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In trying to understand the boundaries of a game, it is helpful to look at all the possible permutations of events possible.

We shall do this with baccarat – bull-dozing through the math.  The total number of permutations for baccarat is 416!/410! = 4,998,398,275,503,360.

This is because there are 416 cards in an 8-deck shoe, so 52 x 8 = 416. We divide this number in factorials, meaning 416!, by 416 – 6 (for each coup has 6-cards, in relation to calculations). So we get 416! / 410!.

As a point of interest, some of you might be thinking – isn’t there some kind of formula where we can plug in numbers and get the probability of results? Well, as far as my research has gone, there isn’t. Also, the removal of a 2 has a different effect from the removal of a 5. Meaning that each card has a different effect on the probabilities of the Player, Banker and Tie. Also, the effects of this removal are not constant. The removal of one 2 has a different effect from two or three 2s removed. This has been demonstrated in some previous research – here is the article, if you are keen to know more:

https://excelpunks.com/2018/09/20/baccarat-an-analysis/

The steps I took in computing this are rather straight-forward.  I used Excel and you would need some knowledge of the following functions to make this work:

1. SUMIFS
2. COUNTIFS
3. IF, OR and AND
4. LEFT and RIGHT

First, you have to construct a table like this –

Spanning over 4,826,809 rows	Column 1	Column 2	Column 3	Column 4	Column 5	Column 6
	371,293 sets of A to K	28,561 sets of A to K consisting of 13 As followed by 13 2s and so on.	2,197 sets of A to K consisting of 169 As followed by 169 2s and so on.	169 sets of A to K consisting of 2,197 As followed by 2,197 2s and so on.	13 sets of A to K consisting of 28,561 As followed by 28,561 2s and so on.	1 set of A to K consisting of 371,293 As followed by 371,293 2s and so on.

 

Then, you’d have to add the values of the cards together across the columns and program the game rules to effect the 4, 5 and 6-card coup results.  You can learn about this from Excelpunks’ Casino Math Course. As a guide, the results for 4-card, 5-card and 6-card coups would need to be programmed into your calculations.

4-Card coups would include natural wins and ties on 8s and 9s, ties on 2-card 6 and wins on 2-card 7s over 2-card 6s.

5-card and 6-card coups would follow the tableau.

For those of you familiar with Excel, you would know that working with any number of rows above 500,000 makes calculations ponderous and unduly time-consuming. Hence, we would also need to work on reducing the number of rows to make calculations more manageable.

For this, we would need to apply our knowledge of combinations and permutations as well as our knowledge of Baccarat tableau. We do know that the first 2-cards for either the Banker or the Player can be in any order and would yield the same result. A Player hand with a 9 and then a King would give the same result if the King had been the first card and the 9 as the second. The same is true for the Banker. This enables us to reduce the number of rows we need to work with.

With a little bit of thought, you would find that you would need only to look at the Aces to 9s, counting the Jacks, Queens and Kings together with the 10s. You would also find that you would only need to use 302,500 rows, instead of 4,826,809. We get this by taking the number of possible combinations of each card value with one other that can only be equal to or higher than its own. You will find that we get 302,500 by the following calculation (give it some thought and I think you will agree!):

55 x 55 x 100 = 302,500

In any case, here are the permutations possible for 4, 5 and 6-card coups:

	BANKER
PLAYER	4 Card Total	1	2	3	4	5	6	7	8	9	–
	1	–	–	–	–	–	–	–	44,856,843,337,728	45,033,013,641,216	–
	2	–	–	–	–	–	–	–	44,681,366,618,112	44,856,843,337,728	–
	3	–	–	–	–	–	–	–	44,856,843,337,728	45,033,013,641,216	–
	4	–	–	–	–	–	–	–	44,683,447,369,728	44,856,843,337,728	–
	5	–	–	–	–	–	–	–	44,856,843,337,728	45,033,013,641,216	–
	6	–	–	–	–	–	44,487,098,110,464	44,856,843,337,728	44,683,447,369,728	44,856,843,337,728	–
	7	–	–	–	–	–	44,856,843,337,728	44,838,810,157,056	44,856,843,337,728	45,033,013,641,216	–
	8	44,856,843,337,728	44,681,366,618,112	44,856,843,337,728	44,683,447,369,728	44,856,843,337,728	44,683,447,369,728	44,856,843,337,728	44,487,098,110,464	44,858,924,089,344	69,686,799,163,392
	9	45,033,013,641,216	44,856,843,337,728	45,033,013,641,216	44,856,843,337,728	45,033,013,641,216	44,856,843,337,728	45,033,013,641,216	44,858,924,089,344	44,838,810,157,056	69,961,111,584,768
	–	–	–	–	–	–	–	–	69,686,799,163,392	69,961,111,584,768	–

 

 

	BANKER
PLAYER	5 Card Total	1	2	3	4	5	6	7	8	9	–
	1	–	–	3,455,256,403,968	26,104,503,730,176	26,207,637,295,104	26,105,087,889,408	33,094,381,387,776	–	–	–
	2	–	–	3,456,714,276,864	24,074,105,966,592	29,582,414,401,536	29,467,414,155,264	33,011,870,158,848	–	–	–
	3	–	–	3,470,505,148,416	24,155,341,135,872	33,120,035,573,760	33,006,509,193,216	33,135,282,634,752	–	–	–
	4	–	–	–	20,627,229,806,592	27,638,627,082,240	32,923,813,625,856	33,053,237,723,136	–	–	–
	5	–	–	–	17,254,903,812,096	24,206,575,435,776	33,005,365,284,864	33,135,122,706,432	–	–	–
	6	32,964,477,505,536	32,924,126,748,672	33,006,509,193,216	36,380,689,514,496	43,347,217,637,376	39,750,157,000,704	45,276,949,782,528	22,554,580,703,232	22,593,912,127,488	38,525,195,292,672
	7	33,094,381,387,776	33,011,870,158,848	33,135,282,634,752	33,053,237,723,136	40,047,091,703,808	36,483,493,122,048	45,322,716,217,344	22,642,153,242,624	22,682,938,331,136	38,676,532,936,704
	8	–	–	5,398,882,111,488	5,377,630,162,944	8,854,301,810,688	15,694,721,402,880	22,642,153,242,624	–	–	–
	9	–	–	3,470,505,148,416	8,834,344,439,808	8,869,388,943,360	15,734,623,518,720	22,682,938,331,136	–	–	–
	–	–	–	3,456,714,276,864	28,249,492,660,224	28,361,385,246,720	31,665,746,755,584	38,676,532,936,704	–	–	–

 

 

	BANKER
PLAYER	6 Card Total	1	2	3	4	5	6	7	8	9	–
	1	20,499,217,668,352	20,469,715,722,240	19,748,366,587,904	14,488,507,043,840	14,547,627,450,368	14,502,970,056,704	12,931,912,226,816	12,917,506,863,104	12,915,314,716,672	24,639,193,538,560
	2	20,114,458,411,008	20,006,606,104,576	19,309,980,278,784	14,511,225,434,112	13,312,971,792,384	13,300,635,432,960	12,532,563,525,632	12,516,529,606,656	12,515,889,123,328	24,214,066,700,288
	3	19,634,953,029,632	19,563,259,723,776	18,780,004,980,992	13,992,499,290,112	11,991,382,372,352	11,972,706,738,176	12,020,129,447,936	12,018,493,939,712	12,018,044,743,680	23,748,210,556,928
	4	20,535,822,327,808	20,465,430,315,008	20,521,555,795,968	15,666,903,656,448	14,143,619,977,216	12,878,226,755,584	12,922,321,715,200	12,932,705,042,432	12,933,283,790,848	24,642,962,657,280
	5	21,127,406,358,528	21,055,697,346,560	21,104,461,946,880	17,067,284,623,360	15,477,471,308,032	13,430,370,922,496	13,474,926,731,264	13,475,506,679,808	13,502,128,295,936	25,228,523,110,400
	6	16,002,925,191,168	15,945,900,431,360	15,993,364,926,464	15,139,556,573,184	13,581,745,606,656	11,932,746,197,248	10,750,080,249,856	10,747,618,420,736	10,770,964,447,232	18,847,186,571,264
	7	16,850,879,371,264	16,775,675,854,848	16,822,139,883,520	16,782,362,558,464	15,210,341,982,208	13,569,671,933,952	11,556,012,525,568	11,557,152,235,520	11,583,922,028,544	19,674,174,881,792
	8	15,694,516,293,632	15,644,071,108,608	14,433,721,114,624	14,394,763,333,632	13,618,506,100,736	11,994,270,017,536	10,411,449,139,200	10,392,318,564,608	10,420,918,235,136	18,501,777,424,384
	9	15,595,868,119,040	15,545,037,361,152	14,806,678,437,888	13,518,746,501,120	13,540,751,020,032	11,908,310,286,336	10,327,732,097,024	10,307,044,319,232	10,307,243,902,976	18,409,431,764,992
	–	24,291,119,898,624	24,211,866,148,864	23,500,746,743,808	17,784,114,872,320	17,810,227,724,288	16,973,457,645,568	15,407,170,355,200	15,389,778,067,456	15,390,342,909,952	28,979,901,420,544

 

	BANKER
PLAYER	TOTAL	1	2	3	4	5	6	7	8	9	–
	1	20,499,217,668,352	20,469,715,722,240	23,203,622,991,872	40,593,010,774,016	40,755,264,745,472	40,608,057,946,112	46,026,293,614,592	57,774,350,200,832	57,948,328,357,888	24,639,193,538,560
	2	20,114,458,411,008	20,006,606,104,576	22,766,694,555,648	38,585,331,400,704	42,895,386,193,920	42,768,049,588,224	45,544,433,684,480	57,197,896,224,768	57,372,732,461,056	24,214,066,700,288
	3	19,634,953,029,632	19,563,259,723,776	22,250,510,129,408	38,147,840,425,984	45,111,417,946,112	44,979,215,931,392	45,155,412,082,688	56,875,337,277,440	57,051,058,384,896	23,748,210,556,928
	4	20,535,822,327,808	20,465,430,315,008	20,521,555,795,968	36,294,133,463,040	41,782,247,059,456	45,802,040,381,440	45,975,559,438,336	57,616,152,412,160	57,790,127,128,576	24,642,962,657,280
	5	21,127,406,358,528	21,055,697,346,560	21,104,461,946,880	34,322,188,435,456	39,684,046,743,808	46,435,736,207,360	46,610,049,437,696	58,332,350,017,536	58,535,141,937,152	25,228,523,110,400
	6	48,967,402,696,704	48,870,027,180,032	48,999,874,119,680	51,520,246,087,680	56,928,963,244,032	96,170,001,308,416	100,883,873,370,112	77,985,646,493,696	78,221,719,912,448	57,372,381,863,936
	7	49,945,260,759,040	49,787,546,013,696	49,957,422,518,272	49,835,600,281,600	55,257,433,686,016	94,910,008,393,728	101,717,538,899,968	79,056,148,815,872	79,299,874,000,896	58,350,707,818,496
	8	60,551,359,631,360	60,325,437,726,720	64,689,446,563,840	64,455,840,866,304	67,329,651,249,152	72,372,438,790,144	77,910,445,719,552	54,879,416,675,072	55,279,842,324,480	88,188,576,587,776
	9	60,628,881,760,256	60,401,880,698,880	63,310,197,227,520	67,209,934,278,656	67,443,153,604,608	72,499,777,142,784	78,043,684,069,376	55,165,968,408,576	55,146,054,060,032	88,370,543,349,760
	–	24,291,119,898,624	24,211,866,148,864	26,957,461,020,672	46,033,607,532,544	46,171,612,971,008	48,639,204,401,152	54,083,703,291,904	85,076,577,230,848	85,351,454,494,720	28,979,901,420,544

 

For those of you who for sure are wondering, when it comes to the likelihood of the Banker winning over the Player, here are the numbers.

Banker Win	Permutations	Probability
0	0	0
1	24,291,119,898,624	0.0048598
2	44,681,581,871,104	0.0089392
3	72,927,778,568,192	0.0145902
4	163,359,790,133,248	0.0326824
5	216,715,928,915,968	0.0433571
6	269,232,304,455,680	0.0538637
7	384,279,324,919,808	0.0768805
8	529,914,458,673,152	0.1060169
9	586,850,279,002,112	0.1174077

 

And in terms of how much you can push the envelope in terms of payouts, here are the numbers too.

Commission Baccarat

Total	4,998,398,275,503,360	Probability	Break-Even Payout	Exp.
Player	2,230,518,282,592,260	0.44624	1.02767	-0.01235
Banker	2,292,252,566,437,890	0.45859	0.97306	-0.01057
Tie	475,627,426,473,216	0.09515	9.50906	-0.14359

 

Non-Commission Baccarat

The reason why the Banker’s expectation is different from regular baccarat is due to the, you guessed it, the ½ payout on Banker wins on 6.

Wager	Permutations	Probability	Pays	Expected Value
Player	2,230,518,282,592,260	0.44625	(1)	(0.44625)
Banker Win (<> 6)	2,023,020,261,982,210	0.40473	1	0.40473
Banker Win on 6	269,232,304,455,680	0.05386	0.5	0.02693
Tie	475,627,426,473,216	0.09516	–	–
Total	4,998,398,275,503,360			(0.01458)

 

The calculation for which is:

0.40473 + (0.05386*0.5) – 0.44624 = -0.01458

All the figures above are based off of the permutation counts in the tables above.

Total Banker permutations: 2,292,252,566,437,890

Total permutations for Banker wins on 6: 269,232,304,455,680

Total permutations for Banker wins on anything other than 6: 2,292,252,566,437,890 – 269,232,304,455,680 = 2,023,020,261,982,210

Probability of Banker wins on anything other than 6: 2,023,020,261,982,210/4,998,398,275,503,360 = 0.40473

Probability of Banker wins on 6: (269,232,304,455,680/4,998,398,275,503,360) = 0.05386

Add the total expectation for Banker wins to get: 0.40473 + (0.05386*0.5) – 0.44624 = -0.01458

So, if you know what is good for you, avoid non-commission baccarat, at least the Banker wagers – the game looks easier, but you are more likely to lose more in the long run.

Well, that’s all there is. Good luck!

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