Understanding Standard Deviation

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 6:28:47 AM permalink

I just had another thought, which may or may not apply. Since 9/6 JoB is a negative expectation game, should that range result be shifted slightly towards the negative, since the mean result is negative? If so, by how much.

dwheatley

dwheatley

Threads: 25
Posts: 1246

Joined: Nov 16, 2009

January 18th, 2013 at 6:46:02 AM permalink

EDIT: I answered too quickly, this isn't right.

The width of your interval looks correct, but the centre/mean/average of the interval is the EV of the game. So you do need to shift it to the negative by the house edge.

At 99.544% Return over 250*$5 of action, the EV is 1244.3, for an expected loss of -$5.7. Shift the interval by that much

Wisdom is the quality that keeps you out of situations where you would otherwise need it

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 7:00:16 AM permalink

That makes perfect sense to me. Thanks a lot!

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

January 18th, 2013 at 8:11:45 AM permalink

Your calculations assume a normal distribution, or close to it, for VP, especially for a small # of hands played.
Ain't even close. The distribution is not normal or close to it.
(like using a normal distribution confidence interval for a gamma distribution. Just will not work.)

This can be calculated in a few programs,
I think VP for Winners does the distribution but is quite slow so I do not use it that often.

Another program I have, for an exact calculation for 250 hands, Excel can do this too with some code,
shows the probability of the interval -354 to 343 is 97.21%

I do not recall seeing confidence levels for VP before. Maybe BruceZ over at 2+2 has something.

another VP site has this quote.
it is not 100% correct (the math) most times either
The other statements IMO are correct

"By the way, I see many people using the normal distribution as a yard stick to make statements about expected results relating to standard deviation. Remember that for the number of hands you have in a session, the distribution of results is decidedly non-normal.

Instead of estimating the result as 95% confidence of being plus or minus two standard deviations (which would be true for a normal distribution),
an estimate of being within (1- 1/(# std deviation)^2) is more appropriate.
This says you should expect to be within +- two standard deviations 75% of the time,
and within +- 3 standard deviations 88.9% (not 99.7%).

Also remember that the estimate is for ending bankroll.
The estimate assumes you will keep playing through any losing streak, perhaps springing back to get inside the estimated region."

winsome johnny (not Win some johnny)

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 8:25:20 AM permalink

Well that throws a wrench in the works! I'll have to do some research on the points you brought up. Thanks for taking the time to respond.

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 8:28:59 AM permalink

Wouldn't the fact that I'm not looking at a particular set of 250 hands, and just looking at 250 hands in general average this out to be a normal distribution? If I took a sample of 1,000,000,000 sets of 250 hands, it wouldn't average 68% within the range?

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

January 18th, 2013 at 9:26:49 AM permalink

Quote: Scot
Wouldn't the fact that I'm not looking at a particular set of 250 hands, and just looking at 250 hands in general average this out to be a normal distribution?
If I took a sample of 1,000,000,000 sets of 250 hands, it wouldn't average 68% within the range?

No
It looks clear you do not understand normal distributions vs other types of non-normal distributions and there are many.

Calculations are very close to exact probability (to .000001 with rounding) of the interval you are looking for.
here is the data
end bank / prob
it sums to 99.999995%
there is some rounding and the probabilities of losing more than $635 is not included or winning more than $5000
You can check my math for me if you wish.
Or just do the calculations yourself and see.
A lot of multiplying and summing over 250 hands. That is why a program can do it much faster.
2nd column is relative prob in % the 3rd is cumulative(or less) in %
The mode looks to be a loss of $65
A net session loss is about 60% (59.958086%)

-635	0.000001	0.000001
-630	0.000001	0.000002
-625	0.000001	0.000003
-620	0.000002	0.000005
-615	0.000002	0.000007
-610	0.000003	0.00001
-605	0.000004	0.000014
-600	0.000005	0.000019
-595	0.000007	0.000026
-590	0.000009	0.000035
-585	0.000012	0.000047
-580	0.000016	0.000063
-575	0.000021	0.000084
-570	0.000028	0.000112
-565	0.000037	0.000149
-560	0.000048	0.000197
-555	0.000062	0.000259
-550	0.00008	0.000339
-545	0.000102	0.000441
-540	0.00013	0.000571
-535	0.000165	0.000736
-530	0.00021	0.000946
-525	0.000264	0.00121
-520	0.000332	0.001542
-515	0.000415	0.001957
-510	0.000517	0.002474
-505	0.000641	0.003115
-500	0.000793	0.003908
-495	0.000976	0.004884
-490	0.001197	0.006081
-485	0.001462	0.007543
-480	0.00178	0.009323
-475	0.002159	0.011482
-470	0.002608	0.01409
-465	0.00314	0.01723
-460	0.003767	0.020997
-455	0.004503	0.0255
-450	0.005364	0.030864
-445	0.006367	0.037231
-440	0.007532	0.044763
-435	0.00888	0.053643
-430	0.010434	0.064077
-425	0.01222	0.076297
-420	0.014263	0.09056
-415	0.016594	0.107154
-410	0.019244	0.126398
-405	0.022245	0.148643
-400	0.025634	0.174277
-395	0.029447	0.203724
-390	0.033723	0.237447
-385	0.038502	0.275949
-380	0.043826	0.319775
-375	0.049737	0.369512
-370	0.056278	0.42579
-365	0.063494	0.489284
-360	0.071428	0.560712
-355	0.080125	0.640837
-350	0.089627	0.730464
-345	0.099977	0.830441
-340	0.111214	0.941655
-335	0.123377	1.065032
-330	0.136502	1.201534
-325	0.150622	1.352156
-320	0.165764	1.51792
-315	0.181955	1.699875
-310	0.199215	1.89909
-305	0.217559	2.116649
-300	0.236997	2.353646
-295	0.257533	2.611179
-290	0.279165	2.890344
-285	0.301884	3.192228
-280	0.325675	3.517903
-275	0.350514	3.868417
-270	0.376373	4.24479
-265	0.403213	4.648003
-260	0.430992	5.078995
-255	0.459657	5.538652
-250	0.48915	6.027802
-245	0.519406	6.547208
-240	0.550354	7.097562
-235	0.581916	7.679478
-230	0.614007	8.293485
-225	0.64654	8.940025
-220	0.67942	9.619445
-215	0.712551	10.331996
-210	0.74583	11.077826
-205	0.779153	11.856979
-200	0.812415	12.669394
-195	0.845508	13.514902
-190	0.878322	14.393224
-185	0.910749	15.303973
-180	0.942682	16.246655
-175	0.974014	17.220669
-170	1.00464	18.225309
-165	1.034459	19.259768
-160	1.063372	20.32314
-155	1.091285	21.414425
-150	1.118108	22.532533
-145	1.143755	23.676288
-140	1.168149	24.844437
-135	1.191214	26.035651
-130	1.212884	27.248535
-125	1.233097	28.481632
-120	1.251801	29.733433
-115	1.268947	31.00238
-110	1.284497	32.286877
-105	1.298415	33.585292
-100	1.310678	34.89597
-95	1.321266	36.217236
-90	1.330167	37.547403
-85	1.337378	38.884781
-80	1.342898	40.227679
-75	1.346738	41.574417
-70	1.348911	42.923328
-65	1.349439	44.272767
-60	1.348347	45.621114
-55	1.345666	46.96678
-50	1.341435	48.308215
-45	1.335693	49.643908
-40	1.328486	50.972394
-35	1.319863	52.292257
-30	1.309878	53.602135
-25	1.298585	54.90072
-20	1.286044	56.186764
-15	1.272315	57.459079
-10	1.257461	58.71654
-5	1.241546	59.958086
0	1.224635	61.182721
5	1.206795	62.389516
10	1.188092	63.577608
15	1.168594	64.746202
20	1.148367	65.894569
25	1.127477	67.022046
30	1.10599	68.128036
35	1.083972	69.212008
40	1.061485	70.273493
45	1.038592	71.312085
50	1.015354	72.327439
55	0.991829	73.319268
60	0.968076	74.287344
65	0.944148	75.231492
70	0.9201	76.151592
75	0.895982	77.047574
80	0.871842	77.919416
85	0.847728	78.767144
90	0.823682	79.590826
95	0.799748	80.390574
100	0.775963	81.166537
105	0.752364	81.918901
110	0.728986	82.647887
115	0.70586	83.353747
120	0.683017	84.036764
125	0.660482	84.697246
130	0.638282	85.335528
135	0.616438	85.951966
140	0.594972	86.546938
145	0.573901	87.120839
150	0.553242	87.674081
155	0.533009	88.20709
160	0.513215	88.720305
165	0.49387	89.214175
170	0.474983	89.689158
175	0.456562	90.14572
180	0.438612	90.584332
185	0.421137	91.005469
190	0.40414	91.409609
195	0.387623	91.797232
200	0.371586	92.168818
205	0.356029	92.524847
210	0.340948	92.865795
215	0.326341	93.192136
220	0.312205	93.504341
225	0.298534	93.802875
230	0.285324	94.088199
235	0.272568	94.360767
240	0.260259	94.621026
245	0.24839	94.869416
250	0.236953	95.106369
255	0.225939	95.332308
260	0.215341	95.547649
265	0.205148	95.752797
270	0.195353	95.94815
275	0.185944	96.134094
280	0.176913	96.311007
285	0.168249	96.479256
290	0.159942	96.639198
295	0.151982	96.79118
300	0.14436	96.93554
305	0.137064	97.072604
310	0.130085	97.202689
315	0.123413	97.326102
320	0.117037	97.443139
325	0.110948	97.554087
330	0.105135	97.659222
335	0.099589	97.758811
340	0.0943	97.853111
345	0.089259	97.94237
350	0.084456	98.026826
355	0.079883	98.106709
360	0.075531	98.18224
365	0.07139	98.25363
370	0.067453	98.321083
375	0.06371	98.384793
380	0.060155	98.444948
385	0.056778	98.501726
390	0.053573	98.555299
395	0.050532	98.605831
400	0.047648	98.653479
405	0.044914	98.698393
410	0.042322	98.740715
415	0.039868	98.780583
420	0.037544	98.818127
425	0.035344	98.853471
430	0.033263	98.886734
435	0.031295	98.918029
440	0.029434	98.947463
445	0.027675	98.975138
450	0.026014	99.001152
455	0.024445	99.025597
460	0.022963	99.04856
465	0.021566	99.070126
470	0.020247	99.090373
475	0.019004	99.109377
480	0.017832	99.127209
485	0.016727	99.143936
490	0.015687	99.159623
495	0.014707	99.17433
500	0.013784	99.188114
505	0.012916	99.20103
510	0.0121	99.21313
515	0.011331	99.224461
520	0.010609	99.23507
525	0.00993	99.245
530	0.009293	99.254293
535	0.008694	99.262987
540	0.008131	99.271118
545	0.007603	99.278721
550	0.007107	99.285828
555	0.006642	99.29247
560	0.006206	99.298676
565	0.005797	99.304473
570	0.005414	99.309887
575	0.005055	99.314942
580	0.004718	99.31966
585	0.004403	99.324063
590	0.004108	99.328171
595	0.003832	99.332003
600	0.003573	99.335576
605	0.003332	99.338908
610	0.003105	99.342013
615	0.002894	99.344907
620	0.002696	99.347603
625	0.002511	99.350114
630	0.002339	99.352453
635	0.002178	99.354631
640	0.002027	99.356658
645	0.001886	99.358544
650	0.001755	99.360299
655	0.001633	99.361932
660	0.001519	99.363451
665	0.001412	99.364863
670	0.001313	99.366176
675	0.00122	99.367396
680	0.001134	99.36853
685	0.001053	99.369583
690	0.000978	99.370561
695	0.000909	99.37147
700	0.000844	99.372314
705	0.000783	99.373097
710	0.000727	99.373824
715	0.000675	99.374499
720	0.000626	99.375125
725	0.00058	99.375705
730	0.000538	99.376243
735	0.000499	99.376742
740	0.000463	99.377205
745	0.000429	99.377634
750	0.000397	99.378031
755	0.000368	99.378399
760	0.000341	99.37874
765	0.000316	99.379056
770	0.000293	99.379349
775	0.000271	99.37962
780	0.000251	99.379871
785	0.000232	99.380103
790	0.000215	99.380318
795	0.000199	99.380517
800	0.000184	99.380701
805	0.00017	99.380871
810	0.000157	99.381028
815	0.000145	99.381173
820	0.000134	99.381307
825	0.000124	99.381431
830	0.000115	99.381546
835	0.000106	99.381652
840	0.000098	99.38175
845	0.00009	99.38184
850	0.000083	99.381923
855	0.000077	99.382
860	0.000071	99.382071
865	0.000066	99.382137
870	0.000061	99.382198
875	0.000056	99.382254
880	0.000052	99.382306
885	0.000048	99.382354
890	0.000044	99.382398
895	0.00004	99.382438
900	0.000037	99.382475
905	0.000034	99.382509
910	0.000032	99.382541
915	0.000029	99.38257
920	0.000027	99.382597
925	0.000025	99.382622
930	0.000023	99.382645
935	0.000021	99.382666
940	0.000019	99.382685
945	0.000018	99.382703
950	0.000016	99.382719
955	0.000015	99.382734
960	0.000014	99.382748
965	0.000013	99.382761
970	0.000012	99.382773
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980	0.00001	99.382794
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990	0.000008	99.382811
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1000	0.000007	99.382826
1005	0.000007	99.382833
1010	0.000006	99.382839
1015	0.000005	99.382844
1020	0.000005	99.382849
1025	0.000005	99.382854
1030	0.000004	99.382858
1035	0.000004	99.382862
1040	0.000004	99.382866
1045	0.000003	99.382869
1050	0.000003	99.382872
1055	0.000003	99.382875
1060	0.000003	99.382878
1065	0.000002	99.38288
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1075	0.000002	99.382884
1080	0.000002	99.382886
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1090	0.000002	99.38289
1095	0.000001	99.382891
1100	0.000001	99.382892
1105	0.000001	99.382893
1110	0.000001	99.382894
1115	0.000001	99.382895
1120	0.000001	99.382896
1125	0.000001	99.382897
1130	0.000001	99.382898
1135	0.000001	99.382899
1140	0.000001	99.3829
1145	0.000001	99.382901
1150	0.000001	99.382902
3450	0.000001	99.382903
3455	0.000001	99.382904
3460	0.000001	99.382905
3465	0.000001	99.382906
3470	0.000002	99.382908
3475	0.000002	99.38291
3480	0.000002	99.382912
3485	0.000003	99.382915
3490	0.000004	99.382919
3495	0.000005	99.382924
3500	0.000006	99.38293
3505	0.000007	99.382937
3510	0.000009	99.382946
3515	0.000011	99.382957
3520	0.000013	99.38297
3525	0.000015	99.382985
3530	0.000019	99.383004
3535	0.000022	99.383026
3540	0.000027	99.383053
3545	0.000032	99.383085
3550	0.000038	99.383123
3555	0.000045	99.383168
3560	0.000053	99.383221
3565	0.000063	99.383284
3570	0.000073	99.383357
3575	0.000086	99.383443
3580	0.0001	99.383543
3585	0.000116	99.383659
3590	0.000134	99.383793
3595	0.000155	99.383948
3600	0.000178	99.384126
3605	0.000204	99.38433
3610	0.000233	99.384563
3615	0.000266	99.384829
3620	0.000302	99.385131
3625	0.000342	99.385473
3630	0.000386	99.385859
3635	0.000434	99.386293
3640	0.000488	99.386781
3645	0.000546	99.387327
3650	0.000609	99.387936
3655	0.000678	99.388614
3660	0.000753	99.389367
3665	0.000834	99.390201
3670	0.00092	99.391121
3675	0.001014	99.392135
3680	0.001113	99.393248
3685	0.00122	99.394468
3690	0.001333	99.395801
3695	0.001453	99.397254
3700	0.00158	99.398834
3705	0.001713	99.400547
3710	0.001854	99.402401
3715	0.002001	99.404402
3720	0.002154	99.406556
3725	0.002314	99.40887
3730	0.002481	99.411351
3735	0.002653	99.414004
3740	0.00283	99.416834
3745	0.003013	99.419847
3750	0.003201	99.423048
3755	0.003393	99.426441
3760	0.003589	99.43003
3765	0.003788	99.433818
3770	0.00399	99.437808
3775	0.004194	99.442002
3780	0.0044	99.446402
3785	0.004607	99.451009
3790	0.004814	99.455823
3795	0.005021	99.460844
3800	0.005227	99.466071
3805	0.005432	99.471503
3810	0.005633	99.477136
3815	0.005832	99.482968
3820	0.006027	99.488995
3825	0.006218	99.495213
3830	0.006404	99.501617
3835	0.006584	99.508201
3840	0.006758	99.514959
3845	0.006926	99.521885
3850	0.007086	99.528971
3855	0.007238	99.536209
3860	0.007382	99.543591
3865	0.007517	99.551108
3870	0.007643	99.558751
3875	0.00776	99.566511
3880	0.007867	99.574378
3885	0.007964	99.582342
3890	0.008051	99.590393
3895	0.008128	99.598521
3900	0.008194	99.606715
3905	0.00825	99.614965
3910	0.008295	99.62326
3915	0.00833	99.63159
3920	0.008354	99.639944
3925	0.008368	99.648312
3930	0.008371	99.656683
3935	0.008365	99.665048
3940	0.008348	99.673396
3945	0.008322	99.681718
3950	0.008287	99.690005
3955	0.008242	99.698247
3960	0.008188	99.706435
3965	0.008126	99.714561
3970	0.008056	99.722617
3975	0.007978	99.730595
3980	0.007893	99.738488
3985	0.007801	99.746289
3990	0.007702	99.753991
3995	0.007597	99.761588
4000	0.007486	99.769074
4005	0.007369	99.776443
4010	0.007248	99.783691
4015	0.007122	99.790813
4020	0.006993	99.797806
4025	0.006859	99.804665
4030	0.006722	99.811387
4035	0.006582	99.817969
4040	0.00644	99.824409
4045	0.006295	99.830704
4050	0.006149	99.836853
4055	0.006001	99.842854
4060	0.005853	99.848707
4065	0.005703	99.85441
4070	0.005553	99.859963
4075	0.005403	99.865366
4080	0.005253	99.870619
4085	0.005104	99.875723
4090	0.004955	99.880678
4095	0.004807	99.885485
4100	0.004661	99.890146
4105	0.004516	99.894662
4110	0.004372	99.899034
4115	0.00423	99.903264
4120	0.00409	99.907354
4125	0.003952	99.911306
4130	0.003817	99.915123
4135	0.003683	99.918806
4140	0.003553	99.922359
4145	0.003424	99.925783
4150	0.003299	99.929082
4155	0.003176	99.932258
4160	0.003056	99.935314
4165	0.002939	99.938253
4170	0.002824	99.941077
4175	0.002713	99.94379
4180	0.002605	99.946395
4185	0.002499	99.948894
4190	0.002397	99.951291
4195	0.002298	99.953589
4200	0.002201	99.95579
4205	0.002108	99.957898
4210	0.002017	99.959915
4215	0.00193	99.961845
4220	0.001845	99.96369
4225	0.001763	99.965453
4230	0.001684	99.967137
4235	0.001608	99.968745
4240	0.001534	99.970279
4245	0.001463	99.971742
4250	0.001395	99.973137
4255	0.00133	99.974467
4260	0.001267	99.975734
4265	0.001206	99.97694
4270	0.001148	99.978088
4275	0.001092	99.97918
4280	0.001038	99.980218
4285	0.000987	99.981205
4290	0.000938	99.982143
4295	0.000891	99.983034
4300	0.000845	99.983879
4305	0.000802	99.984681
4310	0.000761	99.985442
4315	0.000722	99.986164
4320	0.000684	99.986848
4325	0.000648	99.987496
4330	0.000614	99.98811
4335	0.000581	99.988691
4340	0.00055	99.989241
4345	0.00052	99.989761
4350	0.000492	99.990253
4355	0.000465	99.990718
4360	0.00044	99.991158
4365	0.000415	99.991573
4370	0.000392	99.991965
4375	0.00037	99.992335
4380	0.00035	99.992685
4385	0.00033	99.993015
4390	0.000311	99.993326
4395	0.000293	99.993619
4400	0.000276	99.993895
4405	0.00026	99.994155
4410	0.000245	99.9944
4415	0.000231	99.994631
4420	0.000217	99.994848
4425	0.000205	99.995053
4430	0.000192	99.995245
4435	0.000181	99.995426
4440	0.00017	99.995596
4445	0.00016	99.995756
4450	0.00015	99.995906
4455	0.000141	99.996047
4460	0.000132	99.996179
4465	0.000124	99.996303
4470	0.000117	99.99642
4475	0.00011	99.99653
4480	0.000103	99.996633
4485	0.000096	99.996729
4490	0.00009	99.996819
4495	0.000085	99.996904
4500	0.000079	99.996983
4505	0.000074	99.997057
4510	0.00007	99.997127
4515	0.000065	99.997192
4520	0.000061	99.997253
4525	0.000057	99.99731
4530	0.000053	99.997363
4535	0.00005	99.997413
4540	0.000047	99.99746
4545	0.000044	99.997504
4550	0.000041	99.997545
4555	0.000038	99.997583
4560	0.000036	99.997619
4565	0.000033	99.997652
4570	0.000031	99.997683
4575	0.000029	99.997712
4580	0.000027	99.997739
4585	0.000025	99.997764
4590	0.000023	99.997787
4595	0.000022	99.997809
4600	0.00002	99.997829
4605	0.000019	99.997848
4610	0.000018	99.997866
4615	0.000017	99.997883
4620	0.000015	99.997898
4625	0.000014	99.997912
4630	0.000013	99.997925
4635	0.000012	99.997937
4640	0.000012	99.997949
4645	0.000011	99.99796
4650	0.00001	99.99797
4655	0.000009	99.997979
4660	0.000009	99.997988
4665	0.000008	99.997996
4670	0.000008	99.998004
4675	0.000007	99.998011
4680	0.000007	99.998018
4685	0.000006	99.998024
4690	0.000006	99.99803
4695	0.000006	99.998036
4700	0.000005	99.998041
4705	0.000005	99.998046
4710	0.000005	99.998051
4715	0.000004	99.998055
4720	0.000004	99.998059
4725	0.000004	99.998063
4730	0.000004	99.998067
4735	0.000004	99.998071
4740	0.000004	99.998075
4745	0.000004	99.998079
4750	0.000004	99.998083
4755	0.000004	99.998087
4760	0.000004	99.998091
4765	0.000004	99.998095
4770	0.000004	99.998099
4775	0.000005	99.998104
4780	0.000005	99.998109
4785	0.000005	99.998114
4790	0.000005	99.998119
4795	0.000006	99.998125
4800	0.000006	99.998131
4805	0.000007	99.998138
4810	0.000007	99.998145
4815	0.000008	99.998153
4820	0.000009	99.998162
4825	0.00001	99.998172
4830	0.00001	99.998182
4835	0.000011	99.998193
4840	0.000012	99.998205
4845	0.000014	99.998219
4850	0.000015	99.998234
4855	0.000016	99.99825
4860	0.000017	99.998267
4865	0.000019	99.998286
4870	0.000021	99.998307
4875	0.000022	99.998329
4880	0.000024	99.998353
4885	0.000026	99.998379
4890	0.000029	99.998408
4895	0.000031	99.998439
4900	0.000034	99.998473
4905	0.000036	99.998509
4910	0.000039	99.998548
4915	0.000042	99.99859
4920	0.000045	99.998635
4925	0.000049	99.998684
4930	0.000053	99.998737
4935	0.000056	99.998793
4940	0.000061	99.998854
4945	0.000065	99.998919
4950	0.00007	99.998989
4955	0.000074	99.999063
4960	0.000079	99.999142
4965	0.000085	99.999227
4970	0.00009	99.999317
4975	0.000096	99.999413
4980	0.000103	99.999516
4985	0.000109	99.999625
4990	0.000116	99.999741
4995	0.000123	99.999864
5000	0.000131	99.999995

winsome johnny (not Win some johnny)

dwheatley

dwheatley

Threads: 25
Posts: 1246

Joined: Nov 16, 2009

January 18th, 2013 at 9:46:52 AM permalink

I will revise my first answer, 7craps is right that your results for 250 hands is not normally distributed and you are much more likely to fall in a tighter range.

However, for the most recent question, the distribution of session outcomes IS normally distributed. The big problem is that the variance of this distribution is smaller than the VP poker one hand variance. This makes the math complicated again.

Wisdom is the quality that keeps you out of situations where you would otherwise need it

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 9:54:56 AM permalink

Quote: 7craps

It looks clear you do not understand normal distributions vs other types of non-normal distributions and there are many.

You are correct. I will have to try to do some additional reading on this.

Quote: 7craps

You can check my math for me if you wish.

I wish I could, but I don't understand it well enough to be in a position to check your math.

I did notice using the ranges I had gotten from my calculations that the lower end of the range was an impossible number to reach for low numbers of hands. For example, the range that I came up with using my calculation for 10 hands was -$70.08 to +$69.17. 10 hands at $5 per hand, the absolute worst possible result would be -$50. This was a clue to me that there was more to it. Now I just need to try and understand this additional information.

Thanks again for your input.

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 10:12:45 AM permalink

These are the values I get from the Return and Variance simulator at www.beatingbonuses.com (link to the simulator:
http://www.beatingbonuses.com/simulator_java.htm) when you play 250 $5 hands at 1-play 9/6 JoB with perfect strategy:

-1 SD to + 1 SD: -196 ... 112
-2 SD to + 2 SD: -283 ... 331
-3 SD to + 3 SD: -415 ... 4065

Notice the large value at +3SD mark representing you hitting a royal in 250 hands.

The SD values listed above are not saying that the result is normally distributed. The above ranges -1 SD ... +1 SD, -2 SD ... +2 SD, -3 SD ... +3 SD simply say that your end result will fall within these ranges 68%, 95% and 99.86% of time respectively.

You can try the BB simulator yourself in the above web link. Simply enter these values:

Game: VP (JoB 1-play)
Deposit: 1250 (to have enough bankroll to cover playing through all 250 hands). If you want to simulate the probability of busting your bankroll before completing all 250 hands, then enter a smaller value.
Bonus: 0
Wagering: 1250 (250 x $5)
Bet Size: 5
Runs: 100000 or more (affects accurary)

The BB java simulator also displays the distribution of the end result as a neat graph.

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 10:27:56 AM permalink

Quote: dwheatley
I will revise my first answer, 7craps is right that your results for 250 hands is not normally distributed and you are much more likely to fall in a tighter range.

The actual result will fall into the tighter range than predicted by normal distribution on the negative side.
BUT
The actual result will fall into the wider range than predicted by normal distribution on the positive side.

For example in the OPs case of playing 250 hands at $5 per hand and unit standard deviation of JoB 4.42, the total standard deviation is $5*SQRT(250)*4.42 = $349. This would say that (ignoring the shift due to house edge) that losing $700 would be a -2SD event. However in actual simulation of 100,000 runs the largest loss in 250 hands was -$555, so obviously losing $700 is more rare event than predicted by normal distribution -> the loss will be under a tighter probability envolope than predicted by normal distribution.

On the other hand on the positive side, the SD value of $349 means that winning $4000 would be a $4000/$349 = +11.46 SD event, which corresponds to an extremely small probability value. But in reality hitting a single royal flush in 250 hands in no rarer than 250/40000 = 1 in 160 event, and it wins $4000 -> the win amount will be under wider probability distribution than predicted by normal distribution.

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

January 18th, 2013 at 10:28:12 AM permalink

Quote: Jufo81
These are the values I get from the Return and Variance simulator at www.beatingbonuses.com (link to the simulator:
http://www.beatingbonuses.com/simulator_java.htm) when you play 250 $5 hands at 1-play 9/6 JoB with perfect strategy:

-1 SD to + 1 SD: -196 ... 112
-2 SD to + 2 SD: -283 ... 331
-3 SD to + 3 SD: -415 ... 4065

Nice find.
But Without looking at that site, the 1 SD = 154?

How does one arrive at that by math? yeah.

I think that is what the OP really wants.
Like confidence interval for a geometric distribution. Way more math involved.

Having a program to do it is one thing, being able to do it with pencil and paper... another

just goes to show using a normal distribution standard deviation just does not work for VP

winsome johnny (not Win some johnny)

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 10:39:17 AM permalink

Quote: 7craps
Nice find.
But Without looking at that site, the 1 SD = 154?

How does one arrive at that by math? yeah.

Well technically the standard deviation is still the one calculated by the OP: 349 (=SQRT(250)*5*4.42). Those values listed by the simulator don't actually have anything to do with this definition of standard deviation, they just mark the end points of a 68% confidence interval from a non-normal distribution.

Quote: 7craps

just goes to show using a normal distribution standard deviation just does not work for VP

Yes definitely not.

However if you increase the number of hands so that in the end the number of hands includes hitting many royals, then the distribution will eventually become nearly normally distributed. You will see this by running the simulator and choosing playing 1000,10000,100000 and 1000000 hands, and you will see from the graph how the probability peak around hitting a royal gradually merges into the main mass distribution.

tringlomane

tringlomane

Threads: 8
Posts: 6281

Joined: Aug 25, 2012

January 18th, 2013 at 10:43:18 AM permalink

Quote: Jufo81
The actual result will fall into the tighter range than predicted by normal distribution on the negative side.
BUT
The actual result will fall into the wider range than predicted by normal distribution on the positive side.

For example in the OPs case of playing 250 hands at $5 per hand and unit standard deviation of JoB 4.42, the total standard deviation is $5*SQRT(250)*4.42 = $349. This would say that (ignoring the shift due to house edge) that losing $700 would be a -2SD event. However in actual simulation of 100,000 runs the largest loss in 250 hands was -$555, so obviously losing $700 is more rare event than predicted by normal distribution -> the loss will be under a tighter probability envolope than predicted by normal distribution.

On the other hand on the positive side, the SD value of $349 means that winning $4000 would be a $4000/$349 = 11.46 SD event, which corresponds to an extremely low probability. But in reality hitting a single royal flush in 250 hands in no rarer than 250/40000 = 1 in 160 event -> the win amount will be under wider probability distribution than predicted by normal distribution.

Right, in VP results are skewed to rare, large wins, and frequent, smaller losses since the most you can lose per hand is one betting unit, but can win 800 with a Royal. You're better off simulating games to get more accurate results.

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

January 18th, 2013 at 10:57:03 AM permalink

Quote: Jufo81
Well technically the standard deviation is still the one calculated by the OP: 349 (=SQRT(250)*5*4.42). Those values listed by the simulator don't actually have anything to do with this definition of standard deviation, they just mark the end points of a 68% confidence interval from a non-normal distribution.

Yes definitely not.

However if you increase the number of hands so that in the end the number of hands includes hitting many royals, then the distribution will eventually become nearly normally distributed. You will see this by running the simulator and choosing playing 1000,10000,100000 and 1000000 hands, and you will see from the graph how the probability peak around hitting a royal gradually merges into the main mass distribution.

yes I agree. One trying to USE SD will just get more confused.
(damn java stopped working on my computer, so I can not see what you are talking about at that site)

But JOB has a very large knee the more trials.
here are some graphs
http://www.jazbo.com/videopoker/curves.html

The peak of the graph is not centered at the mean, even my graphs in Excel.
Most still think the mean (average or EV) the mode and the median are about the same.
A normal distribution.
I do not think so for VP.
Example 5,000 hands
ev= 5,000*5*-.0046 = -115
The mean is not even close to the peak.
The graph is done by direct calculation, not simulation in my program.

I could study it more but really have no interest in it.
Maybe someone else will find the answers so this can be done without a program.

Everyone talks about perfect play in VP to get the expected return %
but not the 10s of millions of hands (or more) it takes to get that standard error to be very small
N (number of hands) = (unitSD / error)^2
(22.08771/5/.005)^2 for a .5% standard error. And that error is over a very high handle (total wagered)
or even this one
https://wizardofodds.com/ask-the-wizard/video-poker/deuces-wild/
Q
"I see that the return on a full pay deuces wild video poker machine is 100.76% with the strategy you have on your website. Obviously, this is with infinite play. My question is how many hands would you have to play to get that return with, let�s say 90% certainty?"

Wizards calculations about half way down for those that have an interest.

Good Luck

winsome johnny (not Win some johnny)

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 11:29:32 AM permalink

In short, what I was looking to find out, is how common or uncommon my results from my last 250 hands on the Wizards IOS VP app were. 250 hands. Finished down 335 credits (67 bets). I am very much appreciating the detailed responses, and am trying to learn more. Thanks.

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 11:30:58 AM permalink

Quote: Jufo81
However if you increase the number of hands so that in the end the number of hands includes hitting many royals, then the distribution will eventually become nearly normally distributed. You will see this by running the simulator and choosing playing 1000,10000,100000 and 1000000 hands, and you will see from the graph how the probability peak around hitting a royal gradually merges into the main mass distribution.

I had some extra time so I took screenshots from the sim to show how the shape of the distribution changes with increasing number of hands played and I added some remarks. I put it into .JPG (you need to zoom to see it larger):

http://i.imgur.com/Gf4sb.jpg?1

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

January 18th, 2013 at 11:39:45 AM permalink

Quote: Scot
In short, what I was looking to find out, is how common or uncommon my results from my last 250 hands on the Wizards IOS VP app were. 250 hands. Finished down 335 credits (67 bets). I am very much appreciating the detailed responses, and am trying to learn more. Thanks.

You can see the answer in the data table I posted.
These answers have to be calculated or simulated since using an SD value is useless.

to end up -$335 or more in 250 hands is about 1.065%
added:(assumes you have the bankroll to make it to the end or you could hit that -335 or more about 2% during those 250 hands. This is Risk of Ruin stuff)
(the 3rd column. That is what cumulative means.
That value of less but as a negative number it is to lose more)

winsome johnny (not Win some johnny)

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

January 18th, 2013 at 11:45:21 AM permalink

Quote: Jufo81
I had some extra time so I took screenshots from the sim to show how the shape of the distribution changes with increasing number of hands played and I added some remarks. I put it into .JPG (you need to zoom to see it larger):

http://i.imgur.com/Gf4sb.jpg?1

Nice. Excellent work.
I need to remove old Java and update. I will try that site
That info looks interesting. Thanks.

Now since the house edge is -.0046 and with a standard error of just .001 (+ or -)
(22.08771/5/.001)^2 for a .1% standard error
it looks like we need how many hands played??
19,514,677
How much coin in? * the range.
We bet 5 coins per hand.

Good Luck unless my math is just wrong. Could be.
Maybe the Wizard's is better.

winsome johnny (not Win some johnny)

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 11:46:00 AM permalink

Quote: Scot
In short, what I was looking to find out, is how common or uncommon my results from my last 250 hands on the Wizards IOS VP app were. 250 hands. Finished down 335 credits (67 bets). I am very much appreciating the detailed responses, and am trying to learn more. Thanks.

Notice that the the answer to this depends on whether:

1) Your bankroll is "unlimited" so you are quaranteed to play through all the 250 hands and only check the loss amount at the very end.

The sim says that the probability to end up losing 67 units or more in 250 units played in JoB is around 1.09% or ~1 in 92 event. I am not sure if there is a way to obtain this result other than computer simulation, in other words the standard deviation calculations you did won't help here. [EDIT: 7craps beat me to it].

2) Your bankroll is precisely that 67 units ($335) so there is also the possibility of going broke before finishing all the 250 hands.

Now the probability will be higher because there is the additional possibility of losing 67 units at some point during 250 hands, not just at the very end.

In this case the probability of ending up losing 67 units is 1.72% (or 1 in 58 event).

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 11:53:24 AM permalink

Quote: Jufo81
I had some extra time so I took screenshots from the sim to show how the shape of the distribution changes with increasing number of hands played and I added some remarks. I put it into .JPG (you need to zoom to see it larger):

http://i.imgur.com/Gf4sb.jpg?1

This is very helpful. Thank you.

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 11:56:05 AM permalink

Is there any value to the information here? https://wizardofodds.com/games/video-poker/appendix/3/

I feel like it was a red herring. Of course it's very possible I just misused this information.

7craps

Threads: 18
Posts: 1977

Joined: Jan 23, 2010

January 18th, 2013 at 11:58:23 AM permalink

Quote: Jufo81
I don't think there is no practical way to obtain this result other than computer simulation, in other words the standard deviation calculations you did won't help here.

Sure there is for say up to 100,000 trials.
Excel can calculate it. That is how I do it and if I learned to improve the VBA I could do it way faster.
So does the program Gamblers Odds 1.3 (this one is free)
So does VP for Winners and I think the trial version shows this under bankroll

Just a lot of multiplying and summing of the 10 possible outcomes at each trial.
Good Luck

winsome johnny (not Win some johnny)

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 11:59:07 AM permalink

Quote: Jufo81
Notice that the the answer to this depends on whether:

1) Your bankroll is "unlimited" so you are quaranteed to play through all the 250 hands and only check the loss amount at the very end.
.

For 250 hands, I'm willing to go with the unlimited bankroll scenario, since $1250 would be all that is necessary.

Thanks to everyone for their thoughtful responses.

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 12:01:19 PM permalink

Quote: Scot
Is there any value to the information here? https://wizardofodds.com/games/video-poker/appendix/3/

I feel like it was a red herring. Of course it's very possible I just misused this information.

That page lists the standard deviations of multi-hand pokers. It does not mean playing multiple consecutive hands in 1-play VP but refers to the game variation where you play multiple hands at the same time.

But I understood that your question was about 1-play VP, right?

Scot

Threads: 3
Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 12:02:11 PM permalink

Quote: Jufo81
Quote: Scot
Is there any value to the information here? https://wizardofodds.com/games/video-poker/appendix/3/

I feel like it was a red herring. Of course it's very possible I just misused this information.

That page lists the standard deviations of multi-hand pokers. It does not mean playing multiple consecutive hands in 1-play VP but refers to the game variation where you play multiple hands at the same time.

But I understood that your question was about 1-play VP, right?

Yes. 1 hand. It lists n=1 figures there, though. I saw it more as n hand video poker information.

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 12:06:56 PM permalink

Quote: Scot

Yes. 1 hand. It lists n=1 figures there, though. I saw it more as n hand video poker information.

Well, yes those standard deviation values for 1-handed VP are correct. It's just that those standard deviation values cannot be used in this context. Remember also that playing 5 hands in 1-handed VP will not have the same standard deviation as playing one hand in 5-handed VP. That's because the covariance between hands affects the standard deviation in multi-hand VP.

Scot

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Posts: 96

Joined: Jan 5, 2013

January 18th, 2013 at 12:08:45 PM permalink

Quote: Jufo81
Well, yes those standard deviation values for 1-handed VP are correct. It's just that the standard deviation value cannot be used in this context, because of the non-normality.

In what context is that information usable? What is the value? I'm not saying it has no value, I am trying to understand its importance.

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 12:13:04 PM permalink

Quote: 7craps
Sure there is for say up to 100,000 trials.
Excel can calculate it. That is how I do it and if I learned to improve the VBA I could do it way faster.
So does the program Gamblers Odds 1.3 (this one is free)
So does VP for Winners and I think the trial version shows this under bankroll

Just a lot of multiplying and summing of the 10 possible outcomes at each trial.
Good Luck

Ah ok, I stand corrected, I am not familiar with this approach. Is this precise unlike simulation? The only approach I know is Markov Chain model using matrix algebra, so is this the same?

Jufo81

Threads: 6
Posts: 344

Joined: May 23, 2010

January 18th, 2013 at 12:30:35 PM permalink

Quote: Scot
In what context is that information usable? What is the value? I'm not saying it has no value, I am trying to understand its importance.

Hmm, well if you play enough hands in VP then the distribution of the end result will be normally distributed and you CAN use the standard deviation just like you did in your opening post (also remember to shift the mean of the distribution to the 99.54% theoretical return).

Another use I can think of would be to compare the volatility of different games. So standard deviation would tell you how much larger swings you end up having playing Deuces Wild than Jacks or Better for example. Or you could use the standard deviation to check whether playing 10-play JoB at $10 per deal has larger variance than playing 1-play JoB at $5 per deal (based on Wizard's tables the latter has larger variance).

Other than that I can't of think of a direct use for the standard deviation values of Video Poker. Perhaps someone else knows better where they are used.

7craps

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January 18th, 2013 at 12:36:53 PM permalink

Quote: Jufo81
Ah ok, I stand corrected, I am not familiar with this approach.
Is this precise unlike simulation?
The only approach I know is Markov Chain model using matrix algebra, so is this the same?

Yes, very precise.
Yes, like using matrix algebra, multiplying each outcome's probability at each trial then multiplying that probability at the next trial.
I do not know how those two programs I showed actually carry out the calculations, in Excel I just loop thru each trial
the R program would be a good one to do the matrix algebra in. I have been trying to get away from Excel (using loops is a slow method)
and doing more in R with vectors and matrices

That website I linked to earlier kind of explains it.

For example, let's take the case of starting with just 2 bets at Deuces Wild.
At the beginning, you have a starting stake of 2 bets.
After the first iteration, there is a 54.7% chance you will have only 1 bet (that's the chance of losing your bet on Deuces), a 28.5% chance you will push and still have 2 bets, a 7.3% chance you will have 3 bets, etc.

On the 2nd iteration, there is a 29.9% chance you will have busted (0.549x0.549 = 0.299), a 31.1% chance you will be down by 1, etc.

The 3rd iteration gives a 46.9% chance you will have busted, and the 4th iteration the bust probability goes to 56.5%

winsome johnny (not Win some johnny)

MangoJ

MangoJ

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January 18th, 2013 at 3:25:13 PM permalink

There is no need to use some kind of matrix algebra.
Remember that when you know the probability p(A) and p(B), then p(A+B) is simply the convolution of p(A) and p(B).
This is all it takes. Especially if p(A) and p(B) are the same, most efficient way is the Fourier transform convolution (with enough padding). After Fourier transform, take each frequency component to the N-th power, then transform backwards. Can be done in a split second for any number of N, in a simple spreadsheet.

Jufo81

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January 19th, 2013 at 2:58:15 AM permalink

Quote: MangoJ
There is no need to use some kind of matrix algebra.
Remember that when you know the probability p(A) and p(B), then p(A+B) is simply the convolution of p(A) and p(B).
This is all it takes. Especially if p(A) and p(B) are the same, most efficient way is the Fourier transform convolution (with enough padding). After Fourier transform, take each frequency component to the N-th power, then transform backwards. Can be done in a split second for any number of N, in a simple spreadsheet.

Hi MangoJ! It's been a while.
Interesting. I have studied this stuff at the university but it's been a long time so I don't remember anymore how you do those steps you mentioned. Would it possible that either you or 7craps sends me their Excel spreadsheet or some other template so I can see how these steps are done in practice.

The approach that I knew (Markov chain) is that if you have the vector of probabilities to be at any given balance after k rounds, then if you multiply this vector with a transition matrix (the matrix of probabilities of differens payouts), you will get the probabilities to be at any balance after k+1 rounds, so you can keep multiplying by this matrix to get the odds after next round.

Once I took the probabilities of Blackjack and made it's transition matrix. I tried to solve the general k'th power of this matrix by decomposing it to it's eigenvalues and eigenvectors, in hopes to get closed form expression p(k) (vector) to be at any end balance after general k hands played. But I recall that Matlab crashed when I tried to do this.

MangoJ

MangoJ

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January 19th, 2013 at 3:55:35 AM permalink

The Markovian approach is quite powerful, and the best choice if you vary your betsize with your bankroll.

But to get the probability distribution of a simple series of identical play is much more simpler. Sorry I have no proven code for you, but if you work with matlab first get the probability distribution of a single round and unit bet, say in vector p.

Then simply pad your vector with q = [zeros(1,L*N), p, zeros(1, W*N)], where L is the maximum loss amount (depending on double and splits) and W is the maximum win amount (also depending on double and splits).
Then calculate ifft(fft(q).^N) as the N-th convolution. fft() is the Fourier transform, take it to the N-th power, and transform back with the inverse Fourier transform ifft(). Although some knowledge about Fourier transform might help, there is usually no need to know any more details about it here.

Jufo81