GAM 470

Blackjack side bets part 1

Royal Match version 1

In the Royal Match side bet the player is paid if his first two cards are suited. A "royal match" is a suited king and queen and pays the highest win. I call all matches besides a royal match an "easy match." The following is based on a 6-deck game.

Royal match: There are 4 suits. Once a suit has been determined there are six ways to choose the king and six ways to choose the queen. So there are 4*6*6=144 combinations for a royal match.

Each match: There are two types of easy matches, two different ranks and two of the same card. Either way there are 4 possible suits.

For two different ranks there are combin(13,2)=78 ways to choose 2 ranks out of 13. However one of those ways (a queen and king) results in a royal match so there are 77 ways to choose 2 ranks besides the queen and king. Once the suit and ranks are chosen there are 6 cards of each rank to chose from in the shoe. So there are 4*(combin(13,2)-1)*6^2 = 4*77*36 = 11088 way to make an easy match consisting of 2 different cards.

For two of the same card there are 13 ranks to choose from. Once the rank and suit are determined there are combin(6,2)=15 ways to choose 2 cards out of the six in the shoe. So there are 4*13*15 = 780 combinations for an easy match consisting of a pair of the same card. So the number of each matches is 11088 + 780 = 11868.

All losing hands must have two different suits. There are combin(4,2)=6 ways to choose 2 suits out of 4. Once the suits are chosen there are 13 ranks for each one. Once the suit and ranks are chosen there are six cards to choose from in the shoe for each particular suit and rank chosen. So there are 6*13^2*6^2 = 36504 losing combinations.

The following table shows the return for each possible outcome, which is the product of the number of combinations and what the player wins. The bottom right cell shows that over all 48,516 combinations the player can expect to lose 3234 units.

Hand |
Combinations |
Pays |
Return |

Royal Match |
144 |
25 |
3600 |

Easy Match |
11868 |
2.5 |
29670 |

All other |
36504 |
-1 |
-36504 |

Total |
48516 |
-3234 |

Return = -3234/48516 = -6.67%.

Royal Match version 2

Version 2 of the royal match is the same as version 1, except a suited blackjack pays 5 to 1. Let’s assume a six-deck game again. The number of royal matches would be the same.

Suited blackjack: There are 4 suits to chose from. Once the suit is chosen there are 6 aces to chose from and 4*6=24 10-point cards. So the number of suited blackjacks is 4*6*24 = 576.

Each match: There are two types of easy matches, two different ranks and two of the same card. Either way there are 4 possible suits.

For two different ranks there are combin(13,2)=78 ways to choose 2 ranks out of 13. However 5 of those ways (K/Q, A/10, A/J, A/Q, A/K) result in a royal match or blackjack so there are 78-5=73 ways to choose 2 ranks besides the higher paying hands. Once the suit and ranks are chosen there are 6 cards of each rank to chose from in the shoe. So there are 4*(combin(13,2)-1)*6^2 = 4*73*36 = 10512 way to make an easy match consisting of 2 different cards.

There are still 780 ways to get 2 of the same card (see Royal Match version 1). So the total number of ways to get an easy match is 10512 + 780 = 11292.

There are still 36504 to get two cards of different suits (see Royal Match version 1).

Hand |
Combinations |
Pays |
Return |

Royal Match |
144 |
25 |
3600 |

Suited blackjack |
576 |
5 |
2880 |

Easy match |
11292 |
2.5 |
28230 |

All other |
36504 |
-1 |
-36504 |

Total |
48516 |
-1794 |

Return = -1794/48516 = -3.70%.

Pair Square

Pair Square is a side bet based on the player’s first two cards. If they form a pair the player wins. A suited pair pays more than an unsuited pair. Let’s assume 4 decks for this one.

Suited pair: The two cards must be the same in both suit and rank. There are 52 cards in the deck to choose from. Once the card is chosen there are combin(4,2)=6 ways to choose 2 out of the 4 in the shoe. So there are 52*6=312 combinations for a suited pair.

Unsuited pair: There are combin(4,2)=6 ways to pick 2 suits out of 4. Then there are 13 ranks to chose from. For each card there are 4 in the shoe to chose from. So the total number of unsuited pairs is 6*13*4^2 = 1248.

Non pair: For a non-pair there are combin(13,2)=78 ways to picks two different ranks out of 13. Then there are 4*4=16 of each rank in the shoe. So there are 78*16^2 = 19968 ways to choose a non-pair.

Hand |
Combinations |
Pays |
Return |

Suited pair |
312 |
20 |
6240 |

Non-suited pair |
1248 |
10 |
12480 |

All other |
19968 |
-1 |
-19968 |

Total |
21528 |
-1248 |

Return = -1248/21528 = -5.80%.

Perfect Pairs

Perfect Pairs is a blackjack side bet found in various casinos in Australia. It pays if the player's first two cards are a pair. The following table shows the specifics. A "perfect pair" is two identical cards (like two ace of spades). A "colored pair" is two cards of the same rank and color (like the ace of spades and ace of clubs). Let’s assume 8 decks this time.

Perfect pair: There are 52 different cards in a deck to choose from. Once one in chosen there are combin(8,2)=28 ways to choose 2 out of 8 cards from the shoe. So there are 52*28=1456 combinations for a perfect pair.

Colored pair: There are 13 ranks to choose from for the pair. Then there are 2 colors to choose from. The two cards can not be the same suit by definition. So, the two cards must be different. For each card there are 8 in the shoe to choose from. So the total number of colored pair combinations is 13*2*8^2 = 1664.

Red/black pair: There are 13 ranks to choose from for the pair. For the black card there are two suits to choose from. For the red card there are also two suits to choose from. Then there are 8 cards in the shoe for each specific card. So the total number of red/black pair combinations are 13*2*2*8^2 = 3328.

Non-pair: For a non-pair there are combin(13,2)=78 ways to picks two different ranks out of 13. Then there are 4*8=32 of each rank in the shoe. So there are 78*32^2 = 79872 ways to choose a non-pair.

Hand |
Combinations |
Pays |
Return |

Perfect pair |
1456 |
30 |
43680 |

Colored pair |
1664 |
10 |
16640 |

Red/black pair |
3328 |
5 |
16640 |

Non-pair |
79872 |
-1 |
-79872 |

Total |
86320 |
-2912 |

Return = -2912/86320 = -3.37%.

Jack Magic is a Shufflemaster side bet that has been seen at the Spirit Mountain casino in Grande Ronde, Oregon. It is played on a 5-deck blackjack game with a continuous shuffler. Wins are based on the player's initial two cards and the dealer's up card, thus no basic strategy changes are necessary. Two of the four jacks in a deck are one-eyed.

Three one-eyed jacks: There are 5*2=10 one eyed jacks in the shoe. There are combin(10,3)=120 ways to choose 3 of them.

Three jacks: There are 5*4=20 jacks in the shoe. There are combin(20,3)=1140 ways to choose 3 of them. However 120 of those ways result in three one eyed jacks. So there are 1140-120 = 1020 ways to three jacks where at least one is two-eyed.

Two one-eyed jacks: There are combin(10,2)=45 ways to choose 2 one-eyed jacks out of the 10 in the shoe. The other card must not be a jack. There are 48*5=240 non-jacks in the shoe. So there are 45*240=10800 ways to get two one-eyed jacks.

Two jacks: There are combin(20,2)=190 ways to choose 2 jacks out of the 10 in the shoe. The other card must not be a jack. There are 48*5=240 non-jacks in the shoe. So there are 190*240=45600 ways to get two jacks. However 10800 of those ways are for two one-eyed jacks. So the number of ways to get to jacks, where at least one has two eyes, is 45600-10800 = 34800.

One one-eyed jack: There are 10 one-eyed jacks in the shoe. There are combin(48*5,2) = combin(240,2) = 28680 ways to choose 2 non-jacks out of the 240 in the shoe. So there are 10*28680 = 286800 ways to choose one one-eyed jack.

One jack: There are 10 two-eyed jacks in the shoe. There are combin(48*5,2) = combin(240,2) = 28680 ways to choose 2 non-jacks out of the 240 in the shoe. So there are 10*28680 = 286800 ways to choose one two-eyed jack.

Hand |
Combinations |
Pays |
Return |

Three one-eyed jacks |
120 |
500 |
60000 |

Three jacks |
1020 |
100 |
102000 |

Two one-eyed jacks |
10800 |
30 |
324000 |

Two jacks |
34800 |
10 |
348000 |

One one-eyed jack |
286800 |
2 |
573600 |

One jack |
286800 |
1 |
286800 |

No jacks |
2275280 |
-1 |
-2275280 |

Total |
2895620 |
-580880 |

Return = -580880/2895620 = -20.06%.