Discount Gambling

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.

Six Card Poker Bad Beat Bonus

Posted in six card poker by stephenhow on June 25, 2013

sixcardI end up playing a lot of this game at my local Viejas Casino, mostly because it’s a really cheap game if you stick to the just Ante bet (~1.5% house edge). Of course, everyone else plays the Aces Up and Two-Way Bad Beat Bonuses, and you pretty much get ostracised from the table for not betting them. The other players just shake their head at you, and g-d forbid you should lose with a bad beat without betting the bonus. I don’t know where else you’ll ever experience such negative communal disapproval. It’s about as bad when you hit your 12-15 against a dealer 6 upcard in Spanish 21 (you should). On 3rd base. Every hand.

Anyway, everyone just loves the Two-Way Bad Beat Bonus. They don’t care what the house edge is. That’s why they’re there. They just want to hit a 35:1 or higher payout. And it happens frequently enough, especially when you play it every day. It’s the crack cocaine of bonus bets.

I saw the WOO’s numbers for the Two-Way Bad Beat were a little different than mine, but they’re pretty close. The 11.1% house edge is more than I’m usually willing to pay. I’ll bet it once or twice an hour, and consider it an occasional treat. But, unless I made a mistake, it’s not impossible for your straight flush to get beat (1 in 100 million). So you’re telling me there’s a chance …

Six Card Poker Two-Way Bad Beat Bonus
Beat Hand Combinations Probability Payout
(to-1)
Return
Straight Flush 1,967,920 1.0320E-08 10,000 0.000103
Four-of-a-Kind 150,323,712 7.8830E-07 5,000 0.003941
Full House 18,331,506,888 9.6130E-05 500 0.048065
Flush 57,651,601,832 3.0232E-04 200 0.060465
Straight 185,942,016,336 9.7508E-04 100 0.097508
Three-of-a-Kind 776,263,604,160 4.0707E-03 35 0.142475
Two Pairs 6,590,304,418,608 3.4559E-02 10 0.345595
Pair Aces 2,871,866,305,368 1.5060E-02 9 0.135540
Others 180,194,060,203,056 9.4494E-01 -1 -0.944935
Total 190,694,571,947,880 100% -0.111243

Wild Six Card Draw Poker

Posted in collusion by stephenhow on June 14, 2013

While I was playing Six Card Poker at my local Viejas Casino, another player told be about the Wild Six Card Draw that he plays in Colorado. It’s a poker game with two wild Jokers in the 54-card deck, and the player gets 5 cards plus one free replacement card vs the dealer’s 6 card hand. I ran a Monte Carlo analysis to see if ideal 6-way collusion would yield any edge (you never know, the game has two Jokers after all). But even with 6-way collusion, you can’t get the house edge below 2.2%. I guess that makes sense, since it’s probably rare where you’d chose a weird draw over the more obvious discard. Anyway, it’s really easy to check these things, and you never know what you’ll find.

Super Six Baccarat

Posted in +EV, baccarat by stephenhow on June 14, 2013

Some readers asked about a Baccarat side bet called “Super Six” which pays 15:1 for a dealer wins with a 6 total. It’s really easy to analyze the countability of any Baccarat side bet. The ideal return for this bet with a perfect (computer) count of an 8-deck shoe game with 15 cards behind the cut is only +24% of a fixed bet per shoe (2.6 bets per shoe at an average +9.2% advantage per bet). A simple unbalanced count (six => -2, seven, eight, nine => +1) and betting when the running count is +34 or higher yields only +12.2% of a fixed bet per shoe on 2.77 bets/shoe, and +4.41% edge/bet. It really doesn’t seem worth the effort, even if you had an ideal count (e.g., mobile app). You’d go crazy waiting around for less than 3 bets per shoe.

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