# Discount Gambling

## Lucky Draw Baccarat

Posted in baccarat by stephenhow on October 2, 2013

While 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

If 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 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
EOR
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
EOR
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.

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 ðŸ˜¦

## Free Bet Blackjack @ Viejas Casino

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

For 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

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

I 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.

## Edge Sorting Groups for Mississippi Stud

Posted in +EV, edge sorting, mississippi stud by stephenhow on June 13, 2013

You probably know that I’m not much into advantage play based on edge-sorting cards. That’s the realm of Phil Ivey and Eliot Jacobson. It’s a pretty cool technique, but it’s way too involved for my attention span, regardless of the payoff. However, I did watch Warren Beatty in Kaleidescope, if that counts for anything.

Anyways, a reader who saw Eliot’s post on Edge Sorting (Jacks in) Mississippi Stud asked me if it’d be worthwhile to also sort the Queens, Kings, and Aces. That’s a pretty interesting question, since I can see how Eliot would start out with just the Jacks, as you’d know when you had a sure winner. But, maybe sorting the other “pay” cards would improve the return. You might not know exactly when you had a winner, but you’d have a good idea, and much more often.

I realised a Monte Carlo analysis would easily yield the ideal return for any selected sorting group. I modified a few lines of code, and violá, I simulated the estimated theoretical max return for the following sorted card groups in Mississippi Stud:

Max Return for Known Card Groups
Sorted Card Group Ideal Return
Jacks +39.7%
Jacks & Queens +48.9%
Jacks, Queens, Kings +59.0%
Jacks, Queens, Kings, Aces +63.4%

(I use the paytable that pays 5:1 for a straight.)

So it’s probably worthwhile to sort all the “pay” cards, unless it really complicates the practical strategy (not too likely).

While it’s easy to get the return for an ideal strategy for any sorting group, it takes time to work out a practical strategy. It’s straightforward, but tedious, so I’m not doing it. (Well, I actually did it for a reader, so it’s his now.)

## Lucky Stiff BJ Side Bet @ 7 Cedars, WA

Posted in blackjack sidebets by stephenhow on May 28, 2013

Paigow Dan told me about the new Lucky Stiff side bet his friend recently placed at the 7 Cedars Casino in WA. It looks fun, because you’re paid 5:1 when your initial 12-16 hard total ends up winning the main hand. Also, blackjack pays even-money on the side bet, and an initial pair of 8-8, 7-7, and 6-6 instantly wins 10:1. Anyways, I ran the bet through my BJ analyzer, to see if it was interesting in any way. I understand that 7 Cedars lets you bet \$5 on the main hand, and up to \$25 on the side bet. So I ran the analysis for a 5:1 side-to-main ratio on a 6-deck, H17, SP4, SPA4 game. The return showed a house edge of 3.5% of the combined (main+side) wager. The optimal strategy for the 5:1 side-to-main ratio only has a few differences with basic strategy.

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 S S S S S S S S S
soft 19 S S S S D S S S S S
soft 18 D D D D D S S H H H
soft 17 H D D D D H H H H H
soft 16 H H D D D H H H H H
soft 15 H H D D D H H H H H
soft 14 H H H D D H H H H H
soft 13 H H H D D 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 S H
hard 15 S S S S S H H H S H
hard 14 S S S S S H H H H H
hard 13 S S S S S H H H H H
hard 12 H S S S S H H H H H
hard 11 D D D D D D D D D D
hard 10 D D D D D D D D H H
hard 9 H D D D D H H H H H
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 P P P P P P P P P P
10-10 S S S S S S S S S S
9-9 P P P P P S P P S S
8-8 P P P P P P P P P P
7-7 P P P P P P H H H H
6-6 P P P P P H H H H H
5-5 D D D D D D D D H H
4-4 H H H P P H H H H H
3-3 P P P P P P H H H H
2-2 P P P P P P H H H H

The EORs are fairly small for the 5:1 side-to-main ratio. They’re about only 1/3rd as effective as the EORs for a standard 6-deck shoe main game. So it’s not worth your time to count this side bet. For a single card removed in a 6-deck game, the EORs are as follows:

Lucky Stiff EORs (5:1 side-to-main ratio, 6 decks)
Card Removed Return EOR
None 3.5009%
Ace 3.1324% 0.3685%
Deuce 3.3055% 0.1954%
Trey 3.3723% 0.1285%
Four 3.4423% 0.0585%
Five 3.5247% -0.0238%
Six 4.0443% -0.5434%
Seven 3.7864% -0.2856%
Eight 3.7654% -0.2845%
Nine 3.3903% 0.1106%
Ten/Face 3.4194% 0.0815%

This bet looks like fun. If you bet an equal main and side bet (1:1 side-to-main ratio), the house edge is 4.66% on the combined 2 unit bet (2.33% element-of-risk). That’s not too bad for a carnival-like odds. If you make a small side bet 1/5th of your main bet (e.g., a \$1 side bet to a \$5 main bet), then the house edge on the combined 1.2 unit bet is 1.38%. That’s not bad for a little bit of fun.

## Suit’Em Up BJ Side Bet @ Venetian, LV

Posted in +EV, blackjack sidebets by stephenhow on May 16, 2013

I saw this blackjack side bet in the Venetian last month, and it looked pretty you-know-what. I forgot to post about it until now. I’m pretty sure they use 8-deck shoes at the Venetian.

Suit’Em Up BJ Side Bet (8 Decks)
Hand Combinations Frequency Payout Return
Suited Aces 112 0.001297 60 0.077850
Suited BJs 1,024 0.011863 10 0.118628
Suited Pairs 1,344 0.015570 5 0.077850
Suited 11’s 1,024 0.011863 3 0.035589
Other Suited 17,920 0.207560 2 0.415199
nothing 64,896 0.751807 -1 -0.751807
total 86,320 1.000000 -0.026691
Suit’Em Up EORs (8 Decks)
Removed Card EOR Balanced Count Unbalanced Count
Deuce +0.000767 +1 +1
Trey +0.000767 +1 +1
Four +0.000767 +1 +1
Five +0.000767 +1 +1
Six +0.000767 +1 +1
Seven +0.000767 +1 +1
Eight +0.000767 +1 +1
Nine +0.000767 +1 +1
Ten +0.000116 0 +1
Jack +0.000116 0 0
Queen +0.000116 0 0
King +0.000116 0 0
Ace -0.006601 -8 -8

Using the unbalanced taps, the bet is +EV for RC >= +34 (assuming two decks behind the cut card). This yields 16% betting opportunities, with an average edge of +2.8%/bet. The theoretical max (using full shoe composition, including suits) is 17% opportunities @ +3.0%/bet. It’s not worth much.