Discount Gambling

Triple Attack Ace Count Effectiveness

Posted in +EV, triple attack blackjack by stephenhow on August 31, 2011

To graphically demonstrate the large +EV advantage obtained by a simple Ace count in Triple Attack Blackjack, I compared the EV frequencies at the 1/2 shoe depth to normal blackjack using the hi-lo count. The graph below shows the Triple Attack edge dominates that of normal blackjack. The edges obtained in Triple Attack are both larger, and more frequent than in standard blackjack. The Ace count in Triple Attack is much simpler to track, compared to the hi-lo count, because you only have to identify one card — the Aces. The graph clearly shows that if you’re going to count at a shoe game, you should play Triple Attack. (Compare the areas under the curves, to the right of EV=0.)

And for the last hand in the shoe:

Triple Attack BJ Session Outcome Distributions

Posted in +EV, triple attack blackjack by stephenhow on August 26, 2011

For anyone interested in grinding out a profit from Triple Attack Blackjack, I thought I’d post a few graphs showing the distribution of outcomes for playing 100 shoes. I assumed a penetration of 7/8ths of the shoe, and for a heads-up player shamelessly betting 10x/20x when the Ace count is good, but otherwise playing basic strategy. These graphs should not be used to estimate bankroll requirements, but just give an idea of what to expect in the medium-run, and what kind of average profit rate to expect.

For a 10x big bet when the Ace count is good (+2 after 2 decks):

Cumulative Distribution of 100 Shoe Session Outcomes, 10x Bet for +EV Ace Count

Note the bias in the distribution clearly favors the player counting Aces. The x-axis is the net win/loss in small (1x) bets. The curves clearly cross the 50% cumulative frequency in positive net territory.

For a 20x big bet when the Ace count is good (+2 after 2 decks):

Cumulative Distribution of 100 Shoe Session Outcomes, 20x Bet for +EV Ace Count

Of course, you can also see the potentially large swings, which discourages all but extremely serious gamblers from trying this sort of thing. For example, varying your bet 20x against a unit $5 minimum bet means betting nickels when the count is bad, and black ($100) when the count is good. Over a 100 shoes, there’s a 10% probability of ending up stuck $6000 or more. Of course, there’s a greater chance of winning $6000 or more, and you’ll “average” a win of (250 units)($5/unit) = $1250 for this 100 shoe “session”.

In the next post, I’ll compare Triple Attack to normal 6-deck blackjack, showing how much more effective counting is, as far as shoe games go.

Triple Attack Blackjack (+EV) @ Barona Casino, CA

Posted in +EV, triple attack blackjack by stephenhow on August 14, 2011

My nearby Barona Casino opened up a new Triple Attack Blackjack game last month, and I tried it out this weekend. I really like this game, because of the weird, aggressive hitting like Spanish 21. But more importantly, the game allows you to double your initial bet after seeing your first card, and bet again after seeing the dealer upcard. You’re immediately paid on 21 or any 6-card total, and you can double down at any time. The house gets its advantage by paying even money on blackjack, and pushing on dealer 22.

For the full rules and basic strategy, see the Wizard’s analysis, which yields a 1.18% house edge. But for most cases, his basic strategy charts boil down to the following simple rules:

  • If your first card is an Ace, triple your bet.
  • If your first card is a Face, triple your bet against 2-9, else double your bet.
  • 3rd Attack 2 against a 6.
  • 3rd Attack 8 against 6,7.
  • 3rd Attack 9 against 5,6,7,8.
  • Double 10 against 3-6.
  • Double 11 against 7 and under.
  • The only soft total double is soft-18 against a 6.
  • Double all 5-card 14’s and under.
  • Double-for-less (split aces rescue) soft-17 and under, and soft-18 against 9 and A.
  • Hit all 12’s and 13’s.
  • Hit 14 except against a 6.
  • Hit 15 against a 2 or 3.
  • Hit 16 against a 2.
  • Hit 17 against an Ace, and for some 5-card totals.
  • Never split 4’s, 5’s, 6’s, 10’s.
  • Split 2’s and 3’s against 6,7.
  • Split 7’s against 4-7.
  • Split A’s and 8’s against everything except A.
  • Split 9’s against 6,8,9.

I really enjoyed the game. I also won about $400 in 3 sessions, which is unheard-of for me, since I flat bet for the minimum $5. I never vary my bet. However, in Triple Attack, you often end up betting 3 units, or even 6 or more, if you split hands like Aces. I don’t think I’ve ever won that much money playing blackjack before. The players all loved it, because they make more decisions, and their initial $20 bet can easily become a $120 bet on a good hand, or just remain a $20 bet on a weak hand.

Of course, it immediately occurred to me that card counting should be more effective in Triple Attack compared to standard blackjack. While you might dramatically increase your bet for a good count in regular blackjack, in Triple Attack, you only have to commit 1/3 of this amount on the 1st Attack, then another 1/3 only if you like your first card (an Ace or Face), and another 1/3 only if you like the dealer upcard. So, even though the count may be good, you can get away from a bad hand on the first card, and also if the dealer has a good upcard. Compare that to normal blackjack, where you bet huge on a good count, then get a 6, then the dealer shows an Ace.

Using basic strategy, I looked at the EV sensitivities of each of the card ranks (i.e., the effect of removing one card from the shoe). Interestingly, the effect of removing a Face card was very small. Removing a 5 improved the hand EV the most, but not by much compared to a 2, 3, 4, 6, or even a 7. By far, the Ace was the most powerful card, making the other cards insignificant.

So, I tried out a simple scheme, which only relies on the Ace count, compared to the expected number dealt. You subtract the number of actual Aces seen from the expected number of Aces dealt to get the Ace count. For example, after dealing 1/2 the shoe (192 cards), you should have seen 16 Aces. If you’ve only seen 12 Aces, then the count is +4. The Ace count is a measure of the “extra” Aces left in the shoe. Using a basic strategy simulator, I generated the curves showing the effectiveness of this simple Ace-Count system (including double-1-unit-for-less as Ace-split rescue):

Triple Attack Blackjack EV vs. Ace-Count and Shoe Depth

This effect is huge! Look at the green curve, which shows the effect of the Ace count after 3/4 of the shoe has been dealt (288 cards dealt). At this point, 24 Aces should have been dealt, on average. But if only 20 Aces were dealt, then the count is +4, and the next hand has a 4% player advantage. Similarly, a negative count tells you to bet the minimum, or even Wong (sit out) until the next shoe. The blue curve shows the effect after 1/4 of the shoe dealt (96 cards), and the red curve shows the half-way point (192 cards). You can see that generally speaking, it’s good to increase your bet any time the count is +2 or more (the shoe is “loaded” with 2 or more extra Aces).

We don’t need to look at the distribution of Ace counts to know that +1, +2, +3, +4, +5 counts happen all the time, especially at the end of shoe, where the effect if largest. Note that the green curve shows about a 1.25% EV improvement for each surplus Ace left in the shoe.

A simple counting strategy bets the minimum for the first 2 decks of the shoe, then a large bet if the Ace count is +2 or better. Using basic strategy, and assuming the cut card at 48 cards left in the shoe, the following returns are obtained:

Triple Attack Return vs. Large Bet On +EV Ace Count
Small Bet Large Bet Return
(relative to small bet)
1x 1x -1.10%
1x 5x +0.08%
1x 10x +1.60%
1x 20x +4.40%
1x 30x +8.80%
1x 40x +11.8%

On the other hand, the shoe is +EV about 20% of the time (+2 or better count) after the first 1/4 of the shoe is dealt. So, if you just like varying your bet on a good count, you’ll really enjoy this game. Overall, if you Wong and only bet +EV Ace counts, then you’ll extract about +17% total EV out of a shoe (i.e., +0.17 bets/shoe). The Ace-count system is very easy to implement (e.g., use the height of muck cards to estimate the expected Ace count), and is very fun. Some people have a good idea of when the shoe is Ace-rich or not. That’s all you need to know for this Triple Attack game!

And there’s no time to (possibly) bet big like the last hand of the shoe (purple curve, cut card @ 1 deck left):

Note that unless the Ace count is 0 (unlikely, only 22% of the time), you’re either a big favorite, or a big underdog on the last hand. It’s easy to get the Ace count right on the last hand, assuming the cut card was placed with 48 cards behind it. At the last hand, you should have seen 28 Aces. If you only saw 24, then you have a +4 count, and the EV of the next hand is almost 10% in your favor (a $100 1st Attack bet will return a $9.80 profit on average, including possible 2nd and 3rd Attacks, doubles, etc.). But, if you saw 30, then the count is -2, and the house has a 6.2% edge on the last hand, so bet the minimum, or Wong.

I suggest you simply count the number of Aces seen, even using chips if it helps you keep track. Compare your Ace count to the number of decks (48 cards/deck) seen in the muck rack. There are 4 Aces per deck. If the number of Aces seen is less than 4*(mucked decks), then the Ace count is positive. If your count is +2 or better, you’re +EV, and can increase your 1st Attack bet.

Get an idea of what a deck (48 cards) looks like in the discard rack. Ask the dealer for their estimate of the number of decks in the muck. You’ll find it’s pretty easy to estimate the number of dealt decks. Thus, you should be able to determine the Ace count very easily. Also, by watching the level of cards in the discard rack, you’ll know how powerful your count is (see above graphs for 1/4, 1/2, 3/4, and 7/8 shoe dealt depths).

Example: You look at the discard rack, and it looks like 3 decks have been dealt. This means that 4*3 = 12 Aces should have been dealt. Your actual Ace count is 15. The Ace count is -3, and you’re -EV, so bet the minimum. However, if the actual Ace count is only 9, then the Ace count is +3, and you’re +EV, so you should bet more.

This counting system couldn’t be any easier, or any more powerful. It doesn’t get much better than this.

Finally, I saw that at request, Barona increased the limits on the Triple Attack game to $25-$1000. I was locked out until the guy left, and the table returned to $5-$1000. In general, Barona is very flexible about increasing table limits.

Mississippi Stud @ Barona Casino (Update)

Posted in +EV, mississippi stud by stephenhow on August 9, 2011

Mississippi Stud LogoI’ve been playing more full-exposure Mississippi Stud at my nearby Barona Casino, since it’s the closest casino to me, and the game is nice and full most every night. The game is a lot more fun now, since I’ve learned all the decision points of the +EV strategy. Most every hand is a “no-brainer” now, and I only have to look at my chart for some copied, suited hands.

I’ve fine-tuned the strategy a bit, it now simulates at a +1.85% player’s edge for a full table. I’ll publish the full strategy when I’ve finished optimizing it. I’ve computed the “element-of-risk” (return per average bet risked), and it’s an amazing +0.78%. This results from the ability to 3x bet multiple streets at no risk (i.e., the hand either pushes or wins). The “element-of-risk” figure is the more accurate indicator for games requiring post-Ante bets. This means that after your Ante, you only risk another 1.35 units per hand. That’s pretty low, considering there are 3 streets to bet.

Strategy Details

The discussion below elaborates on the +EV strategy card shown in the original +EV Mississippi Stud @ Barona post. This post clarifies a few of the decision points, and shows you how to use additional information, like high cards seen, and flush cards seen.

3rd Street

Raising Hands

There are a few hands you should 3x raise on 3rd Street, that on average will pay off better than just 1x calling. Notice that the expectation value (EV) is positive and higher for 3x raising vs. 1x calling for the following hands:

3rd Street Raising Hands
Hand 1x EV 3x EV Improvement
6 high outs (suited) +1.93 +2.16 +0.23
6 high outs (offsuit) +1.48 +1.55 +0.07
6 high outs (QJo) +1.58 +1.71 +0.13
5 high outs (possible straight flush) +1.44 +1.49 +0.05
small pair, no copies +10.7 +13.3 +2.6

Those are the only raising hands. Bet other hands 1x at most. I often see people raising 5 offsuit high outs. That’s a losing move, as a 1x bet has a +0.63 return, while raising the bet to 3x decreases the return to +0.43. So you’re risking more and winning less. You win more by betting less. Don’t raise any hands except the ones in the above table!

Piggyback Bets

Often, people don’t 3x bet their hands on 3rd Street in the above table. Barona etiquette allows other players at the table to piggyback on the betting spot, if it’s not maxed out (3x). I take as many of these bets as I can, because they’re +EV. First, I tell the owner of the hand to max out their own bet, that it’s in their favour to do so. Only if they won’t make the bet themselves, do I ask to ride along a 2x bet. Usually, I talk people into taking their own EV. But, if they’re too scared, I’ll take the bet myself. From the above table, we see that the return for a 2x piggyback bet is the Improvement value. E.g., if someone doesn’t raise their uncopied small pair, a 2x piggyback bet returns +2.6 Ante average win, or a (2.6-2)/2 = +30% average profit. The returns are smaller for other hands, but is pretty good for 6 high suited outs (+11.5% for a 2x piggyback).

Calling Hands
3rd Street Hands
Outs 1x EV Notes
High Mid Low
3rd Street (Offsuit)
3 0 -0.47 Call
2 2 -0.79 Call
2 1 -1.09 Fold*
2 0 3 -1.05 Fold*
1 3 -0.99 Call*
1 2 -1.31 Fold
0 5 -0.71 Call
0 4 -1.12 Fold*
0 3 3 -0.95 Call
0 0 6 -1.43 Fold
3rd Street (Suited)
2 0 -0.87 Call
1 2 0 -0.96 Call*
0 4 -0.75 Call
0 3 3 -0.71 Call
0 3 2 -0.98 Call*
0 0 6 -1.20 Fold**

* EV varies +/- 0.20 depending on if the hand “reaches” (can make straight), and if less than 4 high cards are seen, and if suited, whether 3 or more flush cards are seen.

** You should call 54s if you can make a straight flush, and you’re not copied, and at most one flush card is seen.

So, you should use the above table as a guideline for calling or folding. Marginal hand (*) decisions depend on whether it “reaches”, the number of flush cards seen (< 3), the number of high cards seen (<= 4), and whether a straight flush is possible. For simplicity, treat each factor equally in making your decision. (A hand is more valuable when less than 4 high cards are seen, because it increases the probability of making a high pair on board).

4th Street

Raising Hands
4th Street Raising Hands
Hand 1x EV 3x EV Improvement
low pair w/ all 5 outs +1.41 +1.67 +0.26
at least 7 or more high outs (suited) +1.06 +1.22 +0.16
at least 3 high outs and possible straight flush +3.05 +3.85 +0.80
Calling Hands

The 4th Street calling points are pretty simple, and basically boil down to the following minimum hands:

  • 3 high (“pay”) outs
  • 2 high (“pay”) outs, and 3 mid (“push”) outs
  • 1 high (“pay”) outs, and 5 mid (“push”) outs
  • 6 mid (“push”) outs

You should also call a straight draw if just below these minimum requirements.

5th Street

The 5th decision is very easy to compute, since you’re betting on one card, and you know all your outs. Refer to the strategy card for the decision points, but your basic “calling hands” are

  • 5 (or more) “pay” outs
  • 4 “pay” and 2 “push” outs
  • 3 “pay” and 4 “push” outs
  • 2 “pay” and 6 “push” outs
  • 1 “pay” and 8 “push” outs

These minimum calling hands are pretty easy to remember. I know that once I have 5 high (“pay”) outs, I’m never folding. Or, once I have 4 high (“pay”) outs, I only need to pick up 2 mid (“push”) outs to not fold. And, when I only have 3 high (“pay”) outs, I know I need at least 4 mid (“push”) outs to see 5th street. If my hand is worse than these minimums, on average, I’m better off folding the hand, and losing 3 units, than playing and losing more than 3 units.

5th Street Hands
Outs 1x EV Notes
High Mid Low
5 0 -2.72 Call
4 2 -2.95 Call
4 1 -3.05 Fold
3 4 -2.95 Call
3 3 -3.05 Fold
2 6 -2.95 Call
1 8 -2.95 Call

You’re getting a lot of odds for the 5th Street bet, so if you feel like it, you can call with a slightly subpar hand (like short one mid out), since it only costs you 5% of an Ante on average. At a $5 Ante, that’s $0.25, so it won’t kill you to play an occasional underdog. Usually, I make the fold, but offer my hand to anyone at the table. Usually, someone wants it, because they’re getting odds for a cheap price ($0.25).

You should 3x raise any flush. It is extremely rare that any flush draw is -EV, so raise them all.

You should refer to the strategy card to know when to raise a straight draw. If all 8 straight outs are available, 3x raise the draw. Else, check the strategy card, which shows you the minimum requirements in order to raise a draw. E.g., you can raise a gutshot (4 straight outs) when you also have 8 high (“pay”) outs and 3 mid (“push”) outs.

Improved Table Rules

If you didn’t think it could get any better, Barona now lets you bet on multiple hands simultaneously. I guess this is part of their Social Wagering concept. Also, the table limits are $5-$100, with no limit on aggregate payout. So that means you can bet $100 Antes on 6 spots, with an EV of (6)($100)(1.8%) = $10.8/hand. If you get a fast table, you could play 40 hands/hr, for an average profit of $432/hr.

Texas Shootout @ Santa Ysabel Casino, CA

Posted in texas shootout by stephenhow on August 6, 2011

Something seems wrong about dealing poker out of a 6-deck shoe. As much as I like anything arcane and anything gambling, while this is both, it may be too much even for me. Amazingly, my relatively close Santa Ysabel Casino has both Texas Shootout and Mini-Tex. I’m sure I’ll end up playing this game sooner than later, so I made a cheat sheet from the Wizard’s analysis to use at the table.

I just sorted the 2-card hand EVs in a “table” that’s easy to use. The table ranks the hands by EV, from top to bottom, left to right in descending order. You use the table to select the best 2-card hand from the four cards you’re dealt. If you can make a second +EV hand, you should “split” and bet it as well. I put some reference marks for the hand EVs, to give you an idea of the hand strengths.

--- +0.55 EV ---

AAs AAo KKs KKo QQs QQo JJs JJo TTs TTo 99s 99o 88s
AKs 88o 77s
AQs 66s 77o AKo
AJs 55s KQs
ATs AQo 66o

--- 0 EV ---

AJo 44s A9s 55o KTs KQo
ATo A8s QJs 33s A7s KJo QTs K9s 44o A6s A5s
A9o KTo A4s JTs 22s K8s QJo Q9s A3s
A8o K7s 33s A2s
A7o QTo K6s J9s K9o Q8s K5s
A6o A5o T9s K4s JTo 22o J8s
A4o Q7s K8o K3s Q9o Q6s
A3o T8s K2s K7o 98s Q5s J7s
A2o J9o K6o Q4s Q8o 97s T7s 87s

--- -.25 EV ---

K5o Q3s T9o J6s Q2s J5s
K4o J8o 86s 96s 76s T6s Q7o J4s
K3o Q6o T8o 98o J3s 75s 65s 95s 85s
K2o T5s Q5o J7o J2s T4s 54s 97o

Q4o T7o 64s 87o 74s 84s 94s T3s 93s
Q3o J6o T2s 53s 92s 96o 86o J5o 76o
Q2o 63s T6o 73s 83s

--- -.35 EV ---

J4o 43s 82s 52s 75o 65o 95o 85o
J3o 62s 72s T5o 42s
J2o T4o 54o 64o 74o 32s

--- -.40 EV ---

94o 84o T3o
93o T2o 53o
92o 63o 73o 83o 43o 82o 52o 62o 72o 42o 32o

--- -.50 EV ---

You don’t make any decisions after selecting your hole cards. You just make one Ante bet, and you win even-money if you beat the dealer’s hand. Otherwise, you lose (dealer wins ties). Hopefully, they deal the flop, turn, and river separately. It’s be pretty boring if they just dealt the 5-card board all at once.

Mini-Tex 3-Card Hold’Em @ Santa Ysabel Casino, CA

Posted in mini-tex, three card poker by stephenhow on August 3, 2011

It was bound to happen sooner or later that someone would invent a 3-Card Hold’Em carnival game. I first heard about this game from the Wizard Of Odds last week. Since they actually have this game at my nearby Santa Ysabel Casino (min $2 Ante, $1 Bonus), I thought I’d figure out a strategy, and give it a try.

The game is pretty simple, and the Wizard lists the full rules. Like Hold’Em, each player has two hole cards, and combines them with a community board to make his best 3-Card Poker hand. The board consists of the Flop (2 cards), and the River (1 card). To form a hand, the player must use at least one of his hole cards (i.e., he cannot play the board).


  1. The player bets an Ante to start the hand.
  2. Each player and the dealer receive two face-down hole cards.
  3. The player decides to either wager the 1x Flop bet, or fold his hand and Ante.
  4. Two community cards are dealt to the board (the “Flop”).
  5. The player decides to either wager the 1x River bet, or fold his hand and all his bets.
  6. One last community card is dealt to the board (the “River”).
  7. The player decides to either wager the 2x Play bet, or fold his hand and all his bets.
  8. At showdown, the dealer must have a pair to qualify, else the Ante, Flop, and River bets automatically win, and the Play bet pushes.
  9. If the dealer does qualify, then all bets receive even-money action against the dealer hand.

Simple Strategy

The theoretical house edge for this game is 3.22%. The simple strategy described here returns 4.0%, which isn’t bad, considering the complexity of draws, and possible dealer hands against various boards. The simple strategy is extremely easy to remember. The game is fun and easy to play, because you only fold really bad hands (they’re rare).

Flop Bet

According to the Wizard of Odds, the player should play everything except 22 and 25o. Using my simplified strategy, you should also fold 26o.

River Bet

To see the river, you should 1x bet any of the following hands (fold everything else):

  • Any pair (that beats the board), or better.
  • Bet two or more draws (e.g., two flush draws, or two straight draws, or a straight draw and a flush draw).
  • Against a paired flop, bet any straight draw, any flush draw, or 3rd nut kicker.
  • Bet a lone gutshot against an offsuit, gapped flop.
  • Bet a single flush draw against a gapped flop.

* “gapped flop” means the flop isn’t a pair, or connected (e.g., 6-7 is connected, while 6-8, 7-2, A-6 are gapped)

Play Bet

To go to showdown, you should Play (2x) any of the following hands (fold everything else):

  • Mid pair or better.
  • If board is not paired and all different suits, play anything (i.e., bet that dealer doesn’t qualify).
  • If the board shows a flush (3 of same suit), and there are no one-card straights, play anything.
  • If board is paired, bet your kicker if there are less than 21 cards that beat you (the lowest possible kicker is a 9 or T; it increases by about one level for each one-card straight).
  • Bet bottom pair unless both a one-card dealer flush and a one-card dealer straight are possible.


You have 2s3c and the flop is 5dKh. You have a lone gutshot draw (you need a 4 on the river). You bet the gutshot draw, because the board isn’t suited nor connected.

You have 8d5h and the flop is Kh2c. You only have a flush draw. You bet the flush draw, because the board is gapped.

You have garbage. The board is 5d6h7c. You bet 2x hoping the dealer doesn’t qualify, because the board is rainbow and not paired. (There are 5 possible one-card straights, but that’s ok.)

You have 9h2c. The board is 6s6dAh. You can bet 2x your 9, since the board is rainbow, and there are no one-card possible straights.

You have Kh2c. The board is 6s6dAd. You can bet 2x with the K, since there are only 19 cards that beat you.

You have Qh2c. The board is 6s6d5c. You can bet 2x with the Q, since there are 21 cards that beat you.


I can’t believe this game doesn’t lend itself to collusion between players sharing hole card info. I thought there’d be a lot of opportunities for collusion, like the Flop bet decision, if confederates hold your adjacent cards (very important for straights). Or, at showdown, when you call with nothing against an unpaired, rainbow board. But my analysis shows confederate hole card info doesn’t help much at all 😦 I really thought that card info would be important, because in 3-card poker, one card often makes a hand.

I Played It!

I went out to Santa Ysabel Casino, and played the game for a few hours. It was really fun! The minimum Ante bet is only $2! So, the most you could lose in a hand is $10, and the house edge is only about $0.08/hand. The strategy is really easy to implement. You only fold if you have absolutely nothing, so it’s pretty easy to play (sometimes you even 2x Play bet nothing!). I usually only looked at one card to make my Flop and River bet decisions. Sometimes, I didn’t even need to see both cards to make a 2x Play bet decision. It was a lot of fun. The only time I ever needed to think was on the Play bet decision with kickers against a paired board (subtracting from 6th nut kicker for one-card flushes and straights).

While it’s easy to play the strategy, most of the other players probably have difficulty figuring out the right move on the fly. So, you’ll enjoy being the expert at the table.

I recommend the game, because it’s fun, and you’re usually betting 5 units to either win even money, or 3-to-5 if the dealer doesn’t qualify. That makes the house advantage per average bet fairly low (4.0%/4.6 units) < 1%.