Introduction

Baccarat is usually dealt from a 6-deck shoe. All players place their bets before the dealer deals cards. The dealer will flip 2 cards on the Banker side (B1, B2) and 2 cards on the Player side (P1, P2). All numbers are computed in a "mod 10" base. That is, a 7 and an 8 sum to 5 (mod 10). Based on the numbers on 2 sides, the game will be played according to following rules:

(1) If (P1+P2 = 8 or 9) or (B1+B2 = 8 or 9),
    then it is Natural and neither hand draws.
(2) If (P1+P2 = 6 or 7),
    then (a) Player hand stands
         (b) Banker hand draws one card on totals of
             0 to 5 and stands on 6 to 9.
(3) If (P1+P2 <= 5),
    then (a) Player hand draws one card (P3)
         (b) Banker hand draws one card when one of the
             following is true:
               (B1+B2 <= 2)
               (B1+B2= 3) & (P3 <> 8)
               (B1+B2 = 4) & (2 <= P3 <= 7)
               (B1+B2= 5) & (4 <= P3 <= 7)
               (B1+B2= 6) & (6 <= P3 <= 7)
(4) The hand closest to 9 wins (or the hand with higher
    score wins).  If Banker's and Player's hands have the
    same score, Tie bet wins and get paid 9:1 (or 8:1),
    and both Banker's and Player's bets are tie (and players
    keep their bets). 

Bets

There are 3 kinds of bets in this game: Banker, Player, and Tie. The payouts of Banker and Player are 1:1, and the payout of Tie is 9:1 (or 8:1). Winning Banker bets are charged a 5 percent commission (Binion's Horseshoe in Las Vegas charges 4% commission).

Expected Gain/Loss

The following chart shows the probabilities of events:

Number of Decks	Banker Wins	Player Wins	Tie
Infinite	45.8428%	44.6147%	9.5426%
8	45.8597%	44.6247%	9.5156%
6	45.8653%	44.6276%	9.5069%
4	45.8761%	44.6340%	9.4898%
2	45.9073%	44.6508%	9.4419%
1	45.9624%	44.6760%	9.3615%

The following chart shows the house edges on different bets:

Number of Decks	Banker Wins		Player Wins	Tie
	5% Comm.	4% Comm.		9:1	8:1
Infinite	1.0640%	0.6056%	1.2281%	4.5744%	14.1170%
8	1.0579%	0.5993%	1.2351%	4.8440%	14.3596%
6	1.0558%	0.5972%	1.2374%	4.9313%	14.4382%
4	1.0517%	0.5929%	1.2421%	5.1018%	14.5916%
2	1.0389%	0.5798%	1.2565%	5.5809%	15.0228%
1	1.0117%	0.5521%	1.2864%	6.3846%	15.7461%

Counting Method

When the rest of decks are 6-or-7 full (i.e., have more 6s and 7s than as usual), it is better for the Tie bet. But in order to convert the 4.8% house edge, the rest of decks must have at least twice 6s and 7s than the average of other kinds. It is similar as adding 4 6s and 4 7s to a fresh single deck.