Discount Gambling

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

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.

Knockout Baccarat

Posted in baccarat by stephenhow on February 15, 2013

knockOutBaccaratI was browsing the web tonight, and saw one of the game publishers that I follow placed their new Knockout Baccarat game at a casino in the UK. It’s a variation on baccarat, where you post an Ante on the Player or Banker, then they deal the first player card face up. You then decide to 2x raise your Ante, or fold. The hand is then played out following standard drawing rules. If your hand wins, then your Ante is paid even-money, and your Raise bet is paid odds depending on the losing hand total.

Raise Paytable
Losing Total Raise Payout
6-8 2:1
4-5 1:1
0-3 1:2

I decided to analyze the game, to see if any part of it was countable. You never know, people make vulnerable games, and you might find some double-digit exploit. Well, to cut to the chase, this new game isn’t very countable. I did find the game was kind of pointless, and that the Player bet is a lot worse than with regular baccarat.

I ran the numbers, and there’s no decision. You always 2x Raise your Ante. You just pick Banker or Player before the hand begins, and that’s it. Folding is never even a close option. The Banker bet has a 4.72% house edge, and the Player bet has a 10.98% house edge. Before you roll over in shock, you should really divide these edges by 3, since you’re always 2x raising your Ante. So, the element-of-risk is 1.57% for Banker, and 3.66% for Player.

Update: Oh, I see. Sometimes you win more than even-money, sometimes you win less. That’s different, I guess. Still, that could be done without the needless extra step of the 2x raise. The raise becomes silly once everyone realises you always make it. (Even against a player 9, you still make the 2x Banker Raise. Folding is a mistake costing at least 31% of the Ante.)

Banker Outcome Net Frequency Probability Return
win vs. 8 5 55,279,842,324,480 0.011060 0.055298
win vs. 7 5 158,356,022,816,768 0.031681 0.158407
win vs. 6 5 257,091,239,776,256 0.051435 0.257174
win vs. 5 3 209,913,277,599,744 0.041996 0.125988
win vs. 4 3 248,966,126,419,968 0.049809 0.149428
win vs. 3 2 287,320,282,048,512 0.057482 0.114965
win vs. 2 2 307,130,524,108,800 0.061446 0.122892
win vs. 1 2 327,378,644,353,024 0.065497 0.130993
win vs. 0 2 440,816,606,990,336 0.088192 0.176383
lose -3 2,230,518,282,592,260 0.446247

push 0 475,627,426,473,216 0.095156

total 4,998,398,275,503,360 1.000000 -0.047213
Player Outcome Net Frequency Probability Return
win vs. 8 5 55,165,968,408,576 0.011037 0.055184
win vs. 7 5 155,954,129,788,928 0.031201 0.156004
win vs. 6 5 239,782,224,326,656 0.047972 0.239859
win vs. 5 3 246,959,201,783,808 0.049408 0.148223
win vs. 4 3 267,343,809,949,696 0.053486 0.160458
win vs. 3 2 268,582,958,172,160 0.053734 0.107468
win vs. 2 2 280,469,279,004,672 0.056112 0.112224
win vs. 1 2 301,505,544,974,336 0.060320 0.120641
win vs. 0 2 414,755,166,183,424 0.082978 0.165955
lose -3 2,292,252,566,437,890 0.458597

push 0 475,627,426,473,216 0.095156

total 4,998,398,275,503,360 1.000000 -0.109777

I checked the EORs for these bets, and they’re on the order of blackjack EORs relative to the Ante. So you’re not likely to overcome the 4.72% Banker Ante edge. Thus the main bets are not very countable.

There’s a bunch of side bets you can make, but they’re not very good, nor very countable. The Ideal Countability column lists the average return of a fixed bet per shoe, made using an ideal count (perfect shoe composition info), for an 8-deck shoe with 16 cards behind the cut-card.

Knockout Baccarat Sidebets
Sidebet Payout Return Ideal Countability
Player Natural Winner 4:1 -0.187055 ~1.6% bet/shoe
Banker Natural Winner 4:1 -0.187055
Winner w/ 1,2,3,4 7:1 -0.162387 ~30% bet/shoe
Winner w/ 5,6 3.5:1 -0.170433 ~8% bet/shoe
Winner w/ 7 4.5:1 -0.128165 ~16% bet/shoe
Winner w/ 8 3:1 -0.131132 ~16% bet/shoe
Winner w/ 9 2.5:1 -0.159784 ~7% bet/shoe

Two-Person Panda-8 Co-Count

Posted in +EV, baccarat, dragon-7, panda-8 by stephenhow on January 17, 2013

Screen Shot 2012-12-09 at 7.08.26 PMThere are times when you’re at a casino with a friend, and you want to count the EZ-Baccarat Dragon-7. Normally, it’s kind of boring, and you certainly don’t need two people to do it. While it’s a good advantage play, it’d be better and a lot more fun if your friend could help with the Panda-8. I’ve posted a very complicated Panda-8 count that yields about 22% of a fixed bet per shoe. I’ve also posted a super-simple Panda-8 co-count that only yields about 9% of a fixed bet per shoe, but is meant as a single-person add-on to the Dragon-7 count.

In this post, I’ve worked out a better Panda-8 co-count that can be easily tracked by a second person. You add its running count to the Dragon-7 RC to determine when to bet the Panda. The idea exploits the common values between the two counts, resulting in a simple Panda-8 co-count. I worked this out, because I plan to use it.

Here’s the taps for the Panda-8 co-count. You add its running count to the unbalanced Dragon-7 running count, and bet when the total count is +35 or higher. You’ll get about +13.4% of a fixed bet per shoe, on an average of 3.6 bets per shoe.

Panda-8 Co-Count (Add to Dragon-7 Unbalanced RC, Bet When Combined RC >= 35)
Rank Count
Six, Seven, King, Queen +1
Trey -1
Eight -3
Unbalanced Dragon-7 Count (Bet for RC >= 32)
Rank Count
Four, Five, Six, Seven -1
Eight, Nine +2
Ace +1

Panda-8 Co-Count with Dragon-7

Posted in +EV, baccarat, panda-8 by stephenhow on December 9, 2012

Screen Shot 2012-12-09 at 7.08.26 PMWith the simplified unbalanced count for the EZ-Baccarat Dragon-7, it’s occasionally fun to count a shoe and find opportunities to bet $25 – $100, to try to win $1000 – $4000. But overall, counting the Dragon gets pretty boring. It only takes a second to see the value of the hand, and to update your count. Then you spend a lot of time watching everyone think deeply about their next bets. Hopefully, the count gets to +32, so you can finally make a bet.

Counting the Dragon-7 would be pretty good if you could make about twice the +EV it offers (+52% of a fixed bet per shoe). Or at least it’d be fun if you could easily track the Panda-8 as well, to add some variety to the game. (I’ve previously posted a complicated Panda-8 count and a RCmin table that yields +22% of a fixed bet per shoe.)

Well, I can’t double the EV of the Dragon-7, nor can I easily get you the full +22% of the Panda. But, here’s an ultra-simplified Panda-8 co-count that you should be able to track alongside the Dragon-7. It’s an unbalanced count, for simplicity. It only has a few taps. The few (4) taps it has are equal to those of the Dragon-7 unbalanced count. Also, these are key taps (you focus on the +2 Nines for the Dragon; it uses the same +1 unbalanced Aces; finally, the -1 Fours and Fives are easy to remember, because they add up to 9). You should be able to track your main Dragon-7 count, then quickly scan the hand for its Panda-8 value.

Simplified Panda-8 Co-Count
Card Count Value
Ace +1
Four, Five -1
Nine +2

Starting from a running count (RC) of 0, you should bet the Panda-8 when its count gets to +35. You’ll get an average of about 2 bets per shoe (when 16 cards are placed behind the cut card), and a profit of around +9.0% of a fixed bet per shoe. It’s not a whole lot, but it’ll make sitting around the baccarat table a little more fun/tolerable. Also, it’ll give you more cred with the degenerates watching their Player lines, Panda lines, and their second bankers 🙂

Thanks to Linus B for his initial work on the Panda co-count. I greatly simplified it here for us script-kiddies.

Unbalanced Dragon 7 Count

Posted in +EV, baccarat, dragon-7 by stephenhow on October 24, 2012

If you’re ever at an EZ-Baccarat table wondering how to properly count the Dragon-7, here’s an easy-to-use unbalanced count that you won’t forget. Unbalanced counts are very handy, because their running counts (RC) approximate true counts, without any division. They’re a nice little trick that everyone should use. I modified the count from my Dragon-7 tracking sheet post into the unbalanced count below.

You simply start the count at -32 for a new shoe, then update the running count for each card dealt, including the exposed burn card. When the running count is > 0, bet the Dragon-7 side bet. This count scheme simulates at a profit rate of +52% of a fixed bet per 8-deck shoe, when 16 cards are placed behind the cut card. You’ll get about 6.8 betting opportunities per shoe.

Unbalanced Dragon-7 Count (Start at -32, bet when RC >= 0)
Card Rank Count Value
Ten/Face 0
Ace +1
Deuce 0
Trey 0
Four -1
Five -1
Six -1
Seven -1
Eight +2
Nine +2

The variance of the bet is very high, and unless you’re heads-up with the dealer, the hand rate is very slow. If you’re wondering if you can grind out a profit from the bet, look at the outcome distribution below for a 500 unit bankroll with a +1000 unit goal, else playing for 500 shoes. While the risk of ruin is only 3.5%, you still have a 24% chance of losing after 500 shoes. Your average win is +250 units. So, if you have a $50k bankroll, can find a heads-up EZ-Baccarat table with a $100 max Dragon-7 bet, are committed to playing for hundreds of hours, and don’t draw any suspicion from casino personnel, then you can win from $50 to $100 per hour, depending on how fast you play. It might be fun for the first hour or two, but only if you hit a dragon. Try playing my Dragon-7 shoe simulator before you head out to the casino.

Easy Six Baccarat

Posted in +EV, baccarat by stephenhow on September 12, 2012

A reader just asked about a no-commission game called Easy Six Baccarat. I’ll keep this post short, for those in-the-know. Use the simple taps (6 => -7, 7 => +3, 8 => +2, 9 => +2) and a true count threshold of 5.0. For an 8-deck shoe with 52 cards behind the cut card, you’ll net +49% of a fixed bet per shoe, on an average of 12 bets/shoe. For an 8-deck shoe with 16 cards behind the cut card, you’ll net +84% of a fixed bet per shoe, on an average of 15 bets/shoe.

For simplicity, you can use the RCmin thresholds in following table:

hand # Min RC Threshold
1 40 40 39 39 38 38 37 37 36 36 35 35
13 34 34 33 33 32 32 31 31 31 30 30 29
25 29 28 28 27 27 26 26 25 25 24 24 23
37 23 22 22 21 21 21 20 20 19 19 18 18
49 17 17 16 16 15 15 14 14 13 13 12 12
61 11 11 11 10 10 9 9 8 8 7 7 6
73 6 5 5 4 4 3 3 2 2 2 1 1

Lucky 9 @ Harrah’s Rincon

Posted in baccarat by stephenhow on July 29, 2012

There’s a new game Lucky 9 at my nearby Harrah’s Rincon that’s a hybrid between baccarat and blackjack. It plays like simplified blackjack, where you can only hit once, and hand totals are between 0-9. Like blackjack, you start with two cards, and the dealer exposes their top card. Like baccarat, hand values are equal to the last digit of the sum of the cards. The best hand is a Lucky 9, which is a two card 9 without a face/ten. It ties a Natural 9 (two card 9 w/ face/ten), but otherwise pays 3:2. However, Lucky 9’s after splits only pay even-money, and push against all dealer 9’s. The house edge of the game is 1.18%.

Probably, the only point of playing this game would be to see how confused the dealer gets when resolving hands involving Lucky 9, especially after split, or against a dealer Natural 9. I’m sure they’ll probably let you hit some hands after splits too. And people will think you’re crazy for standing on 5’s and 6’s.


The rules for Lucky 9 are as follows:

  1. Bets are placed before the hand is dealt.
  2. Player and dealer each receive 2 cards; the dealer upcard is exposed.
  3. The player may split pairs (up to 3 times) by matching the initial wager. Split hands receive only one card.
  4. The player may hit any non-split hand once, or stand.
  5. After the players action is complete, the dealer turns up his hole card.
  6. The dealer hits once on totals of less than 5.
  7. A non-split Lucky 9 (2-card 9 w/o 0-value card) pays 3:2, except for pushes against a dealer 2-card 9.
  8. A dealer Lucky 9 beats a player 3-card 9.
  9. All other player 9’s vs. dealer 9’s push.
  10. The player automatically loses with a total of 0, 1, or 2.
  11. Otherwise, wagers are resolved by comparing hand point totals.

Basic Strategy

I put the rules into my blackjack analyzer, and came up with a house edge of 1.18%, and the following auto-generated strategy:

Hand Dealer Upcard
2 3 4 5 6 7 8 9 10 A
2-Card Totals
6-9 S S S S S S S S S S
5 S S S S H H H H S S
0-4 H H H H H H H H H H
10-10 P P H H H H H H P P
9-9 S S S S S S S S S S
8-8 S S S S S P S S S S
7-7 P P P P P P H H P P
6-6 P P P P P H H H P P
5-5 P P P P H H H H P P
4-4 S S S S S S S S S S
3-3 S S S S S S S S S S
2-2 H H H H H H H H H H

The EORs for the game are extremely low compared to blackjack. There’s no value in counting this game.

Lucky 9 Bonus

The Lucky 9 Bonus is a side bet on your first 2 cards and the dealer upcard. The bet pays for various 3-card totals of 9, as listed in the paytable below. The house edge for this side bet is 13.1%. The EORs (sensitivities to card removal) are too low, and the initial house edge is too high, to make the side bet very countable. An ideal counter (using perfect knowledge of shoe composition) will find +EV betting opportunities about 2.7% of the time, with an average +10.8% EV/bet.

Hand Payout Combinations Probability Return
Suited 3-3-3 200 224 0.000019 0.003761
Suited 2-3-4 100 2,048 0.000172 0.017193
Any 3-3-3 50 4,736 0.000398 0.019879
Any 2-3-4 40 30,720 0.002579 0.103155
Suited 9 Total 30 70,144 0.005888 0.176653
Other 9 Total 5 1,070,592 0.089874 0.449369
nothing -1 10,733,696 0.901071 -0.901071
Total 11,912,160 1.000000 -0.131061

7 Up Baccarat @ Marina Bay Sands, SG

Posted in baccarat, csm by stephenhow on June 16, 2012

In today’s Grail quest, I took a look at the countability of a Baccarat variant called 7 Up Baccarat, dealt out of a constant shuffle machine (CSM). If you’ve read this blog closely, you know that a CSM does not eliminate all countability in a game. This is because cards are in buffered in the exit chute of the CSM, so recently dealt cards have no chance of coming out soon. A windowed count may be effective against a CSM.

You can browse or download all the code for this post, if you want to see how I roll.

Anyway, here’s what I found for 7-Up Baccarat. Both the banker and player bets have very high sensitivities to removed cards (EORs). (Compare this to normal baccarat, where the EORs are effectively zero.) Simulations show a windowed count is strongly correlated to the EV of the next hand dealt out of the CSM. The figure below shows a 20-card windowed count tells you when its better to bet Player or Banker. Unfortunately, the count almost never gets good enough to be +EV. You can see if they made the game more “fair” (house edge only 1.3% instead of the chosen 2.6%), then you’d often find some +EV opportunities. I doubt they did this kind of analysis, but who knows.

Same thing with the Super-7’s side bet. If they made the nominal house edge closer to 5% than the 8.9% they chose, then it’d be very countable. The count is very simple. Any 7 you see is -12, and any non-7 is +1. I think everyone can imagine that it’s better to bet the Super-7’s when they haven’t seen any 7’s out of the CSM in the last few hands. And I’m sure no one bets Super 7’s just after seeing a bunch of 7’s come out. Simulation of the Super-7’s bet show a perfect linear correlation between the count and the EV of the bet. In all simulations, a minBufferDepth of 20 was used (minimum number of cards in the exit chute buffer).

Power98 Baccarat @ Marina Bay Sands, SG

Posted in baccarat by stephenhow on June 16, 2012

A few months ago, a reader contacted me about the ridiculously beatable Banker/Player Natural 8/9 bets in the Power98 Baccarat game at the Marina Bay Sands in Singapore. They used to pay 10:1 on a Natural 9, and 11:1 on a Natural 8. (And, they’d pay even more on a “Power Hand”.) Needless to say, this was obviously way too countable, and they finally figured it out, and lowered the pays down to 8:1 for both bets. This game was so beatable, I was even thinking about taking a trip out there. Oh well, our reader tells us that players figured it out, followed by management, and now it’s gone. I mean, this one was old school beatable. Throp and Walden wrote about this one in 1966, and that’s with a payout of 9:1 for both bets. What were these Power98 guys thinking? Anyway, even with their dealing procedure of placing 2 decks behind the cut card, the game was still beatable for about 61% of a fixed bet per shoe. It must have been funny to see everyone all of a sudden betting huge on both Player and Banker Natural 9. Oh well, sic transit gloria.