RIP Alan Krigman
Remember the immortal words of Sumner A Ingmark,the poet laureate of the casino scene:
Those great and gaudy gambling halls
Will treat you like you're rich and famous.
Enjoy that million dollar feel,
Or miss it. Be an ignoramus!
from Alan K
Casinos generate their earnings by nibbling off small percentages of the money players wager. Say, for instance, you're a roulette buff and always bet $5 on each of three rows – $15 total per spin. Edge, assuming a double-zero game, is 5.26 percent; the bosses therefore figure your action per spin is worth 5.26 percent of $15 – just under $0.79. During a three-hour session on a certain visit, you'd get about 120 spins, making you a $0.79 x 120 or $95 customer.
The $0.79 and $95 are "theoretical" – averages over large numbers of bets made by hosts of players, each of whom wins or loses some amount.
Individuals really only care about the results of their rounds and sessions.
You don't notice the $0.79 and wouldn't gamble if you always lost $95 in three hours.
In practice, after this much time, your chances are 50 percent of losing more than $95 and 50 percent of losing less, including 37 percent of winning.
This, because edge isn't the only factor affecting particular players' outcomes. Volatility is the other major determinant.
Edge goes one-way.
From bettors to bosses.
On the average, for every spin with $15 at risk on double-zero roulette, players pay the house $0.79.
And those $0.79 increments mount linearly.
After 25 spins, the expected total is 25 x $0.79 or $19.75; after 100 spins, the casino figures 100 x $0.79 or $79; and after 1,000,000 spins, the joint estimates 1,000,000 x $0.79 or $790,000.
Volatility is bidirectional.
When nothing hits in any of the three $5 rows, your bankroll shrinks by $15.
On a hit, you're paid $55 for the row in question minus $10 for the two misses, raising your fortune by $45. For one spin, the characteristic bankroll jump under these conditions (math wonks call it "standard deviation") is $25.50 up or down. For multiple spins, one standard deviation defines the upper and lower ends of a range covering 68 percent of all net bankroll changes.
And it's balanced so half the remaining 32 percent is below and half above this range.
As with edge, the amount represented by volatility gets bigger as you make more bets.
But because bankroll jumps may be up or down, there's some cancellation and the cumulative value is less than simply the number of spins times $25.50.
Standard deviation accordingly grows more slowly than does edge.
After 25 spins, it doesn't reach 25 x $25.50 or $637, but 5 x $25.50 or $128; after 100 spins, it's not up 100-fold to $2,550 but to 10-fold to $255; and after 1,000,000 spins, the value isn't 1,000,000 x $25.50 or $25,500,000 but 1,000 x $25.50 or $25,500.
The higher rate at which edge increases relative to volatility explains why casinos depend on long-term averages while solid citizens pin their hopes on the short run.
For one spin, the effect of volatility is much greater than that of edge.
Minus $15 or plus $45 swamp the $0.79 penalty.
After 1,000,000 spins, edge dominates – $790,000 down as opposed to a standard deviation of $25,500 up or down.
At some number of spins, the two are equal.
For the $15 per spin double-zero roulette example, the crossover is 1,044 spins – 26 hours assuming 40 spins per hour.
At this point, edge represents an average loss of $824.21, and one standard deviation a swing of $824.21 above or below this average.
This means players would have 16 percent chance of being more than twice $824.21 in the hole and an equal 16 percent chance of having risen more than $824.21 above the average to find themselves with a profit.
In case you're wondering,
you needn't play 1044 spins for edge to gain enough on volatility to negatively influence your chances of earning a profit.
It's a matter of degree.
The accompanying table shows the effect for sessions of 40, 80, and 120 spins betting as in the example.
Edge rises faster than volatility with extended play
spins, loss due to edge, standard deviation, probability of being even or ahead
40, -$32, $161 , 42%
80 , -$63, $228, 39%
120 , $95, $279 , 37%
The casino views all of this oppositely.
The bosses want the most action possible, over lots of patrons and long periods, so the effect of volatility will be insignificant compared to that of edge.
After 5,622 spins, the chance of players as a group being ahead is only 1 percent.
After 9,950 spins, it's a mere 0.1 percent. After a million spins, the casino has statistically expected earnings due to edge of $790,000 with 99 percent probability of being between $715,000 and $865,000, and virtually no chance of having lost to the hoi polloi..
If you get behind in a game, you may chase your losses and keep playing.
Or, if you get ahead, you may think you can go on forever.
It may work.
You may get lucky, hit a hot streak, and strike it rich.
But the laws of the universe, as applied to edge and volatility, say that the longer you play, the less your chances of success.
The punter's poet, Sumner A Ingmark, put it like this:
Thank You again Alan
some Craps tables
edit begins (not completed)10/25/2013
A) length of a shooter's hand in Craps Table
(probability of rolling a 7 on any roll = 6/36)
examples:
1) what is the probability of any shooter to not get past the 6th roll?
in other words, to 7out by the 6th roll (includes the 6th roill)
0.50278912957882
2) to 7out on the 3rd roll?
(the highest probability of any one roll to 7out on. roll #2 is second)
0.116769547
3) to Survive the 30th roll or the 50th?
30: 0.014315113 or 1 in 69.86
50: 0.000742575 or 1 in 1,346.67
4) get past the 153rd roll? (to Survive the 153rd roll)
1 in 5,590,264,071.83
5) to 7out on exactly your 154th roll?
1 in 40,648,590,642.69
Roll | Survival by | 1 in Survival | 7out by | Roll | 7out on | 1 in 7out on |
---|---|---|---|---|---|---|
1 | 1 | 1.00 | 0 | 1 | 0 | 0.00 |
2 | 0.888888889 | 1.13 | 0.11111111111111 | 2 | 0.111111111 | 9.00 |
3 | 0.772119342 | 1.30 | 0.22788065843621 | 3 | 0.116769547 | 8.56 |
4 | 0.667352538 | 1.50 | 0.33264746227709 | 4 | 0.104766804 | 9.55 |
5 | 0.576128909 | 1.74 | 0.42387109117005 | 5 | 0.091223629 | 10.96 |
6 | 0.49721087 | 2.01 | 0.50278912957882 | 6 | 0.078918038 | 12.67 |
7 | 0.429044107 | 2.33 | 0.57095589337479 | 7 | 0.068166764 | 14.67 |
8 | 0.370191349 | 2.70 | 0.62980865145883 | 8 | 0.058852758 | 16.99 |
9 | 0.319390699 | 3.13 | 0.68060930134840 | 9 | 0.05080065 | 19.68 |
10 | 0.275546562 | 3.63 | 0.72445343801271 | 10 | 0.043844137 | 22.81 |
11 | 0.237710426 | 4.21 | 0.76228957403871 | 11 | 0.037836136 | 26.43 |
12 | 0.205061925 | 4.88 | 0.79493807470694 | 12 | 0.032648501 | 30.63 |
13 | 0.176891903 | 5.65 | 0.82310809653915 | 13 | 0.028170022 | 35.50 |
14 | 0.152587569 | 6.55 | 0.84741243116016 | 14 | 0.024304335 | 41.14 |
15 | 0.13161956 | 7.60 | 0.86838043965154 | 15 | 0.020968008 | 47.69 |
16 | 0.113530703 | 8.81 | 0.88646929664857 | 16 | 0.018088857 | 55.28 |
17 | 0.097926249 | 10.21 | 0.90207375103577 | 17 | 0.015604454 | 64.08 |
18 | 0.08446541 | 11.84 | 0.91553458977339 | 18 | 0.013460839 | 74.29 |
19 | 0.072854029 | 13.73 | 0.92714597073443 | 19 | 0.011611381 | 86.12 |
20 | 0.062838229 | 15.91 | 0.93716177107751 | 20 | 0.0100158 | 99.84 |
21 | 0.05419892 | 18.45 | 0.94580108000490 | 21 | 0.008639309 | 115.75 |
22 | 0.046747051 | 21.39 | 0.95325294882548 | 22 | 0.007451869 | 134.19 |
23 | 0.040319503 | 24.80 | 0.95968049700960 | 23 | 0.006427548 | 155.58 |
24 | 0.03477554 | 28.76 | 0.96522446029318 | 24 | 0.005543963 | 180.38 |
25 | 0.029993744 | 33.34 | 0.97000625573677 | 25 | 0.004781795 | 209.13 |
26 | 0.025869371 | 38.66 | 0.97413062884098 | 26 | 0.004124373 | 242.46 |
27 | 0.022312061 | 44.82 | 0.97768793922773 | 27 | 0.00355731 | 281.11 |
28 | 0.019243866 | 51.96 | 0.98075613388823 | 28 | 0.003068195 | 325.92 |
29 | 0.01659755 | 60.25 | 0.98340245045075 | 29 | 0.002646317 | 377.88 |
30 | 0.014315113 | 69.86 | 0.98568488722231 | 30 | 0.002282437 | 438.13 |
31 | 0.012346528 | 80.99 | 0.98765347180503 | 31 | 0.001968585 | 507.98 |
32 | 0.010648644 | 93.91 | 0.98935135578848 | 32 | 0.001697884 | 588.97 |
33 | 0.009184241 | 108.88 | 0.99081575929113 | 33 | 0.001464404 | 682.87 |
34 | 0.007921214 | 126.24 | 0.99207878589382 | 34 | 0.001263027 | 791.75 |
35 | 0.006831874 | 146.37 | 0.99316812571178 | 35 | 0.00108934 | 917.99 |
36 | 0.005892338 | 169.71 | 0.99410766193198 | 36 | 0.000939536 | 1,064.35 |
37 | 0.005082006 | 196.77 | 0.99491799405020 | 37 | 0.000810332 | 1,234.06 |
38 | 0.004383111 | 228.15 | 0.99561688923312 | 38 | 0.000698895 | 1,430.83 |
39 | 0.003780328 | 264.53 | 0.99621967166785 | 39 | 0.000602782 | 1,658.97 |
40 | 0.003260442 | 306.71 | 0.99673955841081 | 40 | 0.000519887 | 1,923.50 |
41 | 0.002812051 | 355.61 | 0.99718794908172 | 41 | 0.000448391 | 2,230.20 |
42 | 0.002425324 | 412.32 | 0.99757467574136 | 42 | 0.000386727 | 2,585.81 |
43 | 0.002091782 | 478.06 | 0.99790821842246 | 43 | 0.000333543 | 2,998.12 |
44 | 0.001804109 | 554.29 | 0.99819589103257 | 44 | 0.000287673 | 3,476.17 |
45 | 0.001555998 | 642.67 | 0.99844400169998 | 45 | 0.000248111 | 4,030.46 |
46 | 0.001342009 | 745.15 | 0.99865799107506 | 46 | 0.000213989 | 4,673.13 |
47 | 0.001157448 | 863.97 | 0.99884255161671 | 47 | 0.000184561 | 5,418.28 |
48 | 0.00099827 | 1,001.73 | 0.99900173047753 | 48 | 0.000159179 | 6,282.24 |
49 | 0.000860982 | 1,161.46 | 0.99913901824259 | 49 | 0.000137288 | 7,283.97 |
50 | 0.000742575 | 1,346.67 | 0.99925742546618 | 50 | 0.000118407 | 8,445.43 |
51 | 0.000640451 | 1,561.40 | 0.99935954868441 | 51 | 0.000102123 | 9,792.09 |
52 | 0.000552373 | 1,810.37 | 0.99944762735040 | 52 | 8.80787E-05 | 11,353.49 |
53 | 0.000476407 | 2,099.05 | 0.99952359294022 | 53 | 7.59656E-05 | 13,163.85 |
54 | 0.000410889 | 2,433.75 | 0.99958911130615 | 54 | 6.55184E-05 | 15,262.90 |
55 | 0.000354381 | 2,821.82 | 0.99964561920559 | 55 | 5.65079E-05 | 17,696.64 |
56 | 0.000305644 | 3,271.78 | 0.99969435580674 | 56 | 4.87366E-05 | 20,518.46 |
57 | 0.00026361 | 3,793.48 | 0.99973638986160 | 57 | 4.20341E-05 | 23,790.23 |
58 | 0.000227357 | 4,398.37 | 0.99977264314214 | 58 | 3.62533E-05 | 27,583.71 |
59 | 0.000196089 | 5,099.72 | 0.99980391065356 | 59 | 3.12675E-05 | 31,982.08 |
60 | 0.000169122 | 5,912.89 | 0.99983087806775 | 60 | 2.69674E-05 | 37,081.79 |
61 | 0.000145863 | 6,855.74 | 0.99985413675932 | 61 | 2.32587E-05 | 42,994.68 |
62 | 0.000125803 | 7,948.92 | 0.99987419677374 | 62 | 2.006E-05 | 49,850.41 |
63 | 0.000108502 | 9,216.42 | 0.99989149801220 | 63 | 1.73012E-05 | 57,799.33 |
64 | 9.35801E-05 | 10,686.03 | 0.99990641987813 | 64 | 1.49219E-05 | 67,015.75 |
65 | 8.07104E-05 | 12,389.98 | 0.99991928959722 | 65 | 1.28697E-05 | 77,701.77 |
66 | 6.96106E-05 | 14,365.63 | 0.99993038939317 | 66 | 1.10998E-05 | 90,091.75 |
67 | 6.00373E-05 | 16,656.31 | 0.99993996267658 | 67 | 9.57328E-06 | 104,457.37 |
68 | 5.17806E-05 | 19,312.25 | 0.99994821938281 | 68 | 8.25671E-06 | 121,113.67 |
69 | 4.46594E-05 | 22,391.69 | 0.99995534057563 | 69 | 7.12119E-06 | 140,425.91 |
70 | 3.85176E-05 | 25,962.17 | 0.99996148241783 | 70 | 6.14184E-06 | 162,817.60 |
71 | 3.32204E-05 | 30,101.98 | 0.99996677959575 | 71 | 5.29718E-06 | 188,779.76 |
72 | 2.86517E-05 | 34,901.91 | 0.99997134827283 | 72 | 4.56868E-06 | 218,881.74 |
73 | 2.47114E-05 | 40,467.21 | 0.99997528863701 | 73 | 3.94036E-06 | 253,783.65 |
74 | 2.13129E-05 | 46,919.94 | 0.99997868709779 | 74 | 3.39846E-06 | 294,250.86 |
75 | 1.83818E-05 | 54,401.58 | 0.99998161818107 | 75 | 2.93108E-06 | 341,170.79 |
76 | 1.58538E-05 | 63,076.22 | 0.99998414616352 | 76 | 2.52798E-06 | 395,572.37 |
77 | 1.36735E-05 | 73,134.07 | 0.99998632648207 | 77 | 2.18032E-06 | 458,648.58 |
78 | 1.17931E-05 | 84,795.70 | 0.99998820694961 | 78 | 1.88047E-06 | 531,782.64 |
79 | 1.01712E-05 | 98,316.85 | 0.99998982880353 | 79 | 1.62185E-06 | 616,578.34 |
80 | 8.77239E-06 | 113,994.02 | 0.99999122760997 | 80 | 1.39881E-06 | 714,895.19 |
81 | 7.56596E-06 | 132,171.00 | 0.99999243404383 | 81 | 1.20643E-06 | 828,889.20 |
82 | 6.52544E-06 | 153,246.40 | 0.99999347456137 | 82 | 1.04052E-06 | 961,060.20 |
83 | 5.62802E-06 | 177,682.40 | 0.99999437198045 | 83 | 8.97419E-07 | 1,114,306.60 |
84 | 4.85402E-06 | 206,014.84 | 0.99999514598085 | 84 | 7.74E-07 | 1,291,988.99 |
85 | 4.18646E-06 | 238,865.06 | 0.99999581353588 | 85 | 6.67555E-07 | 1,498,003.84 |
86 | 3.61072E-06 | 276,953.42 | 0.99999638928458 | 86 | 5.75749E-07 | 1,736,868.89 |
87 | 3.11415E-06 | 321,115.19 | 0.99999688585272 | 87 | 4.96568E-07 | 2,013,822.31 |
88 | 2.68587E-06 | 372,318.80 | 0.99999731412971 | 88 | 4.28277E-07 | 2,334,937.50 |
89 | 2.31649E-06 | 431,687.11 | 0.99999768350737 | 89 | 3.69378E-07 | 2,707,256.29 |
90 | 1.99791E-06 | 500,522.02 | 0.99999800208591 | 90 | 3.18579E-07 | 3,138,943.40 |
91 | 1.72315E-06 | 580,333.05 | 0.99999827685154 | 91 | 2.74766E-07 | 3,639,465.41 |
92 | 1.48617E-06 | 672,870.39 | 0.99999851382968 | 92 | 2.36978E-07 | 4,219,798.46 |
93 | 1.28178E-06 | 780,163.32 | 0.99999871821711 | 93 | 2.04387E-07 | 4,892,668.84 |
94 | 1.1055E-06 | 904,564.71 | 0.99999889449590 | 94 | 1.76279E-07 | 5,672,832.16 |
95 | 9.53468E-07 | 1,048,802.60 | 0.99999904653173 | 95 | 1.52036E-07 | 6,577,396.87 |
96 | 8.22341E-07 | 1,216,040.02 | 0.99999917765864 | 96 | 1.31127E-07 | 7,626,199.47 |
97 | 7.09248E-07 | 1,409,944.38 | 0.99999929075217 | 97 | 1.13094E-07 | 8,842,239.48 |
98 | 6.11708E-07 | 1,634,767.87 | 0.99999938829236 | 98 | 9.75402E-08 | 10,252,183.86 |
99 | 5.27582E-07 | 1,895,440.72 | 0.99999947241822 | 99 | 8.41259E-08 | 11,886,951.72 |
100 | 4.55025E-07 | 2,197,679.31 | 0.99999954497456 | 100 | 7.25563E-08 | 13,782,392.46 |
101 | 3.92447E-07 | 2,548,111.53 | 0.99999960755250 | 101 | 6.25779E-08 | 15,980,071.74 |
102 | 3.38476E-07 | 2,954,422.12 | 0.99999966152433 | 102 | 5.39718E-08 | 18,528,183.29 |
103 | 2.91926E-07 | 3,425,521.20 | 0.99999970807362 | 103 | 4.65493E-08 | 21,482,605.41 |
104 | 2.51779E-07 | 3,971,739.66 | 0.99999974822116 | 104 | 4.01475E-08 | 24,908,126.61 |
105 | 2.17153E-07 | 4,605,055.69 | 0.99999978284736 | 105 | 3.46262E-08 | 28,879,866.26 |
106 | 1.87288E-07 | 5,339,357.49 | 0.99999981271155 | 106 | 2.98642E-08 | 33,484,921.92 |
107 | 1.61531E-07 | 6,190,747.81 | 0.99999983846863 | 107 | 2.57571E-08 | 38,824,279.47 |
108 | 1.39317E-07 | 7,177,897.07 | 0.99999986068343 | 108 | 2.22148E-08 | 45,015,027.15 |
109 | 1.20157E-07 | 8,322,452.77 | 0.99999987984312 | 109 | 1.91597E-08 | 52,192,924.42 |
110 | 1.03632E-07 | 9,649,514.27 | 0.99999989636784 | 110 | 1.65247E-08 | 60,515,377.07 |
111 | 8.938E-08 | 11,188,183.12 | 0.99999991061998 | 111 | 1.42521E-08 | 70,164,891.60 |
112 | 7.70879E-08 | 12,972,201.30 | 0.99999992291208 | 112 | 1.22921E-08 | 81,353,074.42 |
113 | 6.64863E-08 | 15,040,691.12 | 0.99999993351369 | 113 | 1.06016E-08 | 94,325,275.81 |
114 | 5.73427E-08 | 17,439,013.17 | 0.99999994265731 | 114 | 9.14361E-09 | 109,365,966.74 |
115 | 4.94566E-08 | 20,219,761.05 | 0.99999995054343 | 115 | 7.88613E-09 | 126,804,979.46 |
116 | 4.2655E-08 | 23,443,914.68 | 0.99999995734501 | 116 | 6.80158E-09 | 147,024,741.85 |
117 | 3.67888E-08 | 27,182,177.59 | 0.99999996321119 | 117 | 5.86618E-09 | 170,468,654.74 |
118 | 3.17294E-08 | 31,516,527.36 | 0.99999996827062 | 118 | 5.05943E-09 | 197,650,832.47 |
119 | 2.73658E-08 | 36,542,013.36 | 0.99999997263424 | 119 | 4.36362E-09 | 229,167,361.69 |
120 | 2.36023E-08 | 42,368,841.13 | 0.99999997639775 | 120 | 3.76351E-09 | 265,709,376.56 |
121 | 2.03563E-08 | 49,124,789.07 | 0.99999997964368 | 121 | 3.24593E-09 | 308,078,214.04 |
122 | 1.75568E-08 | 56,958,010.57 | 0.99999998244321 | 122 | 2.79953E-09 | 357,203,007.84 |
123 | 1.51423E-08 | 66,040,282.92 | 0.99999998485773 | 123 | 2.41452E-09 | 414,161,006.84 |
124 | 1.30598E-08 | 76,570,774.24 | 0.99999998694019 | 124 | 2.08246E-09 | 480,201,304.48 |
125 | 1.12637E-08 | 88,780,411.10 | 0.99999998873625 | 125 | 1.79607E-09 | 556,772,080.02 |
126 | 9.71469E-09 | 102,936,942.64 | 0.99999999028532 | 126 | 1.54906E-09 | 645,552,466.57 |
127 | 8.37866E-09 | 119,350,812.05 | 0.99999999162134 | 127 | 1.33602E-09 | 748,489,447.15 |
128 | 7.22638E-09 | 138,381,964.47 | 0.99999999277363 | 128 | 1.15229E-09 | 867,840,236.02 |
129 | 6.23256E-09 | 160,447,740.23 | 0.99999999376744 | 129 | 9.93816E-10 | 1,006,222,112.75 |
130 | 5.37542E-09 | 186,032,026.97 | 0.99999999462458 | 130 | 8.5714E-10 | 1,166,670,067.35 |
131 | 4.63616E-09 | 215,695,870.88 | 0.99999999536384 | 131 | 7.39261E-10 | 1,352,701,845.36 |
132 | 3.99856E-09 | 250,089,780.08 | 0.99999999600144 | 132 | 6.37593E-10 | 1,568,397,883.09 |
133 | 3.44866E-09 | 289,967,989.87 | 0.99999999655134 | 133 | 5.49908E-10 | 1,818,487,443.33 |
134 | 2.97438E-09 | 336,205,002.54 | 0.99999999702562 | 134 | 4.74281E-10 | 2,108,455,911.43 |
135 | 2.56532E-09 | 389,814,764.67 | 0.99999999743468 | 135 | 4.09055E-10 | 2,444,660,285.98 |
136 | 2.21252E-09 | 451,972,902.27 | 0.99999999778748 | 136 | 3.52799E-10 | 2,834,475,948.16 |
137 | 1.90824E-09 | 524,042,501.47 | 0.99999999809176 | 137 | 3.0428E-10 | 3,286,447,400.40 |
138 | 1.64581E-09 | 607,604,000.07 | 0.99999999835419 | 138 | 2.62433E-10 | 3,810,492,076.38 |
139 | 1.41947E-09 | 704,489,845.52 | 0.99999999858053 | 139 | 2.26342E-10 | 4,418,094,044.28 |
140 | 1.22425E-09 | 816,824,679.21 | 0.99999999877575 | 140 | 1.95214E-10 | 5,122,584,572.95 |
141 | 1.05589E-09 | 947,071,928.43 | 0.99999999894411 | 141 | 1.68367E-10 | 5,939,410,552.58 |
142 | 9.10674E-10 | 1,098,087,827.73 | 0.99999999908933 | 142 | 1.45212E-10 | 6,886,480,147.42 |
143 | 7.85432E-10 | 1,273,184,054.14 | 0.99999999921457 | 143 | 1.25242E-10 | 7,984,567,754.96 |
144 | 6.77415E-10 | 1,476,200,350.09 | 0.99999999932259 | 144 | 1.08018E-10 | 9,257,751,028.58 |
145 | 5.84253E-10 | 1,711,588,726.32 | 0.99999999941575 | 145 | 9.31623E-11 | 10,733,960,793.66 |
146 | 5.03902E-10 | 1,984,511,091.53 | 0.99999999949610 | 146 | 8.03501E-11 | 12,445,541,431.59 |
147 | 4.34603E-10 | 2,300,952,449.52 | 0.99999999956540 | 147 | 6.92998E-11 | 14,430,058,546.81 |
148 | 3.74833E-10 | 2,667,852,146.33 | 0.99999999962517 | 148 | 5.97693E-11 | 16,730,997,177.96 |
149 | 3.23284E-10 | 3,093,256,045.41 | 0.99999999967672 | 149 | 5.15494E-11 | 19,398,856,069.45 |
150 | 2.78824E-10 | 3,586,492,968.00 | 0.99999999972118 | 150 | 4.446E-11 | 22,492,132,184.84 |
151 | 2.40478E-10 | 4,158,379,267.89 | 0.99999999975952 | 151 | 3.83457E-11 | 26,078,570,573.71 |
152 | 2.07406E-10 | 4,821,456,026.79 | 0.99999999979259 | 152 | 3.30721E-11 | 30,236,966,550.21 |
153 | 1.78882E-10 | 5,590,264,071.83 | 0.99999999982112 | 153 | 2.85237E-11 | 35,058,517,488.94 |
154 | 1.54281E-10 | 6,481,662,846.05 | 0.99999999984572 | 154 | 2.46011E-11 | 40,648,590,642.69 |
155 | 1.33064E-10 | 7,515,200,124.74 | 0.99999999986694 | 155 | 2.12177E-11 | 47,130,474,563.30 |
156 | 1.14764E-10 | 8,713,540,684.89 | 0.99999999988524 | 156 | 1.82998E-11 | 54,645,387,700.91 |
157 | 9.89809E-11 | 10,102,963,328.60 | 0.99999999990102 | 157 | 1.5783E-11 | 63,359,143,891.37 |
158 | 8.53684E-11 | 11,713,937,159.43 | 0.99999999991463 | 158 | 1.36124E-11 | 73,462,191,132.38 |
159 | 7.3628E-11 | 13,581,789,749.42 | 0.99999999992637 | 159 | 1.17404E-11 | 85,176,071,932.72 |
160 | 6.35022E-11 | 15,747,481,848.92 | 0.99999999993650 | 160 | 1.01258E-11 | 98,757,735,373.51 |
161 | 5.4769E-11 | 18,258,505,628.28 | 0.99999999994523 | 161 | 8.73324E-12 | 114,505,088,285.84 |
162 | 4.72368E-11 | 21,169,926,149.24 | 0.99999999995276 | 162 | 7.5322E-12 | 132,763,387,399.64 |
163 | 4.07405E-11 | 24,545,588,904.60 | 0.99999999995926 | 163 | 6.49625E-12 | 153,935,010,249.71 |
164 | 3.51376E-11 | 28,459,519,906.98 | 0.99999999996486 | 164 | 5.60285E-12 | 178,480,546,402.35 |
165 | 3.03053E-11 | 32,997,549,029.44 | 0.99999999996970 | 165 | 4.83236E-12 | 206,938,364,534.78 |
166 | 2.61375E-11 | 38,259,192,196.81 | 0.99999999997386 | 166 | 4.16778E-12 | 239,936,048,341.53 |
167 | 2.25429E-11 | 44,359,833,703.00 | 0.99999999997746 | 167 | 3.59457E-12 | 278,197,462,851.44 |
168 | 1.94427E-11 | 51,433,256,510.88 | 0.99999999998056 | 168 | 3.1003E-12 | 322,549,659,972.82 |
169 | 1.67688E-11 | 59,634,576,022.67 | 0.99999999998323 | 169 | 2.67386E-12 | 373,991,000,445.98 |
170 | 1.44626E-11 | 69,143,641,656.27 | 0.99999999998554 | 170 | 2.30616E-12 | 433,622,147,830.78 |
171 | 1.24737E-11 | 80,168,980,822.01 | 0.99999999998753 | 171 | 1.98896E-12 | 502,774,169,954.84 |
172 | 1.07582E-11 | 92,952,371,788.45 | 0.99999999998924 | 172 | 1.71552E-12 | 582,914,784,800.74 |
173 | 9.27866E-12 | 107,774,145,717.03 | 0.99999999999072 | 173 | 1.47948E-12 | 675,911,695,538.12 |
174 | 8.0026E-12 | 124,959,334,135.89 | 0.99999999999200 | 174 | 1.27609E-12 | 783,643,575,321.12 |
175 | 6.90204E-12 | 144,884,796,662.48 | 0.99999999999310 | 175 | 1.10056E-12 | 908,624,962,649.15 |
176 | 5.95282E-12 | 167,987,485,281.42 | 0.99999999999405 | 176 | 9.49241E-13 | 1,053,473,597,045.73 |
177 | 5.13415E-12 | 194,774,026,407.44 | 0.99999999999487 | 177 | 8.18678E-13 | 1,221,480,777,697.45 |
178 | 4.42807E-12 | 225,831,830,861.77 | 0.99999999999557 | 178 | 7.05991E-13 | 1,416,449,010,023.74 |
179 | 3.8191E-12 | 261,841,975,396.10 | 0.99999999999618 | 179 | 6.09068E-13 | 1,641,851,851,028.25 |
180 | 3.29387E-12 | 303,594,138,247.50 | 0.99999999999671 | 180 | 5.25135E-13 | 1,904,270,455,547.78 |
181 | 2.84088E-12 | 352,003,916,250.68 | 0.99999999999716 | 181 | 4.53082E-13 | 2,207,105,918,828.96 |
182 | 2.45018E-12 | 408,132,903,260.49 | 0.99999999999755 | 182 | 3.90687E-13 | 2,559,590,581,057.40 |
183 | 2.11322E-12 | 473,211,970,190.72 | 0.99999999999789 | 183 | 3.36953E-13 | 2,967,775,701,726.85 |
184 | 1.82259E-12 | 548,668,257,184.99 | 0.99999999999818 | 184 | 2.90656E-13 | 3,440,488,638,174.56 |
185 | 1.57194E-12 | 636,156,469,839.69 | 0.99999999999843 | 185 | 2.50577E-13 | 3,990,783,896,650.86 |
186 | 1.35576E-12 | 737,595,165,784.21 | 0.99999999999864 | 186 | 2.1616E-13 | 4,626,193,762,065.22 |
187 | 1.16931E-12 | 855,208,827,358.68 | 0.99999999999883 | 187 | 1.86517E-13 | 5,361,428,127,822.02 |
188 | 1.00849E-12 | 991,576,643,014.74 | 0.99999999999899 | 188 | 1.6076E-13 | 6,220,441,474,268.64 |
189 | 8.698E-13 | 1,149,689,067,182.67 | 0.99999999999913 | 189 | 1.38778E-13 | 7,205,759,403,792.79 |
190 | 7.5018E-13 | 1,333,013,398,924.63 | 0.99999999999925 | 190 | 1.19571E-13 | 8,363,230,505,794.79 |
191 | 6.47011E-13 | 1,545,569,817,469.84 | 0.99999999999935 | 191 | 1.0314E-13 | 9,695,585,850,097.95 |
192 | 5.5803E-13 | 1,792,019,542,039.74 | 0.99999999999944 | 192 | 8.90399E-14 | 11,230,921,764,016.20 |
193 | 4.81286E-13 | 2,077,767,049,248.81 | 0.99999999999952 | 193 | 7.67164E-14 | 13,035,020,629,147.60 |
194 | 4.15096E-13 | 2,409,078,589,639.83 | 0.99999999999959 | 194 | 6.61693E-14 | 15,112,750,427,417.80 |
195 | 3.5801E-13 | 2,793,219,602,341.50 | 0.99999999999964 | 195 | 5.70655E-14 | 17,523,733,958,640.10 |
196 | 3.08774E-13 | 3,238,614,041,259.33 | 0.99999999999969 | 196 | 4.92939E-14 | 20,286,484,807,975.20 |
197 | 2.6631E-13 | 3,755,029,106,716.04 | 0.99999999999973 | 197 | 4.24105E-14 | 23,579,055,640,683.20 |
198 | 2.29685E-13 | 4,353,789,433,581.83 | 0.99999999999977 | 198 | 3.66374E-14 | 27,294,543,196,184.80 |
199 | 1.98097E-13 | 5,048,025,432,896.40 | 0.99999999999980 | 199 | 3.16414E-14 | 31,604,207,911,371.90 |
200 | 1.70854E-13 | 5,852,961,232,947.04 | 0.99999999999983 | 200 | 2.72005E-14 | 36,764,078,590,779.60 |
(can actually be done also by a Markov chain or SN Ethier's closed form formula)
Here is a table from the famous Zumma Craps system tester
(35,097 actual casino dice rolls)
4,183 shooter hands
Length | Count | Freq% | or less | count |
---|---|---|---|---|
2 | 448 | 10.71% | 10.71% | 448 |
3 | 520 | 12.43% | 23.14% | 968 |
4 | 432 | 10.33% | 33.47% | 1400 |
5 | 424 | 10.14% | 43.61% | 1824 |
6 | 333 | 7.96% | 51.57% | 2157 |
7 | 270 | 6.45% | 58.02% | 2427 |
8 | 241 | 5.76% | 63.78% | 2668 |
9 | 205 | 4.90% | 68.68% | 2873 |
10 | 184 | 4.40% | 73.08% | 3057 |
11 | 159 | 3.80% | 76.88% | 3216 |
12 | 142 | 3.39% | 80.28% | 3358 |
13 | 115 | 2.75% | 83.03% | 3473 |
14 | 98 | 2.34% | 85.37% | 3571 |
15 | 97 | 2.32% | 87.69% | 3668 |
16 | 79 | 1.89% | 89.58% | 3747 |
17 | 58 | 1.39% | 90.96% | 3805 |
18 | 44 | 1.05% | 92.02% | 3849 |
19 | 37 | 0.88% | 92.90% | 3886 |
20 | 39 | 0.93% | 93.83% | 3925 |
21 | 43 | 1.03% | 94.86% | 3968 |
22 | 24 | 0.57% | 95.43% | 3992 |
23 | 28 | 0.67% | 96.10% | 4020 |
24 | 21 | 0.50% | 96.61% | 4041 |
25 | 21 | 0.50% | 97.11% | 4062 |
26 | 15 | 0.36% | 97.47% | 4077 |
27 | 19 | 0.45% | 97.92% | 4096 |
28 | 8 | 0.19% | 98.11% | 4104 |
29 | 9 | 0.22% | 98.33% | 4113 |
30 | 9 | 0.22% | 98.54% | 4122 |
31 | 8 | 0.19% | 98.73% | 4130 |
32 | 6 | 0.14% | 98.88% | 4136 |
33 | 7 | 0.17% | 99.04% | 4143 |
34 | 6 | 0.14% | 99.19% | 4149 |
35 | 6 | 0.14% | 99.33% | 4155 |
36 | 7 | 0.17% | 99.50% | 4162 |
37 | 0 | 0.00% | 99.50% | 4162 |
38 | 6 | 0.14% | 99.64% | 4168 |
39 | 0 | 0.00% | 99.64% | 4168 |
40 | 2 | 0.05% | 99.69% | 4170 |
41 | 4 | 0.10% | 99.78% | 4174 |
42 | 1 | 0.02% | 99.81% | 4175 |
43 | 1 | 0.02% | 99.83% | 4176 |
44 | 0 | 0.00% | 99.83% | 4176 |
45 | 1 | 0.02% | 99.86% | 4177 |
46 | 2 | 0.05% | 99.90% | 4179 |
47 | 2 | 0.05% | 99.95% | 4181 |
48 | 0 | 0.00% | 99.95% | 4181 |
49 | 0 | 0.00% | 99.95% | 4181 |
50 | 1 | 0.02% | 99.98% | 4182 |
51 | 0 | 0.00% | 99.98% | 4182 |
52 | 0 | 0.00% | 99.98% | 4182 |
53 | 0 | 0.00% | 99.98% | 4182 |
54 | 0 | 0.00% | 99.98% | 4182 |
55 | 1 | 0.02% | 100.00% | 4183 |
Comments
not sure how the "one hour" thing works out. Do the percentages change with more trials?
That's interesting. So for a hour's play you actually have a better than 50% chance of leaving with the same money you came in with. Pass line is a great bet for the casual short-term player. I wonder at which point the edge kicks in and you have a >50% chance of losing.
9-15-2001
I have more information and tables coming. Just finishing a large unrelated musical project that has been slowing me down.
"I wonder at which point the edge kicks in and you have a >50% chance of losing."
I was also surprised at seeing that information when I made the table.
The answer is:
70 pass line wagers have a 50.03% chance of being even or ahead
at 72 wagers it drops to 49.89% of winning and a 50.11% chance or better of losing.
Up to 70 wagers one needs to make an even number of bets to have the chance of coming out even or ahead.
At an odd number of bets like 69 bets, one has a 45.31% chance of being ahead and an odd number of equal wagers can not end in an even outcome.
The same exists in the game of Baccarat.
Player bet is 74 hands ( about the same bets in a full shoe) at 50.09% of being even or ahead.
Player at 76 hands drops to 49.97% coming out even or ahead
Banker bet is 76 hands at 50.03% to be ahead (only by 25 cents when paying 5% commission on a banker win)
78 hands at 50.15% to be even or ahead
a Baccarat table is coming soon
I'm obviously way late to this party, but what if you place one additional bet of double odds on every passline bet when the point is established? From what I understand, the odds help the player. So if it's 50/50 with just a passline bet after an hour of play, wouldn't it be better than 50/50 if you place odds?
ZCore13
but what if you place one additional bet of double odds on every passline bet when the point is established?
From what I understand, the odds help the player.
So if it's 50/50 with just a passline bet after an hour of play,
wouldn't it be better than 50/50 if you place odds?
ZCore13
nice question
the odds bet reduces the combined house edge per $ bet
but still needs time to play out
some data (calculated)
flat bet pass $5 / $5 with $10 odds
30 wagers
prob loss: 45.859956% / 51.203797%
70 wagers
prob loss: 49.974513% / 51.731364%
100 wagers
prob loss: 51.669330% / 52.039478%
110 wagers
52.128743 / 52.132462
120 wagers
52.552143 / 52.221542
130 wagers
52.945653 / 52.307289
140 wagers
53.313939 / 52.389974
150 wagers
53.660638 / 52.469867
200 wagers
prob loss: 55.154561% / 52.835661%
500 wagers
prob loss: 60.706249% / 54.433524%
a bit more than 110 wagers does the 2X odds bettor
has a lower net losing probability
winsome johnny
1 blogwin some johnny
winsome johnny
look here
http://wizardofvegas.com/forum/gambling/craps/16534-the-math/#post311159
Comments
I didn't know Al died. That's too bad.
I always enjoyed his columns. He will be missed.
This man always made sense. Too bad he isn't required reading for every gambler. His line about "the more you play, the more you will lose" always struck me, and he's a good reason why I limit my play. Edge or not, his words are so true. People should keep that in mind before they choose to count cards or go after the +ev poker machines for a boatload of hours.