Black Jack Appendix 8
This appendix shall explain
and analyze some blackjack side bets I have seen. The list is getting so long I
have provided the following index.
Super SevensThe following is the payoff table for Super
Sevens:
| Super Sevens Payoff Table |
| Hand |
Pays |
| First card a seven |
3-1 |
| First two cards unsuited sevens |
50-1 |
| First two cards suited sevens |
100-1 |
| First three cards unsuited sevens |
500-1 |
| First three cards suited sevens |
5000-1 |
These awards are not cummulative, in other words if you get three sevens you
don't get paid for one and two sevens as well. If the dealer gets a blackjack
the player can still get paid for at least two sevens. At some casinos if the
player has two sevens and the dealer gets a blackjack a third card will be dealt
to the player for the chance to get three sevens.
The following probability table 1 shows the probability, payoff, and expected
return of each hand. This table assumes (1) a third card is not dealt if the
player has two sevens and the dealer gets a blackjack and (2) six decks.
| Super Sevens Probability Table 1 |
| Hand |
Probability |
Pays |
Return |
| 1 seven |
0.071234 |
3 to 1 |
0.213703 |
| 2 unsuited 7's |
0.004151 |
50 to 1 |
0.207569 |
| 2 suited 7's |
0.001153 |
100 to 1 |
0.115316 |
| 3 unsuited 7's |
0.000369 |
500 to 1 |
0.184557 |
| 3 suited 7's |
0.000015 |
5000 to 1 |
0.075924 |
| non-paying hand |
0.923077 |
-1 to 1 |
-0.923077 |
| Total |
1 |
|
-0.126008 |
The following probability table 2 shows the probability, payoff, and expected
return of each hand. This table assumes (1) a third card is dealt if the player
has two sevens and the dealer gets a blackjack and (2) six decks.
| Super Sevens Probability Table 2 |
| Hand |
Permutations |
Probability |
Pays |
Return |
| 1 seven |
2142720 |
0.071234 |
3 to 1 |
0.213703 |
| 2 unsuited 7's |
124416 |
0.004136 |
50 to 1 |
0.206809 |
| 2 suited 7's |
34560 |
0.001149 |
100 to 1 |
0.114894 |
| 3 unsuited 7's |
11664 |
0.000388 |
500 to 1 |
0.193883 |
| 3 suited 7's |
480 |
0.000016 |
5000 to 1 |
0.079787 |
| Non-paying hand |
27766080 |
0.923077 |
-1 to 1 |
-0.923077 |
| Total |
30079920 |
1 |
|
-0.114 |
The tables above show a house edge of 12.61% if the player does not get a
third card if the dealer gets a blackjack and a house edge of 11.40% if the
player is guaranteed to get three cards.
Below are the derivations of the table 1 probabilities where n is the number
of decks. The combin(x,y) function is the number of ways to arrange y cards out
of x. For example combin(52,5)=2598960, the number of possible five card poker
hands from a single deck. Let p2 denote the probability that dealer
will get a blackjack if the player's first two cards are sevens. Let
p3 denote the probability that dealer will get a blackjack if the
player's first three cards are sevens. The combin(x,y) function can be used in
Excel, by the way.
Probability of 1 seven: (1/13)*(48*n/(52*n-1))
Probability of 2 unsuited sevens: [combin(4n,2)-4*combin(n,2)]/combin(52*n,2)
* [(48*n)/(52*n-2) * (1-p2) + p2]
Probability of 2 suited sevens: combin(n,2)/combin(52*n,2) * [(48*n)/(52*n-2)
* (1-p2) + p2]
Probability of 3 unsuited sevens: [combin(4n,3)-4*combin(n,3)]/combin(52*n,3)
* (1-p3)
Probability of 3 suited sevens: 4*combin(n,3)/combin(52*n,3) *
(1-p3)
p2 = 4*(4*n)2 / combin(52*n-2,2)
p3 = 4*(4*n)2 / combin(52*n-3,2)
Below are the probabilties for table 2 where the player is guaranteed to get
a third card.
Probability of 1 seven: (1/13)*(48*n/(52*n-1))
Probability of 2 unsuited sevens: [combin(4n,2)-4*combin(n,2)]/combin(52*n,2)
Probability of 2 suited sevens: combin(n,2)/combin(52*n,2) * (48*n)/(52*n-2)
Probability of 3 unsuited sevens: [combin(4n,3)-4*combin(n,3)]/combin(52*n,3)
Probability of 3 suited sevens: 4*combin(n,3)/combin(52*n,3)
Royal MatchThe royal match is a simple bet that pays a
bonus if the first two cards are suited (an easy match) and a top bonus for a
suited king and queen (a royal match). Below are probability tables for two
versions I have seen based on a single deck game.
| Royal Match - Version 1 |
| Hand |
Probability |
Pays |
Return |
| Easy match |
0.232278 |
2.5 |
0.812971 |
| Royal match |
0.003017 |
25 |
0.078431 |
| Total |
0.235294 |
|
0.891403 |
| Royal Match - Version 2 |
| Hand |
Probability |
Pays |
Return |
| Easy match |
0.232278 |
3 |
0.929110 |
| Royal match |
0.003017 |
10 |
0.033183 |
| Total |
0.235294 |
|
0.962293 |
The following table displays the house edge for each version given the number
of decks used.
| Royal Match - House Edge |
Number
of Decks |
Version 1 |
Version 2 |
| 1 |
0.108597 |
0.037707 |
| 2 |
0.083271 |
0.008215 |
| 4 |
0.070792 |
-0.006317 |
| 6 |
0.066658 |
-0.011130 |
| 8 |
0.064597 |
-0.013531 |
In the unlikely event you ever see version 2 at a table with 4 or more decks
be sure to play it hard because the player will have the advantage.
At the Isle of Capri casino in Natchez, Mississippi, they use version 1 of
the royal match with 6 decks. In the event both the player and dealer have a
royal match the player wins an additional $1000. This lowers the house edge from
6.66% to 6.00%, assuming a $1 bet.
The probabilties for the royal match are easy to derive. Lets use n for the
number of decks of cards. The number of two card combinations is combin(52*n,2).
The number of ways to make a royal match is 4*n2. This is because
there are 4 suits and n ways to choose the queen and n ways to choose the king.
The number of ways to make an easy match is 4*(combin(13*n,2)-n2).
The 4 is the number of suits and combin(13*n,2) is the number of ways to arrange
2 cards from a given suit. You must also subtract the number of ways to make a
royal match.
The probability of an easy match is
4*(combin(13*n,2)-n2)/combin(52*n,2).
The probability of a royal match is 4*n2/combin(52*n,2).
StreakStreak is an optional blackjack side bet I noticed
at Caesars in Atlantic City in April of 2000. This is a simple bet on winning a
specified number of consecutive bets. If the player split it is the net win that
counts toward whether the hand as a whole won or lost. For example if the player
split and won one hand and pushed the other the hand would count as a net win.
In the event of a push or breaking even after a split the hand would not count
for purposes of the side bet, neither advancing the number of consecutive wins
nor breaking the winning streak. The player may bet on a winning streak from 2
to 5, or as many of these as desired. My blackjack
appendix 4 shows the following probabilities of the net result in blackjack:
- Loss: 47.89%
- Tie: 8.80%
- Win: 43.31%
Ignoring ties the probability of a loss is 52.51% and of a win is 47.49%. The
following table shows the payoff for each bet, the probability of winning, and
the house edge.
| Streak |
Number
of Wins |
Pays |
Probability |
House Edge |
| 2 |
3 to 1 |
0.2256 |
9.78% |
| 3 |
7 to 1 |
0.1071 |
14.30% |
| 4 |
17 to 1 |
0.0509 |
8.42% |
| 5 |
37 to 1 |
0.0242 |
8.18% |
Over/Under 13
This pair of side bets pay even money if the player can
correctly bet if the sum of the player's first two cards will be over or under
13. Aces count as 1. The following is the house edge according to the number of
decks.
| Over/Under 13 |
Number
of Decks |
Over |
Under |
| 1 |
6.79% |
10.11% |
| 2 |
6.65% |
10.08% |
| 4 |
6.58% |
10.07% |
| 6 |
6.55% |
10.07% |
| 8 |
6.54% |
10.06% |
Pair Square"Pair Square" is a blackjack side bet I have
seen in Tunica and Reno that wins if the player's first two cards are of the
same rank. A suited pair is best and pays more than an unsuited pair. The
following table displays the pay off according to the number of decks and the
corresponding house edge.
| Pair Square |
Number
of Decks |
Unmatched
Pair Pays |
Matched
Pair Pays |
House
Edge |
| 1 |
15:1 |
n/a |
5.88% |
| 2 |
10:1 |
25:1 |
10.68% |
| 4 |
10:1 |
20:1 |
5.80% |
| 6 |
10:1 |
15:1 |
10.61% |
| 8 |
10:1 |
15:1 |
9.40% |
TieCaesars Palace offers a side bet on a tie at two of
their blackjack tables. If the player and dealer do tie the side bet pays 10 to
1. The player may bet no more than 50% of their original blackjack wager on the
side bet. If the player splits he must also split the side bet. The following
table shows the proper basic strategy assuming the maximum side bet is played.
The combined house edge of the blackjack wager and the side bet is 0.81% of
the blackjack wager. For example if the player bets $100 on the blackjack wager
and $50 on the side bet the total expected loss is 81 cents. This is based on 8
decks and the dealer hitting a soft 17.
21+321+3 is a blackjack game with a side bet based I saw
at the Las Vegas Hilton in April, 2001. The side bet pays based on the player's
first two cards and the dealer's up card. If the three cards equal a flush,
straight, straight flush, or three of a kind the side bet pays 9 to 1. The
following table shows the probability of each hand in a six-deck game, as played
at the Hilton.
| 21+3 - 6 decks |
| Hand |
Combinations |
Probability |
Pays |
Return |
| Straight flush |
10368 |
0.002068 |
9 to 1 |
0.018613 |
| Three of a kind |
26312 |
0.005248 |
9 to 1 |
0.047236 |
| Straight |
155520 |
0.031021 |
9 to 1 |
0.279192 |
| Flush |
236736 |
0.047221 |
9 to 1 |
0.424993 |
| Pair+flush |
56160 |
0.011202 |
9 to 1 |
0.100819 |
| Pair (no flush) |
977184 |
0.194918 |
-1 to 1 |
-0.194918 |
| Nothing |
3551040 |
0.708321 |
-1 to 1 |
-0.708321 |
| Total |
5013320 |
1 |
to 1 |
-0.032386 |
The house edge under these rules is 3.24%.
At the Regent in Las Vegas all hands listed above, plus a pair, pay 5 to 2.
Two decks are used in that game. The following table shows a house edge under
these rules of 2.78%.
| 21+3 - 2 decks |
| Hand |
Combinations |
Probability |
Pays |
Return |
| Straight flush |
384 |
0.002109 |
2.5 to 1 |
0.005272 |
| Three of a kind |
728 |
0.003998 |
2.5 to 1 |
0.009994 |
| Straight |
5760 |
0.03163 |
2.5 to 1 |
0.079076 |
| Flush |
8768 |
0.048148 |
2.5 to 1 |
0.120371 |
| Pair |
34944 |
0.19189 |
2.5 to 1 |
0.479726 |
| Nothing |
131520 |
0.722225 |
-1 to 1 |
-0.722225 |
| Total |
182104 |
1 |
|
-0.027786 |
Sweet SixteenSweet Sixteen is a blackjack side bet I
noticed at the Las Vegas Club in April 2001. It is played with a six-deck shoe
and pays based on the player's first two cards. The following table shows each
paying hand, the probability, payoff, and contribution to the total return.
| Sweet Sixteen |
| Hand |
Probability |
Pays |
Return |
| 16-21 points |
0.31907 |
1 to 1 |
0.63814 |
| One ace |
0.142468 |
1 to 1 |
0.284937 |
| Two aces |
0.005689 |
2 to 1 |
0.017067 |
| Pair 2's-7's |
0.034133 |
push |
0.034133 |
| Total |
0.50136 |
|
0.974277 |
The lower right cell shows a return of 97.43%, for a house edge of 2.57%.
Here is the house edge for other numbers of decks.
- 1 deck: 3.62%
- 2 decks: 2.99%
- 4 decks: 2.68%
- 8 decks: 2.52%
Dare any PairDare any Pair is a side bet I noticed at the
Lady Luck in April 2001. It simply pays 11 to 1 if the player's first two cards
are a pair. Six decks are used. The probability of a pair is 0.073954984 for a
house edge of 11.25%. Here is the house edge for other numbers of decks.
- 1 deck: 29.41%
- 2 decks: 18.45%
- 4 decks: 13.04%
- 8 decks: 10.36%
Lucky LadiesThis is a side bet found at the Hard Rock and
the Wizard's Casino (nice
name) in Seattle. Any player 20-point hand wins something. The following table
shows the payoffs, probability, and return for a 6-deck game. The Hard Rock uses
6 decks and I hear the Wizard's Casino uses eight.
| Lucky Ladies - 6 decks |
| Hand |
Permutations |
Probability |
Pays |
Return |
| Q of hearts pair & dealer has BJ |
135360 |
0.000015 |
1000 to 1 |
0.014563 |
| Q of hearts pair |
2738340 |
0.000295 |
125 to 1 |
0.036827 |
| Matched 20 (same rank and suit) |
43105500 |
0.004638 |
19 to 1 |
0.088115 |
| Suited 20 |
193112640 |
0.020777 |
9 to 1 |
0.18699 |
| Unsuited 20 |
744863040 |
0.080139 |
4 to 1 |
0.320554 |
| Non-20 |
8310740400 |
0.894138 |
-1 to 1 |
-0.894138 |
| Total |
9294695280 |
0 |
|
-0.247089 |
The lower right cell shows a return of 75.29%, or a house edge of 24.71%. The
following list shows the house edge for this and other numbers of decks.
- 2 decks: 30.05%
- 4 decks: 26.04%
- 6 decks: 24.71%
- 8 decks: 24.05%
Bonus BlackjackThis is a simple pair of side bets that the
player and/or dealer will get a blackjack. The player may bet on a player
blackjack, dealer blackjack, or both. If the player bets both and the player
gets a blackjack composed of an ace and jack of spades then the player will win
a progressive bonus.
As the number of decks increases the probability of a blackjack decreases,
making the player's odds worse. The following table shows pertinent information
about this bet as explained below.
First column: Number of decks
Second column: House edge if just one bet is
made
Third column: Overal reduction in house edge for each $100 in meter if
both bets are made
Fourth column: Point meter must reach for bet to have zero
house edge.
| Bonus Blackjack |
| Decks |
House Edge
on One Bet |
Reduction in House
for each $100 in Meter |
Breakeven
Meter |
| 1 |
22.78% |
3.77% |
$604.00 |
| 2 |
23.53% |
3.73% |
$630.00 |
| 4 |
23.89% |
3.72% |
$643.00 |
| 6 |
24.02% |
3.71% |
$647.33 |
| 8 |
24.08% |
3.71% |
$649.50 |
Progressive BlackjackAs the name implies this is a
blackjack side bet with a progressive jackpot. For an optional $1 the blackjack
player may see back $3 to the progressive jackpot, which starts at $25,000. I
saw this side bet at the New York New York casino where they had three tables
tied into the same progressive. On July 30, 2001, the jackpot meter was at
$35537.36. At this time I was told they recently put it in place and nobody had
hit the jackpot yet. On August 11 the meter had risen to $37746.28.
Just like in Caribbean Stud the player puts the $1 for the Progressive side
bet in a slot. Before dealing a new hand the dealer presses a button, the
dollars vanish, and a light designates who made the bet. The following table
shows what each winning hand pays, the probability (based on six decks), and the
contribution to the return.
The following table shows the return based on a meter of $35537.36, the
amount the last time I observed it.
| Progressive Blackjack |
| Hand |
Permutations |
Probability |
Pays |
Return |
| 4 red/black aces |
23760 |
0.000003 |
35537.36 |
0.090844 |
| 4 aces |
231264 |
0.000025 |
2000 |
0.049763 |
| 3 suited aces |
138240 |
0.000015 |
1000 |
0.014873 |
| 3 non-suited aces |
3359232 |
0.000361 |
200 |
0.072283 |
| 2 suited aces |
10679040 |
0.001149 |
50 |
0.057447 |
| 2 non-suited aces |
38444544 |
0.004136 |
15 |
0.062043 |
| 1 ace |
662100480 |
0.071234 |
3 |
0.213703 |
| no aces |
8579718720 |
0.923077 |
0 |
0 |
| Total |
9294695280 |
1 |
0 |
0.560955 |
The above table shows an expected return of 56.10% per dollar bet, or a house
edge of 43.90%. The general formula for the return is 47.01% plus 2.56% for each
$10,000 in the meter. To have no house edge the meter would need to reach
$207287.85. Also note there are no basic strategy deviations for this side bet.
If the player gets two aces he should split anyway, which guarantees two more
cards.
It is unclear to me what events cause the meter to go up and down. Sometimes
the meter goes up by 28 cents for each $1 bet made. According to the Mikohn's
web site the house edge is 22%. If this is
the case then the meter contribution rate is 24.60%. Mikohn also mentions that
part of each dollar goes to a higher reseed of the next jackpot. So 24.60% would
be divided between the current meter and the next one. Based on this
contribution rate the average jackpot when won would be $121,225.86.
Mikohn, the owners of this side bet, keep a list of casinos that offer this
side bet here.
Three Way ActionThis is more of three games in one than a
blackjack side bet. There are three wagers to choose from, the player may make
any number and combination of them. There is also an optional blackjack side
bet. The game is played with a single deck of cards that is shuffled after every
hand. Following are the betting options.
- Blackjack: The player may play blackjack under normal house rules.
If the player reaches 7 cards without busting (which rarely happens) it is an
automatic winner.
- Combat: Player's first card against Dealer's up card, highest card
wins. A player win pays even money. Dealer wins half on a tie. House edge of
2.94%.
- Seven Card Showdown: Additional cards are added to player's and
dealer's final blackjack hands to make seven each. The best poker hand wins.
Dealer must have at least an ace high to qualify. If dealer doesn't qualify
then the players wins 1-2 (half their bet), otherwise a win pays even money.
House edge of 3.23%.
There is also a side bet titled Bonus Action based on the player's
final seven cards. The following table shows the winning hands, their
probability, what they pay, and the return.
| Three Way Action |
| Hand |
Combinations |
Probability |
Pays |
Return |
| Royal flush |
4324 |
0.000032 |
1000-1 |
0.032321 |
| Straight flush |
37260 |
0.000279 |
100-1 |
0.027851 |
| 4 of a kind |
224848 |
0.001681 |
25-1 |
0.042017 |
| Full house |
3473184 |
0.025961 |
7-1 |
0.181727 |
| Flush |
4047644 |
0.030255 |
5-1 |
0.151275 |
| Straight |
6180020 |
0.046194 |
3-1 |
0.138581 |
| 3 of a kind |
6461620 |
0.048299 |
3-1 |
0.144896 |
| 2 pair |
31433400 |
0.234955 |
lose |
-0.234955 |
| Pair |
58627800 |
0.438225 |
lose |
-0.438225 |
| Nothing |
23294460 |
0.174119 |
lose |
-0.174119 |
| Total |
133784560 |
1 |
|
-0.128632 |
The number in the lower right hand corner shows the house edge is 12.86%.
Twin BlackjackTwin blackjack is not a side bet, but a
variation of the game of blackjack. I saw the game at the Stardust in August,
2001. Each position has two betting spots. If the player makes a bet in both of
them he will play out two hands against the dealer's up card. In the event the
player gets two blackjacks (called twin blackjacks) they both shall pay 2-1. If
the player gets two identical blackjacks (called identical twin blackjacks) both
shall pay 4-1.
The following table shows what this is worth to the player.
| Twin Blackjack |
| Event |
Probability |
Pays Extra |
Return |
| Twin BJ |
0.002142 |
0.5 |
0.001071 |
| Identical twin BJ |
0.000025 |
2.5 |
0.000062 |
| total |
0.002167 |
0 |
0.001133 |
The lower right cell in the table shows the twin blackjack rules add about
0.1133% to the players return. However as usual with novelty games you give more
than you get back. In this case the player may NOT double after a split and the
number of splits per hand is lowered from 3 to 2. Under the normal Stardust
6-deck rules the house edge is 0.4066%. Under these rules, not including the
twin blackjack bonuses, the house edge is 0.5527%. Overall the house edge is
0.4394%, 0.0328% higher than the conventional rules.
Perfect PairsPerfect 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). There are two pay tables,
which are referred to as A and B. The following two tables show how the expected
return is calculated for each pay table based on an 8 deck game.
| Pay Table A - 8 decks |
| Hand |
Pays |
Probability |
Return |
| Perfect pair |
25 |
0.016867 |
0.421687 |
| Colored pair |
12 |
0.019277 |
0.231325 |
| Red/black pair |
6 |
0.038554 |
0.231325 |
| Non-pair |
-1 |
0.925301 |
-0.925301 |
| Total |
0 |
1 |
-0.040964 |
| Pay Table B - 8 decks |
| Hand |
Pays |
Probability |
Return |
| Perfect pair |
30 |
0.016867 |
0.506024 |
| Colored pair |
10 |
0.019277 |
0.192771 |
| Red/black pair |
5 |
0.038554 |
0.192771 |
| Non-pair |
-1 |
0.925301 |
-0.925301 |
| Total |
0 |
1 |
-0.033735 |
The lower right cell shows a house edge of 4.10% and 3.37% respectively.
The next table shows the house edge for both pay tables according to the
number of decks.
| Perfect Pairs House Edge |
| Decks |
Pay Table A |
Pay Table B |
| 2 |
22.33% |
25.24% |
| 4 |
10.14% |
10.63% |
| 6 |
6.11% |
5.79% |
| 8 |
4.10% |
3.37% |
For more information visit the web site for www.tablegaming.com.
Bonanza BlackjackBonanza Blackjack is a side bet found on
a fully electronic 6-deck game at the Boulder Station in Las Vegas. If the
player has any 20 (including a soft 20) and the dealer has a 10-point card the
player will win something. This is a $1 side bet, no more and no less.
| Bonanza Blackjack |
| Player's hand |
Dealer's hand |
Permutations |
Probability |
Pays |
Return |
| Same rank and suit |
First two cards match |
5760 |
0.00000062 |
25000 to 1 |
0.01549271 |
| Same rank and suit |
Up card matches |
587520 |
0.00006321 |
2500 to 1 |
0.15802562 |
| Same rank and suit |
Up card any 10 |
13348800 |
0.00143617 |
100 to 1 |
0.1436174 |
| Same rank |
Up card any 10 |
50191488 |
0.00540001 |
30 to 1 |
0.16200043 |
| Same suit |
Up card any 10 |
50191488 |
0.00540001 |
20 to 1 |
0.10800029 |
| Different rank and suit (including soft 20) |
Up card any 10 |
184747392 |
0.01987665 |
10 to 1 |
0.19876649 |
| Loser |
|
8995622832 |
0.96782332 |
-1 to 1 |
-0.96782332 |
| Total |
|
9294695280 |
1 |
|
-0.18192038 |
The lower right cell shows a house edge of 18.19%.
Hi/LowThis is a simple pair of side bets I noticed at the
Casablanca in Mesquite, Nevada. The player simply bets if his first card will be
higher or lower than the dealer's up card. In the event the two cards are the
same rank, except aces, the tie shall go to the dealer. Two aces push. The game
I saw it on was 6-decks but here is the house edge for all numbers of decks.
| Hi/Low |
| Decks |
House Edge |
| 1 |
5.43% |
| 2 |
6.27% |
| 3 |
6.55% |
| 4 |
6.69% |
| 5 |
6.77% |
| 6 |
6.83% |
| 7 |
6.87% |
| 8 |
6.9% |
Go back to blackjack
| 