158
[160,] 167 2.105222937e-07 0.9999981263
[161,] 168 1.89257447e-07 0.9999983156
[162,] 169 1.701405585e-07 0.9999984857
[163,] 170 1.52954664e-07 0.9999986387
[164,] 171 1.375047145e-07 0.9999987762
[165,] 172 1.236153627e-07 0.9999988998
[166,] 173 1.111289732e-07 0.999999011
[167,] 174 9.990383311e-08 0.9999991109
[168,] 175 8.981254385e-08 0.9999992007
[169,] 176 8.074057539e-08 0.9999992814
[170,] 177 7.258496629e-08 0.999999354
[171,] 178 6.525315521e-08 0.9999994192
[172,] 179 5.866193039e-08 0.9999994779
[173,] 180 5.273648528e-08 0.9999995306
[174,] 181 4.740956952e-08 0.9999995781
[175,] 182 4.262072568e-08 0.9999996207
[176,] 183 3.831560314e-08 0.999999659
[177,] 184 3.444534122e-08 0.9999996934
[178,] 185 3.096601465e-08 0.9999997244
[179,] 186 2.783813505e-08 0.9999997522
[180,] 187 2.502620275e-08 0.9999997773
[181,] 188 2.249830392e-08 0.9999997998
[182,] 189 2.022574832e-08 0.99999982
[183,] 190 1.818274372e-08 0.9999998382
[184,] 191 1.634610317e-08 0.9999998545
[185,] 192 1.469498182e-08 0.9999998692
[186,] 193 1.321064037e-08 0.9999998824
[187,] 194 1.187623237e-08 0.9999998943
[188,] 195 1.067661304e-08 0.999999905
[189,] 196 9.598167355e-09 0.9999999146
[190,] 197 8.628655564e-09 0.9999999232
[191,] 198 7.757074245e-09 0.999999931
[192,] 199 6.973531433e-09 0.9999999379
[193,] 200 6.269134352e-09 0.9999999442
avg draws
0 49.33094
1 49.23557
2 49.13923
3 49.04190
4 48.94355
5 48.84417
6 48.74373
7 48.64221
8 48.53959
9 48.43584
10 48.33094
11 48.22486
12 48.11757
13 48.00905
14 47.89927
15 47.78819
16 47.67580
17 47.56205
18 47.44691
19 47.33035
20 47.21234
21 47.09283
22 46.97179
23 46.84917
24 46.72494
25 46.59906
26 46.47148
27 46.34214
28 46.21101
29 46.07804
30 45.94316
31 45.80633
32 45.66749
33 45.52658
34 45.38353
35 45.23828
36 45.09076
37 44.94090
38 44.78862
39 44.63384
40 44.47649
41 44.31647
42 44.15369
43 43.98805
44 43.81946
45 43.64780
46 43.47296
47 43.29482
48 43.11326
49 42.92814
50 42.73931
51 42.54663
52 42.34994
53 42.14906
54 41.94382
55 41.73401
56 41.51944
57 41.29987
58 41.07508
59 40.84481
60 40.60877
61 40.36669
62 40.11824
63 39.86307
64 39.60081
65 39.33106
66 39.05338
67 38.76728
68 38.47224
69 38.16768
70 37.85297
71 37.52741
72 37.19022
73 36.84055
74 36.47742
75 36.09977
76 35.70638
77 35.29589
78 34.86675
79 34.41716
80 33.94510
81 33.44819
82 32.92368
83 32.36831
84 31.77823
85 31.14881
86 30.47443
87 29.74818
88 28.96141
89 28.10312
90 27.15899
91 26.10996
92 24.92980
93 23.58105
94 22.00751
95 20.11925
96 17.75893
97 14.61176
98 9.90000
>
I Heart Vi Hart
gordonm888
Administrator
gordonm888
  • Threads: 60
  • Posts: 5005
Joined: Feb 18, 2015
May 27th, 2018 at 4:42:37 PM permalink
Just wanna say that I am proud to be in the same damn forum as MustangSally. Really.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
Wizard
Administrator
Wizard
  • Threads: 1491
  • Posts: 26435
Joined: Oct 14, 2009
May 27th, 2018 at 5:44:37 PM permalink
Quote: gordonm888

Just wanna say that I am proud to be in the same damn forum as MustangSally. Really.



I too.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Ace2
Ace2
  • Threads: 32
  • Posts: 2672
Joined: Oct 2, 2017
May 27th, 2018 at 9:27:03 PM permalink
This can be approximated pretty well.

If the 80 numbers were selected randomly, one by one , with replacement, then the expected value to get all 80 numbers would be ( Ln (80) + 0.577) * 80 = 397

Since the numbers are selected in groups of 20 with no duplicates in the groups, the numerator of the equation changes. In the first case it’s always 80 so the formula is 80/80 + 80/79 + 80/78 .... + 80/1 = 397 (or very close).

But in the second case the numerator oscillates between 80 and 61 since the balls are selected in groups of 20. Starts at 80, decrements to 61 then resets to 80, for an average of 70.5.

70.5 / 80 x 397 is an expected value of 350. 350 / 20 = 17.5 draws of 20.
It’s all about making that GTA
  • Jump to: