BACCARAT: How many more hands can player win than bank?

I was curious to know if anyone had the maths on the ability of player results in Baccarat to exceed bank results?? If the outcomes must eventually reach the probabilities of the game itself, is there anyway to determine how many more than bank results the player can win before the odds of the game are realised over say a million hands? I know that in theory player could win 400,000 hands in a row and then bank win the rest to make odds add up but what is the maths/probability behind the difference that player and bank can achieve? Mathematically, how many hands can the player win, more than the bank??

jms, i think you have to ask the question a different way, and then someone here can help you..... How does this sound....
What are the odds of player winning baccarat 500,001 or more out of 1,000,000 (count a tie as 1/2 win for player and 1/2 win for banker)?

Quote: jms9smith

I know that in theory player could win 400,000 hands in a row and then bank win the rest to make odds add up...

In theory, the player hand could win 400,000 hands in a row... and then win the next 600,000 in a row. In theory, anything can happen. In reality, the odds of this happening are extremely small. In reality, the odds will, over an infinite number of hands (or let's say 10 billion to be a little more realistic) end up being very close to what the math of the game suggests. The math of the game only tells you what, over the long run, you should expect to see happen. You play in reality though so you'll never be able to know with any certainty what is going to happen next based upon what has happened in the past. To answer your question a little more clearly, you could in your lifetime see a shoe where player wins every hand but it would be extremely unlikely. Using math (and keep in mind, mine is a little fuzzy) the odds of having player win every hand in an 80 hand shoe (excluding ties) are approximately 2.76596E-25.

Hi guys, thanks for the replies. I know my question wasn't worded too well, so i'm sorry for that. what i really was curious on was; is there a way to work out the probability of player winning more hands than bank?
For example, if i wanted to know what the odds of player winning 50 more hands than bank, what would the math be?
Also has anyone experienced player winning say 100 more hands than bank?
thank you for the replies.

Quote: SOOPOO

jms, i think you have to ask the question a different way, and then someone here can help you..... How does this sound....
What are the odds of player winning baccarat 500,001 or more out of 1,000,000 (count a tie as 1/2 win for player and 1/2 win for banker)?

that's a really good question as well...... is there a way to work that out?

Quote: jms9smith

I was curious to know if anyone had the maths on the ability of player results in Baccarat to exceed bank results?? If the outcomes must eventually reach the probabilities of the game itself, is there anyway to determine how many more than bank results the player can win before the odds of the game are realised over say a million hands? I know that in theory player could win 400,000 hands in a row and then bank win the rest to make odds add up but what is the maths/probability behind the difference that player and bank can achieve? Mathematically, how many hands can the player win, more than the bank??

You should ask the question more precisely. Like for example: What is the probability that after 1,000 Baccarat hands there will be more player wins than banker wins in total? It would be easy to find an answer to questions like these.

I thought of the following that might be something that you are seeking:

If we play Baccarat indefinitely long then what is the probability that at some point there will be 10 more player wins than banker wins?

To answer this, I took the player's and banker's winning probabilities from 8 deck game:

P = P(Player wins) = 0.493176
Q = P(Banker wins) = 0.506824

Note that these probabilities have been formed by eliminating ties (non-relevant outcome) already.

The answer to the above question is obtained from a surprisingly simple risk of ruin formula: PP = (P/Q)^T, where T is number of number of surplus player wins we are looking for.

Results in my next post.

[Post edited to make equations simpler].

thanks for that but i see that on the wizard of odds site an example of 282 player and 214 bank (a difference of only 68) has a probability of 0.000393, or 1 in 2,544. That's less than the value you gave for the N=100 example. How come?
Are the odds difference if all you want to know is what is the chance of player winning 100 more hands than bank? Does anyone have a formula for working out what the odds of player winning 50 or 100 or N more hands than bank?

Quote: jms9smith

thanks for that but i see that on the wizard of odds site an example of 282 player and 214 bank (a difference of only 68) has a probability of 0.000393, or 1 in 2,544. That's less than the value you gave for the N=100 example. How come?

That's because in the example it's odds of player winning 68 more hands than banker in 282+214 = 496 hands played. With so low number of hands the probability is low. My formulas on the other hand gave the odds with no constraint for the total number of hands (ie. playing indefinitely long).

Quote: jms9smith

Are the odds difference if all you want to know is what is the chance of player winning 100 more hands than bank? Does anyone have a formula for working out what the odds of player winning 50 or 100 or N more hands than bank?

I gave you the formula in my previous reply. But like said that formula gives the answer for the "infinite case" ie. not restricting the total number of hands played. If you state the question as: "What is the probability for player winning 50 or 100 more hands than bank in XXX hands played" then that requires a different formula.

Actually in my previous post the stopping condition "Banker wins N more hands than player" is not necessary. So one can play infinitely long. The probability that in such infinite Baccarat session there will be T more player wins than banker wins is obtained simply from equation:

(P/Q)^T

Where

P = Prob(Player wins)
Q = Prob(Banker wins)

For T = 1, we get 97.3%, which shows that there is a 97.3% chance that such infinite session will have one more player wins than banker wins at some point in the future.

Values for other T are:

T= 10, PP = 0.7611
T = 100, PP = 0.065
T = 200, PP = 0.0042555
T = 300, PP = 1 / 3602
T = 500, PP = 1 / 846485

So we see that between T = 200 and T = 300 the probabilities start to fall under less than 1 in 1000 chance region. This means that it is extremely unlikely to find a sequence of Baccarat results (no matter how long they are) where player has won more than 300 hands than the banker.

Quote: jms9smith

thanks for that but i see that on the wizard of odds site an example of 282 player and 214 bank (a difference of only 68) has a probability of 0.000393, or 1 in 2,544. That's less than the value you gave for the N=100 example. How come?
Are the odds difference if all you want to know is what is the chance of player winning 100 more hands than bank? Does anyone have a formula for working out what the odds of player winning 50 or 100 or N more hands than bank?

askthewizard/baccarat
Question#4
The Wizard used the normal distribution to answer that question and it is an approximation to the binomial distribution.
Since we know the number of hands (trials) this can be calculated very easily.

Jufo81 solution, a gamblers ruin solution, assumes no exact number of trials and a nice solution I might add.

With n=496 I used Excel BINOMDIST function. A spreadsheet is very handy to use for these type of calculations.
There are even online calculators available.

= 1 - BINOMDIST(281,496,0.49317517006,TRUE)
0.000456209 or 1 in 2,192

Here is a sample of my Binomial Dist table.
You can create your own and/or compare the results to the Wizards using his formula of expected value, standard deviation and z-score.

The AB diff column (absolute difference) is the column to find the difference between Player and Banker wins.
Expectation is 245 wins (the 5th row)

For 298 Player wins in 496 trials, (a 100 win difference) the "1 in (or more) column shows a 1 in 1,046,790.716 of Player being 100 wins or more than the Banker

For 310 Player wins in 496 trials, (a 124 win difference) the "1 in (or more) column shows a 1 in 415,584,611.3 of Player being 124 wins or more than the Banker

The NET column is from the Player wager of $10 every hand.
I hope this helps out.
n=496

wins	Ab diff	losses	(or less)	exact	1 in (exact)	(or more)	1 in (or more)	wins	NET	EDGE
241	-14	255	0.3899233	0.033982991	29.42648562	0.644059691	1.552651118	241	-140	-0.028225806
242	-12	254	0.424767439	0.034844139	28.69923125	0.6100767	1.63913816	242	-120	-0.024193548
243	-10	253	0.460207992	0.035440554	28.21626341	0.575232561	1.73842732	243	-100	-0.02016129
244	-8	252	0.495966099	0.035758106	27.96568661	0.539792008	1.852565406	244	-80	-0.016129032
245	-6	251	0.531755323	0.035789225	27.94137091	0.504033901	1.983993532	245	-60	-0.012096774
246	-4	250	0.567288516	0.035533192	28.14270099	0.468244677	2.135635598	246	-40	-0.008064516
247	-2	249	0.602284692	0.034996177	28.5745501	0.432711484	2.311008688	247	-20	-0.004032258
248	0	248	0.636475675	0.034190983	29.24747749	0.397715308	2.514361354	248	0	0
249	2	247	0.669612221	0.033136546	30.17816036	0.363524325	2.750847553	249	20	0.004032258
250	4	246	0.701469414	0.031857193	31.39008543	0.330387779	3.026746337	250	40	0.008064516
251	6	245	0.731851123	0.03038171	32.91454018	0.298530586	3.349740514	251	60	0.012096774
252	8	244	0.760593394	0.02874227	34.79196271	0.268148877	3.729271634	252	80	0.016129032
253	10	243	0.78756667	0.026973276	37.07373184	0.239406606	4.176994173	253	100	0.02016129
254	12	242	0.812676835	0.025110165	39.82450878	0.21243333	4.707359235	254	120	0.024193548
255	14	241	0.83586509	0.023188256	43.12527946	0.187323165	5.338368052	255	140	0.028225806
256	16	240	0.857106751	0.021241661	47.07729767	0.16413491	6.092549126	256	160	0.032258065
257	18	239	0.876409089	0.019302338	51.80719602	0.142893249	6.998231269	257	180	0.036290323
258	20	238	0.893808375	0.017399286	57.47362272	0.123590911	8.091209898	258	200	0.040322581
259	22	237	0.90936631	0.015557935	64.27588427	0.106191625	9.416938511	259	220	0.044354839
260	24	236	0.923166029	0.013799719	72.46524125	0.09063369	11.03342475	260	240	0.048387097
261	26	235	0.935307885	0.012141856	82.35973245	0.076833971	13.01507642	261	260	0.052419355
262	28	234	0.945905179	0.010597294	94.36371635	0.064692115	15.45783443	262	280	0.056451613
263	30	233	0.955080016	0.009174838	108.9937532	0.054094821	18.48605792	263	300	0.060483871
264	32	232	0.962959427	0.00787941	126.9130537	0.044919984	22.2618069	264	320	0.064516129
265	34	231	0.969671847	0.00671242	148.9775622	0.037040573	26.9974222	265	340	0.068548387
266	36	230	0.975344063	0.005672217	176.2979249	0.030328153	32.97266379	266	360	0.072580645
267	38	229	0.980098649	0.004754586	210.3232607	0.024655937	40.55818326	267	380	0.076612903
268	40	228	0.984051921	0.003953272	252.9550174	0.019901351	50.24784531	268	400	0.080645161
269	42	227	0.987312409	0.003260488	306.7025674	0.015948079	62.7034774	269	420	0.084677419
270	44	226	0.989979808	0.002667399	374.8970291	0.012687591	78.81716896	270	440	0.088709677
271	46	225	0.992144372	0.002164564	461.9867616	0.010020192	99.79848839	271	460	0.092741935
272	48	224	0.99388669	0.001742318	573.9480664	0.007855628	127.2972757	272	480	0.096774194
273	50	223	0.995277783	0.001391093	718.8593174	0.00611331	163.5775123	273	500	0.100806452
274	52	222	0.996379459	0.001101676	907.7082575	0.004722217	211.7649413	274	520	0.10483871
275	54	221	0.997244859	0.000865401	1155.533887	0.003620541	276.2017846	275	540	0.108870968
276	56	220	0.997919145	0.000674286	1483.051303	0.002755141	362.9578868	276	560	0.112903226
277	58	219	0.998440256	0.000521111	1918.977768	0.002080855	480.5716731	277	580	0.116935484
278	60	218	0.998839715	0.000399459	2503.383034	0.001559744	641.1307348	278	600	0.120967742
279	62	217	0.999143432	0.000303716	3292.544813	0.001160285	861.8573352	279	620	0.125
280	64	216	0.999372473	0.000229041	4366.029567	0.000856568	1167.449096	280	640	0.129032258
281	66	215	0.999543791	0.000171318	5837.083709	0.000627527	1593.556053	281	660	0.133064516
282	68	214	0.999670889	0.000127097	7867.980054	0.000456209	2191.978191	282	680	0.137096774
283	70	213	0.999764409	9.35206E-05	10692.82829	0.000329111	3038.483972	283	700	0.141129032
284	72	212	0.999832661	6.82515E-05	14651.69973	0.000235591	4244.646704	284	720	0.14516129
285	74	211	0.999882063	4.94022E-05	20242.01252	0.000167339	5975.878936	285	740	0.149193548
286	76	210	0.999917528	3.54655E-05	28196.41846	0.000117937	8479.088994	286	760	0.153225806
287	78	209	0.99994278	2.52515E-05	39601.6453	8.24717E-05	12125.37133	287	780	0.157258065
288	80	208	0.999960611	1.78313E-05	56081.04681	5.72202E-05	17476.33813	288	800	0.161290323
289	82	207	0.999973099	1.2488E-05	80076.90879	3.93889E-05	25387.8703	289	820	0.165322581
290	84	206	0.999981773	8.67378E-06	115289.9982	2.69009E-05	37173.48541	290	840	0.169354839
291	86	205	0.999987748	5.97484E-06	167368.6378	1.82271E-05	54863.31778	291	860	0.173387097
292	88	204	0.999991829	4.08169E-06	244996.4278	1.22523E-05	81617.46272	292	880	0.177419355
293	90	203	0.999994595	2.76532E-06	361621.202	8.17059E-06	122390.2159	293	900	0.181451613
294	92	202	0.999996453	1.85797E-06	538222.5112	5.40526E-06	185004.8679	294	920	0.185483871
295	94	201	0.999997691	1.23797E-06	807772.7255	3.5473E-06	281904.9006	295	940	0.189516129
296	96	200	0.999998509	8.1801E-07	1222479.314	2.30932E-06	433027.2344	296	960	0.193548387
297	98	199	0.999999045	5.36013E-07	1865626.29	1.49131E-06	670549.6762	297	980	0.197580645
298	100	198	0.999999393	3.48302E-07	2871074.891	9.55301E-07	1046790.716	298	1000	0.201612903
299	102	197	0.999999617	2.24436E-07	4455610.302	6.06999E-07	1647448.673	299	1020	0.205645161
300	104	196	0.999999761	1.43411E-07	6972987.848	3.82563E-07	2613948.286	300	1040	0.209677419
301	106	195	0.999999852	9.08686E-08	11004897.78	2.39152E-07	4181432.485	301	1060	0.213709677
302	108	194	0.999999909	5.70933E-08	17515196.94	1.48284E-07	6743822.249	302	1080	0.217741935
303	110	193	0.999999944	3.55703E-08	28113350.28	9.11906E-08	10966044.56	303	1100	0.221774194
304	112	192	0.999999966	2.19743E-08	45507770.44	5.56203E-08	17979049.11	304	1120	0.225806452
305	114	191	0.99999998	1.34604E-08	74291794.65	3.3646E-08	29721192.95	305	1140	0.22983871
306	116	190	0.999999988	8.1755E-09	122316653.1	2.01856E-08	49540295.29	306	1160	0.233870968
307	118	189	0.999999993	4.92349E-09	203108006.7	1.20101E-08	83263346.38	307	1180	0.237903226
308	120	188	0.999999996	2.93986E-09	340151692.9	7.0866E-09	141111438.1	308	1200	0.241935484
309	122	187	0.999999998	1.74048E-09	574552802.3	4.14673E-09	241153701.1	309	1220	0.245967742
310	124	186	0.999999999	1.02163E-09	978828745.3	2.40625E-09	415584611.3	310	1240	0.25

Note: There are limits to using Excel Binomial distribution function. Large values and small probabilities can easily max out the results and return no values at all.
That is anther good reason to understand how to use the normal distribution solution that the Wizard used.

Quote: guido111

With n=496 I used Excel BINOMDIST function. A spreadsheet is very handy to use for these type of calculations.
There are even online calculators available.

= 1 - BINOMDIST(281,496,0.49317517006,TRUE)
0.000456209 or 1 in 2,192

This would show the probability of there being 281 or more player wins after exactly 496 total hands have been played (66+ more player wins than banker wins) but it could be also the case that the asked event of 66+ extra player wins is achieved at some point before all those 496 hands are played and not necessarily at the very end. So would there be a solution that says, what is the chance to see 66+ surplus player wins at any point within the first 496 hands played?

So, the OP should state whether he wishes to

a) play fixed number of hands and get the probability for certain number of surplus player wins at the end of that sample (guido's solution)

or

b) Play fixed number of hands and get the probability for certain number of surplus player wins at any point within the sample (not answered)

or

c) Play indefinitely long, up until the event of certain number of surplus player wins occurs (or doesn't) and get the overall chance of it occuring (my solution)

Quote: Jufo81

So, the OP should state whether he wishes to

a) play fixed number of hands and get the probability for certain number of surplus player wins at the end of that sample (guido's solution)

or

b) Play fixed number of hands and get the probability for certain number of surplus player wins at any point within the sample (not answered)

or

c) Play indefinitely long, up until the event of certain number of surplus player wins occurs (or doesn't) and get the overall chance of it occuring (my solution)

b) Play fixed number of hands and get the probability for certain number of surplus player wins at any point within the sample (not answered)
B) looks to be a random walk Markov Chain solution (added) or maybe a type of negative binomial distribution.
I am very rusty with those but like good math problems.

Quote: jms9smith

I was curious to know if anyone had the maths on the ability of player results in Baccarat to exceed bank results??

Looks like your thread currently has 2 answers for 3 different possible questions.

Quote: jms9smith

If the outcomes must eventually reach the probabilities of the game itself,

No.
The outcomes do not have to eventually reach the probabilities.
Why?
Because, The Law of Large Numbers says as the number of trials increase the probabilities will converge towards the theoretical value. The actual numbers can get even further away from expectation but the percentages will get closer and closer to the expected probability.

Quote: jms9smith

is there anyway to determine how many more than bank results the player can win before the odds of the game are realised over say a million hands?

Well if your question is about the Player being ahead after 1 million hands, answer follows.
But if it just about at any time during those 1 million hands being up X more times than the Banker, you are now getting into some interesting and challenging math areas.

So, 1,000,000 = n.
0.506824 = q (banker win)
0.493176 = p (player win)
n*q
506,824 expected number of Banker wins in 1 million trials
n*p
493,176 expected number of Player wins in 1 million trials

13,648 Banker surplus (more wins than Player)

Variance = n*p*q
249953.4
Standard Deviation = Square root of Variance
249953.4^0.5 = 499.9534309
Let's round to 500.
The Player needs to overcome 13,648 expected Banker wins.
13,648/500 = 27.3 standard deviations

To overcome 7 SDs (from wikipedia) is about 1 / 390,682,215,445
I do not even want to go past 7 SDs.
(I'm assuming that you know what variance and standard deviation are)

Jufo81 method, without a finite number of trials, is quite simple and easy to work with if that is what you are looking for.
Let's see if we can come up with a solution to his B) question.

Quote: jms9smith

I know that in theory player could win 400,000 hands in a row and then bank win the rest to make odds add up but what is the maths/probability behind the difference that player and bank can achieve? Mathematically, how many hands can the player win, more than the bank??

Edit: Jufo81 pointed out a typo in his below post :)
If this were to happen, the Banker would not have to catch up with more wins in your 1 million trial example.
With more trials the Banker could in fact catch up numbers wise, by actual wins, but will get closer and closer to the expected percentages.

p	p run total	b	b run total	diff	session	total hands	p	b
400,000	400,000	0	0	-400,000	400,000	400,000	100.0000%	0.0000%
197,270	597,270	202,730	202,730	-394,540	400,000	800,000	74.6588%	25.3413%
394,540	991,810	405,460	608,190	-383,620	800,000	1,600,000	61.9881%	38.0119%
789,080	1,780,890	810,920	1,419,110	-361,780	1,600,000	3,200,000	55.6528%	44.3472%
1,578,160	3,359,050	1,621,840	3,040,950	-318,100	3,200,000	6,400,000	52.4852%	47.5148%
3,156,320	6,515,370	3,243,680	6,284,630	-230,740	6,400,000	12,800,000	50.9013%	49.0987%
6,312,640	12,828,010	6,487,360	12,771,990	-56,020	12,800,000	25,600,000	50.1094%	49.8906%
2,023,990	14,852,000	2,080,010	14,852,000	0	4,104,000	29,704,000	50.0000%	50.0000%
197,270	15,049,270	202,730	15,054,730	5,460	400,000	30,104,000	49.9909%	50.0091%
394,540	15,443,810	405,460	15,460,190	16,380	800,000	30,904,000	49.9735%	50.0265%
789,080	16,232,890	810,920	16,271,110	38,220	1,600,000	32,504,000	49.9412%	50.0588%
1,578,160	17,811,050	1,621,840	17,892,950	81,900	3,200,000	35,704,000	49.8853%	50.1147%
3,156,320	20,967,370	3,243,680	21,136,630	169,260	6,400,000	42,104,000	49.7990%	50.2010%
6,312,640	27,280,010	6,487,360	27,623,990	343,980	12,800,000	54,904,000	49.6867%	50.3133%
12,625,280	39,905,290	12,974,720	40,598,710	693,420	25,600,000	80,504,000	49.5693%	50.4307%
						expected	49.3175%	50.6825%

It would take ~29.7 million Player/Banker hands for the Banker to equal the Player wins assuming expectation holds true. I do not think any one player could witness that in their lifetime :)

Quote: 7craps

If this were to happen, the Banker would not have to catch up with more wins. With more trials the Banker could in fact never catch up numbers wise, by actual wins, but will get closer and closer to the expected percentages.

I am not sure if I understood you correctly, but banker wins will exceed player wins, no matter what, if you play long enough. Banker will always catch up numbers wise eventually even if you started at a position where player has 400,000 more wins than banker.

thank you for doing all that.. i was curious to know if there is a formula for working out the probability of player being ahead of bank by x number of wins at any given time over any given number of hands? a formula which had no restrictions on how many hands were played but instead could be used to calculate the probability that after x number of hands the player was ahead n amount of wins. further, it could be used to calculate how many hands the player could win more than the bank.. for example what is the formula for calculating the probability that player will win 1 more hand than bank? do we need to know how many hands this is out of? what are the odds that player will win 100 more hands than bank at any time? im really sorry if the answer has already been given but i can't really follow that well. thanks guys

Quote: Jufo81

This would show the probability of there being 281 or more player wins after exactly 496 total hands have been played (66+ more player wins than banker wins) but it could be also the case that the asked event of 66+ extra player wins is achieved at some point before all those 496 hands are played and not necessarily at the very end. So would there be a solution that says, what is the chance to see 66+ surplus player wins at any point within the first 496 hands played?

So, the OP should state whether he wishes to

a) play fixed number of hands and get the probability for certain number of surplus player wins at the end of that sample (guido's solution)

or

b) Play fixed number of hands and get the probability for certain number of surplus player wins at any point within the sample (not answered)

or

c) Play indefinitely long, up until the event of certain number of surplus player wins occurs (or doesn't) and get the overall chance of it occuring (my solution)

yes i think i seek (c) i want to know what the chance of seeing say 100 (or n) surplus player hands is at any point over infinite trials (hands). thank you for clearly stating that for me :)

Quote: jms9smith

yes i think i seek (c) i want to know what the chance of seeing say 100 (or n) surplus player hands is at any point over infinite trials (hands). thank you for clearly stating that for me :)

I already gave you the answer for that in my previous post in this thread. If you play indefinitely long starting from 0 banker wins, 0 player wins, the chance that at some point in the future there are 100 surplus Player wins is 6.5%. The general formula is (P/Q)^T where T is the number of surplus wins.

Other persons already answered the different question of the probability there being X or more surplus player wins after N total hands played.