# Baccarat card counting - Effects of removing a card

## Introduction

In either blackjack or baccarat a good first step in developing a card counting strategy is to determine the effect of removing any given card from the game. The following table shows the number of banker, player, and tie wins resulting from the removing of one card in an 8-deck shoe. The card removed is indicated in the image--leftolumn.

### Number

Card
Removed
Banker Win Player Win Tie Win
1 2259266202814720 2198201626637560 468838163231312
2 2259390347439480 2198279181695870 468636463548240
3 2259415336955130 2198240411263230 468650244465232
4 2259565639560830 2198132965463160 468607387659600
5 2259056540713470 2198626760121850 468622691848272
6 2259230629854970 2198942636434940 468132726393680
7 2259288625471740 2198847351781120 468170015430736
8 2258880877214840 2198299582316670 469125533152080
9 2259013211112320 2198292198535290 469000583035984
10 2259094649086970 2198163195365880 469048148230736

The next table puts these number is some perspective by indicating the probability of a banker, player, and tie win according to the card removed.

### Probability

Card
Removed
Banker Win Player Win Tie Win
1 0.458613 0.446217 0.09517
2 0.458638 0.446233 0.095129
3 0.458643 0.446225 0.095132
4 0.458673 0.446203 0.095123
5 0.45857 0.446303 0.095127
6 0.458605 0.446367 0.095027
7 0.458617 0.446348 0.095035
8 0.458534 0.446237 0.095229
9 0.458561 0.446235 0.095203
10 0.458578 0.446209 0.095213

The next table shows the house edge of each bet by card removed.

### House Edge

Card
Removed
Banker Player Tie
1 0.010535 0.012396 0.143467
2 0.010527 0.012405 0.143836
3 0.010514 0.012418 0.14381
4 0.010463 0.01247 0.143889
5 0.010662 0.012267 0.143861
6 0.010692 0.012238 0.144756
7 0.010662 0.012269 0.144688
8 0.010629 0.012298 0.142942
9 0.010602 0.012326 0.14317
10 0.01056 0.012369 0.143083

The next table shows the effect on the house edge of each bet according to the card removed. A negative number indicates removal is bad for the player, positive indicates removal is good.

### House Edge

Card
Removed
Banker Player Tie
0 0.000019 -0.000018 0.000513
1 0.000044 -0.000045 0.000129
2 0.000052 -0.000054 -0.000239
3 0.000065 -0.000067 -0.000214
4 0.000116 -0.000120 -0.000292
5 -0.000083 0.000084 -0.000264
6 -0.000113 0.000113 -0.001160
7 -0.000083 0.000082 -0.001091
8 -0.00005 0.000053 0.000654
9 -0.000023 0.000025 0.000426

The next table multiplies the above numbers by ten million.

Card
Removed
Banker Player Tie
0 188 -178 5129
1 440 -448 1293
2 522 -543 -2392
3 649 -672 -2141
4 1157 -1195 -2924
5 -827 841 -2644
6 -1132 1128 -11595
7 -827 817 -10914
8 -502 533 6543
9 -231 249 4260
Average 0 0 0

To adapt this information to a card counting strategy, the player should start with three running counts of zero. As each card is seen as it leaves the shoe the player should add the point values of that card to each running count. For example if the first card to be played is an 8 then the three running counts would be: banker=-502, player=533, tie=6543. Of course the player does not have to keep a running track of all three counts. In fact the point values for the banker and player are nearly opposite of each other. A high running count for the banker would mean a corresponding low count for the player, and vise versa.

In order for any given bet to become advantageous the player should divide the running count by the ratio of cards left in the deck to get the true count. A bet hits zero house edge at the following true counts:

• Banker: 105791
• Player: 123508
• Tie: 1435963

Assuming you were able to actually play this strategy perfectly you would notice that the true counts seldom passed the point of zero house edge. The next table shows the ratio of hands played, based on a sample of 100 million, in which the true count passes the break even points above. The image--leftolumn indicates the ratio of cards dealt before the cards are shuffled.

### Positive Expectation

Penetration Banker Player Tie
90 percent 0.000131 0.000024 0.000002
95 percent 0.001062 0.000381 0.000092
98 percent 0.005876 0.003700 0.002106

The final table indicates the expected revenue per 100 bets and a \$1000 wager every time a positive expected value occured. Please remember that this table assumes the player is able to keep a perfect count and the casino is not going to mind the player only making a bet once every 475 hands of less.

### Expected Profit

Penetration Banker Player Tie
90 percent \$0.01 \$0.00 \$0.00
95 percent \$0.20 \$0.06 \$0.15
98 percent \$2.94 \$1.77 \$11.93

I hope this section shows that for all practical purposes baccarat is not a countable game. For more information on a similar experiment I would recomment The Theory of Blackjack by Peter A. Griffin. Although the book is mainly devoted to blackjack he has part of a chapter titled 'Can Baccarat Be Beaten?' on pages 216 to 223. Griffin concludes by saying that even in Atlantic City, with a more liberal shuffle point than Las Vegas, the player betting \$1000 in positive expectation hands can expect to profit 70 cents an hour.

For your further consideration I would recommend this baccarat odds calculator. You can put in any deck composition and it will give the house edge on all three bets.

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