atrain
• Posts: 20
Joined: Nov 14, 2010
November 14th, 2010 at 2:58:47 PM permalink
I am trying to figure out how many spins would u say is it where a number statistically would have to come up
PapaChubby
• Posts: 495
Joined: Mar 29, 2010
November 14th, 2010 at 3:39:11 PM permalink
There is no number of spins that guarantees that a particular number will come up. The probability never goes to 1, it will only approach it.

For 300 spins the probability that a particular number will not show up is (37/38)^300 = 0.0335%. That's pretty small, but not zero. If you repeated this 300 spin experiment 3000 times, you'd only expect to completely miss your number on one of the trials.
guido111
• Posts: 707
Joined: Sep 16, 2010
November 14th, 2010 at 4:01:26 PM permalink
Quote: atrain

I am trying to figure out how many spins would u say is it where a number statistically would have to come up

It is all about the degree of certainty.

For a 38 number wheel.
From https://wizardofvegas.com/member/nope27/blog/:

"The below table is for the probability of NOT hitting "at least 1" number in x spins.
question #2 at WoO.

Wizards' Example: for "at least 1" number NOT hitting in 200 spins
Sum i=1 to 37 [(-1)^(i+1) × combin(38,i) × ((38-i)/38)^38] = 16.9845715651245%"

I have expanded the table below from the above blog.
I have also run 1 million spin simulations and have seen many numbers that did not appear in 500 spins.

I would say from the table 1000 spins would be a hard one to ever witness.
spinsprob1 inexpected # of spins
5099.9999975%..
6099.9991%..
7099.967896%..
8099.678210%..
9098.452658%..
10095.339700%..
11089.728676%..
12081.746470%..
13072.133375%..
14061.855739%..
15051.774802%1.9290
16042.490296%2.4377
17034.326903%2.9495
18027.391862%3.7657
19021.649355%4.6878
20016.984572%5.91,178
21013.249361%7.51,585
22010.290675%9.72,138
2307.966128%12.62,887
2406.151027%16.33,902
2504.740313%21.15,274
2603.647758%27.47,128
2702.803866%35.79,630
2802.153364%46.413,003
2901.652705%60.517,547
3001.267822%78.923,663
3100.972205%102.931,886
3200.745304%134.242,935
3300.571235%175.157,770
3400.437748%228.477,670
3500.335412%298.1104,349
3600.256975%389.1140,091
3700.196867%508.0187,944
3800.150810%663.1251,973
3900.115523%865.6337,594
4000.088490%1,130.1452,028
5000.006149%16,261.58,130,771
6000.000427%234,067.2140,440,306
7000.000030%3,369,203.42,358,442,356
8000.000002061985%48,496,952.138,797,561,717
9000.000000143251%698,074,378.6628,266,940,697
10000.000000009952%10,048,215,791.010,048,215,791,014
statman
• Posts: 95
Joined: Sep 25, 2011
October 8th, 2011 at 4:50:46 AM permalink
For some number or other not to come up, I would say the probability is zero. At least one number has to come up.

For a particular number not to come up in 300 spins I would agree with PapaChubby: (37/38)^300 = 0.0335%.

For at least one number not to come up, I think that guido111 has a good grip on the problem based on a solution given by the Wizard for the case of 200 spins, however the Wizard's logic may not be easy for everyone to follow. Here is another approach based on counting the ways the balls can be distributed to the numbers.

We assume that both the balls and the numbers are distinguishable. The numbers are distinguishable because they are all different and the balls are distinguished by the order in which they are thrown. There is a function T(m,n) that gives the number of ways that m distinguishable objects may be distributed to n containers such that every container has at least one object. The number of ways of distributing m objects to n containers without restriction is nm so the probability that all numbers will have been hit by at least one ball is T(m,n)/nm. Subtract this from 1 and that is the probability that at least one number will remain unhit. For m = 300 and n = 38 this evaluates to 0.01267822135, which agrees with guido111's result.

It is always possible either for a particular number or for some number or other not to come up based on the inequality T(m,n) < nm.

The T function is discussed in several books on combinatorics. I shall be glad to provide a bibliography if anyone asks.
A fool is someone whose pencil wears out before its eraser does. - Marilyn Vos Savant
FleaStiff
• Posts: 14484
Joined: Oct 19, 2009
October 8th, 2011 at 8:13:28 AM permalink
Its similar to that famed "two people with a common birthday". With 24 people in a room you have a fifty percent chance, simply because you are not designating one particular date that has to be matched.

Its the same with a roulette wheel, if you don't specify a particular number that must hit then there are lots of spins that can go by and there being some surviving number that is unhit is still quite likely.

Ofcourse I don't want to know what number will NOT hit on that next spin, I want to know what number will hit on that next spin.
boymimbo
• Posts: 5994
Joined: Nov 12, 2009
October 8th, 2011 at 10:47:20 AM permalink
If you were to use the binomial distribution to approximate the answer over 300 spins, the answer would be 38 x (37/38)^300 = .012742

You are using a different (correct) formula to account for all possibilities.
----- You want the truth! You can't handle the truth!
mrjjj
• Posts: 1579
Joined: Sep 4, 2010
October 29th, 2011 at 1:55:08 PM permalink
(lol) I dont need to look at a chart, I do all the work on my own. TRIAL & ERROR.

I use to play a method (it did quite well for sometime), I tracked all numbers until only ONE left unhit. When I got to that point, I started a 110 progression on that one number. It was around a \$3,600 BR. Like I said, I made ALOT of money BUT as usual, it slowly tanked. The 110 combined with how far back it last hit was well over 300 spins and on multiple occasions I might add so I stopped playing it. That particular method slowly lured me away from playing sleepers/due.

Ken
Keyser
• Posts: 2112
Joined: Apr 16, 2010
October 29th, 2011 at 3:51:49 PM permalink
That's why several gambler's like you have become victims of the 'gambler's fallacy'.
mrjjj
• Posts: 1579
Joined: Sep 4, 2010
October 29th, 2011 at 4:15:33 PM permalink
Quote: Keyser

That's why several gambler's like you have become victims of the 'gambler's fallacy'.

.......and you Keyser are low on gas, better stop. Which city this week? I heard the 21/33 is hitting ALOT at a casino in Tucson. Pack up the car and away you go!! (LMAO) Talk about putting all your eggs in ONE basket.

Ken
Keyser
• Posts: 2112
Joined: Apr 16, 2010
October 29th, 2011 at 5:58:45 PM permalink
There's no difference between a cold number sleeping and a hot number if there are no biased wheels. It's simply gambler's fallacy to believe otherwise.
mrjjj
• Posts: 1579
Joined: Sep 4, 2010
October 29th, 2011 at 6:29:14 PM permalink
Quote: Keyser

There's no difference between a cold number sleeping and a hot number if there are no biased wheels. It's simply gambler's fallacy to believe otherwise.

I respect your opinion, thank you sir. Which city this week? (lol)

Ken
ncfatcat
• Posts: 363
Joined: Jun 25, 2011
October 29th, 2011 at 7:11:49 PM permalink
Life is random but there are also streaks so we won't be bored to death. The thing is the streaks are random.
Gambling is a metaphor for life. Hang around long enough and it's all gone.
mustangsally
• Posts: 2463
Joined: Mar 29, 2011
June 8th, 2018 at 6:56:40 AM permalink
Quote: PapaChubby

For 300 spins the probability that a particular number will not show up is (37/38)^300 = 0.0335%. That's pretty small, but not zero. If you repeated this 300 spin experiment 3000 times, you'd only expect to completely miss your number on one of the trials.

and when it happened, you really would hate that number.
As to the Wizard's solution of any number not hitting (using inclusion exclusion)
this is also easily solved using a Markov chain (in R of course for speed and less waiting)
for 300 spins and at least 1 # not hit
`0.01267822135690134`

a web page is here

for those after info on any number still sleeping after X number of spins or if you have a desire to run the code yourself for other values you can come up with.
btw, this is a simple case of the coupon collecting problem, btw.

I show data below of how many numbers have hit (u=X; meaning unique numbers)/not hit
should help those so interested (now and in the future)

example: 38 number wheel and 38 spins
`> cc.draws(38,38)[1] "Number of coupons:38, draws:38, mean number drawn:24.2066, mean number NOT drawn:13.7934"          not drawn       Probability           cumulative         u=1                  37 3.531888016291733e-59 3.531888016291733e-59u=2                  36 1.796050273122837e-46 1.79605027312319e-46 u=3                  35 1.059174005964773e-38 1.059174023925276e-38u=4                  34 5.183421957365635e-33 5.183432549105874e-33u=5                  33 1.695453232853687e-28 1.695505067179178e-28u=6                  32 9.471698925052215e-25 9.473394430119394e-25u=7                  31 1.493713472970262e-21 1.494660812413274e-21u=8                  30 8.972474437993078e-19 8.987421046117211e-19u=9                  29 2.488592099786068e-16 2.497579520832186e-16u=10                 28 3.632399739616234e-14 3.657375534824556e-14u=11                 27 3.06158131074875e-12  3.098155066096996e-12u=12                 26 1.595200485306436e-10 1.626182035967406e-10u=13                 25 5.409183238270684e-09 5.571801441867425e-09u=14                 24 1.241901364424703e-07 1.297619378843378e-07u=15                 23 1.991224329578271e-06 2.120986267462609e-06u=16                 22 2.284767205332572e-05 2.496865832078833e-05u=17                 21 0.0001912806706747635 0.0002162493289955518u=18                 20 0.001186529878631787  0.001402779207627339 u=19                 19 0.005519546761596696  0.006922325969224035 u=20                 18 0.01943459301293786   0.02635691898216189  u=21                 17 0.05215229288172268   0.07850921186388458  u=22                 16 0.1071593392085864    0.185668551072471    u=23                 15 0.1690424972911288    0.3547110483635998   u=24                 14 0.2048643366386519    0.5595753850022518   u=25                 13 0.1904903213341322    0.7500657063363839   u=26                 12 0.1354368760652057    0.8855025824015896   u=27                 11 0.07321147122465538   0.9587140536262451   u=28                 10 0.02983819924633686   0.9885522528725819   u=29                  9 0.009063960063884168  0.9976162129364661   u=30                  8 0.002020712814747093  0.9996369257512132   u=31                  7 0.0003238875908091962 0.9999608133420224   u=32                  6 3.630885882510717e-05 0.9999971222008475   u=33                  5 2.741719853095042e-06 0.9999998639207006   u=34                  4 1.322553699688062e-07 0.9999999961760705   u=35                  3 3.767712200861788e-09 0.9999999999437827   u=36                  2 5.58749156318279e-11  0.9999999999996576   u=37                  1 3.417426032527703e-13 0.9999999999999994   u=38                  0 4.86120346021012e-16                  1    [1] "Number of coupons:38, draws:38, mean number drawn:24.2066, mean number NOT drawn:13.7934"`

a bunch of results
from 38 to 300 spins
`> cc.draws(38,38)[1] "Number of coupons:38, draws:38, mean number drawn:24.2066, mean number NOT drawn:13.7934"          not drawn       Probability           cumulative         u=1                  37 3.531888016291733e-59 3.531888016291733e-59u=2                  36 1.796050273122837e-46 1.79605027312319e-46 u=3                  35 1.059174005964773e-38 1.059174023925276e-38u=4                  34 5.183421957365635e-33 5.183432549105874e-33u=5                  33 1.695453232853687e-28 1.695505067179178e-28u=6                  32 9.471698925052215e-25 9.473394430119394e-25u=7                  31 1.493713472970262e-21 1.494660812413274e-21u=8                  30 8.972474437993078e-19 8.987421046117211e-19u=9                  29 2.488592099786068e-16 2.497579520832186e-16u=10                 28 3.632399739616234e-14 3.657375534824556e-14u=11                 27 3.06158131074875e-12  3.098155066096996e-12u=12                 26 1.595200485306436e-10 1.626182035967406e-10u=13                 25 5.409183238270684e-09 5.571801441867425e-09u=14                 24 1.241901364424703e-07 1.297619378843378e-07u=15                 23 1.991224329578271e-06 2.120986267462609e-06u=16                 22 2.284767205332572e-05 2.496865832078833e-05u=17                 21 0.0001912806706747635 0.0002162493289955518u=18                 20 0.001186529878631787  0.001402779207627339 u=19                 19 0.005519546761596696  0.006922325969224035 u=20                 18 0.01943459301293786   0.02635691898216189  u=21                 17 0.05215229288172268   0.07850921186388458  u=22                 16 0.1071593392085864    0.185668551072471    u=23                 15 0.1690424972911288    0.3547110483635998   u=24                 14 0.2048643366386519    0.5595753850022518   u=25                 13 0.1904903213341322    0.7500657063363839   u=26                 12 0.1354368760652057    0.8855025824015896   u=27                 11 0.07321147122465538   0.9587140536262451   u=28                 10 0.02983819924633686   0.9885522528725819   u=29                  9 0.009063960063884168  0.9976162129364661   u=30                  8 0.002020712814747093  0.9996369257512132   u=31                  7 0.0003238875908091962 0.9999608133420224   u=32                  6 3.630885882510717e-05 0.9999971222008475   u=33                  5 2.741719853095042e-06 0.9999998639207006   u=34                  4 1.322553699688062e-07 0.9999999961760705   u=35                  3 3.767712200861788e-09 0.9999999999437827   u=36                  2 5.58749156318279e-11  0.9999999999996576   u=37                  1 3.417426032527703e-13 0.9999999999999994   u=38                  0 4.86120346021012e-16                  1    [1] "Number of coupons:38, draws:38, mean number drawn:24.2066, mean number NOT drawn:13.7934"> cc.draws(38,40)[1] "Number of coupons:38, draws:40, mean number drawn:24.923, mean number NOT drawn:13.077"          not drawn       Probability           cumulative         u=1                  37 2.445905828456878e-62 2.445905828456878e-62u=2                  36 4.975208512832792e-49 4.975208512833037e-49u=3                  35 6.601502276020392e-41 6.601502325772477e-41u=4                  34 5.743583538392731e-35 5.743590139895056e-35u=5                  33 2.936439904333339e-30 2.936497340234738e-30u=6                  32 2.365627901547091e-26 2.365921551281115e-26u=7                  31 5.095994590767757e-23 5.098360512319038e-23u=8                  30 4.024885925250708e-20 4.029984285763027e-20u=9                  29 1.427741091283564e-17 1.431771075569327e-17u=10                 28 2.611012305810631e-15 2.625330016566324e-15u=11                 27 2.714764007950449e-13 2.741017308116112e-13u=12                 26 1.724348332716008e-11 1.751758505797169e-11u=13                 25 7.063637319202279e-10 7.238813169781996e-10u=14                 24 1.945716998622009e-08 2.018105130319829e-08u=15                 23 3.723734896551473e-07 3.925545409583456e-07u=16                 22 5.081232650939704e-06 5.47378719189805e-06 u=17                 21 5.046750917346794e-05 5.594129636536599e-05u=18                 20 0.0003709020762543536 0.0004268433726197196u=19                 19 0.002043577820503691  0.002470421193123411 u=20                 18 0.008528187471882529  0.01099860866500594  u=21                 17 0.02716729625820908   0.03816590492321502  u=22                 16 0.06643727959518979   0.1046031845184048   u=23                 15 0.1251825685325609    0.2297857530509657   u=24                 14 0.1820602907923615    0.4118460438433272   u=25                 13 0.2043575555395325    0.6162035993828596   u=26                 12 0.1766867870726062    0.7928903864554658   u=27                 11 0.1171921785515023    0.910082565006968    u=28                 10 0.05925467862926021   0.9693372436362283   u=29                  9 0.02263422840640629   0.9919714720426346   u=30                  8 0.006452245331970358  0.9984237173746049   u=31                  7 0.001350394011747838  0.9997741113863527   u=32                  6 0.0002030295128875364 0.9999771408992403   u=33                  5 2.129464451232586e-05 0.9999984355437527   u=34                  4 1.496280556249304e-06 0.9999999318243089   u=35                  3 6.644897965328748e-08 0.9999999982732886   u=36                  2 1.704986862235516e-09 0.9999999999782754   u=37                  1 2.16287094953398e-11  0.9999999999999041   u=38                  0 9.562498911860692e-14                 1    [1] "Number of coupons:38, draws:40, mean number drawn:24.923, mean number NOT drawn:13.077"> cc.draws(38,50)[1] "Number of coupons:38, draws:50, mean number drawn:27.9841, mean number NOT drawn:10.0159"          not drawn       Probability           cumulative         u=1                  37 3.895863802749243e-78 3.895863802749243e-78u=2                  36 8.114752481285795e-62 8.114752481285795e-62u=3                  35 6.20896875196668e-52  6.208968752778155e-52u=4                  34 9.5931972844765e-45   9.593197905373375e-45u=5                  33 4.570281276377452e-39 4.570290869575357e-39u=6                  32 2.286190923452339e-34 2.286236626361035e-34u=7                  31 2.319649365836473e-30 2.319877989499109e-30u=8                  30 7.083797162255486e-27 7.086117040244985e-27u=9                  29 8.399600882236821e-24 8.406686999277067e-24u=10                 28 4.599915288493157e-21 4.608321975492434e-21u=11                 27 1.315989460008252e-18 1.320597781983745e-18u=12                 26 2.153065427287628e-16 2.166271405107465e-16u=13                 25 2.156766223665314e-14 2.178428937716389e-14u=14                 24 1.394263852948971e-12 1.416048142326135e-12u=15                 23 6.062090022704067e-11 6.20369483693668e-11 u=16                 22 1.831954173169448e-09 1.893991121538815e-09u=17                 21 3.950998076633495e-08 4.140397188787376e-08u=18                 20 6.212992354330983e-07 6.62703207320972e-07 u=19                 19 7.248586372017541e-06 7.911289579338513e-06u=20                 18 6.363476242070619e-05 7.15460520000447e-05 u=21                 17 0.0004251679372204709 0.0004967139892205157u=22                 16 0.002181543237810482  0.002678257227030997 u=23                 15 0.008655931663796607  0.0113341888908276   u=24                 14 0.0266931261253937    0.03802731501622131  u=25                 13 0.06418568547470954   0.1022130004909308   u=26                 12 0.1205297327608814    0.2227427332518122   u=27                 11 0.1767114290150352    0.3994541622668474   u=28                 10 0.2018505993464849    0.6013047616133324   u=29                  9 0.1788789139554827    0.7801836755688151   u=30                  8 0.1221661869765338    0.9023498625453489   u=31                  7 0.06367709033345331   0.9660269528788021   u=32                  6 0.02498667892683492   0.9910136318056371   u=33                  5 0.007240830227813003  0.9982544620334501   u=34                  4 0.001507738334938373  0.9997622003683885   u=35                  3 0.0002165918064271544 0.9999787921748157   u=36                  2 2.011177821573497e-05 0.9999989039530314   u=37                  1 1.071533839809168e-06 0.9999999754868711   u=38                  0 2.451312857572011e-08                 1    [1] "Number of coupons:38, draws:50, mean number drawn:27.9841, mean number NOT drawn:10.0159"> cc.draws(38,60)[1] "Number of coupons:38, draws:60, mean number drawn:30.3287, mean number NOT drawn:7.67134"          not drawn       Probability           cumulative         u=1                  37 6.205371683973392e-94 6.205371683973392e-94u=2                  36 1.323546694828295e-74 1.323546694828295e-74u=3                  35 5.839774228826861e-63 5.839774228840096e-63u=4                  34 1.602242514767703e-54 1.602242520607477e-54u=5                  33 7.109428194174103e-48 7.109429796416623e-48u=6                  32 2.203074610454639e-42 2.203081719884435e-42u=7                  31 1.046266258472442e-37 1.04628828928964e-37 u=8                  30 1.220591678583641e-33 1.22069630741257e-33 u=9                  29 4.746917383218712e-30 4.748138079526124e-30u=10                 28 7.581514977448452e-27 7.586263115527978e-27u=11                 27 5.768899311037464e-24 5.776485574152992e-24u=12                 26 2.332641708610509e-21 2.338418194184662e-21u=13                 25 5.442890776345501e-19 5.466274958287347e-19u=14                 24 7.810295205699577e-17 7.86495795528245e-17 u=15                 23 7.245443753107571e-15 7.324093332660396e-15u=16                 22 4.521896101256436e-13 4.59513703458304e-13 u=17                 21 1.960565668494546e-11 2.006517038840376e-11u=18                 20 6.061631474273508e-10 6.262283178157546e-10u=19                 19 1.365273055407227e-08 1.427895887188803e-08u=20                 18 2.27966374186562e-07  2.4224533305845e-07  u=21                 17 2.862573210129439e-06 3.104818543187889e-06u=22                 16 2.73474579779513e-05  3.045227652113919e-05u=23                 15 0.0002006198548370605 0.0002310721313581997u=24                 14 0.001138259149159849  0.001369331280518048 u=25                 13 0.005021061058583047  0.006390392339101095 u=26                 12 0.01727881417611137   0.02366920651521246  u=27                 11 0.04645970331888044   0.07012890983409291  u=28                 10 0.09757335450736968   0.1677022643414626   u=29                  9 0.1596664715091173    0.3273687358505799   u=30                  8 0.2025826845366208    0.5299514203872007   u=31                  7 0.1977337545345196    0.7276851749217204   u=32                  6 0.1467406159137689    0.8744257908354892   u=33                  5 0.08139161441700314   0.9558174052524924   u=34                  4 0.03290472056006815   0.9887221258125605   u=35                  3 0.009332435821000618  0.9980545616335611   u=36                  2 0.001744711456900689  0.9997992730904618   u=37                  1 0.0001914634275327707 0.9999907365179945   u=38                  0 9.263482005009225e-06                 1    [1] "Number of coupons:38, draws:60, mean number drawn:30.3287, mean number NOT drawn:7.67134"> cc.draws(38,70)[1] "Number of coupons:38, draws:70, mean number drawn:32.1244, mean number NOT drawn:5.8756"          not drawn       Probability            cumulative          u=1                  37 9.883979442270366e-110 9.883979442270366e-110u=2                  36 2.158754512144196e-87  2.158754512144196e-87 u=3                  35 5.492532536571576e-74  5.492532536571792e-74 u=4                  34 2.676037843410646e-64  2.676037843959899e-64 u=5                  33 1.105863998570025e-56  1.105864025330404e-56 u=6                  32 2.121995805714823e-50  2.121996911578849e-50 u=7                  31 4.709953220585482e-45  4.709974440554598e-45 u=8                  30 2.091626410963303e-40  2.091673510707709e-40 u=9                  29 2.650422921584741e-36  2.650632088935812e-36 u=10                 28 1.221907495459336e-32  1.222172558668229e-32 u=11                 27 2.437506392812673e-29  2.438728565371341e-29 u=12                 26 2.390694960254796e-26  2.393133688820167e-26 u=13                 25 1.269820377421524e-23  1.272213511110344e-23 u=14                 24 3.935759010892948e-21  3.948481146004051e-21 u=15                 23 7.548932206275237e-19  7.588417017735278e-19 u=16                 22 9.389962317359031e-17  9.465846487536383e-17 u=17                 21 7.866756061976922e-15  7.961414526852287e-15 u=18                 20 4.577961837504567e-13  4.65757598277309e-13  u=19                 19 1.897920097068131e-11  1.944495856895862e-11 u=20                 18 5.723430068573269e-10  5.917879654262855e-10 u=21                 17 1.277185942521362e-08  1.336364739063991e-08 u=22                 16 2.138831394452419e-07  2.272467868358818e-07 u=23                 15 2.718794299991473e-06  2.946041086827355e-06 u=24                 14 2.647216519094763e-05  2.941820627777499e-05 u=25                 13 0.0001988050758384531  0.0002282232821162281 u=26                 12 0.001157261646427439   0.001385484928543667  u=27                 11 0.00523708165409327    0.006622566582636938  u=28                 10 0.01844215382694246    0.0250647204095794    u=29                  9 0.0504732371808132     0.0755379575903926    u=30                  8 0.1069588824370598     0.1824968400274524    u=31                  7 0.1743185946748213     0.3568154347022737    u=32                  6 0.216177974712569      0.5729934094148428    u=33                  5 0.2007478900590928     0.7737412994739356    u=34                  4 0.1362741643763473     0.9100154638502829    u=35                  3 0.06515913341960795    0.9751745972698908    u=36                  2 0.02064184192717225    0.9958164391970631    u=37                  1 0.003862522827008437   0.9996789620240715    u=38                  0 0.0003210379759282154                  1     [1] "Number of coupons:38, draws:70, mean number drawn:32.1244, mean number NOT drawn:5.8756"> cc.draws(38,80)[1] "Number of coupons:38, draws:80, mean number drawn:33.4998, mean number NOT drawn:4.50022"          not drawn       Probability            cumulative          u=1                  37 1.574330347810349e-125 1.574330347810349e-125u=2                  36 3.521009921231e-100    3.521009921231e-100   u=3                  35 5.165938353233378e-85  5.165938353233381e-85 u=4                  34 4.469471793332122e-74  4.469471793383781e-74 u=5                  33 1.720149192755896e-65  1.720149197225368e-65 u=6                  32 2.043747888859852e-58  2.043748060874772e-58 u=7                  31 2.119386848872945e-52  2.119388892621005e-52 u=8                  30 3.579079822100949e-47  3.579101015989875e-47 u=9                  29 1.474400454448409e-42  1.474436245458569e-42 u=10                 28 1.954246625121784e-38  1.954394068746329e-38 u=11                 27 1.015717978731245e-34  1.015913418138119e-34 u=12                 26 2.395630121681826e-31  2.396646035099964e-31 u=13                 25 2.864034370429222e-28  2.866431016464321e-28 u=14                 24 1.890804147225001e-25  1.893670578241465e-25 u=15                 23 7.374364380497755e-23  7.39330108628017e-23  u=16                 22 1.793195728981654e-20  1.800589030067935e-20 u=17                 21 2.839827702567925e-18  2.857833592868604e-18 u=18                 20 3.035127038367567e-16  3.063705374296253e-16 u=19                 19 2.254202276491186e-14  2.284839330234148e-14 u=20                 18 1.1918941051022e-12    1.214742498404542e-12 u=21                 17 4.577143210755795e-11  4.698617460596249e-11 u=22                 16 1.297865990126534e-09  1.344852164732496e-09 u=23                 15 2.754322906984974e-08  2.888808123458224e-08 u=24                 14 4.422697012178971e-07  4.711577824524794e-07 u=25                 13 5.419563040806481e-06  5.89072082325896e-06  u=26                 12 5.100514208363642e-05  5.689586290689538e-05 u=27                 11 0.0003702413579791929  0.0004271372208860882 u=28                 10 0.00207727799865773    0.002504415219543818  u=29                  9 0.009006730460039641   0.01151114567958346   u=30                  8 0.0300957684973874     0.04160691417697086   u=31                  7 0.07704906382622738    0.1186559780031982    u=32                  6 0.1496546435363517     0.2683106215395499    u=33                  5 0.2171960049732974     0.4855066265128473    u=34                  4 0.2301092340901066     0.7156158606029538    u=35                  3 0.1716054561756792     0.887221316778633     u=36                  2 0.08479377169297933    0.9720150884716123    u=37                  1 0.02476700839889994    0.9967820968705123    u=38                  0 0.003217903129487141                   1     [1] "Number of coupons:38, draws:80, mean number drawn:33.4998, mean number NOT drawn:4.50022"> cc.draws(38,90)[1] "Number of coupons:38, draws:90, mean number drawn:34.5532, mean number NOT drawn:3.44679"          not drawn       Probability            cumulative          u=1                  37 2.507609468952249e-141 2.507609468952249e-141u=2                  36 5.742899804338209e-113 5.742899804338209e-113u=3                  35 4.858763947537586e-96  4.858763947537586e-96 u=4                  34 7.464833889694114e-84  7.464833889698974e-84 u=5                  33 2.675655694949218e-74  2.675655695695701e-74 u=6                  32 1.968361538936074e-66  1.968361565692631e-66 u=7                  31 9.535977150062042e-60  9.535979118423608e-60 u=8                  30 6.122010001013378e-54  6.122019536992496e-54 u=9                  29 8.192631487060863e-49  8.192692707256234e-49 u=10                 28 3.117180825396421e-44  3.117262752323494e-44 u=11                 27 4.21017534832548e-40   4.210487074600712e-40 u=12                 26 2.378373257554012e-36  2.378794306261472e-36 u=13                 25 6.364484448082871e-33  6.366863242389132e-33 u=14                 24 8.885059367015558e-30  8.891426230257947e-30 u=15                 23 6.982912772645691e-27  6.991804198875948e-27 u=16                 22 3.283670046849203e-24  3.290661851048079e-24 u=17                 21 9.706641186987555e-22  9.739547805498036e-22 u=18                 20 1.878019064140003e-19  1.887758611945501e-19 u=19                 19 2.458668006210965e-17  2.47754559233042e-17  u=20                 18 2.238886673884381e-15  2.263662129807685e-15 u=21                 17 1.450865066784481e-13  1.473501688082558e-13 u=22                 16 6.818993196324223e-12  6.96634336513248e-12  u=23                 15 2.361055818224224e-10  2.430719251875549e-10 u=24                 14 6.099942286773565e-09  6.343014211961121e-09 u=25                 13 1.187953175467035e-07  1.251383317586646e-07 u=26                 12 1.757579177620905e-06  1.88271750937957e-06  u=27                 11 1.986421398726907e-05  2.174693149664864e-05 u=28                 10 0.0001720591618347414  0.0001938060933313901 u=29                  9 0.00114314237106186    0.001336948464393251  u=30                  8 0.005814895553797971   0.007151844018191221  u=31                  7 0.022533313265721      0.02968515728391222   u=32                  6 0.06592062431665729    0.09560578160056951   u=33                  5 0.1434885677890287     0.2390943493895983    u=34                  4 0.2271861140588167     0.466280463448415     u=35                  3 0.2524513291286919     0.7187317925771068    u=36                  2 0.1854296492612008     0.9041614418383077    u=37                  1 0.08036513949329337    0.9845265813316011    u=38                  0 0.01547341866839833                    1     [1] "Number of coupons:38, draws:90, mean number drawn:34.5532, mean number NOT drawn:3.44679"> cc.draws(38,100)[1] "Number of coupons:38, draws:100, mean number drawn:35.36, mean number NOT drawn:2.63995"          not drawn       Probability            cumulative          u=1                  37 3.994145991992575e-157 3.994145991992575e-157u=2                  36 9.366885893674846e-126 9.366885893674846e-126u=3                  35 4.569854590524564e-107 4.569854590524564e-107u=4                  34 1.246763545163958e-93  1.246763545164004e-93 u=5                  33 4.161925560954717e-83  4.161925561079393e-83 u=6                  32 1.895752212470496e-74  1.895752216632422e-74 u=7                  31 4.290539358909213e-67  4.290539548484435e-67 u=8                  30 1.047064395922847e-60  1.047064824976802e-60 u=9                  29 4.550717981033151e-55  4.550728451681401e-55 u=10                 28 4.967542448361708e-50  4.967587955646225e-50 u=11                 27 1.74158252677738e-45   1.741632202656937e-45 u=12                 26 2.35213216794488e-41   2.352306331165146e-41 u=13                 25 1.405018150352919e-37  1.405253380986036e-37 u=14                 24 4.132055712517859e-34  4.133460965898845e-34 u=15                 23 6.511796986012881e-31  6.51593044697878e-31  u=16                 22 5.885413781982201e-28  5.89192971242918e-28  u=17                 21 3.223463304999225e-25  3.229355234711654e-25 u=18                 20 1.119289750354748e-22  1.122519105589459e-22 u=19                 19 2.557531242892246e-20  2.56875643394814e-20  u=20                 18 3.966356712129982e-18  3.992044276469463e-18 u=21                 17 4.283897522542899e-16  4.323817965307594e-16 u=22                 16 3.292114408822892e-14  3.335352588475968e-14 u=23                 15 1.832415653939385e-12  1.865769179824145e-12 u=24                 14 7.496109026557715e-11  7.68268594454013e-11  u=25                 13 2.280545979806504e-09  2.357372839251906e-09 u=26                 12 5.207666290054342e-08  5.443403573979532e-08 u=27                 11 8.986663350977509e-07  9.531003708375463e-07 u=28                 10 1.177073833704264e-05  1.272383870788018e-05 u=29                  9 0.0001172365737487515  0.0001299604124566317 u=30                  8 0.000887102027084915   0.001017062439541547  u=31                  7 0.005078243591551884   0.00609530603109343   u=32                  6 0.02181112231756904    0.02790642834866247   u=33                  5 0.06931848109431271    0.09722490944297518   u=34                  4 0.1594654414967347     0.2566903509397098    u=35                  3 0.2563562242939999     0.5130465752337098    u=36                  2 0.2713859871559937     0.7844325623897034    u=37                  1 0.1689644396552439     0.9533970020449474    u=38                  0 0.04660299795505186                    1     [1] "Number of coupons:38, draws:100, mean number drawn:35.36, mean number NOT drawn:2.63995"> cc.draws(38,110)[1] "Number of coupons:38, draws:110, mean number drawn:35.978, mean number NOT drawn:2.02198"          not drawn       Probability            cumulative          u=1                  37 6.361916559525538e-173 6.361916559525538e-173u=2                  36 1.527774370690689e-138 1.527774370690689e-138u=3                  35 4.298124215135454e-118 4.298124215135454e-118u=4                  34 2.082322742190263e-103 2.082322742190267e-103u=5                  33 6.473786697110853e-92  6.473786697131677e-92 u=6                  32 1.825820737982187e-82  1.825820738629566e-82 u=7                  31 1.930442218094299e-74  1.930442236352506e-74 u=8                  30 1.790776396095108e-67  1.790776589139332e-67 u=9                  29 2.527492037597842e-61  2.527493828374431e-61 u=10                 28 7.913721520948484e-56  7.913746795886767e-56 u=11                 27 7.198599834397637e-51  7.198678971865596e-51 u=12                 26 2.322429583525562e-46  2.32250157031528e-46  u=13                 25 3.092574024616496e-42  3.092806274773527e-42 u=14                 24 1.912232347962469e-38  1.912541628589947e-38 u=15                 23 6.026538405280983e-35  6.028450946909573e-35 u=16                 22 1.043236438491348e-31  1.043839283586039e-31 u=17                 21 1.054089545328017e-28  1.055133384611603e-28 u=18                 20 6.534478610491893e-26  6.545029944338009e-26 u=19                 19 2.58998079824493e-23   2.596525828189268e-23 u=20                 18 6.792588823006343e-21  6.818554081288236e-21 u=21                 17 1.213013558100816e-18  1.219832112182104e-18 u=22                 16 1.510702494056831e-16  1.522900815178652e-16 u=23                 15 1.338549271137239e-14  1.353778279289025e-14 u=24                 14 8.57817995271538e-13   8.713557780644282e-13 u=25                 13 4.029969713571376e-11  4.117105291377819e-11 u=26                 12 1.402785550030796e-09  1.443956602944575e-09 u=27                 11 3.64726589719659e-08   3.791661557491048e-08 u=28                 10 7.122318891881173e-07  7.501485047630278e-07 u=29                  9 1.047603503921354e-05  1.122618354397657e-05 u=30                  8 0.0001160606129600151  0.0001272867965039916 u=31                  7 0.0009652025441122243  0.001092489340616216  u=32                  6 0.005980172654800023   0.007072661995416239  u=33                  5 0.02724233049434175    0.03431499248975799   u=34                  4 0.08931328897928323    0.1236282814690412    u=35                  3 0.2035570765666486     0.3271853580356899    u=36                  2 0.3040783346281022     0.6312636926637921    u=37                  1 0.2660230699804053     0.8972867626441974    u=38                  0 0.102713237355802                      1     [1] "Number of coupons:38, draws:110, mean number drawn:35.978, mean number NOT drawn:2.02198"> cc.draws(38,120)[1] "Number of coupons:38, draws:120, mean number drawn:36.4513, mean number NOT drawn:1.54867"          not drawn       Probability            cumulative          u=1                  37 1.013332572006809e-188 1.013332572006809e-188u=2                  36 2.491857543941542e-151 2.491857543941542e-151u=3                  35 4.042551333479776e-129 4.042551333479776e-129u=4                  34 3.477859149360674e-113 3.477859149360674e-113u=5                  33 1.006983751916606e-100 1.006983751916954e-100u=6                  32 1.758468841821549e-90  1.758468841922247e-90 u=7                  31 8.68563038570674e-82   8.685630403291429e-82 u=8                  30 3.062713230628904e-74  3.062713317485208e-74 u=9                  29 1.403735595495735e-67  1.403735901767066e-67 u=10                 28 1.260581693750088e-61  1.26058309748599e-61  u=11                 27 2.97454849790703e-56   2.974561103738005e-56 u=12                 26 2.291554654213147e-51  2.291584399824184e-51 u=13                 25 6.79805426879421e-47   6.798283427234192e-47 u=14                 24 8.828838539858111e-43  8.829518368200834e-43 u=15                 23 5.556371861034281e-39  5.557254812871101e-39 u=16                 22 1.838597448801662e-35  1.839153174282949e-35 u=17                 21 3.418381845836541e-32  3.420220999010824e-32 u=18                 20 3.771261426279216e-29  3.774681647278227e-29 u=19                 19 2.582936182137094e-26  2.586710863784372e-26 u=20                 18 1.14041071082594e-23   1.142997421689724e-23 u=21                 17 3.349789165126533e-21  3.36121913934343e-21  u=22                 16 6.721270428347227e-19  6.754882619740661e-19 u=23                 15 9.417966624964981e-17  9.485515451162388e-17 u=24                 14 9.386716005880669e-15  9.481571160392293e-15 u=25                 13 6.755795700241924e-13  6.850611411845846e-13 u=26                 12 3.553960364810378e-11  3.622466478928837e-11 u=27                 11 1.379377246523718e-09  1.415601911313006e-09 u=28                 10 3.976257469611837e-08  4.117817660743138e-08 u=29                  9 8.546191873232981e-07  8.957973639307295e-07 u=30                  8 1.370769390947192e-05  1.460349127340265e-05 u=31                  7 0.0001636587857464236  0.0001782622770198263 u=32                  6 0.00144455833467422    0.001622820611694046  u=33                  5 0.009309295739343979   0.01093211635103803   u=34                  4 0.04289985134194571    0.05383196769298373   u=35                  3 0.1366315854771961     0.1904635531701799    u=36                  2 0.2836975011332268     0.4741610543034067    u=37                  1 0.3433036476453676     0.8174647019487743    u=38                  0 0.1825352980512247                     1     [1] "Number of coupons:38, draws:120, mean number drawn:36.4513, mean number NOT drawn:1.54867"> cc.draws(38,130)[1] "Number of coupons:38, draws:130, mean number drawn:36.8138, mean number NOT drawn:1.18615"          not drawn       Probability            cumulative          u=1                  37 1.614046477790517e-204 1.614046477790517e-204u=2                  36 4.064313512793911e-164 4.064313512793911e-164u=3                  35 3.802175196861802e-140 3.802175196861802e-140u=4                  34 5.808659732577519e-123 5.808659732577519e-123u=5                  33 1.566341808864703e-109 1.566341808864761e-109u=6                  32 1.693601446129118e-98  1.693601446144781e-98 u=7                  31 3.907921220653608e-89  3.907921222347209e-89 u=8                  30 5.238060614694662e-81  5.238060653773874e-81 u=9                  29 7.796082176567837e-74  7.796082700373902e-74 u=10                 28 2.007909638192816e-67  2.007910417801086e-67 u=11                 27 1.228976572112825e-61  1.228978580023243e-61 u=12                 26 2.260451304537793e-56  2.260463594323593e-56 u=13                 25 1.493453922652545e-51  1.493476527288488e-51 u=14                 24 4.071790293001042e-47  4.071939640653771e-47 u=15                 23 5.113201887927717e-43  5.113609081891783e-43 u=16                 22 3.230612551024488e-39  3.231123911932677e-39 u=17                 21 1.103588633785362e-35  1.103911746176555e-35 u=18                 20 2.162562575335009e-32  2.163666487081185e-32 u=19                 19 2.553276367239386e-29  2.555440033726467e-29 u=20                 18 1.892348814638675e-26  1.894904254672401e-26 u=21                 17 9.111886425211969e-24  9.130835467758693e-24 u=22                 16 2.934005110236599e-21  2.943135945704358e-21 u=23                 15 6.472698019691768e-19  6.502129379148811e-19 u=24                 14 9.983328043452971e-17  1.004834933724446e-16 u=25                 13 1.094717180563111e-14  1.104765529900355e-14 u=26                 12 8.650941140487971e-13  8.761417693478006e-13 u=27                 11 4.979435557564841e-11  5.067049734499621e-11 u=28                 10 2.103954590277398e-09  2.154625087622394e-09 u=29                  9 6.557860837082117e-08  6.773323345844356e-08 u=30                  8 1.510576846451862e-06  1.578310079910306e-06 u=31                  7 2.567001090030283e-05  2.724832098021313e-05 u=32                  6 0.0003198695610449542  0.0003471178820251673 u=33                  5 0.002888294039495216   0.003235411921520384  u=34                  4 0.01852118774654595    0.02175659966806634   u=35                  3 0.08156369304515529    0.1033202927132216    u=36                  2 0.2328091457437557     0.3361294384569773    u=37                  1 0.3852043118324718     0.7213337502894491    u=38                  0  0.27866624971055                      1     [1] "Number of coupons:38, draws:130, mean number drawn:36.8138, mean number NOT drawn:1.18615"> cc.draws(38,140)[1] "Number of coupons:38, draws:140, mean number drawn:37.0915, mean number NOT drawn:0.908493"          not drawn       Probability            cumulative          u=1                  37 2.570869726716399e-220 2.570869726716399e-220u=2                  36 6.62904842631995e-177  6.62904842631995e-177 u=3                  35 3.576092184137358e-151 3.576092184137358e-151u=4                  34 9.701522241080361e-133 9.701522241080361e-133u=5                  33 2.43641136960249e-118  2.436411369602499e-118u=6                  32 1.631126913541777e-106 1.631126913544214e-106u=7                  31 1.758288918304143e-96  1.758288918467256e-96 u=8                  30 8.958483487669118e-88  8.958483505252007e-88 u=9                  29 4.329783113303861e-80  4.329783202888696e-80 u=10                 28 3.198242449144408e-73  3.198242882122728e-73 u=11                 27 5.07746217805736e-67   5.077465376300242e-67 u=12                 26 2.229506255454393e-61  2.229511332919769e-61 u=13                 25 3.280073117158739e-56  3.280095412272068e-56 u=14                 24 1.87688730212654e-51   1.876920103080663e-51 u=15                 23 4.700922911237207e-47  4.701110603247515e-47 u=16                 22 5.667613384241342e-43  5.668083495301667e-43 u=17                 21 3.554111998413183e-39  3.554678806762713e-39 u=18                 20 1.23561168421064e-35   1.235967152091317e-35 u=19                 19 2.511111353506624e-32  2.512347320658716e-32 u=20                 18 3.118343717536271e-29  3.12085606485693e-29  u=21                 17 2.455937318804851e-26  2.459058174869708e-26 u=22                 16 1.265764485112735e-23  1.268223543287604e-23 u=23                 15 4.38313202091396e-21   4.395814256346836e-21 u=24                 14 1.042599185469201e-18  1.046994999725548e-18 u=25                 13 1.735147291924133e-16  1.745617241921388e-16 u=26                 12 2.051022770664394e-14  2.068478943083608e-14 u=27                 11 1.74260924208328e-12   1.763294031514116e-12 u=28                 10 1.073761121293884e-10  1.091394061609026e-10 u=29                  9 4.826911287295775e-09  4.936050693456678e-09 u=30                  8 1.587340277225494e-07  1.636700784160061e-07 u=31                  7 3.815202395030286e-06  3.978872473446292e-06 u=32                  6 6.666501736592105e-05  7.064388983936735e-05 u=33                  5 0.0008374591867000157  0.0009081030765393831 u=34                  4 0.007416846286993623   0.008324949363533006  u=35                  3 0.0448074466479248     0.05313239601145781   u=36                  2 0.1743626662416429     0.2274950622531007    u=37                  1 0.391062330679651      0.6185573929327517    u=38                  0 0.3814426070672478                     1     [1] "Number of coupons:38, draws:140, mean number drawn:37.0915, mean number NOT drawn:0.908493"> cc.draws(38,150)[1] "Number of coupons:38, draws:150, mean number drawn:37.3042, mean number NOT drawn:0.695829"          not drawn       Probability            cumulative          u=1                  37 4.094907577131537e-236 4.094907577131537e-236u=2                  36 1.081222767391451e-189 1.081222767391451e-189u=3                  35 3.363452404824344e-162 3.363452404824344e-162u=4                  34 1.620331335063632e-142 1.620331335063632e-142u=5                  33 3.789786066059777e-127 3.789786066059779e-127u=6                  32 1.570956976678549e-114 1.570956976678928e-114u=7                  31 7.911059919551071e-104 7.911059919708167e-104u=8                  30 1.532139893415887e-94  1.532139894206993e-94 u=9                  29 2.404669843879117e-86  2.404669859200516e-86 u=10                 28 5.094206288062753e-79  5.094206528529739e-79 u=11                 27 2.097694738679303e-72  2.097695248099956e-72 u=12                 26 2.198875838671371e-66  2.198877936366619e-66 u=13                 25 7.203164940348603e-61  7.203186929127966e-61 u=14                 24 8.649318551875189e-56  8.649390583744481e-56 u=15                 23 4.319834715897163e-51  4.319921209803e-51    u=16                 22 9.934773512613155e-47  9.935205504734135e-47 u=17                 21 1.143080283681395e-42  1.143179635736443e-42 u=18                 20 7.045511535720874e-39  7.04665471535661e-39  u=19                 19 2.462348151143994e-35  2.463052816615529e-35 u=20                 18 5.117419738579322e-32  5.119882791395938e-32 u=21                 17 6.582641188762665e-29  6.587761071554062e-29 u=22                 16 5.420771522675395e-26  5.427359283746949e-26 u=23                 15 2.940413338340055e-23  2.945840697623802e-23 u=24                 14 1.076103566770627e-20  1.079049407468251e-20 u=25                 13 2.710776733407836e-18  2.721567227482519e-18 u=26                 12 4.778429962890855e-16  4.805645635165681e-16 u=27                 11 5.972654624613075e-14  6.020711080964732e-14 u=28                 10 5.347137209149052e-12  5.407344319958699e-12 u=29                  9 3.452780042809885e-10  3.506853486009472e-10 u=30                  8 1.613971544413744e-08  1.649040079273839e-08 u=31                  7 5.460984079152173e-07  5.625888087079557e-07 u=32                  6 1.331392944639599e-05  1.387651825510395e-05 u=33                  5 0.0002314461104393751  0.000245322628694479  u=34                  4 0.002814979335581469   0.003060301964275948  u=35                  3 0.02319061998319668    0.02625092194747262   u=36                  2 0.1222593546924203     0.1485102766398929    u=37                  1 0.3692377475323197     0.5177480241722127    u=38                  0 0.4822519758277869                     1     [1] "Number of coupons:38, draws:150, mean number drawn:37.3042, mean number NOT drawn:0.695829"> cc.draws(38,160)[1] "Number of coupons:38, draws:160, mean number drawn:37.4671, mean number NOT drawn:0.532947"          not drawn       Probability            cumulative          u=1                  37 6.522410642201722e-252 6.522410642201722e-252u=2                  36 1.76351505909063e-202  1.76351505909063e-202 u=3                  35 3.163456504197357e-173 3.163456504197357e-173u=4                  34 2.706249153634593e-152 2.706249153634593e-152u=5                  33 5.894931621847827e-136 5.894931621847827e-136u=6                  32 1.513006622599799e-122 1.513006622599858e-122u=7                  31 3.559418948981843e-111 3.559418948996973e-111u=8                  30 2.620368256149154e-101 2.620368256505096e-101u=9                  29 1.335502307558351e-92  1.335502310178719e-92 u=10                 28 8.114111077207453e-85  8.114111210757684e-85 u=11                 27 8.666325343229005e-78  8.666326154640126e-78 u=12                 26 2.16862121428679e-71   2.168622080919405e-71 u=13                 25 1.581757506251202e-65  1.581759674873283e-65 u=14                 24 3.98541551696417e-60   3.985431334560919e-60 u=15                 23 3.968695311053344e-55  3.968735165366689e-55 u=16                 22 1.740717703329567e-50  1.74075739068122e-50  u=17                 21 3.673731430749923e-46  3.673905506488991e-46 u=18                 20 4.012782290633305e-42  4.013149681183954e-42 u=19                 19 2.410387805992268e-38  2.410789120960386e-38 u=20                 18 8.377356742494067e-35  8.379767531615028e-35 u=21                 17 1.758332733223738e-31  1.7591707099769e-31   u=22                 16 2.310906320019586e-28  2.312665490729563e-28 u=23                 15 1.960828238905166e-25  1.963140904395896e-25 u=24                 14 1.102265374163678e-22  1.104228515068074e-22 u=25                 13 4.194924788408187e-20  4.205967073558868e-20 u=26                 12 1.100365747934766e-17  1.104571715008325e-17 u=27                 11 2.018460514329917e-15  2.02950623148e-15     u=28                 10 2.618495256027226e-13  2.638790318342026e-13 u=29                  9 2.421580776781914e-11  2.447968679965334e-11 u=30                  8 1.603791689804447e-09  1.628271376604101e-09 u=31                  7 7.612524298559633e-08  7.775351436220044e-08 u=32                  6 2.579764857950106e-06  2.657518372312306e-06 u=33                  5 6.180827344988249e-05  6.446579182219479e-05 u=34                  4 0.001027947875640624   0.001092413667462819  u=35                  3 0.01149554208694935    0.01258795575441217   u=36                  2 0.08170847138393711    0.09429642713834928   u=37                  1 0.330606532808544      0.4249029599468933    u=38                  0 0.5750970400531065                     1     [1] "Number of coupons:38, draws:160, mean number drawn:37.4671, mean number NOT drawn:0.532947"> cc.draws(38,170)[1] "Number of coupons:38, draws:170, mean number drawn:37.5918, mean number NOT drawn:0.408193"          not drawn       Probability            cumulative          u=1                  37 1.038896233533715e-267 1.038896233533715e-267u=2                  36 2.876359486160795e-215 2.876359486160795e-215u=3                  35 2.97535265835617e-184  2.97535265835617e-184 u=4                  34 4.519930166789216e-162 4.519930166789216e-162u=5                  33 9.169440760119396e-145 9.169440760119396e-145u=6                  32 1.457193974120181e-130 1.45719397412019e-130 u=7                  31 1.601487459106108e-118 1.601487459107565e-118u=8                  30 4.481529243938885e-108 4.481529244099034e-108u=9                  29 7.417094096112601e-99  7.417094100594131e-99 u=10                 28 1.292424307463878e-90  1.292424314880972e-90 u=11                 27 3.580358989005946e-83  3.580359118248377e-83 u=12                 26 2.138764265018536e-76  2.138764623054448e-76 u=13                 25 3.473329531572073e-70  3.473331670336696e-70 u=14                 24 1.836286966329524e-64  1.836290439661194e-64 u=15                 23 3.64566321940757e-59   3.645681582311966e-59 u=16                 22 3.049302405583942e-54  3.049338862399765e-54 u=17                 21 1.180229414385042e-49  1.180259907773666e-49 u=18                 20 2.284010650776934e-45  2.284128676767711e-45 u=19                 19 2.357168843585523e-41  2.3573972564532e-41   u=20                 18 1.369377643709549e-37  1.369613383435194e-37 u=21                 17 4.687001021828701e-34  4.688370635212136e-34 u=22                 16 9.823380756135008e-31  9.82806912677022e-31  u=23                 15 1.302618535405658e-27  1.303601342318335e-27 u=24                 14 1.123500137254357e-24  1.124803738596675e-24 u=25                 13 6.451078963990304e-22  6.462327001376271e-22 u=26                 12 2.514202974624245e-19  2.520665301625621e-19 u=27                 11 6.756568112406597e-17  6.781774765422853e-17 u=28                 10 1.267601431174547e-14  1.27438320593997e-14  u=29                  9 1.675256267834204e-12  1.688000099893604e-12 u=30                  8 1.568221929263028e-10  1.585101930261963e-10 u=31                  7 1.04148304604003e-08   1.05733406534265e-08  u=32                  6 4.89193774671137e-07   4.997671153245635e-07 u=33                  5 1.610403168594443e-05  1.6603798801269e-05   u=34                  4 0.0003650280008586745  0.0003816317996599435 u=35                  3 0.00552176383090093    0.005903395630560873  u=36                  2 0.05271815838878487    0.05862155401934575   u=37                  1 0.2846474779543352     0.343269031973681     u=38                  0 0.6567309680263188                     1     [1] "Number of coupons:38, draws:170, mean number drawn:37.5918, mean number NOT drawn:0.408193"> cc.draws(38,180)[1] "Number of coupons:38, draws:180, mean number drawn:37.6874, mean number NOT drawn:0.312641"          not drawn       Probability            cumulative          u=1                  37 1.654764539152364e-283 1.654764539152364e-283u=2                  36 4.691450663252891e-228 4.691450663252891e-228u=3                  35 2.798433747971911e-195 2.798433747971911e-195u=4                  34 7.549108582708746e-172 7.549108582708746e-172u=5                  33 1.426287008007457e-153 1.426287008007457e-153u=6                  32 1.403440174347408e-138 1.40344017434741e-138 u=7                  31 7.205563936755128e-126 7.205563936756532e-126u=8                  30 7.66461136568138e-115  7.664611365753436e-115u=9                  29 4.119295262039513e-105 4.119295262805974e-105u=10                 28 2.058586869722315e-96  2.058586873841611e-96 u=11                 27 1.479168770416319e-88  1.479168791002187e-88 u=12                 26 2.109310692437553e-81  2.109310840354432e-81 u=13                 25 7.62688812420366e-75   7.626890233514501e-75 u=14                 24 8.460493401864668e-69  8.460501028754902e-69 u=15                 23 3.348723855981109e-63  3.348732316482138e-63 u=16                 22 5.340982377058279e-58  5.341015864381444e-58 u=17                 21 3.790808933201925e-53  3.790862343360569e-53 u=18                 20 1.299548112522574e-48  1.299586021146007e-48 u=19                 19 2.303786185590083e-44  2.303916144192197e-44 u=20                 18 2.236438732512023e-40  2.236669124126443e-40 u=21                 17 1.247766886233125e-36  1.247990553145538e-36 u=22                 16 4.168323621617776e-33  4.169571612170922e-33 u=23                 15 8.632541392591644e-30  8.636710964203816e-30 u=24                 14 1.141470851280778e-26  1.142334522377199e-26 u=25                 13 9.879630523237711e-24  9.891053868461483e-24 u=26                 12 5.714650172495218e-21  5.724541226363679e-21 u=27                 11 2.247041750838498e-18  2.252766292064861e-18 u=28                 10 6.087939447566702e-16  6.110467110487351e-16 u=29                  9 1.147943074251691e-13  1.154053541362178e-13 u=30                  8 1.516160233926254e-11  1.527700769339876e-11 u=31                  7 1.406031333042159e-09  1.421308340735558e-09 u=32                  6 9.134014809906825e-08  9.276145643980381e-08 u=33                  5 4.12175780703298e-06   4.214519263472784e-06 u=34                  4 0.0001270106360081335  0.0001312251552716063 u=35                  3 0.002591810769193645   0.002723035924465251  u=36                  2 0.03314120928679522    0.03586424521126047   u=37                  1 0.2380543763847624     0.2739186215960229    u=38                  0 0.7260813784039769                     1     [1] "Number of coupons:38, draws:180, mean number drawn:37.6874, mean number NOT drawn:0.312641"> cc.draws(38,190)[1] "Number of coupons:38, draws:190, mean number drawn:37.7605, mean number NOT drawn:0.239457"          not drawn       Probability            cumulative          u=1                  37 2.635725871025857e-299 2.635725871025857e-299u=2                  36 7.65193274054674e-241  7.65193274054674e-241 u=3                  35 2.632034700086521e-206 2.632034700086521e-206u=4                  34 1.260838957474639e-181 1.260838957474639e-181u=5                  33 2.218559105652999e-162 2.218559105652999e-162u=6                  32 1.351669275300991e-146 1.351669275300992e-146u=7                  31 3.241995518011247e-133 3.241995518011383e-133u=8                  30 1.310853151991432e-121 1.310853151994674e-121u=9                  29 2.287768375143337e-111 2.287768375274423e-111u=10                 28 3.278938330487449e-102 3.278938332775217e-102u=11                 27 6.110949832558073e-94  6.110949865347456e-94 u=12                 26 2.080259558972793e-86  2.080259620082291e-86 u=13                 25 1.674737145755925e-79  1.674737353781887e-79 u=14                 24 3.8980302452823e-73    3.898031920019654e-73 u=15                 23 3.075877850171848e-67  3.075881748203768e-67 u=16                 22 9.354375236348732e-62  9.354405995166215e-62 u=17                 21 1.217436554701092e-56  1.217445909107087e-56 u=18                 20 7.392600959734733e-52  7.392722704325644e-52 u=19                 19 2.250851378396872e-47  2.250925305623915e-47 u=20                 18 3.650581184307165e-43  3.650806276837727e-43 u=21                 17 3.319183416868478e-39  3.319548497496162e-39 u=22                 16 1.766745847339412e-35  1.767077802189162e-35 u=23                 15 5.711951555524948e-32  5.713718633327137e-32 u=24                 14 1.157304710262439e-28  1.157876082125772e-28 u=25                 13 1.508888382969457e-25  1.510046259051582e-25 u=26                 12 1.294345050421828e-22  1.295855096680879e-22 u=27                 11 7.440001842197857e-20  7.452960393164666e-20 u=28                 10 2.907907468654215e-17  2.91536042904738e-17  u=29                  9 7.813856394537911e-15  7.843009998828386e-15 u=30                  8 1.454144909460769e-12  1.461987919459597e-12 u=31                  7 1.880252265534547e-10  1.894872144729143e-10 u=32                  6 1.686581723816931e-08  1.705530445264223e-08 u=33                  5 1.041392145973605e-06  1.058447450426247e-06 u=34                  4 4.354007351961648e-05  4.459852097004273e-05 u=35                  3 0.001196043525911724   0.001240642046881767  u=36                  2 0.02043664814032888    0.02167729018721065   u=37                  1 0.1948162549585596     0.2164935451457703    u=38                  0 0.7835064548542298                     1     [1] "Number of coupons:38, draws:190, mean number drawn:37.7605, mean number NOT drawn:0.239457"> cc.draws(38,200)[1] "Number of coupons:38, draws:200, mean number drawn:37.8166, mean number NOT drawn:0.183404"          not drawn       Probability            cumulative          u=1                  37 4.198211107412155e-315 4.198211107412155e-315u=2                  36 1.248059051850993e-253 1.248059051850993e-253u=3                  35 2.475529987972786e-217 2.475529987972786e-217u=4                  34 2.105831250496224e-191 2.105831250496224e-191u=5                  33 3.450921502925237e-171 3.450921502925237e-171u=6                  32 1.301808130611776e-154 1.301808130611776e-154u=7                  31 1.458669304861897e-140 1.45866930486191e-140 u=8                  30 2.241908824637661e-128 2.24190882463912e-128 u=9                  29 1.270577560328177e-117 1.270577560350596e-117u=10                 28 5.222726555094276e-108 5.222726556364853e-108u=11                 27 2.52464107603269e-99   2.524641081255417e-99 u=12                 26 2.051607229532814e-91  2.051607254779224e-91 u=13                 25 3.677434809281644e-84  3.677435014442369e-84 u=14                 24 1.79594088206979e-77   1.795941249813291e-77 u=15                 23 2.825220131919505e-71  2.825221927860755e-71 u=16                 22 1.638303149638541e-65  1.638305974860469e-65 u=17                 21 3.909605695975998e-60  3.909622079035747e-60 u=18                 20 4.204861896326384e-55  4.204900992547174e-55 u=19                 19 2.198700111790809e-50  2.198742160800735e-50 u=20                 18 5.957034701305324e-46  5.957254575521404e-46 u=21                 17 8.825109984668983e-42  8.825705710126536e-42 u=22                 16 7.483086409408148e-38  7.48396897997916e-38  u=23                 15 3.775704034576675e-34  3.776452431474673e-34 u=24                 14 1.171758459808469e-30  1.172136105051616e-30 u=25                 13 2.300293499699335e-27  2.301465635804387e-27 u=26                 12 2.924698618559932e-24  2.927000084195736e-24 u=27                 11 2.455963197673213e-21  2.458890197757408e-21 u=28                 10 1.383714516721352e-18  1.386173406919109e-18 u=29                  9 5.294011558026385e-16  5.307873292095576e-16 u=30                  8 1.386791891626898e-13  1.392099764918994e-13 u=31                  7 2.497417197602163e-11  2.511338195251353e-11 u=32                  6 3.089332172661289e-09  3.114445554613802e-09 u=33                  5 2.606508592810835e-07  2.637653048356973e-07 u=34                  4 1.476372035250276e-05  1.502748565733846e-05 u=35                  3 0.0005450487602642354  0.0005600762459215739 u=36                  2 0.01242295712656632    0.01298303337248789   u=37                  1 0.1568626822787534     0.1698457156512413    u=38                  0 0.8301542843487586                     1     [1] "Number of coupons:38, draws:200, mean number drawn:37.8166, mean number NOT drawn:0.183404"> cc.draws(38,300)[1] "Number of coupons:38, draws:300, mean number drawn:37.9873, mean number NOT drawn:0.0127415"          not drawn       Probability            cumulative          u=1                  37                 0                      0     u=2                  36                 0                      0     u=3                  35                 0                      0     u=4                  34 3.556829418667059e-289 3.556829418667059e-289u=5                  33 2.861382077311971e-259 2.861382077311971e-259u=6                  32 8.939482043069444e-235 8.939482043069444e-235u=7                  31 4.959080792835434e-214 4.959080792835434e-214u=8                  30 4.80017404103927e-196  4.80017404103927e-196 u=9                  29 3.547255440553925e-180 3.547255440553925e-180u=10                 28 5.489561180123812e-166 5.489561180123847e-166u=11                 27 3.656861593019237e-153 3.656861593019786e-153u=12                 26 1.785908927011541e-141 1.785908927015198e-141u=13                 25 9.583434554139785e-131 9.583434554318377e-131u=14                 24 7.739950169321291e-121 7.739950170279635e-121u=15                 23 1.207327673145613e-111 1.207327673919608e-111u=16                 22 4.44682530316511e-103  4.446825315238387e-103u=17                 21 4.557276492212105e-95  4.557276536680358e-95 u=18                 20 1.488425491557719e-87  1.488425537130484e-87 u=19                 19 1.735129058646098e-80  1.735129207488651e-80 u=20                 18 7.942783852508284e-74  7.942785587637492e-74 u=21                 17 1.548151534289554e-67  1.548152328568113e-67 u=22                 16 1.376737820120136e-61  1.376739368272464e-61 u=23                 15 5.926279197522296e-56  5.926292964915979e-56 u=24                 14 1.299182795224427e-50  1.299188721517392e-50 u=25                 13 1.515186035343944e-45  1.51519902723116e-45  u=26                 12 9.758930626470046e-41  9.759082146372768e-41 u=27                 11 3.583358842963966e-36  3.58345643378543e-36  u=28                 10 7.704152193861348e-32  7.704510539504726e-32 u=29                  9 9.912918140432359e-28  9.913688591486309e-28 u=30                  8 7.76494541652329e-24   7.765936785382439e-24 u=31                  7 3.747810937924624e-20  3.748587531603162e-20 u=32                  6 1.12179998971635e-16   1.122174848469511e-16 u=33                  5 2.081386431745599e-13  2.082508606594068e-13 u=34                  4 2.370474016764953e-10  2.372556525371547e-10 u=35                  3 1.617809507859614e-07  1.620182064384985e-07 u=36                  2 6.299172026237903e-05  6.315373846881753e-05 u=37                  1 0.01261506761843252    0.01267822135690134   u=38                  0 0.9873217786430989                     1     [1] "Number of coupons:38, draws:300, mean number drawn:37.9873, mean number NOT drawn:0.0127415"`

thinking...
one must be very careful (and quiet) when Roulette numbers are sleeping
even at 200 spins there could be 1 or 2 number(s) still sleeping quietly

that could cause major damage to a bankroll
trying to get those 2 numbers to awaken, I would think
Sally
Last edited by: mustangsally on Jun 8, 2018
I Heart Vi Hart