# Games 3.1: Baccarat – An Analysis on Calculations

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!