Discount Gambling

Lucky Win Baccarat

Posted in +EV, baccarat sidebets by stephenhow on September 30, 2013

luckyWinBaccaratIf 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 in Spades 500 500
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
(spade)
EOR
(non-spade)
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
(spade)
EOR
(non-spade)
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.

Screen Shot 2013-09-30 at 1.48.46 PM

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 :(

Push 22 Side Bet @ Viejas Casino

Posted in +EV, blackjack sidebets, free bet blackjack by stephenhow on August 22, 2013

a91d79570d5b39e5076b8950689c832a_XL 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.

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

Mississippi Stud LogoYou 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.)

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.

High Card Flush

Posted in +EV, collusion by stephenhow on April 22, 2013

GGsite_HCF_ProdBanner_041713A couple of readers have asked about Galaxy Gaming’s new High Card Flush game, which has a few placements now, and may be picking up some steam. The game is pretty simple, where each player and the dealer receive 7 cards. Each hand is measured by its highest flush, where a flush is first ranked by its length (number of cards of same suit), then by its card values. Each player must Ante before the hand, then wagers a 1x-3x Play bet (depending on flush size), or folds. The dealer qualifies with a three-card, 9-high flush. If the dealer doesn’t qualify, the Play bets push, and the remaining Antes are paid even-money. If the dealer qualifies, the Ante and Play bets receive even-money action against the dealer hand.

As you would expect, collusion helps in this game. A Monte Carlo analysis shows that with 6 confederates, perfect knowledge of the dealt cards gives each spot at least a +7.3% edge over the house. But practically, you’d be lucky if you could even communicate the suit counts (number of cards of each suit) dealt. If you figure out a non-suspicious way of doing this, then the following simple strategy yields a +3.1% edge over the house:

High Card Flush Advanced Strategy (6 Player Hands Seen)
Flush Size Play Bet
1 or 2 cards 1x for suit counts (9, 11, 11, 11) or (10, 10, 11, 11), else
fold others
3 card, Jack-high or lower 1x for suit counts (9, 11, 11, 11) or (10, 10, 11, 11), else
fold others
3 card, Queen-high 1x if lowest suit count is 9 or higher,
else fold.
3 card, King-high or better 1x if lowest suit count is 8 or higher, else fold.
4 cards 1x
5 cards 2x
6 or 7 cards 3x

where the suit counts 4-tuple is the sorted number of cards of each suit.

Double Attack Blackjack

Posted in +EV, counting, double attack blackjack by stephenhow on March 29, 2013

wagerworks-double-attackThanks to reader John A. for pointing out this game to me. The game has been around (mostly in Atlantic City), but it’s new to me. It looks like the predecessor to Triple Attack Blackjack, as it’s based on a Spanish deck (10′s removed, J/Q/K’s remain) and the player may double his bet after the first card is dealt face up to the dealer. After this initial double attack option, the hand plays out normally with the total amount bet as the hand wager. (I.e., doubles and splits are based on the total amount bet after any double attack.)

The rules following the double-attack option are as follows:

  • Dealer stands on soft-17
  • Double-down at any time (no re-doubles)
  • Surrender at any time, including double-down rescue and after splits
  • No re-splitting of Aces
  • Blackjack pays even money

The house edge for the game is a reasonable 0.50% on the initial bet. The element-of-risk is even lower, as you double your wager 58% of the time (i.e., you double-attack vs. a dealer 2-8). The return is even lower still if they allow you to surrender after splitting Aces. The EORs are listed in the following table for removing a single card from a 8-deck shoe.

Double Attack Blackjack EORs (8 deck)
Removed Card EOR Balanced Unbalanced
Deuce +0.0832% +1 +1
Trey +0.1127% +1 +1
Four +0.1514% +1 +1
Five +0.1917% +1 +1
Six +0.1184% +1 +1
Seven +0.0341% +1
Eight -0.0560%
Nine -0.0895% -1 -1
Face -0.1466% -1 -1
Ace -0.0937% -1 -1

Basic Strategy

The basic strategy for the game was auto-generated by my analyzer program. You should double-down rescue 16 and lower against a dealer 8-thru-A, and 17 against an Ace. The strategy simulates at a -0.53% return, averaged over the whole shoe, very close to the analyzer’s calculated -0.50% return.

The unbalanced count in the above table yields 23.8% +EV betting opportunities (count >= +23) in an 6-deck shoe game with 52 cards behind the cut card. The average +EV hand returns +0.52%/bet. Compare this to the “Knockout” unbalanced count for 6-deck standard blackjack with cut card @ 5th deck, where 21.3% of the hands are +EV (count >= +17) with an average yield of +0.30%/bet.

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 S S S S S S
soft 18 S S S D D S S H H H
soft 17 H H D 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 R
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 H H S S S H H H H H
hard 13 H H H H H H H H H H
hard 12 H H H H H H H H H H
hard 11 D D D D D D D D H D
hard 10 D D D D D D D H H H
hard 9 H H H H H 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 H
10-10 S S S S S S S S S S
9-9 S S 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 H H P P P H H H H H
5-5 D D D D D D D H H H
4-4 H H H H H H H H H H
3-3 H P P P P P P H H H
2-2 H P P P P P H H H H

Dealer Bluff Collusion Strategy (+EV)

Posted in +EV, dealer bluff by stephenhow on January 28, 2013

dealerBluffWhen you play ShuffleEntertainment’s Dealer Bluff 6-Card Poker, you can feel at a disadvantage. The dealer makes the first bet (1x to 3x the Ante), and of course, you don’t know what he has. He might be bluffing. It’s easy enough to fold your weak hand to a strong bet, or to raise your strong hand against a weak bet. But the in-between decisions aren’t obvious, and you’re left blindly following basic strategy.

Interestingly, full-table collusion (6 players) makes this game +EV against the dealer. After all, the confederates’ 36 cards give some indication of what the dealer holds. (E.g., the dealer can’t have a pair of Aces when the confederates hold three of them.) So, you can get better idea of when to call, raise, or fold your hand. I worked out the collusion strategy details, hoping for a big edge (some games, ahem, yield near double-digit edges with collusion; you never know). Alas, I only came up with a +0.66% +EV 6-way collusion strategy :(

The full rules and game details are available from the WoO. Briefly, you post the familiar ShuffleEntertainment Ante = Blind bets before the hand starts, and each player and the dealer receives 6 cards. The shuffler reads the dealer hand, and bets 1x to 3x against the players. The player, in turn, must either call (wager a Play bet equal to the dealer bet), raise (wager a Play bet twice the amount of the dealer bet), or fold his Ante and Blind. The dealer will always call any raise. The hands are then turned over, and the bets are resolved. The remaining Antes push if the dealer doesn’t qualify with a pair or better. The Play bets always receives even-money action against the dealer hand. The Blind bets only pay for winning player hands of trips or better, according to a paytable.

The dealer follows a simple table that dictates the 1x, 2x, and 3x betting frequencies for each type of hand (nothing, low pair (2-5), mid pair (6-9), high pair (T-A), two pairs, etc.). This betting table completely describes “how the dealer plays”, and basic strategy is a nearly optimal counter-strategy (based on your hand only).

My collusion strategy tracks the “strong ranks” available to the dealer. Strong ranks are defined as card ranks (2 thru A) that the confederates only hold 0 or 1 copies of. These ranks are “strong”, because of the dealer’s chance of holding a pair of them. For example, the Seven is a strong rank for the dealer if the 6 confederates hold 1 or less Seven’s in total. But if the confederates hold 2 Aces, then the Ace is not a strong rank for the dealer. When you hold a pair, you’re usually interested in the number of strong ranks that are higher than your pair. When you hold 22′s or less, you’re interested in the total number of strong ranks.

Dealer Bluff 6-Way Collusion Strategy
Dealer Bet Basic Strategy Collusion Strategy
1x 2x pair 3′s or better 2x two pairs or better
2x pair 7′s thru A’s when 0-2 higher strong ranks
2x pair 3′s thru 6′s when 0-1 higher strong ranks
2x pair 2′s when 0 strong ranks
1x pair when 3 or less higher strong ranks
fold pair when 4+ higher strong ranks
1x KJ8 or higher 2x AK when 0 strong ranks
1x A-high when 0-2 strong ranks
1x K-high when all Aces seen and 0-2 strong ranks
1x K-high when 3 Aces seen and 0-1 strong ranks
fold others fold others
2x 4x pair J’s or better
4x pair T’s w/ 0-2 cards under T
4x two pairs or better
4x pair 9′s thru A’s when 0 higher strong ranks
2x pair 7′s thru T’s
2x pair 6′s w/ 0 cards under 6
2x pair 8′s thru K’s when 1 higher strong ranks
2x pair 5′s thru 7′s when 0 higher strong ranks
fold other pairs fold other pairs
3x 6x with pair K’s with A-kicker, or better 6x two pairs or better
6x pair A’s or K’s when 0 higher strong ranks
3x with pair T’s thru K’s 3x pair 8′s thru Q’s when 0 higher strong ranks
fold pair of 9′s or less fold others

For each dealer bet (1x-3x), the strategy is listed in priority from the top down. Yes, the strategy says to fold a pair of K’s against a 3x dealer bet if 1 or less Aces are held among the 6 confederates. There are undoubtedly better collusion strategies out there. As I said, I was hoping for a big edge, especially since you have 6x, 4x, and 2x raise opportunities. But I couldn’t find much more than the above +0.66% strategy, so I kept it simple and published it for reference’s sake.

How would you use this strategy in practise? Well, I guess you’d find a table full of friendly, helpful players. Then you start betting black ($100 Antes), and start asking questions when you need help. Say you’re holding AK, and the dealer bet is 1x. You start asking around if anyone has any deuces, treys, fours, etc. You count the dealer strong ranks (when the players have 1 or less cards of the rank), and play accordingly. When the floorman asks you not to discuss your hands during play, just tell him it’s not going to help much. You should be able to play for an hour before they ask you to leave.

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

Bust It Blackjack Side Bet

Posted in +EV, blackjack sidebets by stephenhow on December 24, 2012

I ran across the Bust It blackjack side bet last weekend at the Palazzo in Las Vegas. It seemed countable, so I ran the numbers today. The bet is simple. You make the side bet before the hand begins, and if the dealer busts on 3 cards, you win according to the paytable. If the dealer doesn’t bust on 3 cards, you lose. The basic house edge for a 6-deck shoe game is -6.91%. The EORs are fairly high, as listed below.

EORs for 6-Deck Bust It Side Bet
Card EOR Balanced Count Unbalanced Count Simplified Count
Deuce +0.006589 +2 +2 +2
Trey +0.005042 +2 +2 +2
Four +0.002963 +1 +2 +2
Five +0.000256 0 0 0
Six -0.006910 -2 -2 -1
Seven -0.001608 -1 0 0
Eight -0.003443 -1 -1 -1
Nine -0.003001 -1 -1 -1
Ten/Face -0.002231 -1 -1 -1
Ace +0.009038 +3 +3 +2

If the cut card is placed after the 5th deck, then an ideal count (using perfect shoe composition) yields 14.7% betting opportunities, with an average +6.73% advatange per bet. That’s an average return of about 1.0% per dealt hand.

Practically, you’d use the unbalanced count in the table above and bet with a running count of +25 or more. This practical count yields 14.4% betting opportunities, with an average +6.1% edge per bet. That works out to an average return of +0.88% per dealt hand.

Depending on the side bet limits, counting this bet could be profitable. But, more likely, they’ll limit you to a $25 max bet. So your profit rate would be (100 hands/hr)(14.4% bets/hand)(+6.1% profit/bet)($25/bet) = $22/hr. Of course, you’ll almost certainly have to make the main bet too (e.g., the Cosmopolitan wouldn’t let me make bonus bets on my friend’s blackjack hand). If it’s only $5, and you get good rules @ -0.6%, then your cost would be (100 hands/hr)($5/hand)(-0.6%) = $3/hr, leaving you with a $19/hr job.

The unbalanced count is fairly complicated, with its multi-level taps. Unless your a very skilled counter, you’ll be better off using the simplified count above. It only uses +2 and -1 taps, and it still performs well, yielding 13.5% betting opportunities, with an average +5.3% edge per bet. Bet when the running count is +24 or more.

Also, the standard blackjack counts don’t work for this bet (there’s no correlation, I checked). You can tell that blackjack counts are very different than this specialised count, because Aces are +3 and Sixes are -2. Those are opposite to blackjack values, and they make sense. Ace-rich shoes are bad for 3-card busts. Also, sixes are valuable because of the 15:1 payouts.

Note: a reader says the Palazzo/Venetian deals out of 8-deck shoes. If that’s the case, and they place the cut card @ 6 decks, then the ideal return decreases to 10.7% frequency at an average +4.7% edge. The simplified count return decreases to 8.9% opportunities @ +3.5% edge per bet. You would bet for an RC of +32 or higher.

6-Deck Bust It Blackjack Side Bet
Dealer Outcome
Frequency
Probability
Payout
Return
bust with 888 suited
240
0.001596%
200
0.003191
bust with 888 coloured
1,080
0.007181%
50
0.003590
bust with 6
73,440
0.488299%
15
0.073245
bust with 7
157,536
1.047450%
9
0.094270
bust with 8
245,232
1.630536%
7
0.114138
bust with 9
342,720
2.278729%
5
0.113936
bust with 10
1,782,144
11.849393%
3
0.355482
no 3 card bust
12,437,568
82.696816%
-1
-0.826968
total
15,039,960
100.000000%
-0.069115
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