Discount Gambling

Chase The Flush

Posted in Uncategorized by stephenhow on May 21, 2018

Chase-the-Flush-thumb

Ok, I’ll just come out and say it.  AGS’s (relatively) new Chase-the-Flush game “Makes Gambling Great Again” 🙂  I actually did the game development math on this a few years ago, when I wasn’t gambling much.  But this weekend I discovered a table at my local Hollywood Jamul Casino, so I worked out the Basic Strategy.  And guess what?  It’s a fun and elegant game!  It has a similar structure to Ultimate Texas Hold’Em, with an equal Ante and “Blind” (called the X-Tra Bonus), and 3x/2x/1x Play decisions.  But, the game is much more fun, because of the novel flush decisions, and because it’s a lot less frustrating and dread-inducing than the loved/hated UTH.  (Hint: the X-Tra pays off more frequently than the Blind, and a 3x pre-flop raise doesn’t miss the board as often as a 4x UTH bet.)

The Rules

The layout below shows the betting spots and payouts of the main game, and the pay table of the optional, independent Same Suit Bonus.  The game is played with a standard 52-card deck, where each player and the dealer receives three hole cards and shares a four-card community board to make their best-of-seven flush hand.

Chase-the-Flush-layout

The Player wagers an Ante and equal X-Tra Bonus bet before the hand begins.  Each Player and the Dealer receive three hole cards.  A Player may wager a 3x All-In bet based on his hole cards, or he may check and see the two-card flop.  The flop community cards are dealt, and a previously checking Player may now wager a 2x All-In bet, or check again.  The final two community board cards are dealt, and a previously checking Player must now wager a 1x All-In bet, or else fold his hand.

Each Player and the Dealer forms the highest flush made from their three hole cards and the four card community board.   The Dealer qualifies with a 3-card Nine-high flush or better.  If the Dealer doesn’t qualify, the remaining Antes are pushed back to the Player.  The qualified Antes and the All-In bet then receive even-money action against the Dealer hand.  The Player must beat the Dealer to receive the X-Tra Bonus payout listed in the pay table.  If the Dealer’s hand beats the Player’s hand, the X-Tra Bonus loses.  All bets push on a tie.

The House Edge

The house advantage for the main Chase-the-Flush game is only 2.65% of an Ante.  That’s very reasonable, and is comparable to the UTH house edge.  The Basic Strategy yields a practical -2.99% return to the player.  The Same Suit Bonus for the pay table in the above layout has a reasonable house edge of 5.67%.

Basic Strategy

I crafted out as simple a Basic Strategy as possible, in terms of how people intuitively view their hands during the game.  The following strategy shows the betting conditions for each of the 3x (pre-flop), 2x (flop), and 1x (river) decision points.  Check your hand for any of the betting requirements listed per decision point.  If your hand doesn’t match any of the listed conditions for the decision point, then you shouldn’t bet it.

 

Decision Betting Requirements
3x

(“pre-flop”)

Pair of Aces with Four+ kicker, or K-K-K
Three suited cards
Suited Ace
Suited King w/ Six+ singleton
Suited Queen w/ Queen+ singleton
Suited Jack w/ King+ singleton
Suited Ten w/ Ace singleton
A-Q-T or higher ranks
Check all others
2x

(“flop”)

4-card flush or better
3-card flush w/ 2-card flush
3-card flush vs offsuit board
3-card flush using suited board with Six+ kicker (hole card)
Nut 2-card flush using hole A or K plus another 2-card flush
Two 2-card flushes using board w/ Jack+ average hole cards
Check all others
1x

(“river”)

4-card flush or better
3-card flush vs rainbow board
3-card flush w/ less than 15 one-card beats
2-card flush w/ less than 10 one-card beats
Fold all others
Chase-the-Flush Basic Strategy

For optimal play, you’ll 3x raise about 23.8% of your hands, bet 2x on the flop about 24.9% of the time, 1x call on the river about 35.2% of the time, and otherwise fold about 16.1% of the time.

3x Pre-flop Examples

You should 3x Play any suited Ace.  For example, Ac-2c-2d has a EV(3x) of +68.6% and an EV(check pre-flop) of +59.9%.  So, it’s still worth +8.7% to 3x raise the hand instead of checking it.

You should 3x any pair of Aces with a Four or better kicker.  However, the hand is only +EV for A-A-6 or higher.

You should 3x a suited King with a Six or higher singleton (i.e., the offsuit card).  For example, Kd-9d-6c has a EV(3x) of +45.2% and an EV(check pre-flop) of +43.3%, showing it’s slightly better to 3x raise the hand than check it.

You should 3x raise a rainbow hand if each of the ordered ranks are higher than, or equal to, A-Q-T.  This means you should 3x A-Q-T, A-K-T, A-A-T, A-Q-J, etc.  You should check A-J-T, A-K-6, A-J-J, etc.

2x Flop Examples

The Basic Strategy bets almost all 3-card flushes on the flop.  The only exception is when the board is suited, and you’re using a hole card less than a Six to make the flush, AND you don’t have another 2-card flush.  Otherwise, you’re betting all other 3-card flushes (or better).  For example, say you’re holding Kh-7d-5c and the flop is Ac-2c.  You shouldn’t bet your 3-card flush, because your 5c kicker is less than a Six.  Notice however, if you were instead holding Kh-7h-5c, you’d 2x Play your 3-card club flush with Five kicker, because you also have a two card Kh-7h flush.

You can 2x bet a 2-card “nut” flush when you have any another 2-card flush.  For example, if you’re holding Ac-6d-5h and the flop is 5c 7d, you have the 2-card “nut” (i.e., highest possible) flush in clubs, along with another 2-card flush (7d-6d).  You should 2x Play this hand, because one of your hole cards makes the “nut” 2-card flush with a board card, and your hand makes another 2-card flush.  Note you shouldn’t bet your Ac-7d-6d hand with a board of 8c-Ah, because many single dealer heart cards (Nine or higher) beat your 2-card flush.

You can also bet two 2-card flushes that use both offsuit board cards with two hole cards averaging a Jack or higher.  For example, you can 2x Play your Kh-Qs-2d when the board flops a heart and a spade.

1x River Examples

The Basic Strategy bets almost all 3-card flushes on the river.  The only exception is when you’re playing a single small hole card to make your hand, and the board is double-suited.  In most of these cases, there are 15 or more single dealer cards that’ll beat your hand.

Otherwise, if the board is rainbow, you’ll always 1x Play any 3-card flush.

If the board has only two cards of one suit, and you have any 3-card flush, there are always less than 15 single dealer cards that’ll beat your hand, so you’ll always 1x play any 3-card flush.

If the board has a 3-card flush on board, you’ll 1x Play the board since Basic Strategy says to always call when there are less than 15 single dealer cards that’ll beat your hand (there are only 10 remaining cards of the flush suit).  However, you can get a little fancy, and fold if the board singleton is higher than the 3-card flush AND you don’t hold any cards of the singleton suit.

You can play a very high 2-card flush against a rainbow board if there are less than 10 single dealer cards that’ll beat your hand.  This usually means you can play a very high 2-card flush using the highest board card if it’s not paired on board.  For example, if the board is 9s-7h-6d-5c, you can 1x Play a Kh in the hole, since the only single dealer cards that will beat your Kh-7h is an As, Ks, Ah, Ad, Ac (5 of them).

Same Suit Bonus

While straight flushes don’t have any meaning in the main game, they are included in the pay table (along with 4+ card regular flushes) in the optional Same Suit Bonus bet.  The resulting payouts are very attractive, and add a nice dimension to the game.  The breakdown of the 7-card hand outcomes is listed in the table below, and show a total house edge of 5.67% (good as far as bonuses go).

Outcome Combinations Frequency Payout Return
6-or-7 Card Straight Flush 1,624 0.000012 2000 0.024278
5 Card Straight Flush 39,312 0.000294 100 0.029385
4 Card Straight Flush 636,272 0.004756 20 0.095119
7 Card Flush 6,664 0.000050 300 0.014899
6 Card Flush 256,620 0.001918 50 0.095908
5 Card Flush 3,550,872 0.026542 10 0.265417
4 Card Flush 25,735,424 0.192365 1 0.192365
Nothing 103,557,792 0.774064 -1 -0.774064
Total 133,784,560  1.000000 -0.056694

Optimal Play Statistics

The following table breaks down the total outcomes for the main Chase-the-Flush game, over all possible starting hands, using optimal decisions.  The total return in the lower right corner shows a house edge of 2.65% of the Ante.

Outcome Combinations Frequency Net Return
Win 3x Play w/ 7-card flush against qualified dealer

20,439,619,200

0.000051

204

0.010459

Win 3x Play w/ 6-card flush against qualified dealer

534,992,418,432

0.001342

24

0.032207

Win 3x Play w/ 5-card flush against qualified dealer

4,296,578,849,136

0.010777

9

0.096997

Win 3x Play w/ 4-card flush against qualified dealer

16,130,726,914,176

0.040462

5

0.202309

Win 3x Play w/ 3-card flush against qualified dealer

16,796,416,174,704

0.042132

4

0.168527

Lose 3x Play  against qualified dealer

30,809,847,740,400

0.077283

-5

-0.386413

Push 3x Play against qualified dealer

2,751,669,318,312

0.006902

0

0.000000

Win 3x Play w/ 6-card flush against unqualified dealer

24,404,889,600

0.000061

23

0.001408

Win 3x Play w/ 5-card flush against unqualified dealer

1,075,217,004,000

0.002697

8

0.021576

Win 3x Play w/ 4-card flush against unqualified dealer

6,377,470,048,800

0.015997

4

0.063988

Win 3x Play w/ 3-card flush against unqualified dealer

12,970,988,479,440

0.032536

3

0.097608

Win 3x Play w/ 2-card flush against unqualified dealer

1,600,580,385,168

0.004015

3

0.012045

Lose 3x Play  against unqualified dealer

1,162,087,560,552

0.002915

-4

-0.011660

Push 3x Play against unqualified dealer

478,678,665,600

0.001201

0

0.000000

Win 2x Play w/ 6-card flush against qualified dealer

227,291,635,008

0.000570

23

0.013113

Win 2x Play w/ 5-card flush against qualified dealer

4,704,150,904,080

0.011800

8

0.094398

Win 2x Play w/ 4-card flush against qualified dealer

21,499,155,021,948

0.053928

4

0.215712

Win 2x Play w/ 3-card flush against qualified dealer

14,714,103,160,440

0.036908

3

0.110725

Lose 2x Play  against qualified dealer

32,751,544,964,688

0.082153

-4

-0.328613

Push 2x Play against qualified dealer

622,124,227,116

0.001561

0

0.000000

Win 2x Play w/ 5-card flush against unqualified dealer

187,837,403,616

0.000471

7

0.003298

Win 2x Play w/ 4-card flush against unqualified dealer

6,488,002,635,144

0.016274

3

0.048823

Win 2x Play w/ 3-card flush against unqualified dealer

16,304,458,158,816

0.040898

2

0.081795

Win 2x Play w/ 2-card flush against unqualified dealer

987,169,878,672

0.002476

2

0.004952

Lose 2x Play  against unqualified dealer

710,513,189,700

0.001782

-3

-0.005347

Push 2x Play against unqualified dealer

79,383,252,492

0.000199

0

0.000000

Win 1x Play w/ 5-card flush against qualified dealer

393,192,506,064

0.000986

7

0.006904

Win 1x Play w/ 4-card flush against qualified dealer

10,828,061,228,676

0.027161

3

0.081482

Win 1x Play w/ 3-card flush against qualified dealer

20,718,789,206,988

0.051970

2

0.103941

Lose 1x Play  against qualified dealer

68,485,489,408,332

0.171787

-3

-0.515362

Push 1x Play against qualified dealer

7,086,006,696,552

0.017774

0

0.000000

Win 1x Play w/ 5-card flush against unqualified dealer

5,385,180,384

0.000014

6

0.000081

Win 1x Play w/ 4-card flush against unqualified dealer

1,985,444,394,456

0.004980

2

0.009960

Win 1x Play w/ 3-card flush against unqualified dealer

26,514,857,520,000

0.066509

1

0.066509

Win 1x Play w/ 2-card flush against unqualified dealer

1,746,004,992,372

0.004380

1

0.004380

Lose 1x Play  against unqualified dealer

2,094,365,166,192

0.005253

-2

-0.010507

Push 1x Play against unqualified dealer

362,165,402,664

0.000908

0

0.000000

folds

64,139,016,142,080

0.160885

-2

-0.321769

total

398,664,610,344,000

1.000000

-0.026470

expected

398,664,610,344,000

Ultimate Casino War

Posted in Uncategorized by stephenhow on July 24, 2015

UCW_table cardI saw this new variant of Casino War at Barona Casino, where they player gets an option to swap his card and make a 1x Raise bet. Of course, the catch is the dealer gets two cards, and gets to use the highest one. I wanted to see what the strategy and house edge were, and to check if it was at all countable out of the One-2-Six CSM they use.

The rules are pretty simple. You’re dealt one card face up, and the dealer is dealt two cards face down. The dealer will use his highest card. You have the option to replace your card with the next card out of the shoe (CSM), but you must wager an additional 1x bet to do make this swap. Finally, you may wager an optional 1x Raise on your final hand.

The dealer reveals his hand, and all your bets receive action against the dealer high card. Wins win a Six or lower pay 2:1, else it pays even-money. Ties push, and there’s no “going to War”.

For a 6-deck CSM game, the house edge is a fair 2.56%.

The basic strategy is pretty simple. You should swap an Eight or lower card. You should Raise a Jack or higher final card.

I checked the countability in a CSM by assuming perfect play given 16 known cards before every hand. The EV barely changes by +/- 0.3%, and thus is never +EV.

Ultimate Casino War Optimal Outcomes (6 Decks)
Outcome Combinations Frequency Net Return
Win 3x bet with drawn A 165,477,312 0.035607 3 0.106820
Win 3x bet with drawn K 138,914,496 0.029891 3 0.089673
Win 3x bet with drawn Q 114,674,112 0.024675 3 0.074026
Win 3x bet with drawn J 92,756,160 0.019959 3 0.059877
Lose 3x bet with drawn card 163,441,152 0.035169 -3 -0.105506
Tie 3x bet with drawn card 97,187,328 0.020912 0 0.000000
Win 2x bet with drawn T 73,160,640 0.015742 2 0.031485
Win 2x bet with drawn 9 55,887,552 0.012026 2 0.024051
Win 2x bet with drawn 8 40,772,160 0.008773 2 0.017546
Win 2x bet with drawn 7 28,274,400 0.006084 2 0.012168
Win 2x bet with drawn 6 18,057,600 0.003886 4 0.015542
Win 2x bet with drawn 5 10,121,760 0.002178 4 0.008712
Win 2x bet with drawn 4 4,466,880 0.000961 4 0.003845
Win 2x bet with drawn 3 1,092,960 0.000235 4 0.000941
Lose 2x bet with drawn card 1,409,976,288 0.303394 -2 -0.606788
Tie 2x bet with drawn card 88,157,160 0.018969 0 0.000000
Win 2x bet with original A 306,488,448 0.065949 2 0.131898
Win 2x bet with original K 257,453,856 0.055398 2 0.110796
Win 2x bet with original Q 212,690,880 0.045766 2 0.091532
Win 2x bet with original J 172,199,520 0.037053 2 0.074107
Lose 2x bet with original card 301,682,880 0.064915 -2 -0.129830
Tie 2x bet with original card 179,437,536 0.038611 0 0.000000
Win 1x bet with original T 135,979,776 0.029260 1 0.029260
Win 1x bet with original 9 104,031,648 0.022385 1 0.022385
Lose 1x bet with original card 409,808,160 0.088181 -1 -0.088181
Tie 1x bet with original card 65,156,976 0.014020 0 0.000000
Total 4,647,347,640 1.000000 -0.0256406
Expected 4,647,347,640

According to Dan Lubin, there’s a version that pays 2:1 for a win with a Six, 3:1 for a win with a Five, 5:1 for a win with a Four, and 8:1 for a win with a Trey. For a 6-deck game, these payouts reduce the house edge to 1.27%. The basic strategy remains the same. Still, the game never gets +EV with only 16 known cards.

Ultimate Casino War Optimal Outcomes (6 Decks, 2-3-5-8 Pay Table)
Outcome Combinations Frequency Net Return
Win 3x bet with drawn A 165,477,312 0.035607 3 0.106820
Win 3x bet with drawn K 138,914,496 0.029891 3 0.089673
Win 3x bet with drawn Q 114,674,112 0.024675 3 0.074026
Win 3x bet with drawn J 92,756,160 0.019959 3 0.059877
Lose 3x bet with drawn card 163,441,152 0.035169 -3 -0.105506
Tie 3x bet with drawn card 97,187,328 0.020912 0 0.000000
Win 2x bet with drawn T 73,160,640 0.015742 2 0.031485
Win 2x bet with drawn 9 55,887,552 0.012026 2 0.024051
Win 2x bet with drawn 8 40,772,160 0.008773 2 0.017546
Win 2x bet with drawn 7 28,274,400 0.006084 2 0.012168
Win 2x bet with drawn 6 18,057,600 0.003886 4 0.015542
Win 2x bet with drawn 5 10,121,760 0.002178 6 0.013068
Win 2x bet with drawn 4 4,466,880 0.000961 10 0.009612
Win 2x bet with drawn 3 1,092,960 0.000235 16 0.003763
Lose 2x bet with drawn card 1,409,976,288 0.303394 -2 -0.606788
Tie 2x bet with drawn card 88,157,160 0.018969 0 0.000000
Win 2x bet with original A 306,488,448 0.065949 2 0.131898
Win 2x bet with original K 257,453,856 0.055398 2 0.110796
Win 2x bet with original Q 212,690,880 0.045766 2 0.091532
Win 2x bet with original J 172,199,520 0.037053 2 0.074107
Lose 2x bet with original card 301,682,880 0.064915 -2 -0.129830
Tie 2x bet with original card 179,437,536 0.038611 0 0.000000
Win 1x bet with original T 135,979,776 0.029260 1 0.029260
Win 1x bet with original 9 104,031,648 0.022385 1 0.022385
Lose 1x bet with original card 409,808,160 0.088181 -1 -0.088181
Tie 1x bet with original card 65,156,976 0.014020 0 0.000000
Total 4,647,347,640 1.000000 -0.0126955
Expected 4,647,347,640

DJ Wild Poker

Posted in collusion by stephenhow on June 23, 2015

DJ_Wild_PrintDJ Wild is a new “deuces wild” poker game against the dealer, using a standard 52-card deck plus one additional Joker. The game is pretty simple, where you wager an Ante and equal Blind bet before receiving a five card hand. You then decide to either 2x Play the hand, or fold. The Dealer also receives a five card hand, and always qualifies. The Ante and Play bets receive even money action against the Dealer hand, but the Blind only pays for a straight or better. The Blind pays nice odds for rare hands, but only pays about 6% of your hands.

The full analysis of the game shows a house edge of about 3.5% of an Ante.

When I first looked at this game, it looked like easy pickings for a table full of colluding advantage players. The confederates would silently share the number of Deuces or Jokers they held in their hands (using simple chip signaling). The whole table would know the number of outstanding Wild cards seen. Each player would 2x Play if they had better than the minimum hand needed for the given Wild count. It looked like the game was toast.

So, I quickly coded it all up to find the theoretical 6-way collusion edge. I was shocked to find that even perfect info sharing only yielded +0.5% between 6 players. You’d expect more of an edge on a perfect 2x Play decision and the always-qualifying Ante. Plus, you get the chance to “save” the Blind bet with a weak hand when the Wild count is high.

Anyways, I worked out a simple 6-way collusion strategy, just in case it turned out to be slighly +EV. The strategy just uses separate minimum calling hands for each Wild count (0 thru 5). The strategy below only decreases the house edge to 1.1%.

6-Way Collusion Strategy for DJ Wild Poker
Wild Count Minimum Play Hand
0 Pair Jacks
1 Pair Nines
2 Pair Sevens
3 Pair Fours
4 Ace-King high
5 Ace high

Well, at least we know now. No one needs to lose any sleep over this game.

Arizona Stud @ Red Wind Casino, WA

Posted in Uncategorized by stephenhow on July 30, 2014

Screen Shot 2014-07-30 at 9.21.14 PMArizona Stud is a new poker-based table game debuting at the Red Wind Casino in Olympia, WA next week (6 Aug 2014). In this game, both the Dealer and the player each receive three hole cards. The player must discard one of his hole cards before the flop, while the Dealer must use exactly two hole cards to make a hand. After the player discards, he may wager a Play bet of 2x-4x the Ante, or check pre-flop. The two card flop is then revealed, as well as one of the Dealer’s hole cards. If the player checked pre-flop, he must then make a 1x Play bet, or fold. Finally, the community river card and all Dealer hole cards are revealed. The Dealer qualifies with a hand of AK-high or better. The Ante pushes if the Dealer doesn’t qualify. The Play bet always receives even-money action against the Dealer hand.

The set of all possible outcomes for the optimal player is listed in the table below. The total in the lower right corner shows a house edge of 1.34% of the Ante. Note that you should either 4x bet pre-flop, or check. You should never only bet 2x.

Optimal Outcomes for Arizona Stud
Outcome Combinations Frequency Net Return
Win 4x Play w/ ROYAL_FLUSH against qualified dealer 59,240,916 0.000001 5 0.000005
Win 4x Play w/ FULL_HOUSE against qualified dealer 89,284,476,240 0.001605 5 0.008025
Win 4x Play w/ FLUSH against qualified dealer 10,295,059,284 0.000185 5 0.000925
Win 4x Play w/ STRAIGHT against qualified dealer 13,761,723,420 0.000247 5 0.001237
Win 4x Play w/ THREE_OF_A_KIND against qualified dealer 674,048,087,712 0.012117 5 0.060586
Win 4x Play w/ TWO_PAIRS against qualified dealer 1,233,004,030,272 0.022165 5 0.110827
Win 4x Play w/ ONE_PAIR against qualified dealer 3,533,244,131,304 0.063516 5 0.317580
Win 4x Play w/ HIGH_CARD against qualified dealer 44,095,696,596 0.000793 5 0.003963
Lose 4x Play against qualified dealer 5,323,636,585,296 0.095701 -5 -0.478507
Push 4x Play against qualified dealer 90,869,346,720 0.001634 0 0.000000
Win 4x Play w/ ROYAL_FLUSH against unqualified dealer 20,545,164 0.000000 4 0.000001
Win 4x Play w/ FLUSH against unqualified dealer 4,736,232,972 0.000085 4 0.000341
Win 4x Play w/ STRAIGHT against unqualified dealer 5,959,832,148 0.000107 4 0.000429
Win 4x Play w/ THREE_OF_A_KIND against unqualified dealer 350,836,147,584 0.006307 4 0.025227
Win 4x Play w/ TWO_PAIRS against unqualified dealer 28,557,204,480 0.000513 4 0.002053
Win 4x Play w/ ONE_PAIR against unqualified dealer 2,530,675,447,344 0.045493 4 0.181973
Win 4x Play w/ HIGH_CARD against unqualified dealer 1,146,771,919,728 0.020615 4 0.082461
Lose 4x Play against unqualified dealer 25,248,339,684 0.000454 -4 -0.001816
Push 4x Play against unqualified dealer 25,026,495,696 0.000450 0 0.000000
Win 1x Play w/ ROYAL_FLUSH against qualified dealer 94,841,496 0.000002 2 0.000003
Win 1x Play w/ STRAIGHT_FLUSH against qualified dealer 541,732,152 0.000010 2 0.000019
Win 1x Play w/ FOUR_OF_A_KIND against qualified dealer 6,309,658,080 0.000113 2 0.000227
Win 1x Play w/ FULL_HOUSE against qualified dealer 45,729,841,680 0.000822 2 0.001644
Win 1x Play w/ FLUSH against qualified dealer 51,704,956,552 0.000929 2 0.001859
Win 1x Play w/ STRAIGHT against qualified dealer 90,255,233,808 0.001622 2 0.003245
Win 1x Play w/ THREE_OF_A_KIND against qualified dealer 591,850,723,248 0.010640 2 0.021279
Win 1x Play w/ TWO_PAIRS against qualified dealer 1,068,681,540,840 0.019211 2 0.038423
Win 1x Play w/ ONE_PAIR against qualified dealer 4,443,972,518,832 0.079888 2 0.159776
Win 1x Play w/ HIGH_CARD against qualified dealer 188,023,085,280 0.003380 2 0.006760
Lose 1x Play against qualified dealer 10,848,202,319,420 0.195015 -2 -0.390029
Push 1x Play against qualified dealer 161,798,077,992 0.002909 0 0.000000
Win 1x Play w/ ROYAL_FLUSH against unqualified dealer 14,941,584 0.000000 1 0.000000
Win 1x Play w/ STRAIGHT_FLUSH against unqualified dealer 220,863,744 0.000004 1 0.000004
Win 1x Play w/ FLUSH against unqualified dealer 23,880,526,848 0.000429 1 0.000429
Win 1x Play w/ STRAIGHT against unqualified dealer 48,873,031,380 0.000879 1 0.000879
Win 1x Play w/ THREE_OF_A_KIND against unqualified dealer 34,291,273,536 0.000616 1 0.000616
Win 1x Play w/ TWO_PAIRS against unqualified dealer 301,557,935,124 0.005421 1 0.005421
Win 1x Play w/ ONE_PAIR against unqualified dealer 3,348,961,937,952 0.060203 1 0.060203
Win 1x Play w/ HIGH_CARD against unqualified dealer 2,842,212,690,936 0.051094 1 0.051094
Lose 1x Play against unqualified dealer 1,090,180,312,308 0.019598 -1 -0.019598
Push 1x Play against unqualified dealer 100,426,510,008 0.001805 0 0.000000
folds 15,186,972,477,600 0.273011 -1 -0.273011
total 55,627,620,048,000 1.000000 -0.013402
expected 55,627,620,048,000

The basic strategy for the game is listed in the table below, which returns a 1.70% house edge. The player should 4x his hand about 27% of the time, 1x call about 46% of the time, and fold the remaining 27% of the time.

The game looks like fun. The strategy is actually pretty simple, but you get to make the occasional decision. I’ll actually be in Seattle next week (my first time), so I’ll try to check out the game. Maybe I could hit a nice bad beat for once.

Arizona Stud Basic Strategy
Decision Strategy
discard Hold pair, else
hold two highest cards.
Advanced exception: hold highest and lowest cards when suited, AND
highest two cards aren’t suited, AND highest card is Eight or better, AND
middle card is Six or lower, AND lowest card is only one rank below middle
card.
4x / check 4x raise any pair, else
4x raise suited Ace and Nine or better, else
4x raise Ace and Ten or better, else
check.
1x / fold 1x call three-of-a-kind, else
1x call any pair beating the dealer by more than a kicker, else
1x call same pair as dealer plus Ten or better kicker, else
1x call open-ended straight flush draw, else
1x call flush draw or any straight draw when beating the dealer, else
1x call flush draw or open-ended straight draw when dealer has no pair, else
1x call with higher hand (Jack or better kicker), else
fold all others.

The optional 2 Pair Plus Bonus bet pays for the final hand made by the player. The house edges for the various offered paytables are listed below.

2 Pair Plus Paytables
Player Hand Paytable A Paytable B Paytable C Paytable D
Royal Flush 500-to-1 500-to-1 500-to-1 500-to-1
Straight Flush 200-to-1 200-to-1 200-to-1 200-to-1
Four-of-a-Kind 100-to-1 100-to-1 100-to-1 80-to-1
Full House 50-to-1 50-to-1 40-to-1 40-to-1
Flush 30-to-1 25-to-1 30-to-1 30-to-1
Straight 20-to-1 20-to-1 20-to-1 20-to-1
Three-of-a-Kind 6-to-1 6-to-1 6-to-1 6-to-1
Two Pairs 4-to-1 4-to-1 4-to-1 4-to-1
others lose lose lose lose
House Edge 3.03% 5.12% 5.53% 6.58%

The Player Bad Beat Bonus bet pays when a player’s Jacks-or-Better hand is beat by the Dealer. The following table shows the optimal outcomes for the strategy maximizing the Bad Beat Bonus return. The house edge for the optimal Bad Beat Bonus strategy is 8.00%.

Player Bad Beat Bonus
Player Beat Hand Combinations Frequency Payout Return
Straight Flush 142,560 0.000000 1000 0.000008
Full House 1,324,642,176 0.000071 500 0.035719
Flush 2,719,437,696 0.000147 300 0.043998
Staight 1,597,456,728 0.000086 200 0.017230
Three-of-a-Kind 49,285,841,520 0.002658 30 0.079740
Two Pairs 257,968,615,536 0.013912 20 0.278245
Jacks-or-Better 923,384,598,264 0.049798 8 0.398385
other 17,306,259,281,520 0.933327 -1 -0.9333273
total 18,542,540,016,000 1.000000 -0.080002

Screen Shot 2014-07-30 at 9.51.37 PM

1 Bet Threat @ Casino Pauma

Posted in Uncategorized by stephenhow on June 15, 2014

1bet_smI saw a new Hold’Em type game at Casino Pauma last week, and I thought I’d work out the numbers and give it a try. The game is pretty simple. You bet an Ante before the hand begins. After seeing your two hole cards, you may bet 2x preflop, or check. After the flop, you may 1x bet or check. The turn, river, and the dealer’s hole cards are then revealed. The dealer qualifies with a pair of 6’s or better. If the dealer doesn’t qualify, all post-Ante wagers push. If the dealer beats your hand, you lose all your remaining bets. If you beat a qualified dealer hand, you win all your bets. If you beat a non-qualified dealer, you only win 1/2 your Ante.

The game is a bit calmer than Ultimate Texas Hold’Em, since you only have a single Ante, and you can check it down to showdown (in fact, this happens 69.8% of the time). Plus, players may like the fact that they can make the 2x and 1x bets only when they have an advantage. (I.e., all properly made 2x and 1x bets are +EV.) And the Ante is only a -11.4% loser, on average. The optimal player makes a 2x preflop bet 11.2% of the time, and a 1x flop bet on 25.5% of the time. The dealer qualifies 69.1% of the time. The game has relatively low variance, and I found myself increasing the Ante from the $5 minimum, to $10, and $15. (I’d never do that with UTH.)

The total outcomes for the optimal player strategy are listed in the table below, and show a house edge of 3.2% of the Ante.

1 Bet Threat Optimal Outcomes
Outcome Combinations Frequency Net Return
Bet 2x and 1x and beat qualified dealer 884,580,718,240 0.031804 4 0.127215
Bet 2x and 1x and beat non-qualified dealer 505,981,246,728 0.018192 0.5 0.009096
Bet 2x and 1x and lose to qualified dealer 374,729,986,984 0.013473 -4 -0.053891
Bet 2x and 1x and lose to non-qualified dealer 5,856,935,220 0.000211 -1 -0.000211
Bet 2x and 1x and tie dealer 25,182,150,868 0.000905 0 0.000000
Bet 2x only and beat qualified dealer 293,907,701,760 0.010567 3 0.031701
Bet 2x only and beat non-qualified dealer 387,449,913,432 0.013930 0.5 0.006965
Bet 2x only and lose to qualified dealer 524,307,039,216 0.018851 -3 -0.056552
Bet 2x only and lose to non-qualified dealer 76,858,269,780 0.002763 -1 -0.002763
Bet 2x only and tie dealer 25,553,189,772 0.000919 0 0.000000
Bet 1x only and beat qualified dealer 2,434,367,467,360 0.087524 2 0.175047
Bet 1x only and beat non-qualified dealer 1,467,870,962,280 0.052775 0.5 0.026387
Bet 1x only and lose to qualified dealer 1,215,166,965,412 0.043689 -2 -0.087379
Bet 1x only and lose to non-qualified dealer 17,931,292,692 0.000645 -1 -0.000645
Bet 1x only and tie dealer 164,852,060,176 0.005927 0 0.000000
Bet ante only and beat qualified dealer 3,363,692,256,360 0.120936 1 0.120936
Bet ante only and beat non-qualified dealer 4,003,403,426,760 0.143936 0.5 0.071968
Bet ante only and lose to qualified dealer 9,229,633,097,868 0.331836 -1 -0.331836
Bet ante only and lose to non-qualified dealer 1,896,770,105,748 0.068195 -1 -0.068195
Bet ante only and tie dealer 915,715,237,344 0.032923 0 0.000000
Total 27,813,810,024,000 1.000000 -0.032157

I worked out the basic strategy for the game, just in case anyone wants to play the game. The strategy is actually pretty simple. Since the dealer qualifies with a pair of 6’s or better, you generally only bet the flop if there’s a qualified hand to beat. You can bet kickers and draws against a qualified flop, otherwise you should only bet a qualifying pair when there’s a board card lower than your pair, but 6 or higher.

The basic strategy below has an error rate of 4.5%, that only results in a cost of 0.23% to the player. So the practical house edge is 3.5% for the game.

1 Bet Threat Basic Strategy
Wager Player Hand Rules
2x Pairs 2x bet a pocket pair of 7’s or better, else
check pocket 2’s thru 6’s.
Suited Bet QJs, KTs, KJs, KQs, and A8s or better, else
check all others.
Offsuit Bet KQo, and ATo or better, else
check all others.
1x Straight or better Always bet.
Three-of-a-Kind Always bet, except if trips on flop and less than 2nd nut kicker.
Two Pairs Bet if flop not paired, else
bet if flop qualified (pair 6’s or better), else
bet if board has undercard to pairs, else
bet 9’s up or better, else
check all others.
One Pair
(qualified board has pair 6’s or better)
Bet nut kicker, else
bet flush draw, else
bet open-ended straight draw with both holecards > 8, else
check all others.
One Pair
(small pair on board)
Always check.
One Pair
(unpaired board)
Bet if board has any qualifying undercards to pair, else
bet pair w/ flush draw, else
bet pair 9’s or better, else
check all others.
No Pair Bet 1st or 2nd nut flush draw, else
check all others.

There’s not much opportunity for collusion in the game. Knowledge of the hole cards of all 6 players will modify some of the preflop 2x decisions, but the frequency and value of these counter-(basic)strategy decisions aren’t enough to overcome the 3.2% house edge. Trust me, I’d have worked it out if it was worthwhile.

There’s two bonus bets offered, where the Pocket Bonus pays when your hole cards make a pocket pair, and the Final Hand bonus on your final 7-card hand. The paytables offered at Casino Pauma aren’t very good.

Pocket Bonus
Outcome Combinations Frequency Payout (to-1) Return
Pocket A’s 6 0.004525 50 0.226244
Pocket J’s – K’s 18 0.013575 20 0.271493
Pocket 2’s – T’s 54 0.040724 8 0.325792
no pair 1,248 0.941176 -1 -0.941176
Total 1,326 1.000000 -0.117647
Final Hand Bonus
Outcome Combinations Frequency Payout (to-1) Return
Royal Flush 4,324 0.000032 250 0.008080
Straight Flush 37,260 0.000279 50 0.013925
Four-of-a-Kind 224,848 0.001681 15 0.025210
Full House 3,473,184 0.025961 5 0.129805
Flush 4,047,644 0.030255 4 0.121020
Straight 6,180,020 0.046194 3 0.138581
Three-of-a-Kind 6,461,620 0.048299 2 0.096597
Jacks Up 17,385,408 0.129951 1 0.129951
others 95,970,252 0.717349 -1 -0.717349
Total 133,784,560 1.000000 -0.054179

Flush Rush @ The D Casino, Las Vegas

Posted in Uncategorized by stephenhow on May 3, 2014

Screen Shot 2014-05-03 at 3.32.29 PM

A reader told me about a new ShuffleMaster game at The D Casino, where you try to make a 4- to 7- card flush, starting with 4 hole cards, and paying to see 5th/6th and 7th Streets on a community board. The betting structure is similar to Mississippi Stud, where you post an Ante, then make a 1x Play decision to see 6th Street, and a final 1x Play decision to see 7th Street. The game pays odds on the Ante if you make a 4-card flush or better, and even-money on the 1x Play bets. Otherwise, if you fold or don’t make a hand, you lose your bets.

Ante Pay Table
Length Flush Straight Flush
7 300-to-1 1000-to-1
6 20-to-1 500-to-1
5 9-to-1 100-to-1
4 5-to-1 15-to-1

I believe The D will award the highest possible payout for a given hand. So, if you make a 5-card flush that contains a 4-card straight flush, they’ll pay you 15-to-1 (instead of 9-to-1). With this liberal rules interpretation, the house edge is 3.75%. The total possible outcomes for an optimal player are listed below.

Optimal Play Outcomes (Liberal Rules)
Outcome Combinations Frequency Net Return
7-card Straight Flush 3,360 2.3919E-07 1002 0.000240
6-card Straight Flush 167,160 1.1900E-05 502 0.005974
7-card Flush 697,620 4.9662E-05 302 0.014998
5-card Straight Flush 4,127,760 0.000294 102 0.029972
6-card Flush 26,945,100 0.001918 22 0.042119
4-card Straight Flush 65,648,544 0.004673 17 0.079447
5-card Flush 372,841,560 0.026542 11 0.291959
4-card Flush 2,627,978,496 0.187080 7 1.309557
Nothing 5,035,629,456 0.358475 -3 -1.075424
Fold before river 4,431,366,576 0.315459 -2 -0.630917
Fold before flop 1,481,973,168 0.105498 -1 -0.105498
Total 14,047,378,800 1.000000 -0.037493

If the rules are interpreted strictly, and you must make a straight flush with all your cards of the same suit, then the house edge is 5.41%.

Optimal Play Outcomes (Strict Rules)
Outcome Combinations Frequency Net Return
7-card Straight Flush 3,360 2.3919E-07 1002 0.000240
7-card Flush 717,360 5.1067E-05 302 0.015422
6-card Straight Flush 147,420 1.0494E-05 502 0.005268
6-card Flush 27,960,660 0.001990 22 0.043790
5-card Straight Flush 3,112,200 0.000222 102 0.022598
5-card Flush 397,427,940 0.028292 11 0.311212
4-card Straight Flush 41,062,164 0.002923 17 0.049693
4-card Flush 2,627,978,496 0.187080 7 1.309557
Nothing 5,035,629,456 0.358475 -3 -1.075424
Fold before river 4,431,366,576 0.315459 -2 -0.630917
Fold before flop 1,481,973,168 0.105498 -1 -0.105498
Total 14,047,378,800 1.000000 -0.054059
All-Or-Nothing Side Bet
Outcome Combinations Frequency Net Return
All hole cards same suit 2,860 0.010564 30 0.316927
All hole cards different suits 28,561 0.105498 5 0.527491
Others 239,304 0.883938 -1 -0.883938
Total 270,725 1.000000 -0.039520

Of course, the only reason why I analyzed the game was to Monte Carlo the 6-way collusion edge. The return is about +4.07% for 6 players sharing perfect info under strict rule interpretation, and +5.67% under the liberal rules. That’s not much, considering it’s pretty hard to convey suit information between confederates. It’s probably not worth anyone’s trouble to attack the game. I didn’t bother working out a practical strategy.

(FYI, I’m spending a lot more time outside of the casino these days. Before, I used to practically live in the casino. About 9 months ago, I changed obsessions. You can read about my current mania on my other blog.)

Lucky Draw Baccarat

Posted in baccarat by stephenhow on October 2, 2013

Screen Shot 2013-10-02 at 9.23.33 AMWhile visiting the TCSJohnHuxley booth at G2E last week, I played at the Lucky Draw Baccarat demo table. It’s a fun game that plays like midi-baccarat, where you can squeeze your draw card. Each player wagers an initial bet, and receives their own 2-card starting hand. Everyone plays against the bank hand, whose first card is exposed. Each player may wager an optional 1x Draw bet to receive a 3rd card, or otherwise stand pat. After the action is complete, the banker reveals his hole card. The bank draws a 3rd card when his two-card total is less than five points. Otherwise the bank stands with 5 points or more.

The game is fun, because winning hands pay odds for drawn 7, 8, and 9 totals. The player makes a decision based on his 2-card total, and the exposed bank upcard. Winning hands pay even-money on the initial wager, and odds on the 1x Draw bet according to the paytable:

Draw Bet Paytable
Outcome Paytable 1 Paytable 2
Lucky 9 (3-card 9) 3-to-1 3-to-1
Lucky 8 (3-card 8) 2-to-1 2-to-1
Lucky 7 (3-card 7) 2-to-1 3-to-2
6 or less 1-to-1 1-to-1

I analyzed the game to check the house edge, and to run EORs. The outcomes for an 8-deck shoe and optimal player decisions are listed below.

Lucky Draw Baccarat Outcomes (Paytable 1)
Outcome Combinations Frequency Net Return
Win w/ Lucky 9 134,129,168,192,512 0.053669 4 +0.214675
Win w/ Lucky 8 98,365,258,946,560 0.039359 3 +0.118076
Win w/ Lucky 7 93,724,551,243,776 0.037502 3 +0.112506
Win on draw 260,100,744,978,432 0.104074 2 +0.208147
Lose on draw 976,828,113,772,544 0.390856 -2 -0.781713
Tie on draw 165,885,343,716,480 0.066375 0 +0.000000
Win on stand 471,832,788,590,592 0.188794 1 +0.188794
Lose on stand 195,977,691,906,048 0.078416 -1 -0.078416
Tie on stand 102,355,476,404,736 0.040955 0 +0.000000
Total 2,499,199,137,751,680 1.000000 -0.017931

The house edge for Paytable 1 is 1.79%, and 3.34% for Paytable 2.

The basic strategy for the Paytable 1 game is shown in the table below.

Basic Strategy (Paytable 1)
Total Upcard
0 1 2 3 4 5 6 7 8 9
9 D S S S S S S S S S
8 S S S S S S S S S S
7 D D S S S S S S S S
6 D S S S S S S S S S
5 D D D D D D D D S S
4 D D D D D D D D D S
3 D D D D D D D D D S
2 D D D D D D D D D D
1 D D D D D D D D D D
0 D D D D D D D D D D

The computed single-card EORs for an 8-deck game with Paytable 1 are fairly low. Still, I checked the countability of an 8-deck shoe, assuming only 15 cards cut off the end. For the simple count below, the game gets advantageous only 3.2% of the time (count is +40 or better), and for an average of only +0.23%/bet. That’s essentially worthless. You might pick up some additional edge with indexed plays, or better yet, using a computer and full knowledge of shoe composition. But overall, this game is surprisingly uncountable, given the options and the odds on the Draw bet.

Lucky Draw Baccarat EORs
(Paytable 1)
Removed EOR Unbalanced
Deuce -0.020302% -1
Trey -0.012306% -1
Four -0.027603% -1
Five +0.025698% +1
Six +0.023507% +1
Seven +0.015953% +1
Eight +0.033915% +2
Nine +0.050682% +2
Ten/Face -0.016883% -1
Ace +0.017983% +1

Lucky Win Baccarat

Posted in +EV, baccarat sidebets by stephenhow on September 30, 2013

luckyWinBaccaratIf I haven’t been posting a lot lately, it’s either because I’m gambling too much, or the edges are too good to post publicly. While both these reasons would normally apply to Galaxy Gaming‘s new Lucky Win Baccarat Side bet, Eliot Jacobson just found out about it, so it’s a free-for-all while it lasts. Hopefully, you can still find a placement in the UK.

I picked up the literature for the game at Galaxy’s booth @ G2E last week. Lucky Win is a baccarat side bet that pays out for wins on low totals. When you bet on Lucky Player, you’re paid when the player wins with 5 points or less. If you bet on Lucky Banker, you’re paid for a banker win with 4 points or less. The top end of the paytable is very nice.

Lucky Win Baccarat Paytable
With With Lucky Banker
(to-1)
Lucky Player
(to-1)
1 in Spades 500 500
1 Suited 200 200
1 Offsuit 30 30
2 points 20 20
3 points 12 12
4 points 8 8
5 points lose 5

The basic house edge is computed in the following tables (8 deck shoe). The Lucky Player has a nominal 12.04% house edge, and the Lucky Banker has a nominal 10.46% house edge.

Lucky Banker Baccarat Side Bet
Outcome Combinations Frequency Payout Return
Banker win w/ 1 in Spades 373,248,411,648 0.000075 500 +0.037337
Banker win w/ 1 Suited 1,119,745,234,944 0.000224 200 +0.044804
Banker win w/ 1 Offsuit 22,798,126,252,032 0.004561 30 +0.136833
Banker win w/ 2 44,681,581,871,104 0.008939 20 +0.178784
Banker win w/ 3 72,927,778,568,192 0.014590 12 +0.175083
Banker win w/ 4 163,359,790,133,248 0.032682 8 +0.261459
Others 4,693,138,005,032,192 0.938928 -1 -0.938928
Total 4,998,398,275,503,360 1.000000 -0.104629
Lucky Player Baccarat Side Bet
Outcome Combinations Frequency Payout Return
Player win w/ 1 in Spades 378,622,455,808 0.000076 500 +0.037874
Player win w/ 1 Suited 1,135,867,367,424 0.000227 200 +0.045449
Player win w/ 1 Offsuit 23,124,703,715,328 0.004626 30 +0.138793
Player win w/ 2 44,328,525,111,296 0.008869 20 +0.177371
Player win w/ 3 62,946,423,310,336 0.012593 12 +0.151120
Player win w/ 4 86,165,771,096,064 0.017239 8 +0.137909
Player win w/ 5 122,838,277,197,824 0.024576 5 +0.122878
Others 4,657,480,085,249,280 0.931795 -1 -0.931795
Total 4,998,398,275,503,360 1.000000 -0.120400

The calculated EORs are pretty high, and lend to a very simple unbalanced count. One count nicely fits both the Lucky Player and Lucky Banker bets, for spade and non-spade cards.

Lucky Banker EORs (8 Deck)
Removed EOR
(spade)
EOR
(non-spade)
Unbalanced
Count
Deuce -0.065642% -0.018710%
Trey 0.046125% 0.091348%
Four 0.098900% 0.136220% +1
Five 0.292038% 0.334057% +1
Six 0.288705% 0.333087% +1
Seven 0.254152% 0.296456% +1
Eight 0.136562% 0.201645% +1
Nine 0.113699% 0.180474% +1
Ten/Face -0.357683% -0.277855% -1
Ace -0.368121% -0.231719% -1
Lucky Player EORs (8 Deck)
Removed EOR
(spade)
EOR
(non-spade)
Unbalanced
Count
Deuce -0.078258% -0.033211%
Trey 0.086812% 0.130836%
Four 0.207816% 0.256991% +1
Five 0.323979% 0.372323% +1
Six 0.451908% 0.523600% +1
Seven 0.260662% 0.331159% +1
Eight 0.201067% 0.260858% +1
Nine 0.092613% 0.153953% +1
Ten/Face -0.462411% -0.390156% -1
Ace -0.340367% -0.221416% -1

Using the simple unbalanced count above (Four thru Nine => +1, Ten thru Ace => -1), and starting at 0 for a new shoe (don’t forget to count the burn card!), you should bet both the Lucky Player and Lucky Banker side bets when the count is +34 or better. For an 8-deck shoe with 15 cards behind the cut card, you’ll be able to bet 6.0% of the hands. The Lucky Player bet has an average edge of +14.0%, and the Lucky Banker bet has an average edge of +10.5%. That’s a whopping combined +1.47% edge per dealt hand. That’s insane. You can see how good the bet gets in the graph below.

Screen Shot 2013-09-30 at 1.48.46 PM

Normally, I would never post about something this good. But Eliot is posting today, so the cat’s out of the bag. My apologies to any APs already hitting this game 😦

Free Bet Blackjack @ Viejas Casino

Posted in free bet blackjack by stephenhow on August 23, 2013

a91d79570d5b39e5076b8950689c832a_XLFor now, Viejas is offering a liberal interpretation of the Free Bet Blackjack rules, where they’ll now let you free double on two card soft 19, 20, and 21. The rules printed on the felt say free doubles only on hard 9, 10, and 11, but they’ve interpreted A-8 to mean hard 9, A-9 to mean hard 10, and A-Ten/Face to mean hard 11. This is only slightly helpful to the player.

The Viejas rules are:

  • Free double on two card hard 9, 10, 11
  • Free double on two card soft-19, soft-20, and Blackjack
  • No re-split of Aces
  • No surrender
  • Free double after free split

The house edge is 0.88%, which isn’t too bad. Without the free doubles on the soft totals, the house edge would be 1.10%. I guess the free doubles on the soft totals make up for the lack of surrender. The basic strategy for the non-free-split hand is listed below. Refer to the original post for the basic strategy for free-split hands.

Hand Dealer Upcard
2 3 4 5 6 7 8 9 10 A
Soft Totals
soft 21 S S S S S S S S S S
soft 20 S FD FD FD FD S S S S S
soft 19 FD FD FD FD FD FD S FD FD FD
soft 18 S S S D D S S H H H
soft 17 H H H D D H H H H H
soft 16 H H H H D H H H H H
soft 15 H H H H H H H H H H
soft 14 H H H H H H H H H H
soft 13 H H H H H H H H H H
Hard Totals
hard 20 S S S S S S S S S S
hard 19 S S S S S S S S S S
hard 18 S S S S S S S S S S
hard 17 S S S S S S S S S S
hard 16 S S S S S H H H H H
hard 15 S S S S S H H H H H
hard 14 S S S S S H H H H H
hard 13 H S S S S H H H H H
hard 12 H H H H S H H H H H
hard 11 FD FD FD FD FD FD FD FD FD FD
hard 10 FD FD FD FD FD FD FD FD FD FD
hard 9 FD FD FD FD FD FD FD FD FD FD
hard 8 H H H H H H H H H H
hard 7 H H H H H H H H H H
hard 6 H H H H H H H H H H
hard 5 H H H H H H H H H H
Pairs
A-A FP FP FP FP FP FP FP FP FP FP
10-10 S S S S S S S S S S
9-9 FP FP FP FP FP FP FP FP FP FP
8-8 FP FP FP FP FP FP FP FP FP FP
7-7 FP FP FP FP FP FP FP FP FP FP
6-6 FP FP FP FP FP FP FP FP FP FP
5-5 FD FD FD FD FD FD FD FD FD FD
4-4 FP FP FP FP FP FP FP FP FP FP
3-3 FP FP FP FP FP FP FP FP FP FP
2-2 FP FP FP FP FP FP FP FP FP FP

Push 22 Side Bet @ Viejas Casino

Posted in +EV, blackjack sidebets, free bet blackjack by stephenhow on August 22, 2013

a91d79570d5b39e5076b8950689c832a_XL My local Viejas Casino just installed Free Bet Blackjack, a game I worked on for Geoff Hall and ShuffleMaster. There’s a side bet on the game called “Push 22”, that I did not work on (until I got back from Viejas last night). I figured I’d check if it was you-know-what.

The bet has some nice payouts, and is a natural match for a game where the dealer pushes all bets on a 22 bust. For a 6-deck shoe, the game has a 5.85% house edge. Not too bad, considering the odds it pays.

Push-22 Side Bet (6 Decks)
Outcome Frequency Payout Return
Suited Dealer 22 0.003327 50-to-1 0.166345
Same Colour Dealer 22 0.011659 20-to-1 0.233174
Other Dealer 22 0.058551 8-to-1 0.468405
Dealer Not 22 0.926464 lose -0.926464
Total 1.000000 -0.058540

Of course, I had to check the EORs (for a single removed card), which showed promise:

Push-22 Side Bet EORs (6 Decks)
Removed EOR Balanced Unbalanced
Deuce -0.44% -1 -1
Trey +0.07%
Four +0.11%
Five +0.13% +1
Six -0.32% -1 -1
Seven -0.12%
Eight -0.06%
Nine -0.03%
Ten/Face +0.03%
Ace +0.52% +2 +2

With the unbalanced blah, you should blah for +24 or better. This will happen 5.1% of the time, with an average +3.6% blah.