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Introduction
Sometimes in blackjack there will be a premium if the player gets a 6,7,8 or 7,7,7 in his initial hand. Rules vary about what happens if the dealer also gets 21 points. Sometimes the entire win pays, sometimes it is a push, and sometimes the original wager will push but a bonus is still paid. However, in all cases, the player never has the chance to hit (U.S.), or the hand loses (Europe), against a dealer blackjack.
What prompted me to write this appendix was a visit to the Winnavegas casino, south of Sioux City, Iowa. When I was there on March 1, 2006, they paid bonuses of $50 for an unsuited 678 or 777, and $100 for a suited 678 or 777, not to mention other bonuses for 2 to 4 consecutive blackjacks. A $10 minimum bet was required. The dealer hit a soft 17. The reason for the generous bonuses was they removed a Caribbean Stud game with $170,000 in the meter, and had to return the money to the players somehow.
This appendix is based on the WinnaVegas situation, where a qualifying hand pays an immediate bonus, and the original wager still stands as 21 points. The following tables show the value independently of the four types of bonuses across the bonus amount. The analysis is based on a six-deck game.
Value of 678 and 777 Bonuses — Dealer Hits on Soft 17
| Hand | 1 Unit Bonus | 2 Unit Bonus | 3 Unit Bonus | 5 Unit Bonus | 10 Unit Bonus |
|---|---|---|---|---|---|
| Any 678 | 0.16% | 0.36% | 0.61% | 1.13% | 2.44% |
| Any 777 | 0.02% | 0.04% | 0.06% | 0.11% | 0.28% |
| Suited 678 | 0.01% | 0.02% | 0.03% | 0.05% | 0.11% |
| Suited 777 | 0% | 0% | 0% | 0% | 0.01% |
Value of 678 and 777 Bonuses — Dealer Stands on Soft 17
| Hand | 1 Unit Bonus | 2 Unit Bonus | 3 Unit Bonus | 5 Unit Bonus | 10 Unit Bonus |
|---|---|---|---|---|---|
| Any 678 | 0.17% | 0.37% | 0.62% | 1.15% | 2.46% |
| Any 777 | 0.02% | 0.04% | 0.06% | 0.11% | 0.28% |
| Suited 678 | 0.01% | 0.02% | 0.03% | 0.05% | 0.11% |
| Suited 777 | 0% | 0% | 0% | 0% | 0.01% |
In the case of WinnaVegas, the combined bonuses were worth 1.26%. After considering these and other bonuses, the player had an advantage of 1.89% on a $10 bet!
The next tables show the point at which the player should deviate from basic strategy, and hit as opposed to stand, for an any 678 bonus. For example, if the player had a 7,8 against a dealer 5, then the bonus would need to be at least 3 times the bet for the player to hit.
Strategy Deviation Points for any 678 Bonus — Dealer Hits on Soft 17
| Hand | Dealer 2 | Dealer 3 | Dealer 4 | Dealer 5 | Dealer 6 |
|---|---|---|---|---|---|
| 6,7 | 1 | 1 | 2 | 2 | 2 |
| 6,8 | 2 | 2 | 2 | 3 | 3 |
| 7,8 | 2 | 3 | 3 | 3 | 4 |
Strategy Deviation Points for any 678 Bonus — Dealer Stands on Soft 17
| Hand | Dealer 2 | Dealer 3 | Dealer 4 | Dealer 5 | Dealer 6 |
|---|---|---|---|---|---|
| 6,7 | 1 | 1 | 1 | 2 | 2 |
| 6,8 | 1 | 2 | 2 | 3 | 2 |
| 7,8 | 2 | 2 | 3 | 3 | 3 |
The next tables show the point at which the player should deviate from basic strategy, and hit as opposed to stand, for a suited 678 bonus.
Strategy Deviation Points for Suited 678 Bonus — Dealer Hits on Soft 17
| Hand | Dealer 2 | Dealer 3 | Dealer 4 | Dealer 5 | Dealer 6 |
|---|---|---|---|---|---|
| 6,7 | 2 | 3 | 5 | 6 | 7 |
| 6,8 | 5 | 6 | 7 | 9 | 10 |
| 7,8 | 7 | 9 | 10 | 12 | 13 |
Strategy Deviation Points for Suited 678 Bonus — Dealer Stands on Soft 17
| Hand | Dealer 2 | Dealer 3 | Dealer 4 | Dealer 5 | Dealer 6 |
|---|---|---|---|---|---|
| 6,7 | 2 | 3 | 4 | 6 | 5 |
| 6,8 | 4 | 6 | 7 | 9 | 8 |
| 7,8 | 7 | 8 | 10 | 12 | 11 |
The next tables show the point at which the player should deviate from basic strategy, and hit as opposed to split, for an any 777 bonus.
Strategy Deviation Points for Any 777 Bonus — Dealer Hits Soft 17
| Hand | Dealer 2 | Dealer 3 | Dealer 4 | Dealer 5 | Dealer 6 | Dealer 7 |
|---|---|---|---|---|---|---|
| 7,7 | 4 | 5 | 6 | 7 | 8 | 5 |
Strategy Deviation Points for Any 777 Bonus — Dealer Stands on Soft 17
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| Hand | Dealer 2 | Dealer 3 | Dealer 4 | Dealer 5 | Dealer 6 | Dealer 7 |
|---|---|---|---|---|---|---|
| 7,7 | 4 | 5 | 6 | 7 | 8 | 5 |
The next tables show the point at which the player should deviate from basic strategy, and hit as opposed to split, for a suited 777 bonus.
Strategy Deviation Points for Suited 777 Bonus — Dealer Hits Soft 17
| Hand | Dealer 2 | Dealer 3 | Dealer 4 | Dealer 5 | Dealer 6 | Dealer 7 |
|---|---|---|---|---|---|---|
| 7,7 | 20 | 26 | 33 | 38 | 44 | 22/301 |
Strategy Deviation Points for Suited 777 Bonus — Dealer Stands on Soft 17
| Hand | Dealer 2 | Dealer 3 | Dealer 4 | Dealer 5 | Dealer 6 | Dealer 7 |
|---|---|---|---|---|---|---|
| 7,7 | 20 | 26 | 32 | 38 | 44 | 22/301 |
Note 1: Breakeven point is 22 if dealer's 7 a different suit, and 30 if matching suit to player's sevens.
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Acknowledgment
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MGP caught and corrected some minor mistakes in this page. He is one of the best minds in blackjack combinatorial analysis. When MGP says I made a mistake somewhere, I always listen.