Discount Gambling

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

Counting CSM Blackjack (+EV)

Posted in +EV, blackjack, csm by stephenhow on July 27, 2012

A few people have asked me about the countability of blackjack dealt from a constant shuffle machine (CSM). I’m a big advocate of counting the CSM, especially for card craps, because of the ease of the windowed count. Even if the dealer collects no muck (i.e., immediately shuffles completed hands back into the CSM), you’ll still be +EV more than 8% of the time for good BJ rules. It’s a lot easier to count a CSM than a shoe. I call it counting for the ADHD crowd. All you have to do is pay attention to the last 16 cards (or the collected muck) fed into the CSM before the hand is dealt. Plus, you’ll probably never get backed off from CSM blackjack, even when wildly varying your bets.

EV vs. Windowed Count

I use my model of the ShuffleMaster 126 (source) CSM in the blackjack simulations for this post. I’ve talked in detail about this model before, in my posts on card craps. Basically, there’s a buffer of at least 16 cards in the chute (so the dealer never waits for a card), which introduces state into the system. If the dealer collects no muck, you simply use the running count of the last 16 cards fed into the shuffler. Use the simple hi-lo count (2-6 are +1, 10-A are -1). If the dealer collects a lot of muck, and feeds it all into the shuffler right before the next hand starts, then use the count of the entire muck.

For these simulations, I used 6 decks in the CSM, and typical-good H17 blackjack rules (3:2 BJ, late surrender, re-split Aces 3-times, double-after-split). My blackjack analyzer calculates the ideal EV for these rules at -0.445% for 6 decks. I ran the simulator head’s up against the dealer, and kept track of the 16-card windowed count and the subsequent hand outcome. I plotted the next-hand EV vs. the windowed hi-lo count in the graph below.

The graph shows a very linear relationship between the 16-card windowed hi-lo count and the EV of the next hand. When the running windowed count is +5 or more, the next hand from the CSM is +EV. The windowed count is ≥ 5 about 8.2% of the time.

Frequencies of 16-Card Windowed Counts
Count Frequency Approx. BJ EV
0 11.5% -0.44%
1 10.8% -0.34%
2 9.36% -0.24%
3 7.46% -0.16%
4 5.46% -0.05%
5 3.67% +0.04%
6 2.25% +0.14%
7 1.25% +0.22%
8 0.627% +0.29%
9 0.280% +0.43%
10 0.111% +0.49%


Serious card counters will tell you you can’t count a CSM. But the data above shows that a CSM goes +EV more than 8% of the time. Plus, it’s infinitely easier to count a CSM than it is to count a shoe. You can lose track of the count for a hand or two. As soon as you regain attention, you’ll know what the count is. You can probably vary your bets wildly without attracting any attention or interest from the floor. You can probably even Wong hands when the count is bad. Or less than +5.

Counting a CSM is great for the casual counter. It’s basically short-attention span counting. If you see the last 16 cards into the CSM prior to the deal are low (have a running count of 5 or better), then you’re +EV for the next hand. Even if you just see a net +3 count for the last 16 cards, you still know the next hand will be better than average. You can start/stop paying attention on a per-hand basis (unlike a shoe, where you have to wait for the next shoe if you lose the count).

At it’s simplest, CSM counting will tell you when the next hand will be better-than-average (half the time), or worse-than-average (half the time). So, if you Wong half of the time, you’ll only play the better-than-average hands (EV better than -0.45%; the above curve to the right of count=0), and miss the bad hands. That’s a quick way to reduce the house edge from 0.45% to 0.22% (only play 53% of the hands; wait until the count is ≥ 0).

No-Bust 21st Century BJ (Ver 4.01) @ Ocean’s 11 Cardroom

Posted in blackjack by stephenhow on July 24, 2012

They’ve changed the No-Bust 21st Century BJ rules a few times at the Ocean’s 11 Cardroom in Oceanside, CA. They’re on something called Version 4.01, with the following rules:

  • blackjack pays 6:5
  • re-split pairs up to 3 times, including Aces
  • push on 3-card player busts of 23, 24, 25 when dealer busts with a higher total
  • late surrender on first 2 cards

I ran the rules in my blackjack analyzer, which calculates the EV at -1.34%. If that’s not crappy enough, they charge an additional ≥ 1% collection on every hand. I have no idea why they would create a worse game for the player, especially when they also face the house collection. I guess the corporation bank wasn’t making enough after paying for collection and employee costs.

The WoOs lists the player cost of 6:5 blackjacks at 1.39%, which is not overcome by the “No-Bust” rule. The strategy below looks a lot like regular blackjack, with some subtle exceptions that make sense for the alternate rules.

My analyzer now outputs HMTL, which is cut-and-paste here:

Hand Dealer Upcard
2 3 4 5 6 7 8 9 10 A
Soft Totals
soft 20/21 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
2-Card Hard Totals
hard 18-20 S S S S S S S S S S
hard 17 S S S S S S S S S R
hard 16 S S S S S H H H R R
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 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 5-8 H H H H H H H H H H
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 P 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 P H H H
2-2 P P P P P P H H H H

Ultimate Three Card Poker (Face Up)

Posted in three card poker by stephenhow on July 23, 2012

Thanks to reader Jet for pointing out this new ShuffleMaster ™ game. I ran a quick analysis this morning to check his numbers, and of course to check for any collusion possibilities. An optimal strategy yields a -3.8807% return, and we pretty much came up with the same basic strategy (differences arise from suit assumptions). You should 3x raise all pairs or better, and 1x call according to the following table.

An ideal collusion simulation with 6 confederates yielded no practical gain 😦 This probably makes sense, because you can’t really take advantage of collusion information when they limit 3x raises to pairs. Perhaps you can get a slight edge on some 1x/fold borderline cases, but it doesn’t amount to anything significant.

Jet has a hole-carding strategy that yields 9%, which you can contact him for.

Ultimate Three Card Poker (Face Up)
Dealer Upcard Min Calling Hand
2 98x
3 Txx
4 T4x
5 T5x
6 T6x
7 T7x
8 T6x
9 Txx
T T84
J J84
Q Q84
K K84
A A85