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Blackjack Odds of a Winning Session after X Amount of Hands Blackjack Odds of a Winning Session after X Amount of Hands
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08-13-2014 , 01:02 AM
#1
SJM3000
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centurion
Join Date: Dec 2013 Posts: 162
I'm not math expert, but I am wondering, in a typical blackjack game with 0.66083% house edge, no counting or anything, if somebody is flat betting what are his odds of being ahead after 5 hands? After 10 hands? 25, 50, 100, 200, 500 hands? Could be easier if anybody can easily come of up with a forumula and I could just plug in the amount of hands. Can't find anything about this on google or searching this forum. Thanks!
Last edited by SJM3000; 08-13-2014 at 01:07 AM.
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-13-2014 , 03:03 AM
#2
nickthegeek
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journeyman
Join Date: Sep 2011 Posts: 273
First of all, we need to calculate the probability of winning a single bet. An edge of 0.66083% means that we are losing on average 0.0066083*B per bet, where B is the bet amount. So our EV=-0.0066083*B. Say we win B with probability p. This means that we lose B with probability 1-p. So we have:

-Edge*B = B*p - (1-p)*B

We can now solve for p (B cancels out) and we obtain:

p = (1-Edge)/2 = 0.4966958

The probability of winning Nw bets out of N trials is dictated by the binomial distribution. We don't need to do the math with pencil and paper; rather we can use some software like R (free and easy download here):

Code:
edge<-0.0066083
Nhands<-c(5,10,25,50,100,200,500)
p<-(1-edge)/2
probToBeWinning<-function(x,p) 1-pbinom(x%/%2L,x,p)
probToBeWinning(Nhands,p)
The results:

[1] 0.4938049 0.3688492 0.4866882 0.4253814 0.4340134 0.4347654 0.4236535

You can edit the first two lines and modify the house edge and the number of hands and then rerun the code.
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-13-2014 , 03:39 AM
#3
BruceZ's Avatar
BruceZ
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Carpal \'Tunnel
Join Date: Sep 2002 Posts: 11,877
That's no way to do it. You don't just win or lose 1 bet. There are blackjacks, doubles, splits, insurance, and about 9% pushes. Your edge doesn't give you the probability of winning a hand that way. It's about 43%. Winning doesn't mean you have to win over half your hands.

If you have enough hands, you can just use the edge and the standard deviation which depends on the number of decks. For 6 deck it's about 1.16 bets per hand. Multiply that by the square root of the number of hands to get your standard deviation for all the hands. Your EV for all the hands is your negative edge times the number of hands. Then use a normal distribution. If you want to be more accurate for a very small number of hands like 5, you can use the actual probabilities for each of the results which you could find online or from a simulation. You would produce the actual probability distribution for 1 hand, and then convolve those with each other to get the overall distribution. Or you can do it on a blackjack simulator.

To get the results just using the normal distribution for all the hands you asked about and a standard deviation of 1.16, just enter this in R:

Code:
Nhands = c(5,10,25,50,100,200,500)
edge = .0066083
sd = 1.16
ev = -edge*Nhands
sd = sd*sqrt(Nhands)
1-pnorm(0,ev,sd)
[1] 0.4949182 0.4928135 0.4886380 0.4839339 0.4772853 0.4678939 0.4493180
Last edited by BruceZ; 08-14-2014 at 11:46 PM.
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-13-2014 , 05:16 AM
#4
nickthegeek
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journeyman
Join Date: Sep 2011 Posts: 273
Ouch. Never played blackjack (!) and just assumed that "flat betting" meant that you just win or lose your bet.
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-13-2014 , 01:22 PM
#5
BruceZ's Avatar
BruceZ
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Flat betting means you make the same initial bet each hand. As opposed to say a card counter who would vary his bets with the composition of the deck to gain an advantage, or just a gambler playing a progression system, hunch, etc. Blackjacks pay 3:2, or in terrible games 6:5. You must double and split correctly to minimize the house edge. If you never doubled or split, the house edge would be far greater than 0.6%. You'd have to be giving up more than an additional 2% to the house because because the 3:2 blackjack payoff is worth about 2.3%, and if you never doubled or split and BJs paid like any other hand as you imagined, the house edge would be over 5%. You can split multiple times, and if the rules allow it, double on the individual hands that result from the split, so you could possible win or lose several bets. Insurance wouldn't matter since you would never take it if you're not counting. Some games allow surrender though where you can sacrifice half your bet.
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-13-2014 , 09:19 PM
#6
sallymustang
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centurion
Join Date: Jun 2011 Posts: 123
Quote:
Originally Posted by BruceZ
If you want to be more accurate for a very small number of hands like 5, you can use the actual probabilities for each of the results which you could find online or from a simulation.
You would produce the actual probability distribution for 1 hand, and then convolve those with each other to get the overall distribution. Or you can do it on a blackjack simulator.
This is also very easy to do using Matrix Algebra, Excel can do it once the transition matrix is set up
easier than doing a Risk of Ruin Matrix in my opinion.
I would gather with some more effort R could be set to do this too.

Using the normal dist did you not answer the question of being even or ahead,
I think you did without looking closer.
OP asked for being ahead after X rounds played
I used the prob dist from here
probability of a net win, loss, and push

and I gets this (values are %)
hope it all fits, I did not format it
Code:
round	    win	        push	    loss
1	43.338158	8.779258	47.882584
2	35.912942	29.210131	34.876927
3	42.225318	12.442896	45.331786
4	41.96775	15.803355	42.228895
5	43.416708	11.065677	45.517615
6	43.983913	11.036439	44.979648
7	44.507817	9.287889	46.204294
8	44.982167	8.708694	46.309139
9	45.285114	7.846453	46.868433
10	45.608773	7.300012	47.091215
11	45.826858	6.763045	47.410097
12	46.047893	6.332115	47.619992
13	46.213894	5.950243	47.835863
14	46.371207	5.619012	48.009781
15	46.499142	5.329329	48.171529
16	46.615726	5.071843	48.312431
17	46.715056	4.844517	48.440427
18	46.804114	4.640453	48.555433
19	46.881928	4.458071	48.660001
20	46.951423	4.293225	48.755352
21	47.012989	4.144326	48.842685
22	47.068055	4.008864	48.923081
23	47.117242	3.88544	48.997318
24	47.161381	3.772434	49.066185
25	47.201027	3.668713	49.13026
26	47.236733	3.573159	49.190108
27	47.268938	3.484891	49.246171
28	47.298038	3.403102	49.29886
29	47.324369	3.327113	49.348518
30	47.348229	3.256318	49.395453
31	47.369874	3.190198	49.439928
32	47.389532	3.12829	49.482178
33	47.4074	3.07019	49.52241
34	47.423655	3.015539	49.560806
35	47.438451	2.964021	49.597528
36	47.451925	2.915357	49.632718
37	47.464199	2.869295	49.666506
38	47.475382	2.825614	49.699004
39	47.485568	2.784118	49.730314
40	47.494846	2.744625	49.760529
41	47.503292	2.706978	49.78973
42	47.510975	2.671033	49.817992
43	47.517957	2.636663	49.84538
44	47.524295	2.603749	49.871956
45	47.530039	2.572188	49.897773
46	47.535234	2.541884	49.922882
47	47.539922	2.512751	49.947327
48	47.54414	2.484711	49.971149
49	47.547923	2.457691	49.994386
50	47.5513	2.431628	50.017072
51	47.5543	2.406461	50.039239
52	47.55695	2.382136	50.060914
53	47.559271	2.358604	50.082125
54	47.561286	2.335818	50.102896
55	47.563015	2.313736	50.123249
56	47.564475	2.292321	50.143204
57	47.565683	2.271535	50.162782
58	47.566654	2.251347	50.181999
59	47.567402	2.231725	50.200873
60	47.567941	2.212641	50.219418
61	47.568283	2.194068	50.237649
62	47.568438	2.175983	50.255579
63	47.568416	2.158364	50.27322
64	47.568229	2.141186	50.290585
65	47.567883	2.124433	50.307684
66	47.567388	2.108085	50.324527
67	47.566752	2.092125	50.341123
68	47.565981	2.076536	50.357483
69	47.565083	2.061303	50.373614
70	47.564063	2.046413	50.389524
71	47.562927	2.031851	50.405222
72	47.561682	2.017605	50.420713
73	47.560332	2.003663	50.436005
74	47.558882	1.990014	50.451104
75	47.557337	1.976646	50.466017
76	47.555701	1.96355	50.480749
77	47.553978	1.950717	50.495305
78	47.552173	1.938136	50.509691
79	47.550287	1.925801	50.523912
80	47.548326	1.913701	50.537973
81	47.546292	1.901831	50.551877
82	47.544189	1.890181	50.56563
83	47.542019	1.878745	50.579236
84	47.539785	1.867517	50.592698
85	47.53749	1.85649	50.60602
86	47.535135	1.845659	50.619206
87	47.532725	1.835016	50.632259
88	47.53026	1.824557	50.645183
89	47.527743	1.814276	50.657981
90	47.525177	1.804167	50.670656
91	47.522562	1.794228	50.68321
92	47.519901	1.784451	50.695648
93	47.517196	1.774833	50.707971
94	47.514448	1.76537	50.720182
95	47.511659	1.756057	50.732284
96	47.508831	1.74689	50.744279
97	47.505965	1.737865	50.75617
98	47.503062	1.72898	50.767958
99	47.500123	1.720231	50.779646
100	47.497151	1.711612	50.791237
101	47.494146	1.703122	50.802732
102	47.49111	1.694757	50.814133
103	47.488044	1.686514	50.825442
104	47.484948	1.678391	50.836661
105	47.481824	1.670384	50.847792
106	47.478673	1.66249	50.858837
107	47.475496	1.654707	50.869797
108	47.472294	1.647033	50.880673
109	47.469067	1.639464	50.891469
110	47.465817	1.631999	50.902184
111	47.462544	1.624635	50.912821
112	47.45925	1.617369	50.923381
113	47.455935	1.610199	50.933866
114	47.452599	1.603125	50.944276
115	47.449244	1.596143	50.954613
116	47.44587	1.589251	50.964879
117	47.442478	1.582447	50.975075
118	47.439069	1.575729	50.985202
119	47.435642	1.569098	50.99526
120	47.4322	1.562547	51.005253
121	47.428741	1.55608	51.015179
122	47.425268	1.54969	51.025042
123	47.42178	1.54338	51.03484
124	47.418278	1.537145	51.044577
125	47.414763	1.530985	51.054252
126	47.411234	1.524899	51.063867
127	47.407693	1.518884	51.073423
128	47.40414	1.512939	51.082921
129	47.400575	1.507064	51.092361
130	47.396999	1.501257	51.101744
131	47.393412	1.495516	51.111072
132	47.389815	1.48984	51.120345
133	47.386207	1.484229	51.129564
134	47.38259	1.47868	51.13873
135	47.378964	1.473193	51.147843
136	47.375329	1.467766	51.156905
137	47.371685	1.462399	51.165916
138	47.368033	1.45709	51.174877
139	47.364373	1.451838	51.183789
140	47.360706	1.446642	51.192652
141	47.357031	1.441501	51.201468
142	47.353349	1.436415	51.210236
143	47.349661	1.431382	51.218957
144	47.345966	1.426402	51.227632
145	47.342264	1.421473	51.236263
146	47.338557	1.416595	51.244848
147	47.334844	1.411766	51.25339
148	47.331126	1.406986	51.261888
149	47.327402	1.402255	51.270343
150	47.323674	1.39757	51.278756
151	47.319941	1.392932	51.287127
152	47.316203	1.38834	51.295457
153	47.312461	1.383793	51.303746
154	47.308715	1.37929	51.311995
155	47.304965	1.374831	51.320204
156	47.301211	1.370414	51.328375
157	47.297453	1.366041	51.336506
158	47.293693	1.361707	51.3446
159	47.289929	1.357415	51.352656
160	47.286162	1.353163	51.360675
161	47.282392	1.348951	51.368657
162	47.27862	1.344778	51.376602
163	47.274845	1.340643	51.384512
164	47.271067	1.336546	51.392387
165	47.267288	1.332486	51.400226
166	47.263506	1.328463	51.408031
167	47.259722	1.324476	51.415802
168	47.255937	1.320524	51.423539
169	47.25215	1.316607	51.431243
170	47.248361	1.312726	51.438913
171	47.244571	1.308878	51.446551
172	47.24078	1.305063	51.454157
173	47.236987	1.301282	51.461731
174	47.233194	1.297532	51.469274
175	47.229399	1.293816	51.476785
176	47.225604	1.290131	51.484265
177	47.221807	1.286478	51.491715
178	47.218011	1.282854	51.499135
179	47.214213	1.279262	51.506525
180	47.210416	1.275698	51.513886
181	47.206618	1.272165	51.521217
182	47.202819	1.268661	51.52852
183	47.199021	1.265185	51.535794
184	47.195222	1.261738	51.54304
185	47.191424	1.258318	51.550258
186	47.187625	1.254927	51.557448
187	47.183827	1.251562	51.564611
188	47.180029	1.248224	51.571747
189	47.176231	1.244913	51.578856
190	47.172434	1.241628	51.585938
191	47.168637	1.238368	51.592995
192	47.164841	1.235134	51.600025
193	47.161046	1.231925	51.607029
194	47.157251	1.228741	51.614008
195	47.153457	1.225581	51.620962
196	47.149663	1.222446	51.627891
197	47.145871	1.219334	51.634795
198	47.142079	1.216246	51.641675
199	47.138289	1.213181	51.64853
200	47.134499	1.210139	51.655362
201	47.130711	1.20712	51.662169
202	47.126924	1.204123	51.668953
203	47.123138	1.201148	51.675714
204	47.119353	1.198196	51.682451
205	47.11557	1.195264	51.689166
206	47.111788	1.192354	51.695858
207	47.108007	1.189465	51.702528
208	47.104228	1.186597	51.709175
209	47.10045	1.18375	51.7158
210	47.096674	1.180923	51.722403
211	47.0929	1.178115	51.728985
212	47.089127	1.175328	51.735545
213	47.085356	1.17256	51.742084
214	47.081586	1.169812	51.748602
215	47.077818	1.167083	51.755099
216	47.074052	1.164373	51.761575
217	47.070288	1.161681	51.768031
218	47.066526	1.159008	51.774466
219	47.062766	1.156353	51.780881
220	47.059007	1.153717	51.787276
221	47.055251	1.151098	51.793651
222	47.051496	1.148497	51.800007
223	47.047743	1.145914	51.806343
224	47.043993	1.143347	51.81266
225	47.040245	1.140797	51.818958
226	47.036498	1.138266	51.825236
227	47.032754	1.13575	51.831496
228	47.029012	1.133251	51.837737
229	47.025272	1.130768	51.84396
230	47.021535	1.128301	51.850164
231	47.017799	1.125851	51.85635
232	47.014066	1.123416	51.862518
233	47.010335	1.120997	51.868668
234	47.006607	1.118593	51.8748
235	47.002881	1.116204	51.880915
236	46.999157	1.113831	51.887012
237	46.995435	1.111474	51.893091
238	46.991716	1.10913	51.899154
239	46.987999	1.106802	51.905199
240	46.984285	1.104488	51.911227
241	46.980573	1.102188	51.917239
242	46.976864	1.099902	51.923234
243	46.973157	1.097631	51.929212
244	46.969452	1.095374	51.935174
245	46.965751	1.09313	51.941119
246	46.962051	1.0909	51.947049
247	46.958354	1.088684	51.952962
248	46.95466	1.086481	51.958859
249	46.950968	1.084291	51.964741
250	46.947279	1.082114	51.970607
251	46.943593	1.07995	51.976457
252	46.939909	1.0778	51.982291
253	46.936227	1.075662	51.988111
254	46.932549	1.073536	51.993915
255	46.928873	1.071423	51.999704
256	46.925199	1.069323	52.005478
257	46.921529	1.067234	52.011237
258	46.917861	1.065158	52.016981
259	46.914195	1.063094	52.022711
260	46.910533	1.061041	52.028426
261	46.906873	1.059001	52.034126
262	46.903216	1.056972	52.039812
263	46.899561	1.054955	52.045484
264	46.89591	1.052949	52.051141
265	46.892261	1.050954	52.056785
266	46.888615	1.048971	52.062414
267	46.884971	1.046999	52.06803
268	46.88133	1.045038	52.073632
269	46.877693	1.043087	52.07922
270	46.874057	1.041148	52.084795
271	46.870425	1.039219	52.090356
272	46.866796	1.037301	52.095903
273	46.863169	1.035394	52.101437
274	46.859545	1.033497	52.106958
275	46.855924	1.03161	52.112466
276	46.852306	1.029733	52.117961
277	46.84869	1.027867	52.123443
278	46.845078	1.02601	52.128912
279	46.841468	1.024164	52.134368
280	46.837861	1.022328	52.139811
281	46.834257	1.020501	52.145242
282	46.830656	1.018684	52.15066
283	46.827057	1.016877	52.156066
284	46.823462	1.015078	52.16146
285	46.819869	1.01329	52.166841
286	46.81628	1.011511	52.172209
287	46.812693	1.009741	52.177566
288	46.809109	1.00798	52.182911
289	46.805527	1.00623	52.188243
290	46.801949	1.004487	52.193564
291	46.798374	1.002753	52.198873
292	46.794801	1.001029	52.20417
293	46.791232	0.999313	52.209455
294	46.787665	0.997606	52.214729
295	46.784101	0.995908	52.219991
296	46.78054	0.994218	52.225242
297	46.776982	0.992537	52.230481
298	46.773427	0.990864	52.235709
299	46.769874	0.9892	52.240926
300	46.766325	0.987544	52.246131
301	46.762778	0.985896	52.251326
302	46.759235	0.984256	52.256509
303	46.755694	0.982625	52.261681
304	46.752156	0.981001	52.266843
305	46.748621	0.979386	52.271993
306	46.745089	0.977778	52.277133
307	46.74156	0.976178	52.282262
308	46.738034	0.974586	52.28738
309	46.73451	0.973002	52.292488
310	46.73099	0.971425	52.297585
311	46.727472	0.969857	52.302671
312	46.723958	0.968294	52.307748
313	46.720446	0.966741	52.312813
314	46.716937	0.965194	52.317869
315	46.713431	0.963655	52.322914
316	46.709928	0.962123	52.327949
317	46.706428	0.960598	52.332974
318	46.70293	0.959081	52.337989
319	46.699436	0.95757	52.342994
320	46.695944	0.956067	52.347989
321	46.692456	0.95457	52.352974
322	46.68897	0.953081	52.357949
323	46.685487	0.951598	52.362915
324	46.682007	0.950123	52.36787
325	46.67853	0.948654	52.372816
326	46.675056	0.947192	52.377752
327	46.671584	0.945737	52.382679
328	46.668116	0.944288	52.387596
329	46.66465	0.942846	52.392504
330	46.661188	0.941409	52.397403
331	46.657728	0.93998	52.402292
332	46.654271	0.938558	52.407171
333	46.650817	0.937141	52.412042
334	46.647365	0.935732	52.416903
335	46.643917	0.934328	52.421755
336	46.640471	0.932931	52.426598
337	46.637029	0.931539	52.431432
338	46.633589	0.930155	52.436256
339	46.630152	0.928776	52.441072
340	46.626718	0.927403	52.445879
341	46.623286	0.926037	52.450677
342	46.619858	0.924675	52.455467
343	46.616432	0.923321	52.460247
344	46.61301	0.921971	52.465019
345	46.60959	0.920628	52.469782
346	46.606173	0.919291	52.474536
347	46.602758	0.91796	52.479282
348	46.599347	0.916634	52.484019
349	46.595938	0.915314	52.488748
350	46.592532	0.914	52.493468
351	46.589129	0.912691	52.49818
352	46.585729	0.911387	52.502884
353	46.582332	0.910089	52.507579
354	46.578937	0.908798	52.512265
355	46.575546	0.90751	52.516944
356	46.572157	0.906229	52.521614
357	46.568771	0.904952	52.526277
358	46.565387	0.903682	52.530931
359	46.562007	0.902416	52.535577
360	46.558629	0.901157	52.540214
361	46.555254	0.899902	52.544844
362	46.551882	0.898652	52.549466
363	46.548512	0.897408	52.55408
364	46.545146	0.896168	52.558686
365	46.541782	0.894933	52.563285
366	46.538421	0.893704	52.567875
367	46.535062	0.89248	52.572458
368	46.531707	0.89126	52.577033
369	46.528354	0.890046	52.5816
370	46.525004	0.888837	52.586159
371	46.521656	0.887633	52.590711
372	46.518312	0.886432	52.595256
373	46.51497	0.885238	52.599792
374	46.511631	0.884047	52.604322
375	46.508294	0.882863	52.608843
376	46.504961	0.881681	52.613358
377	46.50163	0.880505	52.617865
378	46.498301	0.879335	52.622364
379	46.494976	0.878168	52.626856
380	46.491653	0.877006	52.631341
381	46.488333	0.875848	52.635819
382	46.485016	0.874695	52.640289
383	46.481701	0.873547	52.644752
384	46.478389	0.872403	52.649208
385	46.47508	0.871263	52.653657
386	46.471773	0.870128	52.658099
387	46.468469	0.868998	52.662533
388	46.465168	0.867871	52.666961
389	46.461869	0.866749	52.671382
390	46.458573	0.865632	52.675795
391	46.45528	0.864518	52.680202
392	46.45199	0.863408	52.684602
393	46.448702	0.862303	52.688995
394	46.445416	0.861203	52.693381
395	46.442134	0.860106	52.69776
396	46.438854	0.859014	52.702132
397	46.435577	0.857925	52.706498
398	46.432302	0.856841	52.710857
399	46.42903	0.855761	52.715209
400	46.425761	0.854685	52.719554
401	46.422494	0.853613	52.723893
402	46.41923	0.852544	52.728226
403	46.415968	0.851481	52.732551
404	46.412709	0.850421	52.73687
405	46.409453	0.849364	52.741183
406	46.406199	0.848312	52.745489
407	46.402948	0.847263	52.749789
408	46.3997	0.846218	52.754082
409	46.396454	0.845177	52.758369
410	46.39321	0.844141	52.762649
411	46.38997	0.843107	52.766923
412	46.386731	0.842078	52.771191
413	46.383496	0.841052	52.775452
414	46.380263	0.84003	52.779707
415	46.377032	0.839012	52.783956
416	46.373804	0.837997	52.788199
417	46.370579	0.836986	52.792435
418	46.367356	0.835978	52.796666
419	46.364136	0.834974	52.80089
420	46.360918	0.833974	52.805108
421	46.357703	0.832977	52.80932
422	46.354491	0.831984	52.813525
423	46.35128	0.830995	52.817725
424	46.348073	0.830008	52.821919
425	46.344868	0.829025	52.826107
426	46.341665	0.828047	52.830288
427	46.338465	0.827071	52.834464
428	46.335268	0.826098	52.838634
429	46.332073	0.825129	52.842798
430	46.32888	0.824164	52.846956
431	46.32569	0.823201	52.851109
432	46.322503	0.822242	52.855255
433	46.319318	0.821286	52.859396
434	46.316135	0.820334	52.863531
435	46.312955	0.819385	52.86766
436	46.309778	0.818439	52.871783
437	46.306603	0.817496	52.875901
438	46.30343	0.816557	52.880013
439	46.30026	0.815621	52.884119
440	46.297092	0.814688	52.88822
441	46.293927	0.813758	52.892315
442	46.290764	0.812832	52.896404
443	46.287604	0.811908	52.900488
444	46.284446	0.810987	52.904567
445	46.28129	0.810071	52.908639
446	46.278137	0.809156	52.912707
447	46.274987	0.808244	52.916769
448	46.271838	0.807337	52.920825
449	46.268693	0.806431	52.924876
450	46.265549	0.80553	52.928921
451	46.262408	0.804631	52.932961
452	46.25927	0.803734	52.936996
453	46.256134	0.802841	52.941025
454	46.253	0.801951	52.945049
455	46.249868	0.801064	52.949068
456	46.246739	0.80018	52.953081
457	46.243613	0.799298	52.957089
458	46.240489	0.798419	52.961092
459	46.237367	0.797543	52.96509
460	46.234247	0.796671	52.969082
461	46.23113	0.795801	52.973069
462	46.228015	0.794934	52.977051
463	46.224903	0.794069	52.981028
464	46.221793	0.793207	52.985
465	46.218685	0.792349	52.988966
466	46.21558	0.791492	52.992928
467	46.212477	0.790639	52.996884
468	46.209376	0.789788	53.000836
469	46.206278	0.78894	53.004782
470	46.203182	0.788095	53.008723
471	46.200088	0.787253	53.012659
472	46.196997	0.786413	53.01659
473	46.193908	0.785575	53.020517
474	46.190821	0.784741	53.024438
475	46.187737	0.783909	53.028354
476	46.184655	0.783079	53.032266
477	46.181575	0.782253	53.036172
478	46.178497	0.781429	53.040074
479	46.175422	0.780607	53.043971
480	46.172349	0.779788	53.047863
481	46.169279	0.778971	53.05175
482	46.16621	0.778158	53.055632
483	46.163144	0.777346	53.05951
484	46.16008	0.776538	53.063382
485	46.157019	0.775731	53.06725
486	46.15396	0.774926	53.071114
487	46.150902	0.774126	53.074972
488	46.147848	0.773326	53.078826
489	46.144795	0.77253	53.082675
490	46.141745	0.771736	53.086519
491	46.138697	0.770944	53.090359
492	46.135651	0.770155	53.094194
493	46.132607	0.769368	53.098025
494	46.129566	0.768584	53.10185
495	46.126527	0.767801	53.105672
496	46.12349	0.767022	53.109488
497	46.120455	0.766245	53.1133
498	46.117423	0.765469	53.117108
499	46.114393	0.764696	53.120911
500	46.111365	0.763925	53.12471
a chart to 3000 rounds


Sally
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-13-2014 , 10:07 PM
#7
BruceZ's Avatar
BruceZ
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Carpal \'Tunnel
Join Date: Sep 2002 Posts: 11,877
Quote:
Using the normal dist did you not answer the question of being even or ahead.
I'm just using a continuous distribution rather than doing a correction for continuity with the discrete distribution. For the discrete distribution, you would need to model BJs, double's and splits. We already know this will be approximate for a small number of hands, so this is just another source of approximation.
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-13-2014 , 11:06 PM
#8
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BruceZ
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More to your point, we should at least consider being ahead by 0.25 bets rather than 0. If you could only win or lose 1 bet, then anything over 0.5 ahead would be considered 1 bet ahead when you correct for continuity. Because of BJs, we could actually only be 0.5 ahead, so we would consider anything over 0.25 bets ahead to be 0.5 bets ahead. Whether we use 0.5, 0.25, or 0 makes a significant difference which gets smaller as the hands increase, but it's significant even up to 500. If you were talking a very large number of hands, it wouldn't matter. Here are the results of using 0.5, 0.25, and 0:

0.5:
[1] 0.4185896 0.4386787 0.4543450 0.4596800 0.4601438 0.4557908 0.4416998

0.25:
[1] 0.4565534 0.4656667 0.4714651 0.4717939 0.4687073 0.4618379 0.4455064

0:
[1] 0.4949182 0.4928135 0.4886380 0.4839339 0.4772853 0.4678939 0.4493180


Which to use? Well, you shouldn't use 0 if you want to be ahead rather than ahead or even. At a sufficient number of hands, using 0.25 should be more accurate than 0.5, but perhaps not for a very small number of hands. Imagine if the probability of BJ were small enough, you could ignore it. You could split the difference between the two and be within about half a percent at 200. The real way to do it is to use a BJ simulator that takes everything into account if you're interested in a small number of hands like this.
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-14-2014 , 05:32 PM
#9
sallymustang
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centurion
Join Date: Jun 2011 Posts: 123
Quote:
Originally Posted by BruceZ
The real way to do it is to use a BJ simulator that takes everything into account if you're interested in a small number of hands like this.
I did just that too and the values are about .5% off from the Matrix calculations I show below for 5 to 50 rounds played.
the large difference is in the push values

I noticed the Wizards' sim data was changed and I had some old stuff
so here is using new stuff
of course, game rules and BS could move the values 1% high or low as I noticed with the Dealer hitting on soft 17.

But I think it gives the OP a ball park answer as there is no one simple formula fits all is what he/she really wanted.
They still sell Skittles!

I do wonder what the OP thought his/her winning chances were.
I find in casinos players really have no clue and when pushed for an educated guess they are so far off it really is great comedy!
Sally

data not formatted again for those that care in future reading
Code:
round	win	       push	    loss
1	42.423888	8.481647	49.094465
2	37.635512	26.675903	35.688585
3	42.323496	11.487638	46.188866
4	43.239856	13.516454	43.24369
5	44.014364	9.76373	46.221906
6	44.939134	9.214479	45.846387
7	45.2278	7.958355	46.813845
8	45.761803	7.265674	46.972523
9	45.975792	6.654919	47.369289
10	46.275252	6.14874	47.576008
11	46.445924	5.761864	47.792212
12	46.628292	5.412894	47.958814
13	46.759345	5.137076	48.103579
14	46.881687	4.888899	48.229414
15	46.980962	4.681783	48.337255
16	47.069403	4.49664	48.433957
17	47.145386	4.335774	48.51884
18	47.212652	4.191514	48.595834
19	47.272057	4.062852	48.665091
20	47.325058	3.946404	48.728538
21	47.372607	3.84072	48.786673
22	47.415488	3.744031	48.840481
23	47.454388	3.655158	48.890454
24	47.489818	3.573027	48.937155
25	47.522245	3.496784	48.980971
26	47.552027	3.425714	49.022259
27	47.579483	3.359212	49.061305
28	47.604874	3.296773	49.098353
29	47.628424	3.237964	49.133612
30	47.650323	3.18242	49.167257
31	47.670735	3.129824	49.199441
32	47.689803	3.079903	49.230294
33	47.70765	3.03242	49.25993
34	47.724382	2.987168	49.28845
35	47.740095	2.943965	49.31594
36	47.754872	2.90265	49.342478
37	47.768786	2.863081	49.368133
38	47.781902	2.825132	49.392966
39	47.794281	2.788688	49.417031
40	47.805974	2.753649	49.440377
41	47.817029	2.719921	49.46305
42	47.827489	2.687421	49.48509
43	47.837392	2.656075	49.506533
44	47.846774	2.625814	49.527412
45	47.855667	2.596573	49.54776
46	47.864101	2.568296	49.567603
47	47.872102	2.540931	49.586967
48	47.879696	2.514426	49.605878
49	47.886906	2.488738	49.624356
50	47.893752	2.463826	49.642422
51	47.900255	2.439649	49.660096
52	47.906432	2.416172	49.677396
53	47.912301	2.393362	49.694337
54	47.917876	2.371188	49.710936
55	47.923174	2.349619	49.727207
56	47.928207	2.328629	49.743164
57	47.932989	2.308192	49.758819
58	47.93753	2.288285	49.774185
59	47.941843	2.268885	49.789272
60	47.945939	2.249969	49.804092
61	47.949826	2.231519	49.818655
62	47.953514	2.213515	49.832971
63	47.957012	2.195941	49.847047
64	47.960328	2.178779	49.860893
65	47.96347	2.162013	49.874517
66	47.966446	2.145627	49.887927
67	47.969261	2.129609	49.90113
68	47.971924	2.113944	49.914132
69	47.97444	2.098619	49.926941
70	47.976814	2.083623	49.939563
71	47.979053	2.068944	49.952003
72	47.981162	2.054571	49.964267
73	47.983146	2.040492	49.976362
74	47.98501	2.026699	49.988291
75	47.986757	2.013182	50.000061
76	47.988394	1.999931	50.011675
77	47.989922	1.98694	50.023138
78	47.991348	1.974197	50.034455
79	47.992674	1.961696	50.04563
80	47.993903	1.94943	50.056667
81	47.99504	1.937391	50.067569
82	47.996088	1.925572	50.07834
83	47.997049	1.913966	50.088985
84	47.997927	1.902568	50.099505
85	47.998725	1.89137	50.109905
86	47.999444	1.880369	50.120187
87	48.000089	1.869556	50.130355
88	48.00066	1.858929	50.140411
89	48.001162	1.848479	50.150359
90	48.001595	1.838205	50.1602
91	48.001963	1.828099	50.169938
92	48.002267	1.818158	50.179575
93	48.002509	1.808378	50.189113
94	48.002691	1.798754	50.198555
95	48.002816	1.789281	50.207903
96	48.002885	1.779956	50.217159
97	48.002899	1.770775	50.226326
98	48.00286	1.761736	50.235404
99	48.002771	1.752832	50.244397
100	48.002631	1.744064	50.253305
101	48.002444	1.735424	50.262132
102	48.00221	1.726912	50.270878
103	48.001931	1.718524	50.279545
104	48.001608	1.710256	50.288136
105	48.001242	1.702107	50.296651
106	48.000835	1.694073	50.305092
107	48.000387	1.686152	50.313461
108	47.999901	1.67834	50.321759
109	47.999376	1.670636	50.329988
110	47.998815	1.663037	50.338148
111	47.998217	1.655541	50.346242
112	47.997585	1.648144	50.354271
113	47.996918	1.640847	50.362235
114	47.996219	1.633645	50.370136
115	47.995487	1.626538	50.377975
116	47.994724	1.619522	50.385754
117	47.993931	1.612596	50.393473
118	47.993109	1.605757	50.401134
119	47.992257	1.599006	50.408737
120	47.991377	1.592339	50.416284
121	47.99047	1.585754	50.423776
122	47.989537	1.579249	50.431214
123	47.988577	1.572825	50.438598
124	47.987593	1.566478	50.445929
125	47.986584	1.560207	50.453209
126	47.985551	1.554011	50.460438
127	47.984494	1.547888	50.467618
128	47.983415	1.541837	50.474748
129	47.982314	1.535856	50.48183
130	47.981191	1.529944	50.488865
131	47.980047	1.5241	50.495853
132	47.978882	1.518322	50.502796
133	47.977698	1.512609	50.509693
134	47.976494	1.50696	50.516546
135	47.975271	1.501374	50.523355
136	47.974029	1.49585	50.530121
137	47.97277	1.490386	50.536844
138	47.971492	1.484982	50.543526
139	47.970198	1.479635	50.550167
140	47.968886	1.474347	50.556767
141	47.967559	1.469113	50.563328
142	47.966215	1.463936	50.569849
143	47.964855	1.458813	50.576332
144	47.963481	1.453743	50.582776
145	47.962091	1.448726	50.589183
146	47.960687	1.44376	50.595553
147	47.959269	1.438844	50.601887
148	47.957837	1.433979	50.608184
149	47.956391	1.429163	50.614446
150	47.954932	1.424395	50.620673
151	47.953461	1.419673	50.626866
152	47.951977	1.414998	50.633025
153	47.95048	1.41037	50.63915
154	47.948971	1.405787	50.645242
155	47.947451	1.401248	50.651301
156	47.945919	1.396752	50.657329
157	47.944376	1.3923	50.663324
158	47.942823	1.387889	50.669288
159	47.941258	1.38352	50.675222
160	47.939683	1.379193	50.681124
161	47.938098	1.374905	50.686997
162	47.936503	1.370657	50.69284
163	47.934898	1.366448	50.698654
164	47.933284	1.362278	50.704438
165	47.93166	1.358146	50.710194
166	47.930027	1.354051	50.715922
167	47.928386	1.349992	50.721622
168	47.926736	1.34597	50.727294
169	47.925077	1.341984	50.732939
170	47.92341	1.338033	50.738557
171	47.921735	1.334116	50.744149
172	47.920052	1.330234	50.749714
173	47.918361	1.326385	50.755254
174	47.916663	1.32257	50.760767
175	47.914957	1.318787	50.766256
176	47.913245	1.315036	50.771719
177	47.911525	1.311317	50.777158
178	47.909798	1.30763	50.782572
179	47.908065	1.303972	50.787963
180	47.906324	1.300347	50.793329
181	47.904578	1.296751	50.798671
182	47.902825	1.293184	50.803991
183	47.901066	1.289647	50.809287
184	47.899301	1.286139	50.81456
185	47.897531	1.282658	50.819811
186	47.895754	1.279206	50.82504
187	47.893972	1.275782	50.830246
188	47.892185	1.272384	50.835431
189	47.890392	1.269014	50.840594
190	47.888593	1.265672	50.845735
191	47.88679	1.262354	50.850856
192	47.884982	1.259063	50.855955
193	47.883169	1.255797	50.861034
194	47.881351	1.252556	50.866093
195	47.879528	1.249341	50.871131
196	47.877701	1.24615	50.876149
197	47.87587	1.242983	50.881147
198	47.874034	1.23984	50.886126
199	47.872194	1.236721	50.891085
200	47.870349	1.233626	50.896025
201	47.868501	1.230553	50.900946
202	47.866649	1.227503	50.905848
203	47.864793	1.224476	50.910731
204	47.862933	1.221471	50.915596
205	47.861069	1.218489	50.920442
206	47.859202	1.215527	50.925271
207	47.857332	1.212587	50.930081
208	47.855457	1.209669	50.934874
209	47.85358	1.206771	50.939649
210	47.851699	1.203894	50.944407
211	47.849815	1.201038	50.949147
212	47.847928	1.198201	50.953871
213	47.846038	1.195385	50.958577
214	47.844145	1.192588	50.963267
215	47.84225	1.18981	50.96794
216	47.840351	1.187053	50.972596
217	47.838449	1.184314	50.977237
218	47.836545	1.181594	50.981861
219	47.834639	1.178892	50.986469
220	47.832729	1.17621	50.991061
221	47.830817	1.173545	50.995638
222	47.828903	1.170898	51.000199
223	47.826987	1.168269	51.004744
224	47.825068	1.165658	51.009274
225	47.823147	1.163064	51.013789
226	47.821224	1.160486	51.01829
227	47.819298	1.157927	51.022775
228	47.817371	1.155384	51.027245
229	47.815441	1.152858	51.031701
230	47.81351	1.150348	51.036142
231	47.811577	1.147854	51.040569
232	47.809641	1.145378	51.044981
233	47.807704	1.142916	51.04938
234	47.805766	1.14047	51.053764
235	47.803825	1.13804	51.058135
236	47.801883	1.135626	51.062491
237	47.799939	1.133226	51.066835
238	47.797994	1.130842	51.071164
239	47.796047	1.128473	51.07548
240	47.794099	1.126118	51.079783
241	47.792149	1.123779	51.084072
242	47.790198	1.121453	51.088349
243	47.788246	1.119142	51.092612
244	47.786292	1.116845	51.096863
245	47.784337	1.114562	51.101101
246	47.782381	1.112293	51.105326
247	47.780423	1.110039	51.109538
248	47.778465	1.107797	51.113738
249	47.776505	1.105569	51.117926
250	47.774544	1.103355	51.122101
251	47.772583	1.101153	51.126264
252	47.77062	1.098965	51.130415
253	47.768656	1.09679	51.134554
254	47.766692	1.094627	51.138681
255	47.764726	1.092477	51.142797
256	47.76276	1.09034	51.1469
257	47.760792	1.088216	51.150992
258	47.758824	1.086104	51.155072
259	47.756855	1.084004	51.159141
260	47.754886	1.081915	51.163199
261	47.752916	1.079839	51.167245
262	47.750945	1.077775	51.17128
263	47.748973	1.075723	51.175304
264	47.747001	1.073682	51.179317
265	47.745028	1.071653	51.183319
266	47.743055	1.069635	51.18731
267	47.741081	1.067629	51.19129
268	47.739106	1.065634	51.19526
269	47.737132	1.063649	51.199219
270	47.735156	1.061677	51.203167
271	47.73318	1.059715	51.207105
272	47.731204	1.057764	51.211032
273	47.729228	1.055823	51.214949
274	47.727251	1.053893	51.218856
275	47.725274	1.051973	51.222753
276	47.723296	1.050064	51.22664
277	47.721318	1.048166	51.230516
278	47.71934	1.046277	51.234383
279	47.717362	1.044399	51.238239
280	47.715383	1.042531	51.242086
281	47.713404	1.040673	51.245923
282	47.711425	1.038825	51.24975
283	47.709446	1.036986	51.253568
284	47.707467	1.035157	51.257376
285	47.705488	1.033337	51.261175
286	47.703508	1.031528	51.264964
287	47.701529	1.029727	51.268744
288	47.699549	1.027937	51.272514
289	47.697569	1.026155	51.276276
290	47.69559	1.024382	51.280028
291	47.69361	1.022619	51.283771
292	47.69163	1.020865	51.287505
293	47.689651	1.01912	51.291229
294	47.687671	1.017384	51.294945
295	47.685692	1.015656	51.298652
296	47.683712	1.013937	51.302351
297	47.681733	1.012227	51.30604
298	47.679754	1.010525	51.309721
299	47.677775	1.008832	51.313393
300	47.675796	1.007148	51.317056
301	47.673817	1.005472	51.320711
302	47.671838	1.003804	51.324358
303	47.66986	1.002144	51.327996
304	47.667881	1.000494	51.331625
305	47.665903	0.99885	51.335247
306	47.663925	0.997215	51.33886
307	47.661948	0.995587	51.342465
308	47.659971	0.993968	51.346061
309	47.657994	0.992356	51.34965
310	47.656017	0.990753	51.35323
311	47.65404	0.989158	51.356802
312	47.652064	0.987569	51.360367
313	47.650088	0.985989	51.363923
314	47.648113	0.984415	51.367472
315	47.646138	0.982849	51.371013
316	47.644163	0.981291	51.374546
317	47.642188	0.979741	51.378071
318	47.640214	0.978197	51.381589
319	47.638241	0.976661	51.385098
320	47.636267	0.975132	51.388601
321	47.634294	0.97361	51.392096
322	47.632322	0.972095	51.395583
323	47.63035	0.970587	51.399063
324	47.628378	0.969087	51.402535
325	47.626407	0.967593	51.406
326	47.624437	0.966105	51.409458
327	47.622466	0.964626	51.412908
328	47.620497	0.963152	51.416351
329	47.618527	0.961686	51.419787
330	47.616559	0.960225	51.423216
331	47.61459	0.958772	51.426638
332	47.612623	0.957325	51.430052
333	47.610656	0.955884	51.43346
334	47.608689	0.954451	51.43686
335	47.606723	0.953023	51.440254
336	47.604757	0.951602	51.443641
337	47.602792	0.950188	51.44702
338	47.600828	0.948779	51.450393
339	47.598864	0.947376	51.45376
340	47.596901	0.94598	51.457119
341	47.594938	0.94459	51.460472
342	47.592976	0.943206	51.463818
343	47.591015	0.941828	51.467157
344	47.589054	0.940456	51.47049
345	47.587094	0.93909	51.473816
346	47.585134	0.937731	51.477135
347	47.583175	0.936377	51.480448
348	47.581217	0.935028	51.483755
349	47.579259	0.933686	51.487055
350	47.577302	0.932349	51.490349
351	47.575346	0.931018	51.493636
352	47.57339	0.929693	51.496917
353	47.571435	0.928373	51.500192
354	47.56948	0.92706	51.50346
355	47.567527	0.92575	51.506723
356	47.565574	0.924447	51.509979
357	47.563622	0.92315	51.513228
358	47.56167	0.921858	51.516472
359	47.559719	0.920571	51.51971
360	47.557769	0.91929	51.522941
361	47.55582	0.918013	51.526167
362	47.553871	0.916743	51.529386
363	47.551923	0.915477	51.5326
364	47.549976	0.914217	51.535807
365	47.548029	0.912962	51.539009
366	47.546083	0.911712	51.542205
367	47.544138	0.910467	51.545395
368	47.542194	0.909227	51.548579
369	47.540251	0.907992	51.551757
370	47.538308	0.906762	51.55493
371	47.536366	0.905537	51.558097
372	47.534425	0.904317	51.561258
373	47.532484	0.903103	51.564413
374	47.530545	0.901892	51.567563
375	47.528606	0.900687	51.570707
376	47.526668	0.899486	51.573846
377	47.52473	0.898291	51.576979
378	47.522794	0.8971	51.580106
379	47.520858	0.895914	51.583228
380	47.518923	0.894732	51.586345
381	47.516989	0.893555	51.589456
382	47.515056	0.892383	51.592561
383	47.513124	0.891215	51.595661
384	47.511192	0.890052	51.598756
385	47.509261	0.888893	51.601846
386	47.507331	0.887739	51.60493
387	47.505402	0.88659	51.608008
388	47.503474	0.885444	51.611082
389	47.501546	0.884304	51.61415
390	47.49962	0.883167	51.617213
391	47.497694	0.882035	51.620271
392	47.495769	0.880907	51.623324
393	47.493845	0.879784	51.626371
394	47.491921	0.878665	51.629414
395	47.489999	0.87755	51.632451
396	47.488078	0.876439	51.635483
397	47.486157	0.875332	51.638511
398	47.484237	0.87423	51.641533
399	47.482318	0.873132	51.64455
400	47.4804	0.872038	51.647562
401	47.478483	0.870948	51.650569
402	47.476566	0.869862	51.653572
403	47.474651	0.86878	51.656569
404	47.472736	0.867702	51.659562
405	47.470823	0.866628	51.662549
406	47.46891	0.865558	51.665532
407	47.466998	0.864492	51.66851
408	47.465087	0.86343	51.671483
409	47.463177	0.862372	51.674451
410	47.461267	0.861318	51.677415
411	47.459359	0.860267	51.680374
412	47.457451	0.859221	51.683328
413	47.455545	0.858178	51.686277
414	47.453639	0.857139	51.689222
415	47.451734	0.856104	51.692162
416	47.44983	0.855072	51.695098
417	47.447927	0.854044	51.698029
418	47.446025	0.85302	51.700955
419	47.444124	0.851999	51.703877
420	47.442224	0.850982	51.706794
421	47.440325	0.849968	51.709707
422	47.438426	0.848959	51.712615
423	47.436529	0.847953	51.715518
424	47.434632	0.846951	51.718417
425	47.432736	0.845952	51.721312
426	47.430841	0.844957	51.724202
427	47.428947	0.843965	51.727088
428	47.427055	0.842975	51.72997
429	47.425163	0.84199	51.732847
430	47.423271	0.841009	51.73572
431	47.421381	0.840031	51.738588
432	47.419492	0.839056	51.741452
433	47.417604	0.838084	51.744312
434	47.415716	0.837117	51.747167
435	47.41383	0.836151	51.750019
436	47.411944	0.83519	51.752866
437	47.41006	0.834232	51.755708
438	47.408176	0.833277	51.758547
439	47.406293	0.832326	51.761381
440	47.404411	0.831377	51.764212
441	47.40253	0.830432	51.767038
442	47.400651	0.82949	51.769859
443	47.398772	0.828551	51.772677
444	47.396893	0.827616	51.775491
445	47.395016	0.826684	51.7783
446	47.39314	0.825754	51.781106
447	47.391265	0.824828	51.783907
448	47.389391	0.823904	51.786705
449	47.387517	0.822985	51.789498
450	47.385645	0.822068	51.792287
451	47.383773	0.821154	51.795073
452	47.381903	0.820243	51.797854
453	47.380033	0.819335	51.800632
454	47.378165	0.81843	51.803405
455	47.376297	0.817528	51.806175
456	47.37443	0.81663	51.80894
457	47.372564	0.815734	51.811702
458	47.3707	0.81484	51.81446
459	47.368836	0.81395	51.817214
460	47.366973	0.813063	51.819964
461	47.365111	0.812178	51.822711
462	47.36325	0.811297	51.825453
463	47.361389	0.810419	51.828192
464	47.35953	0.809543	51.830927
465	47.357672	0.80867	51.833658
466	47.355815	0.8078	51.836385
467	47.353959	0.806932	51.839109
468	47.352103	0.806068	51.841829
469	47.350249	0.805206	51.844545
470	47.348395	0.804347	51.847258
471	47.346543	0.803491	51.849966
472	47.344691	0.802638	51.852671
473	47.342841	0.801786	51.855373
474	47.340991	0.800938	51.858071
475	47.339142	0.800093	51.860765
476	47.337295	0.799249	51.863456
477	47.335448	0.798409	51.866143
478	47.333602	0.797572	51.868826
479	47.331757	0.796737	51.871506
480	47.329913	0.795905	51.874182
481	47.32807	0.795075	51.876855
482	47.326228	0.794248	51.879524
483	47.324387	0.793423	51.88219
484	47.322547	0.792601	51.884852
485	47.320708	0.791781	51.887511
486	47.31887	0.790964	51.890166
487	47.317033	0.790149	51.892818
488	47.315196	0.789338	51.895466
489	47.313361	0.788528	51.898111
490	47.311527	0.787721	51.900752
491	47.309693	0.786917	51.90339
492	47.307861	0.786114	51.906025
493	47.306029	0.785315	51.908656
494	47.304199	0.784517	51.911284
495	47.302369	0.783723	51.913908
496	47.30054	0.782931	51.916529
497	47.298713	0.78214	51.919147
498	47.296886	0.781353	51.921761
499	47.29506	0.780568	51.924372
500	47.293235	0.779785	51.92698
600	47.115486	0.711657	52.172857
700	46.946638	0.658686	52.394676
800	46.785824	0.615973	52.598203
900	46.632177	0.580581	52.787242
1000	46.484921	0.550632	52.964447
2000	45.253827	0.388251	54.357922
3000	44.282927	0.316124	55.400949
give it a spin if so inclined
gas tank is full and looks like all the tires have air in them
Blackjack Odds of a Winning Session after X Amount of Hands Quote
08-15-2014 , 01:06 AM
#10
BruceZ's Avatar
BruceZ
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Carpal \'Tunnel
Join Date: Sep 2002 Posts: 11,877
Sorry, I didn't realize what you did before. That's how to do it, except that there is a very large difference in the long term win rates between the game the Wizard analyzed and the game the OP wants. The Wizard's game only has a 0.29% house edge compared to the OP's 0.66083%. I suspect that's because the dealer hits soft 17 in the OP game, right OP? If I analyze the Wizard's game with the normal distribution, I agree with your win rate at 3000 hands of 44.3%, but for the OP game, I get only 37.4%. They are about 3% off at 500 hands, and they get closer below that.

Also, if you take my values for using 0.25 and 0.5 as the cutoff, your value for 5 hands of 44.0% agrees exactly with simply averaging my 2 results. When you get out to 10 hands, your value is about 0.6% from my 0.25 value. At 25 hands, it's only about 0.2% away. Then they're within 0.1% after that. So that works just like I predicted. Using 0.25 is the better estimate except for 5 hands, and then the average of the 2 estimates is perfect. There is a rise in the probabilities up to 200 when they start to decrease, and my numbers capture that. You can see that if you plot your second set of numbers and blow that part up. So there is an optimal number of hands to optimize the probability of being ahead. That's because the edge is much smaller than the standard deviation, so even though the edge increases faster with the number of hands (linear vs. square root), initially the increase in standard deviation will increase your win rate until the edge catches up.

Here are my values for the probability of being ahead for the same game as your sim (not the OP game):

Code:
5     0.4394
10    0.4692
25    0.4775
50    0.4805
100   0.4811
200   0.4795
500   0.4735
3000  0.4431
These compare well with your numbers as described above.

Here are my values for the OP game:

Code:
5     0.4376
10    0.4657
25    0.4715
50    0.4718
100   0.4687
200   0.4618
500   0.4455
3000  0.376
If we had data like the Wizard's for this game, and we used Sally's method, the results should agree nicely just as they did for the Wizard's game.
Last edited by BruceZ; 08-15-2014 at 02:05 PM.
Blackjack Odds of a Winning Session after X Amount of Hands Quote
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