Last open number Roulette

Quote: masterj

Hello,
I am new in this forum and would like to ask a question regarding European Roulette. (Single Zero Roulette)
In the Roulette Lexikon from Kurt von Haller he writes, that the last open number should show up on average after 132 spins.
Theoretically after 169 spins (132+37).

Quote: masterj

I could not find any more information on the internet about this topic. Can you confirm this numbers?
And why is it 37+132 = 169 spins? I don't understand this.

Quote: mustangsally

never heard of the guy. Is this a conditional probability problem? given the 37 numbers, how long on average does it take to get 36 of them?
then add 37 to that for the last number?
If that is what the secret is

169 is then not right.

118.4586903 is the average number of spins to get 36 of the 37 numbers
add 37 to that 155.4586903

that is what I get calculating getting all 37 numbers starting at 0
> an(37)
[1] 155.4587

"last open number" could have different meanings to different people

Sally

Quote: masterj

Quote: mustangsally

never heard of the guy. Is this a conditional probability problem? given the 37 numbers, how long on average does it take to get 36 of them?
then add 37 to that for the last number?
If that is what the secret is

169 is then not right.

118.4586903 is the average number of spins to get 36 of the 37 numbers
add 37 to that 155.4586903

that is what I get calculating getting all 37 numbers starting at 0
> an(37)
[1] 155.4587

"last open number" could have different meanings to different people

Sally

for me "last open number" means if for example after 120 spins we got 36 out of 37 numbers, then on average how many spins we have to wait for the last one to show up?
I know we can get this last number in the next spin, and it can't show up for the next x spins.

Quote: masterj

) says that the average is 169.

Quote: masterj

sure it can come up in the very next spin.
But the calculation from Kurt von Haller (he wrote a well known Roulette-Book) says that the average is 169.
I just would like to know if there are more sources which can confirm this Statement.

Quote: michael99000

I bet Kurt Von Haller made more money selling his book than he did playing roulette

Quote: Wizard

If the question is how long will it take for every number to appear in double-zero roulette, the answer is 160.6602765, on average. This is the sum of the inverse of every integer from 1 to 38.

Quote: Wizard

If the question is how long will it take for every number to appear in double-zero roulette, the answer is 160.6602765, on average. This is the sum of the inverse of every integer from 1 to 38.

Quote: TomG

(160.66 is the harmonic series up to 38 times 38; using 37 for single zero roulette it is is 155)

(19:17) gp > a=sum(k=1,37,37/(37-(k-1)));
(19:20) gp > a
%6 = 2040798836801833/13127595717600
(19:20) gp > a=sum(k=1,37,37./(37-(k-1)));
(19:20) gp > a
%8 = 155.45869028140164699369367483727361613

(19:17) gp > a=sum(k=1,38,38/(38-(k-1)));
(19:17) gp > a
%2 = 2053580969474233/12782132672400
(19:17) gp > a=sum(k=1,38,38./(38-(k-1)));
(19:17) gp > a
%4 = 160.66027650522331312865836875179452467

Quote: TomG

The sum of the inverse of every integer from 1 to 38 is 4.2279020133. Using the inverse of every integer, it would take like 10^70 of them to get to 160. (160.66 is the harmonic series up to 38 times 38; using 37 for single zero roulette it is is 155)

Quote: mustangsally

the OP questions 37. Where it came from

that is just 1/p where p=1/37
The average is not the mode (The "mode" is the value that occurs most often)
or median (The "median" is the "middle" value or close to 50%)
median = spin 147 @ 0.501522154
mode = 133 @ 0.0106293156

Quote: Wizard

I agree on the median. Here is my transition matrix.

> tMax.dist.cum(37, 160)
 Row        Draw X  Draw X Prob   cumulative: (X or less)
  [1,]         36          0              0             
  [2,]         37 1.30398646e-15 1.30398646e-15         
  [3,]         38 2.34717563e-14 2.47757428e-14         
  [4,]         39 2.18963997e-13 2.4373974e-13          
  [5,]         40 1.41020849e-12 1.65394823e-12         
  [6,]         41 7.04758973e-12 8.70153796e-12         
  [7,]         42 2.91274839e-11 3.78290219e-11         
  [8,]         43 1.0362069e-10  1.41449712e-10         
  [9,]         44 3.26113904e-10 4.67563616e-10         
 [10,]         45 9.26201149e-10 1.39376477e-09         
 [11,]         46 2.40983412e-09 3.80359888e-09         
 [12,]         47 5.81187695e-09 9.61547584e-09         
 [13,]         48 1.31152608e-08 2.27307366e-08         
 [14,]         49 2.79063823e-08 5.0637119e-08          
 [15,]         50 5.63461642e-08 1.06983283e-07         
 [16,]         51 1.08539739e-07 2.15523022e-07         
 [17,]         52 2.00382556e-07 4.15905578e-07         
 [18,]         53 3.55945139e-07 7.71850717e-07         
 [19,]         54 6.10432194e-07 1.38228291e-06         
 [20,]         55 1.01371507e-06 2.39599799e-06         
 [21,]         56 1.63439194e-06 4.03038993e-06         
 [22,]         57 2.56428073e-06 6.59467066e-06         
 [23,]         58 3.92320027e-06 1.05178709e-05         
 [24,]         59 5.86384932e-06 1.63817202e-05         
 [25,]         60 8.57655591e-06 2.49582762e-05         
 [26,]         61 1.22936442e-05 3.72519204e-05         
 [27,]         62 1.72931556e-05 5.4545076e-05          
 [28,]         63 2.39016655e-05 7.84467415e-05         
 [29,]         64 3.24959609e-05 0.000110942702         
 [30,]         65 4.35033766e-05 0.000154446079         
 [31,]         66 5.74006425e-05 0.000211846721         
 [32,]         67 7.47111458e-05 0.000286557867         
 [33,]         68 9.60005844e-05 0.000382558452         
 [34,]         69 0.000121871045 0.000504429497         
 [35,]         70 0.000152953612 0.000657383109         
 [36,]         71 0.000189899666 0.000847282775         
 [37,]         72 0.000233371083 0.00108065386          
 [38,]         73 0.000284029587 0.00136468345          
 [39,]         74 0.000342525541 0.00170720899          
 [40,]         75 0.000409486459 0.00211669545          
 [41,]         76 0.000485505566 0.00260220101          
 [42,]         77 0.000571130666 0.00317333168          
 [43,]         78 0.000666853627 0.0038401853           
 [44,]         79 0.000773100711 0.00461328601          
 [45,]         80 0.000890223971 0.00550350999          
 [46,]         81 0.0010184939   0.00652200389          
 [47,]         82 0.00115809345  0.00768009734          
 [48,]         83 0.00130911356  0.0089892109           
 [49,]         84 0.00147155014  0.010460761            
 [50,]         85 0.0016453027   0.0121060637           
 [51,]         86 0.00183017435  0.0139362381           
 [52,]         87 0.00202587342  0.0159621115           
 [53,]         88 0.00223201628  0.0181941278           
 [54,]         89 0.00244813147  0.0206422593           
 [55,]         90 0.00267366499  0.0233159243           
 [56,]         91 0.00290798647  0.0262239107           
 [57,]         92 0.0031503962   0.0293743069           
 [58,]         93 0.00340013282  0.0327744397           
 [59,]         94 0.00365638142  0.0364308212           
 [60,]         95 0.00391828208  0.0403491032           
 [61,]         96 0.00418493855  0.0445340418           
 [62,]         97 0.00445542691  0.0489894687           
 [63,]         98 0.00472880426  0.053718273            
 [64,]         99 0.00500411709  0.0587223901           
 [65,]        100 0.00528040947  0.0640027995           
 [66,]        101 0.00555673069  0.0695595302           
 [67,]        102 0.00583214262  0.0753916728           
 [68,]        103 0.00610572634  0.0814973992           
 [69,]        104 0.0063765884   0.0878739876           
 [70,]        105 0.00664386631  0.0945178539           
 [71,]        106 0.00690673349  0.101424587            
 [72,]        107 0.00716440359  0.108588991            
 [73,]        108 0.00741613413  0.116005125            
 [74,]        109 0.00766122955  0.123666355            
 [75,]        110 0.00789904366  0.131565398            
 [76,]        111 0.00812898147  0.13969438             
 [77,]        112 0.00835050049  0.14804488             
 [78,]        113 0.00856311152  0.156607992            
 [79,]        114 0.00876637894  0.165374371            
 [80,]        115 0.00895992057  0.174334291            
 [81,]        116 0.00914340706  0.183477698            
 [82,]        117 0.00931656105  0.192794259            
 [83,]        118 0.00947915584  0.202273415            
 [84,]        119 0.00963101392  0.211904429            
 [85,]        120 0.00977200518  0.221676434            
 [86,]        121 0.0099020449   0.231578479            
 [87,]        122 0.0100210917   0.241599571            
 [88,]        123 0.010129145    0.251728716            
 [89,]        124 0.010226243    0.261954959            
 [90,]        125 0.01031246     0.272267419            
 [91,]        126 0.0103879038   0.282655323            
 [92,]        127 0.0104527134   0.293108036            
 [93,]        128 0.0105070562   0.303615092            
 [94,]        129 0.0105511257   0.314166218            
 [95,]        130 0.0105851391   0.324751357            
 [96,]        131 0.0106093345   0.335360692            
 [97,]        132 0.0106239689   0.345984661            
 [98,]        133 0.0106293156   0.356613976            
 [99,]        134 0.0106256625   0.367239639            
[100,]        135 0.0106133093   0.377852948            
[101,]        136 0.0105925662   0.388445514            
[102,]        137 0.0105637515   0.399009266            
[103,]        138 0.0105271901   0.409536456            
[104,]        139 0.0104832116   0.420019667            
[105,]        140 0.0104321489   0.430451816            
[106,]        141 0.0103743366   0.440826153            
[107,]        142 0.0103101097   0.451136263            
[108,]        143 0.0102398025   0.461376065            
[109,]        144 0.0101637472   0.471539812            
[110,]        145 0.0100822729   0.481622085            
[111,]        146 0.00999570494  0.49161779             
[112,]        147 0.00990436365  0.501522154            
[113,]        148 0.00980856384  0.511330718            
[114,]        149 0.00970861406  0.521039332            
[115,]        150 0.00960481601  0.530644148            
[116,]        151 0.00949746401  0.540141612            
[117,]        152 0.00938684459  0.549528456            
[118,]        153 0.00927323608  0.558801692            
[119,]        154 0.00915690833  0.567958601            
[120,]        155 0.00903812243  0.576996723            
[121,]        156 0.00891713057  0.585913854            
[122,]        157 0.00879417581  0.594708029            
[123,]        158 0.00866949209  0.603377522            
[124,]        159 0.00854330407  0.611920826            
[125,]        160 0.00841582721  0.620336653

Quote: Wizard

If the question is how long will it take for every number to appear in double-zero roulette, the answer is 160.6602765, on average.
This is the sum of the inverse of every integer from 1 to 38.

Quote: Wizard

You're right. I forgot to say to multiply by 38.

Quote: mustangsally

the OP questions 37. Where it came from

that is just 1/p where p=1/37

handy tables
0 Roulette (155.4586903)

# of numbers	average # of spins	cumulative sum
1	1	1
2	1.027777778	2.027777778
3	1.057142857	3.084920635
4	1.088235294	4.173155929
5	1.121212121	5.29436805
6	1.15625	6.45061805
7	1.193548387	7.644166437
8	1.233333333	8.877499771
9	1.275862069	10.15336184
10	1.321428571	11.47479041
11	1.37037037	12.84516078
12	1.423076923	14.2682377
13	1.48	15.7482377
14	1.541666667	17.28990437
15	1.608695652	18.89860002
16	1.681818182	20.58041821
17	1.761904762	22.34232297
18	1.85	24.19232297
19	1.947368421	26.13969139
20	2.055555556	28.19524694
21	2.176470588	30.37171753
22	2.3125	32.68421753
23	2.466666667	35.1508842
24	2.642857143	37.79374134
25	2.846153846	40.63989519
26	3.083333333	43.72322852
27	3.363636364	47.08686488
28	3.7	50.78686488
29	4.111111111	54.897976
30	4.625	59.522976
31	5.285714286	64.80869028
32	6.166666667	70.97535695
33	7.4	78.37535695
34	9.25	87.62535695
35	12.33333333	99.95869028
36	18.5	118.4586903
37	37	155.4586903

The average is not the mode (The "mode" is the value that occurs most often)
or median (The "median" is the "middle" value or close to 50%)
median = spin 147 @ 0.501522154
mode = 133 @ 0.0106293156

00 Roulette (160.6602765)

# of numbers	average # of spins	cumulative sum
1	1	1
2	1.027027027	2.027027027
3	1.055555556	3.082582583
4	1.085714286	4.168296868
5	1.117647059	5.285943927
6	1.151515152	6.437459079
7	1.1875	7.624959079
8	1.225806452	8.85076553
9	1.266666667	10.1174322
10	1.310344828	11.42777702
11	1.357142857	12.78491988
12	1.407407407	14.19232729
13	1.461538462	15.65386575
14	1.52	17.17386575
15	1.583333333	18.75719908
16	1.652173913	20.409373
17	1.727272727	22.13664572
18	1.80952381	23.94616953
19	1.9	25.84616953
20	2	27.84616953
21	2.111111111	29.95728064
22	2.235294118	32.19257476
23	2.375	34.56757476
24	2.533333333	37.1009081
25	2.714285714	39.81519381
26	2.923076923	42.73827073
27	3.166666667	45.9049374
28	3.454545455	49.35948285
29	3.8	53.15948285
30	4.222222222	57.38170508
31	4.75	62.13170508
32	5.428571429	67.56027651
33	6.333333333	73.89360984
34	7.6	81.49360984
35	9.5	90.99360984
36	12.66666667	103.6602765
37	19	122.6602765
38	38	160.6602765

median = spin 152 @ 0.501599171
mode = 138 @ 0.010333952

still interesting one brings up this question
Sally

Quote: masterj

One more question:
On the single 0 Roulette if we get 36 out of 37 numbers and start to bet the open number then it is exactly the same if I bet a random number?

Quote: masterj

Because on average you need 37 spins that the last open number shows up.

Quote: masterj

I know that this open number can not show up in the next x spins

Quote: masterj

but it should show up very often on average 155 spins. Right or wrong?

gp > p=1/37;
gp > q=1-p;
gp > sum(k=1,155,a=q^(k-1.)*p)
%3 = 0.98569063411562596101969728657385450278
gp > 1-%3
%4 = 0.014309365884374038980302713426145497223
gp > 1/%4
%5 = 69.884298722979005501155771181951345195

Quote: masterj

Hello Johnny,

you mean after 36 numbers showed up there is still a chance to have a 1 in 70 chance that this last number is still sleeping after 155 spins more? This count starts at 0, after the 36 different numbers occured?

masterj

Quote: Keyser

In other words, simply count the number of pockets on the wheel and realize that the ball can land in any one of them.

Quote: masterj

I agree!
But if 1000 Players start after 36 numbers showed up (average after 118 spins) then the biggest part of this group should hit the last open number at around 155 spins. Is this assumtion right or wrong?

Quote: masterj

Let's say 1000 Players start immediately, then the biggest part of these group will get 36 different numbers around 118 spins and 37 different numbers around 155 spins. Right or wrong?

not drawn       Probability            cumulative 
u=32                  5 5.792644277504847e-05  6.034500425211893e-05 
u=33                  4 0.0009855780328734401  0.001045923037125559  
u=34                  3 0.01121913540428637    0.01226505844141193   
u=35                  2 0.08080438752582932    0.09306944596724125   
u=36                  1 0.3299338309788963     0.4230032769461376    
u=37                  0 0.5769967230538622                     1    

Quote: Keyser

In other words, simply count the number of pockets on the wheel and realize that the ball can land in any one of them.

Quote: masterj

Hello 7 Craps,
it takes only on average 15 spins to get the last open number? I don't understand this.

Quote:

If this is true, then we should wait until 36 numbers show up, bet 15x the open number.

Quote:

The average win ist higher then the loss of 15 bets. So we are profitable.

But I can not believe that this is correct.

Quote: EvenBob

Maybe it will. I've seen the ball fly
out of the wheel many times and
end up on the floor. No bet for
that, though.