Discount Gambling

Free Bet Blackjack @ Golden Nugget, LV

Posted in blackjack, free bet blackjack by stephenhow on June 22, 2012

Well, the tables are turned around, and now it’s me that’s on the hook for the house edge in a new game. I did the analysis for Geoff Hall’s (the inventor of Blackjack Switch) new Free Bet Blackjack. The game just went live at the Golden Nugget in downtown Las Vegas this week, and everyone is on edge that the game performs as calculated. There were a lot of winners on Wednesday, and because the game seems so crazily liberal, people were concerned (including me). So, I double checked all my work today, and while I found a few small things in the report (**ahem**), everything seems to check out. Basic strategy simulations are running at a 0.64% house edge, vs. my calculated optimal strategy 0.625%.

Ok, here’s why the game is crazy. The game is called “Free Bet” Blackjack, because you get free doubles on any hard 9, 10, or 11, and free splits on all pairs except 10’s and 4’s. What that means is that instead of paying for a double (say on a 3-card 11), the dealer will give you “free play” chips, matching your original bet, to wager. The same applies for “free” splits. Plus, you get up to 4 free re-splits, and you also get free doubles after free splits. So you can have like 7 free bets in the game (and only 1 real bet at risk). If you win the hand, the dealer pays all the free play with real money. If you lose the hand, you only lose your original bet. The dealer retrieves all the free bets at the end of the hand. Of course, the “catch” is that it’s a 22-push game. But overall, it works out very well for the player, since you are generally not at risk for your doubles and splits. You might push more hands than you like, but it beats the dreaded “losing (all) your doubles”.

You can see how it’s possible to have a nice run in the game.

Here’s the specific rules:

  • Free double anytime on hard 9, 10, or 11.
  • Free splits on all pairs except Tens and 4’s
  • Up to 4 free re-splits, including Aces
  • Free double after (free) split
  • Normal double on any two cards, including after (free) split
  • Late surrender (no longer?) available on first two cards
  • Blackjacks pay 3:2
  • Dealer 22’s push
  • 6 deck shoe

They initially offered late surrender, but we told them it brought the house edge down by 0.21% (huge). I don’t know if they took it away yet. If allowed, late surrender on 15 vs. T or A, on 16 vs. 9, T, A, and on 17 vs. dealer A.

Here’s the basic strategy generated by my analysis program. Note you treat free splits differently than normal hands. This is because you can’t lose any real money by hitting hard-17 against a dealer 7 upcard, so you might as well try to improve your hand (given push-22, and push-17 yields nothing on the free bet). Also note the regular doubles for free-split soft totals. It’s worth it to risk a real-money double rather than just hitting a free hand. These exceptions are listed in the bottom part of the strategy chart.

Basic Strategy for Free Bet Blackjack (6 Decks).

where FD = free double, FS = free split, DD = double down. Note that you take all your free doubles, and all your free splits.

I have to admit I overlooked a small rule until this morning (**ahem**), but everything is now fixed. Luckily, the numbers still worked out. This is my first time doing the math for a live game on the floor. I was slightly nervous today 🙂

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.

Field of Gold (+EV)

Posted in +EV, blackjack sidebets by stephenhow on June 12, 2012

Everybody thank Eliot Jacobson for working for you. He emailed me this morning telling me the Field of Gold blackjack side bet was crushable. And it is. For real. The count is simple, and for a 6-deck shoe game, you’ll be able to bet 19% of the time, with an average edge of +6.5%/bet. That’s a profit of ($25 bet)(19% bet/hand)(+6.5% EV/bet) = $0.31/hand. For double-deck, the numbers are even better, yielding 27% betting opportunities with an average edge of +8.2%/bet. For $25 bets, that yields a profit of $0.55 per hand dealt. That adds up pretty quickly. It’s more than $100/hr.

This is about as good as it gets, as it’s a fast blackjack game. Oh, by the way, Eliot also brought you the equally crushable Lucky Lucky bet, which I probably understated in my post. These are probably the best two countable side bets out there, and nobody but Eliot has noticed them. Oh, and he’s the one that pointed out the Dragon-7 vulnerability.

Anyway, here’s what you’re looking for:

Use the following count:

Field of Gold Count
Card Value
Ace -3
Deuce -1
Nine, Ten, Queen, King +1
others 0

Bet when the true count (= runningCount/decksRemaining) is 2.2 or better. You can see from the graph below that the EV is strongly correlated with the true count.

This side bet becomes highly profitable when the shoe is rich in aces and deuces, and lean in high cards. During some shoes, the deck can easily get really good, or really bad for the Field of Gold bet. The distribution of the side bet EV is plotted below over the course of the 6-deck shoe and the double-deck game. Note the peak of the distribution is centered about the nominal -5.66% return of the bet right after the shuffle. Then, especially towards the end of the shoe, the side bet return can vary wildly. Note all the area under the long tail of the distributions to the right of the y-axis (EV>0). This is why the game is crushable.

(5 deck penetration of 6-deck shoe; 29 cards behind cut card in double-deck.)

Card Craps Simple Explanation

Posted in +EV, card craps, csm by stephenhow on June 10, 2012

I love the card craps at Viejas, not because I’m ever going to win any money there, but because it’s so obviously countable. However, it’s almost impossible to explain to anyone why the odds are different than dice, or why the game is countable. After all, they use a Constant Shuffle Machine (CSM) with 312 cards, right? So, once again, I’m going to explain how the card buffering in the exit chute of the CSM makes the game easily countable.

A picture is worth a thousand words. Example code and simulations are the proof of the pudding. All the code used in this example is available on github, where you can browse or download it.

You can read up on the details of card craps @ Viejas. Here’s how they play it. They use a normal craps layout, but replace the dice with two cards (1 thru 6), dealt out of a 312-card CSM. They take two cards out of the shuffler, call the roll, then muck the two cards back into the CSM. They allow 10x pass/dont odds on all points.

The reason why the CSM screws up the game (favors the dont’s) is that on the comeout, the two cards that just made the point have no chance of coming out on the next roll. Nor do they have any realistic chance of coming out in the next few rolls. This is because a CSM buffers a dozen or more cards in the chute where the dealer pulls the cards from. This buffer is necessary to deal blackjack. (Imagine the dealer waiting for the machine to drop one shuffled card at a time.)

Dice Baseline

Ok, so download the example code, compile and run it with the -d option for normal dice. The results are just as you’d expect. The pass line returns -1.42%, and the dont pass returns -1.36%, and odds and counting don’t make any difference:

>./cardcraps -d
using normal dice
1665000000 games:
pass flat: -0.0142, pass10x: -0.0144, pass w/count: -0.0142,
dont flat: -0.0136, dont10x: -0.0134, dont w/count: -0.0135

It takes billions of games to settle out the averages (especially when playing 10x odds), so don’t worry about the 1/100th of percents.

A) 36-Card Deck Is Same As Dice

At Pala Casino, they use a 36-card deck (one card per roll), and a simple deck shuffler. No buffer. Each card has a picture of two dice. The shuffler spits out one card from the red deck, one card from the blue deck. The player “roll” chooses between the blue or red card. Exact same odds as craps. At Pala, no one ever says anything like “How many cards are in there?”, or “This machine deals a lot of sevens!”.

B) 2-Card Roll Hurts Pass Odds

Now, let’s try the case B in the above diagram. We use the -c option to select an ideal shuffler, and -m 0 option to indicate no buffered cards in the chute.

>./cardcraps -m 0 -c
using CSM with 52 dice sets, and minBufferDepth of 0 cards
1265000000 games:
pass flat: -0.0137, pass10x: -0.0266, pass w/count: -0.0170,
dont flat: -0.0137, dont10x: -0.0053, dont w/count: -0.0075

This shows that even without a buffer, making a dice roll from two cards out of a perfectly shuffled 312-card shoe favors the don’t pass odds. You can use a simple spreadsheet to show this. The point is that you’ll distort the well-known dice roll distribution by using 2 cards dealt from a shoe. It’s a simple exercise to prove (a simple spreadsheet will give you the exact numbers).

Note the pass line player loses more by taking odds. The don’t pass player improves his return by laying 10x odds. That doesn’t happen in a regular dice game. In a dice game, taking or laying odds is fair (0 EV).

C) CSM Is Countable

At Viejas, they use a ShuffleMaster 126 CSM loaded with 312 cards. If you ever open the top (used to happen a lot when they had jams), you’ll see a buffer of approximately 16 cards in the exit chute. This distorts the game, and in general favors the Don’t Pass odds. Sometimes, a good count makes the pass odds +EV.

We’ll run the simulator for the CSM with a minimum buffer depth of 16 cards:

>./cardcraps -m 16 -s
using model of ShuffleMaster 126 CSM with 52 dice sets, and minBufferDepth of 16 cards
using window size of 6 rolls
2083000000 games:
pass flat: -0.0147, pass10x: -0.0420, pass w/count: -0.0011,
dont flat: -0.0126, dont10x: +0.0042, dont w/count: +0.0130

Now you see the pass line player is severely penalised for taking odds. I don’t think someone taking 10x odds on every point would think they’ve increased the house edge from a nominal 1.4% to a whopping 4.2% (of the flat bet). And we see that a don’t pass player laying 10x odds on every point now has a small 0.4% advantage over the house. Of course, there’s a lot of variance laying 10x odds to win an average (0.4%)(flat bet). Using a simple (and fun!) count, the don’t player has a 1.3% advantage over the house.

You can use the -v option in the cardcraps program to generate the statistics on the odds bet vs the count for each point. I ran the program, and plotted the results (don’t pass odds advantage; pass odds are inverted):

The correlation between the count and the next roll out of the CSM is clear. The count is simple and important! Quite often, you have a +/- 1-2% advantage in laying odds or taking odds. Where else can you play a craps game where the previous 6 rolls have a significant effect on the next roll?! The graph was generated with a fair simulator (using a Mersenne Twister 64-bit PRNG with a period of 2^19937-1).

Even though the game is +EV, the edge is small relative to the variance. No one will grind out any money from this game. However, it is a lot of fun to watch the rolls, know the count, and guess the outcome. Plus, the game is dealt on a table, so you get to sit and watch the rolls. And it’s probably 10x faster than a craps game with dice. You could get a roll every 5 seconds if you’re heads up with the dealer.

The count provides a fun, small predictor of the next roll out of the CSM. If you like counting, and/or predicting the next roll in craps, then you have to check out the card craps game. Here’s a video that shows how I play the game @ Viejas:

Bad Beat Bonus @ Ultimate Texas Hold’Em

Posted in ultimate texas hold'em by stephenhow on June 8, 2012

A reader recently asked about the new two-way Bad Beat Bonus in Ultimate Texas Hold’Em. It’s pretty easy these days to crunch out the numbers, so I took an hour to work this one out. After all, I might see the bet somewhere, and I’m becoming a sucker for bonus bets lately 🙂

Well, at a 14.8% house edge, I’ll probably look for a better paytable before I play this bet. Has anyone seen any other paytables out there?

UTH 2-Way Bad Beat Bonus
Losing Hand Frequency Probability Payout Return
Royal Flush 0 0.000000% 0 0.000000
Straight Flush 10,300,592 0.000370% 7500 0.027776
Four-of-a-Kind 471,040,512 0.016935% 500 0.084677
Full House 8,435,225,376 0.303275% 50 0.151637
Flush 19,434,208,592 0.698725% 30 0.209618
Straight 18,271,076,976 0.656907% 20 0.131381
Three-of-a-Kind 64,049,759,448 2.302804% 9 0.207252
Two Pairs 399,099,149,640 14.348956% -1 -0.143490
One Pair 1,366,512,556,968 49.130722% -1 -0.491307
High Card 791,967,420,480 28.473892% -1 -0.284739
push 113,130,263,816 4.067413% -1 -0.040674
Total 2,781,381,002,400 100.0000% -0.147868