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Card Counting the Panda 8 Bet
Introduction
The Panda 8 is a side bet in EZ Baccarat, along with the Dragon 7 bet. The rules of EZ Baccarat are the same as those of conventional baccarat, except there is no 5% commission on winning Banker bets. However, a winning Banker bet with a 3-card 7 will push. To cover this contingency, they also offer the player the Dragon 7 bet, which pays 40 to 1 for a 3-card Banker total of 7. Later came the Panda 8 bet, which pays 25 to 1 for a 3-card winning Player total of 8.
Odds
The following table shows the odds of the Panda 8 for the non-counter. The lower right cell shows a house edge of 10.19%.
Panda 8 Odds
| Event | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Win | 25 | 172,660,763,262,976 | 0.034543 | 0.863580 |
| Lose | -1 | 4,825,737,512,240,380 | 0.965457 | -0.965457 |
| Total | 4,998,398,275,503,360 | 1.000000 | -0.101876 |
Effect of Card Removal
The standard way to start a card counting vulnerability study is to first look at the effect of removing just one card, one at a time for each rank. The following table shows the number of winning combinations and probability in a 415-card shoe, which is a full eight decks, less one card of the given rank. The column for change in probability shows the difference compared to a full shoe, with a positive change meaning an increase in probability of the Panda 8 winning. These are very small effects, so the right column shows the effect multiplied by 100,000.
Panda 8 — Effect of Card Removal
| Card Removed | Combinations | Probability | Change in Probability | Change X100,000 |
|---|---|---|---|---|
| 0 | 170,399,271,999,488 | 0.034590 | 0.000046 | 4.6 |
| 1 | 170,420,880,433,408 | 0.034594 | 0.000051 | 5.1 |
| 2 | 170,444,015,883,264 | 0.034599 | 0.000056 | 5.6 |
| 3 | 169,628,395,374,464 | 0.034433 | -0.000110 | -11.0 |
| 4 | 169,689,936,212,224 | 0.034446 | -0.000098 | -9.8 |
| 5 | 169,667,461,501,952 | 0.034441 | -0.000102 | -10.2 |
| 6 | 170,001,539,425,792 | 0.034509 | -0.000034 | -3.4 |
| 7 | 169,997,271,604,224 | 0.034508 | -0.000035 | -3.5 |
| 8 | 169,745,936,511,104 | 0.034457 | -0.000086 | -8.6 |
| 9 | 171,023,504,362,496 | 0.034716 | 0.000173 | 17.3 |
| Weighted Avg. | 170,170,463,792,837 | 0.034543 | 0.000000 | 0.0 |
The next table multiplies the effect of removal from the table above by 21,530. This factor was carefully chosen so that the effect of removing a 0-point card, the most frequent rank, would be close to 1. Next, I rounded the effects of the other cards to the nearest integer, as shown in the right column.
Weighting Factors
| Card Removed | Effect X 21,530 | Rounded Effect |
|---|---|---|
| 0 | 0.999987 | 1 |
| 1 | 1.094425 | 1 |
| 2 | 1.195536 | 1 |
| 3 | -2.369064 | -2 |
| 4 | -2.100105 | -2 |
| 5 | -2.198329 | -2 |
| 6 | -0.738270 | -1 |
| 7 | -0.756922 | -1 |
| 8 | -1.855360 | -2 |
| 9 | 3.728141 | 4 |
If we add the effect of each rank, counting the 1 for 0 points four times (for the 10, J, Q, and K), we end with a sum of zero. That means it is a balanced count and the total count for an entire shoe is zero.
Next, I did a simulation of actual results using the above weighting factors. The following table shows the frequency of each count, average advantage, and contribution to the total return of the game, for true counts of 1 to 25. The true count is based on the following standard baccarat rules:
- Eight decks
- The first card in the deck is exposed.
- If the exposed card is an ace one card is burned. If it is a 2 to 9, then the number of burns is according to the pip value of the first card. If it is any 0-point card, then 10 cards are burned.
- The cut card is placed between the 14th and 15th cards from the end of the shoe.
- When the cut card is reached, that hand shall be finished and a final hand will be dealt. If the cut card is exposed between hands, then one hand more will be dealt.
The true count defined as INT(RC/ND), where RC=Running Count and ND=Number of Decks of unseen cards.
Player Advantage by True Count
| True Count | Frequency | Player Advantage | Contribution to Return |
|---|---|---|---|
| 20 | 0.16% | 9.18% | 0.01% |
| 19 | 0.17% | 8.24% | 0.01% |
| 18 | 0.22% | 7.43% | 0.02% |
| 17 | 0.24% | 6.48% | 0.02% |
| 16 | 0.30% | 5.51% | 0.02% |
| 15 | 0.33% | 4.57% | 0.02% |
| 14 | 0.41% | 3.67% | 0.02% |
| 13 | 0.54% | 2.68% | 0.01% |
| 12 | 0.55% | 1.69% | 0.01% |
| 11 | 0.70% | 0.82% | 0.01% |
| 10 | 0.88% | -0.14% | 0.00% |
| 9 | 1.07% | -1.06% | -0.01% |
| 8 | 1.36% | -2.06% | -0.03% |
| 7 | 1.65% | -2.98% | -0.05% |
| 6 | 2.14% | -3.93% | -0.08% |
| 5 | 2.72% | -4.92% | -0.13% |
| 4 | 3.62% | -5.86% | -0.21% |
| 3 | 4.67% | -6.84% | -0.32% |
| 2 | 6.38% | -7.79% | -0.50% |
| 1 | 8.66% | -8.74% | -0.76% |
What the table above shows is that the odds swing to the player's favor at a true count of 11 or more.
The next table shows various statistics over all Panda 8 bets made at a true count of 11 or more. The Panda 8 is accompanied by another side bet, the Dragon 7, so I show the results side by side as a basis of comparison.
Summary for Panda 8 and Dragon 7
| Statistic | Panda 8 | Dragon 7 |
|---|---|---|
| Bet frequency | 4.61% | 9.16% |
| Advantage per bet made | 6.34% | 8.04% |
| Advantage per hand dealt | 0.29% | 0.74% |
| Expected units won per shoe | 0.238 | 0.599 |
| Expected units won per hour | 0.178 | 0.449 |
| Standard deviation | 5.150 | 6.567 |
Units won per shoe is based on 81.3 hands per shoe. Units won per hour is based on an average of 80 minutes per shoe.
The bottom line shows if the counter were to bet $100 every time the Panda count rose to 11 or higher, he could expect to earn $17.80 per hour. Meanwhile, he could earn $44.90 per hour counting the Dragon 7 bet. The weightings are different for the two counts, so it would require twice the effort to count both bets. If the game goes too fast and only one count is possible, obviously, go with the Dragon 7, which is 2.5 times as profitable. However, if you are counting the Dragon 7 anyway and have the ability to count the Panda as well, doing so will increase the expected win by 40%, to $62.70 for the $100 player.
Some casinos do not place the cut card so deeply into the shoe. For them, the following tables show the same statistics as the table above for various cut card placement points, from 14 to 104 cards from the end of the shoe. The next table is for the Panda 8, and the following for the Dragon 7.
Panda 8 at Various Penetrations
| Statistic | 14 Cards | 26 Cards | 39 Cards | 52 Cards | 78 Cards | 104 Cards |
|---|---|---|---|---|---|---|
| Bet frequency | 4.61% | 3.87% | 3.20% | 2.63% | 1.74% | 1.10% |
| Advantage per bet made | 6.34% | 5.15% | 4.39% | 3.81% | 3.08% | 2.51% |
| Advantage per hand dealt | 0.29% | 0.20% | 0.14% | 0.10% | 0.05% | 0.03% |
| Expected units won per shoe | 0.24 | 0.16 | 0.11 | 0.08 | 0.04 | 0.02 |
| Expected units won per hour | 0.18 | 0.12 | 0.09 | 0.06 | 0.03 | 0.02 |
Dragon 7 at Various Penetrations
| Statistic | 14 Cards | 26 Cards | 39 Cards | 52 Cards | 78 Cards | 104 Cards |
|---|---|---|---|---|---|---|
| Bet frequency | 9.16% | 8.32% | 7.49% | 6.73% | 5.37% | 4.20% |
| Advantage per bet made | 8.04% | 6.96% | 6.23% | 5.66% | 4.82% | 4.19% |
| Advantage per hand dealt | 0.74% | 0.58% | 0.47% | 0.38% | 0.26% | 0.18% |
| Expected units won per shoe | 0.60 | 0.47 | 0.38 | 0.31 | 0.21 | 0.14 |
| Expected units won per hour | 0.45 | 0.35 | 0.28 | 0.23 | 0.16 | 0.11 |
For more information on counting the Dragon 7, please read Card Counting the Dragon Side Bet in EZ Baccarat by Dr. Eliot Jacobson.
Outside Links
The always outstanding site Discount Gambling has a page on counting the Panda and the Dragon.