# Baccarat Combinatorial Analysis the Easy Way

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The intention of this post is to put the gaming mathematicians out of business. Not really. But I do want to tell you how you can easily figure out the house edge for baccarat, its variants and most of its side bets. You can do this without needing a computer program or fancy spread sheet analysis. All you need to know is one advanced Excel function, “sumifs”. You will then be able to compute the combinatorial analysis (CA) for the baccarat wager you are considering (see this post for an introduction to CA). You will then be able to duplicate the CA results I give in my baccarat posts as well as those at the Wizard of Odds website. You will also be able to investigate you own baccarat wagering ideas. In other words, with a few minutes study, you can become a professional baccarat mathematician. Who wouldn’t want to do that? What makes CA easy for baccarat is the simple observation that every hand in baccarat is completely determined by a sequence of six cards. While it is true that the 5th and 6th cards may not be used, by always including them as a possibility, the baccarat universe can be fully written out: there are exactly 302500 unique baccarat hands.

To make CA possible, we need to compute the exact number of ways each of the 302500 six-card hands can occur. To explain how to count these hands, I am going to give an example. You will never need to do this type of computation and you don’t really need to understand this example. However, for those who are motivated, this example will help explain the spread sheet provided below.

To count the number of ways a hand can occur, first recall that a baccarat shoe has 416 cards. There are 32 each of the cards Ace through 9 and 128 total ten-valued cards (Ten, Jack, Queen, King). Using this information, the total number of ways a six-card sequence of cards can occur is just a matter of simple multiplication.

Here is an example hand, the sequence of cards (1, 4, 2, 4, 2, 7):

• Player’s two cards: 1, 4
• Banker’s two cards: 2, 4
• Card 5: 2 (dealt to Player)
• Card 6: 7 (not used).

The final result of this sequence of six cards is Player = 7, Banker = 6, with Player winning the hand. The Player’s hand uses three cards. The Banker’s hand uses two cards.

Now, let’s count the ways this hand can occur.

First consider the Player hand. The first card is a 1, there are 32 of those. The second card is a 4, there are 32 of those, for a total of 32x32 = 1024. The Player hand can occur in the order 4, 1 or 1, 4, so this number must be doubled. We see that there are 2048 ways the Player hand can occur.

Now consider the Banker hand. The first card is a 2, there are 32 of those left. The second card is a 4, there are 31 of those, for a total of 32x31 = 992. Again, the order of the cards dealt to the Banker hand doesn’t matter, so this number must be doubled. We see that there are 1984 ways the Banker hand can occur.

In total so far, between the Player hand and Banker hand, we have 2048x1984 ways these two hands can be dealt.

For card 5, there are 31 2’s left in the shoe. The total ways the hand can be dealt including this card is now 2048x1984x31.

Finally, for card 6, there are 32 7’s left in the shoe. It follows that the total ways the hand can be dealt including all six cards is 2048x1984x31x32 = 4030726144.

The following spread sheet lists each of the 302500 unique six-card sequences, the number of ways each hand can occur, and other information that is useful for doing CA. Note that this spread sheet is a big file (16M).

Baccarat_Easy_Edge

Here are the details for each column in this spread sheet:

• Columns A and B give the Player’s starting two cards.
• Columns C and D give the Banker’s starting two cards.
• Column E gives the fifth card in the sequence of six cards.
• Column F gives the sixth (last) card in the sequence of six cards.
• Column G gives the baccarat total for the Player hand.
• Column H gives the baccarat total for the Banker hand.
• Column I has the value 0 if the Player’s hand uses two cards. The value 1 is given in this column if the Player’s hand requires three cards.
• Column J has the value 0 if the Banker’s hand uses two cards. The value 1 is given in this column if the Banker’s hand requires three cards.
• Column K gives the winner of the hand (P, B, T).
• Column L gives the number of ways the six-card sequence can occur.

In particular, the example presented above is shown in row 17718 (highlighted in yellow). Note the value 1 in column I (the Player’s hand uses 3 cards) and the value 0 in column J (the Banker’s hand uses 2 cards). As expected, Column L has the value 4030726144 that we computed above.

Starting at the top of the sheet, in columns O, P, Q, R and S, I give some examples of using the spread sheet. These examples start with the easiest computations and get trickier the further down you go. I suggest that you start with the first computation at the top and work your way down the list.

Here is a quick summary:

• Column O gives all the possible outcomes for the wager. Each event that can happen in the wager should be described, including those when the player loses the wager.
• Column P gives the number of ways each event can occur. This is where you will use the "sumifs" function. You must sum the values in column L that correspond to each event.
• Column Q gives the probability for each event. This is simple division. You divide the value in column N by the total number of things that can happen, which is always 4998398275503360.
• Column R gives the payout for each event.
• Column S just multiplies the values from columns Q and R.
• Always include a row called "Total" where the values in columns P, Q and S are summed. Column P should always sum to 4998398275503360. Column Q should always sum 1. And finally, the sum of the values in Column S gives the house edge.

Here are the example computations I give in the spread sheet:

• Baccarat Player bet.
• Baccarat Banker bet
• Baccarat Tie bet.
• Commission free: Banker 6 pays 1:2, Banker bet. This event occurs when column H has the value 6 and column K has the value "B".
• Commission free; Banker three Card 7 is a push, Banker bet This event occurs when column H has the value 7, column J has the value 1 and column K has the value "B."
• Player natural 9 side bet This event occurs when column G has the value 9 and column I has the value 0.
• Dragon 7 side bet This event occurs when column H has the value 7, column J has the value 1 and column K has the value "B."
• Panda 8 side bet This event occurs when column G has the value 8, column I has the value 1 and column K has the value "P."
• SHFL Dragon Bonus, Player. This is complicated.

I have also included the outline of the Player Natural 8 side bet for you to try. See this post for the answer.

The hardest part in these computations is figuring out the values that go into column P. That’s where the Excel function “sumifs” is used. Before you look at any of these examples make sure you understand the Excel “sumifs” function. Here are two links to get you started:

I debated explaining one of the examples in the spread sheet in detail, but for now I am going to pass on this extra effort. Either you are motivated or you aren’t. But, I hope you go for it! Go forth and compute!