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

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%.