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
Royal Match
Easy Match
All other
Total
Combinations
Pays
144
11868
36504
48516
Return = -3234/48516 = -6.67%.
Return
25
2.5
-1
3600
29670
-36504
-3234
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
Royal Match
Suited blackjack
Easy match
All other
Total
Combinations
Pays
144
576
11292
36504
48516
Return
25
5
2.5
-1
3600
2880
28230
-36504
-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
Suited pair
Non-suited pair
All other
Total
Combinations
Pays
312
1248
19968
21528
Return
20
10
-1
6240
12480
-19968
-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
Perfect pair
Colored pair
Red/black pair
Combinations
Pays
1456
1664
3328
Return
30
10
5
43680
16640
16640
Non-pair
Total
79872
86320
-1
-79872
-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
Three one-eyed jacks
Three jacks
Two one-eyed jacks
Combinations
Pays
120
1020
10800
Return
500
100
30
60000
102000
324000
Two jacks
One one-eyed jack
One jack
No jacks
Total
34800
286800
286800
2275280
2895620
Return = -580880/2895620 = -20.06%.
10
2
1
-1
348000
573600
286800
-2275280
-580880