Discount Gambling

Counting The Royal Match Sidebet @ Barona Casino

Posted in +EV, blackjack sidebets by stephenhow on December 20, 2011

My local Barona Casino offers the Royal Match blackjack side bet which pays 75-to-1 when you’re dealt a suited K-Q on your first two cards, and 2-1 for any other suited hand. Normally, the house edge is 4.06% for this side bet, but as readers of this blog know by now, sucker bets are often countable. Using a simple true K-Q count, the bet yields an average 5.4% player edge.

I always play this bet for $1, and get really excited when I hit it for $75. The last time I hit it, I decided to analyze it’s countability.

I first looked at the bet’s distributions of EVs at the last hand of the shoe. This would show me if there was any potential for a counting scheme. I’d see how often and how strongly the bet went +EV. Here’s what it looks like with one deck remaining:

I could tell from the graph that the bet was exploitable. So I calculated the average profit per shoe, assuming heads-up play with the dealer. This theoretical limit comes out to a profit of +0.504 bets/shoe. That’s a really good profit rate (compare to the Dragon-7 profit rate of 0.54 bets/shoe). That means a player betting a fixed amount on the Royal Match will net an average profit equal to half his bet per shoe, when heads up with the dealer. On a per bet basis, the average Royal Match bet EV is +5.9%, assuming perfect knowledge of the dealt cards.

I then looked for a simple (practical) counting scheme that would capture most of the theoretical edge. As it worked out, the second idea that came to mind happened to be very effective. A simple true count of excess Kings and Queens yields about +0.43/shoe (85% of the theoretical edge):

trueCount = (8*decksDealt - countedKingsAndQueens)/decksRemaining

where decksDealt and decksRemaining are floating-point numbers.

The relationship between the true KQ-count and the Royal Match side bet EV is shown in the following graph:

Note that you should bet the Royal Match when the true count is >= 0.8.

In heads-up play, you’ll get an average of 8.0 bets/shoe, with an average bet EV of +5.4%. The scheme is very comparable with the highly profitable Dragon-7 counting scheme (EV +0.54/shoe), but it’s much faster, and has much lower variance.

So if you don’t find me at the full-exposure Mississippi Stud game, look for me in the i-Table pit counting the Royal Match.

Instant-18 Blackjack Side Bet Counting

Posted in +EV, blackjack sidebets by stephenhow on November 2, 2011

I recently saw the Instant-18 blackjack side bet, and I thought it was funny. It’s an even-money side bet that plays against the dealer as a hand of value 18. Hence, the name “Instant-18”. You put up a bet, and it’s an 18. It has a 2.036% house edge for a 6-deck shoe, but it’s still kind of interesting. It’s an optional side bet where (typically) you can bet an amount less than or equal to your main blackjack bet. Of course, I had to see if this bet was countable.

First, I looked at the distribution of EVs after 5 of 6 decks were dealt from the shoe:

This looked promising, so I checked out the EORs for the bet:

Card ΔEV
Deuce -0.001031
Trey -0.000634
Four -0.000252
Five +0.000038
Six -0.000371
Seven -0.001946
Eight -0.000649
Nine +0.000606
Ten/Face +0.000347
Ace +0.002849

From the EORs, I made a simple count system, where Aces are +3, Sevens are -2, and Deuces are -1. Then I plotted the EVs of the main and side bet vs. the true count:

Unfortunately, when the count for the side bet gets good (true count >= 4), the main bet is -EV. However, for a good enough count (true count >= 7), the combination of (main bet + side bet) is +EV. Also, for a bad enough count (true count <= -3), them main bet is +EV.

So, by itself, the Instant-18 side bet gets good (true count >= 4) about 7.8% of the time, with an average return of +1.53%.

Of course, you can’t just bet the side bet when it gets good. You have to have a main bet to bet the side bet. If you bet both when the combination gets good (true count >= 7), or just the main bet for true count <= -3, and Wong all others, you'll bet 13.9% of the time with an average EV of +0.47%.

As usual, these things are interesting, but not practical.

Blackjack Bad Beat Progressive @ Barona Casino

Posted in blackjack sidebets by stephenhow on October 10, 2011

My local Barona Casino offers a $1 progressive blackjack side bet that pays when your 20 loses to a dealer 21. The payout depends on the number of dealer cards, where the payout increases with the number of cards in the dealer’s hand. The progressive pays out when your 20 is beaten by a 7-or-more card dealer 21.

Blackjack Bad Beat Progressive
Number of Dealer Cards Payout Probability* Return
7 or more cards 100% 4.6827e-6 jackpot/213,500
6 cards 1000-for-1 5.7405e-5 0.057347
5 cards 100-for-1 5.0457e-4 0.049952
4 cards 50-for-1 0.002790 0.136727
3 cards 25-for-1 0.008282 0.198762
2 cards 10-for-1 0.004975 0.044771
magic card** 20-for-1 ~(1/26)(2/52) ~0.028
miss -1 ~98%
total 100% ~0.52 +
jackpot/213,500

*calculated using basic strategy (includes surrenders and other decisions that are not optimal for the sidebet).

**magic card average frequency assumed at 1-every-26 hands, and average 3 cards/hand

So, the break-even point for the jackpot is about $100,000. I used basic simple strategy and a 6 deck shoe in this analysis.

Effect of Shoe Depth on EV

While looking for a counting edge on this side bet, I ran into a very pronounced effect of shoe depth on the bet EV. I was expecting the random distribution of cards at the end of the shoe to distribute the EVs around 0. Well, I did find a roughly bell-shaped distribution of EVs, but I discovered that the average return decreases sharply with the shoe depth.

What this means is that as you reach the end of the shoe, the average return of the bad beat side bet decreases very quickly. Most bets don’t behave like this. You want the bet to have the same average return at the start of the shoe as at the end of the shoe. Imagine if blackjack started at a 0.5% house edge for the first hand, but for the last hand ended up at a 50% house edge, on average. Most people wouldn’t stand for that (if they knew it was happening). [You’d probably sense such a bias on an “even-money” bet, but not for a bet that hits < 2% of the time.]

I've plotted out some curves to shoe the effect of shoe depth on the bad beat EV. I set the jackpot at $100k, so the EV for a full six-deck shoe is near 0 (magic card not included). Then, I plotted EV vs. number of decks in the shoe, assuming no missing cards (i.e., a neutral shoe). So, in the graph below, 0.5 decks means half a deck.

The above graph shows the big disparity between playing the bonus at a 6-deck shoe game vs. a double-deck pitch game. The graph also reflects the bet’s average as a function of decks remaining in the shoe. Taking an extreme example (1/2 deck remaining in shoe; most houses shuffle before this point), the distribution of actual EVs is sampled in the below graph. Note how it’s shaped mostly below the -41.5% average return of the above graph.

This effect is probably due to the fact that you need a lot of perfect cards to make a 7-or-more card dealer 21. When you run low on cards, there’s probably much fewer ways (percentage-wise) to make the perfect hand. Some players might intuitively suspect this, or at least would not be surprised when they see these results.

So be careful with this bet. It gets a lot worse than you think it might, especially at the end of the shoe.

Bet The Set @ Barona i-Table

Posted in blackjack sidebets by stephenhow on March 6, 2010

Another bonus side bet offered on the i-Table at Barona is a version of Bet The Set that pays for various types of pairs made on the player’s first two cards. For an n-deck shoe, the probability of being dealt a pair is 13*C(n*4,2)/C(n*52,2), and the probability of a suited pair is 4*13*C(n,2)/C(n*52,2).

Bet The Set bonus, 6-deck @ Barona
Hand Frequency Payout Return
Suited Pair 1.6077% 20:1 +.321543
Offsuit Pair 5.7878% 10:1 +.578778
nothing 92.6045% -1:1 -.926045
Total -.025724

The pit boss told me they increased the suited payout yesterday from 15:1 to 20:1, which would be an improvement of 8%, changing it from a sucker bet to a fun bet. They’re probably trying to promote the new i-Table games, and it’s an easy thing to change the payouts, since they’re not printed on the mat, but instead displayed on the player touch screens. At only a 2.5% house edge, this is very cheap as bonus bets go.

Royal Match @ Barona i-Table

Posted in blackjack sidebets by stephenhow on March 6, 2010

One of the nice features of the Shuffle Master i-Table is that they can change the odds of the bonus side-bets at any time. Yesterday, they increased the payout on the Royal from 50:1 to 75:1. The change showed up on the display, so I calculated the improvement.

For an n-deck shoe, the probability of hitting a Royal (KQs) is (4)(n^2)/C(52*n,2) and the probability of getting dealt suited cards is 4*C(n*13,2)/C(n*52,2).

Royal Match Bonus for 6-deck shoe @ Barona
Hand Frequency Payout Return
Royal Match (KQs) 0.29680935% 75:1 +0.222607
Suited 24.462033% 2:1 +0.489241
nothing 75.241158% -1:1 -0.752412
Total 100% -0.040564

The change from the 50:1 to the 75:1 payout on the Royal improved the return from -10.2% to -4.1%. So, it went from a sucker bet to a fun $1 bet you can place every hand.

Blackjack Odds Side Bet @ Barona Casino, CA

Posted in blackjack sidebets by stephenhow on March 4, 2010

At Barona Casino near San Diego, they’re spreading a new blackjack table game with an electronic twist. The game uses a human dealer, regular cards, and a modified blackjack table (Shuffle Master i-Table). However, players don’t use chips to bet, but instead use a drag & drop touch-screen. There’s no printing on the mat, and the table has a clean look, uncluttered by chips. This game is offered in a separate “Chipless Blackjack” pit of 4-5 tables.

The decks are shuffled by hand, and loaded into the shoe normally. The shoe reads the cards as they’re dealt, so it knows what everyone’s totals are. Your display prompts you for your actions. The game is a little faster than normal blackjack, since the dealer doesn’t have to mess with any chips, except during buy-ins and cash-outs.

This electronic format offers an interesting odds side bet on your starting two-card hand. The computer offers you particular odds for winning, based on your total vs. the dealer up card, as an optional side bet. For example, say you’re dealt 6,4 (=10), and the dealer is showing a 4. It offers you something like $2.69 on an additional $5 side bet (equal to the original bet) to win the hand. If you lose the hand, you lose the side bet. If your hand pushes, the side bet also pushes. Note that the true odds for winning this double should pay about $2.95, so the vig here is about $.25 😦

I worked out the true odds for a $5 bet, a six deck shoe, and dealer hit on soft-17. (The offer is made after the dealer peeks for blackjack.) The table is presented below.

I played for a few hours, and checked how close the offered side bet was to the true odds. For a $5 bet, the payouts were almost always lower than true odds by about $.25 -$.50 (no surprise here). For the payouts above $10, this might be acceptable. However, this vig really hurts the low payouts. For example, a player 20 against a dealer 10 should win about $.32 for a $5 bet. Their offered odds on this hand were often lower than $.10. That’s really bad, since you’re not even getting 1/2 the fair odds. The only odds that seemed to be true odds were for 14 vs. a dealer 6 and 7. The worst odds seemed to occur for double and split opportunities, and anything against a dealer 10 or A. (I think this is because in some situations, like 7,7 vs. an 8 upcard, the odds are quite different for splitting vs. hitting; to avoid any player edge, they offer you the lower of the two.)

In my session, almost no one took odds. Most people correctly assume the house isn’t offering true odds, and they don’t have a chart for comparison. True odds aren’t very intuitive, since people don’t think of blackjack hands in these terms.

So, as usual, you have to pay some vig to get odds in a blackjack side bet. The vig for $5 bets seems to average between $.25 and $.50 for all payouts. I’m not sure, but I think the vig scales up with the bet size, so its no better for $10, $25, or $100 bets.

Positive Count Shoe

I tried a simple experiment to see if a positive count shoe would swing the odds enough to make the side bet +EV. I removed a 2, 3, 4, 5, and 6 from the deck (+5), then re-calculated the odds. Overall, this made the basic blackjack game +EV, swinging it from about -0.5% to +0.5%. Then I checked a few odds points that would benefit the most from this positive count. I saw that a (T,4) vs. a dealer 6 moved from $6.37 down to $6.20. This doesn’t overcome the vig, and the offered odds are still probably below this. Similarly, for the same +5 shoe, a (5,6) vs. a dealer 6 improves only from $2.34 to $2.25. So, a negative shoe count doesn’t help the odds bet much, and will never overcome the vig.

Negative Count Shoe

Experiments with a -5 count shoe (five Tens removed) show that the higher odds bets improve more in absolute payouts than for improvements from positive count shoes. For example, a 14 vs. a dealer 10 upcard improves from $14.78 down to $14.36. Intuitively, the odds improve because fewer Tens in the shoe mean a higher chance of the dealer not having 20, and a better chance for the player to draw to a hand. Similarly, a 16 vs. a dealer 9 improves from $16.59 down to $16.18. These are significant improvements, but I believe that the vig on the higher odds bet are steeper for a reason: to account for this variation. So overall, it looks like counting isn’t going to ever make the odds bets +EV 😦

True odds for winning a hand, H17, 6-deck shoe ($5 side bet).
upcard 2 3 4 5 6 7 8 9 T A
soft totals
A,2 $4.53 $4.27 $4.01(H) $4.31(D)
$3.73(H)
$4.04(D)
$3.51(H)
$3.84 $4.45 $5.40 $6.29 $6.26
A,3 $4.76 $4.48 $4.19(H) $4.32(D)
$3.91(H)
$4.04(D)
$3.67(H)
$4.23 $4.82 $5.89 $6.80 $6.77
A,4 $5.00 $4.69 $4.66(D)
$4.39(H)
$4.33(D) $4.05(D) $4.63 $5.33 $6.43 $7.37 $7.36
A,5 $5.23 $4.91 $4.68(D) $4.35(D) $4.04(D) $5.09 $5.84 $7.02 $7.94 $8.00
A,6 $5.00 $4.70(D) $4.37(D) $4.03(D) $3.80(D) $4.38 $5.89 $7.00 $7.83 $8.25
A,7 $4.41(D) $4.13(D) $3.83(D) $3.60(D) $3.39(D) $1.83 $3.55 $6.26 $6.93 $7.26
A,8 $1.95 $1.86 $1.79 $1.67 $1.62 $0.99 $0.95 $1.92 $4.33 $3.08
A,9 $0.78 $0.76 $0.74 $0.70 $0.68 $0.44 $0.41 $0.37 $0.32 $0.68
pairs
2,2 $5.84(S) $5.55(S) $5.19(S) $4.84(S) $4.51(S) $5.37(S) $7.09(H)
$6.47(S)
$8.56 $9.61 $9.77
3,3 $6.11(S) $5.75(S) $5.30(S) $4.94(S) $4.60(S) $5.70(S) $8.16(H) $9.78 $10.85 $11.05
4,4 $5.25(H) $4.90(H) $4.53(H) $5.14(S)
$4.19(H)
$4.79(S) $4.10 $5.77 $8.13 $8.79 $9.36
6,6 $6.47(S) $6.04(S) $5.58(S) $5.16(S) $4.86(S) $8.14(H)
$7.09(S)
$9.34 $11.05 $12.07 $12.25
7,7 $6.16(S) $5.77(S) $5.32(S) $4.97(S) $4.63 $5.76(S) $11.82(H)
$8.31(S)
$13.89(H)
$10.70(S)
$15.25(H)
$11.48(S)
$15.55
8,8 $5.01(S) $4.69(S) $4.41(S) $4.10(S) $4.01(S) $3.86(S) $5.50(S) $7.85(S) $8.38(S) $12.28(S)
9,9 $4.29(S) $4.08(S) $3.83(S) $3.57(S) $3.37(S) $1.83 $4.07(S) $6.19(S) $7.42 $8.84
A,A $2.65(S) $2.51(S) $2.38(S) $2.24(S) $2.13(S) $2.59(S) $4.31(S) $4.84(S) $5.15(S) $5.80(S)
hard totals
5 $6.50 $6.07 $5.60 $5.16 $4.80 $6.51 $7.59 $9.13 $10.20 $10.37
6 $6.68 $6.21 $5.74 $5.28 $4.90 $7.02 $8.17 $9.76 $10.67 $11.06
7 $6.36 $5.90 $5.43 $4.98 $4.63 $5.91 $8.20 $9.73 $10.50 $11.35
8 $5.26 $4.92 $4.55 $4.20 $3.94 $4.13 $5.78 $8.16 $8.82 $9.40
9 $4.24 $4.35(D)
$3.98(H)
$4.05(D)
$3.72(H)
$3.75(D) $3.53(D) $3.36(H) $3.98(H) $5.67 $7.06 $6.69
10 $3.34(D) $3.14(D) $2.95(D) $2.76(D) $2.63(D) $3.20(D) $3.64(D) $4.24(D)
$3.83(H)
$4.69(H) $4.63(H)
11 $2.92(D) $2.76(D) $2.60(D) $2.45(D) $2.34(D) $2.98(D) $3.40(D) $3.89(D) $4.12(D)
$3.84(H)
$4.38(D)
$3.92(H)
12 $8.65(H) $8.25(H) $7.46 $6.87 $6.35 $8.12 $9.35 $11.06 $11.99 $12.26
13 $8.93 $8.18 $7.46 $6.88 $6.40 $9.29 $10.65 $12.33 $13.61 $13.83
14 $9.07 $8.27 $7.55 $6.91 $6.37 $10.34 $11.57 $13.52 $14.78 $15.16
15 $9.02 $8.26 $7.54 $6.90 $6.37 $11.49 $13.03 $15.18 $16.51(H) $16.98(H)
16 $9.03 $8.27 $7.55 $6.91 $6.42 $12.67 $14.30 $16.59 $16.78(S)
$18.17(H)
$18.53(H)
17 $7.21 $6.60 $6.02 $5.55 $5.11 $7.08 $12.90 $14.24 $14.04 $17.75(S)
18 $3.87 $3.63 $3.42 $3.18 $3.01 $1.84 $3.60 $7.66 $7.47 $8.93
19 $1.97 $1.89 $1.80 $1.68 $1.63 $1.00 $0.96 $1.96 $4.27 $3.05
20 $0.80 $0.77 $0.74 $0.70 $0.68 $0.44 $0.41 $0.37 $0.32 $0.69
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Perfect Charlie @ Fort McDowell, AZ

Posted in blackjack sidebets by stephenhow on April 4, 2009

There’s a blackjack bonus sidebet at Fort McDowell Casino near Phoenix, AZ that offers huge payouts for a $.25 bet (you can only bet a quarter).  I remember seeing this strange bet a while back, and was always curious about it.  I called across state lines to find out the details, and got out the paper and pencil to calculate the house edge for a six-deck shoe.

It’s called “Perfect Charlie”, and pays jackpots for 5-card 21’s (2-3-4-5-7), with different payouts for suited and/or ordered hands. The bet also pays off hands that start off as Perfect Charlies, but don’t make it (like 2-3-4, 2-3-4-5, etc.).

I wrote up an OpenOffice spreadsheet to calculate the odds. I wanted to include the spreadsheet source in this post, but WordPress.com won’t let me upload the .ods document 😦 Fortunately, OpenOffice exports spreadsheets to html.

This is a pretty obscure sidebet, which is probably only found at Fort McDowell. Nothing popped up immediately on Google for this, so I’m having a millionth-monkey moment here (i.e., when you realise you’re the millionth-monkey pounding at the keyboard, and all-of-a-sudden Shakespeare is on the screen).

The bet is pretty horrible in terms of house edge, but want do you want for a quarter?  Or actually, ($.25)(1-.6478) = 8.8¢.  You get the chance to win up to $70,000, while the house has an equally small chance of getting rich off this sidebet.

  Fort McDowell Blackjack Bonus        
           
           
           
Bet Hand Conditions Payout Probability Return
$0.25 2,3,4 offsuit, any order $10.00 1.98E-03 8.11%
$0.25 2,3,4 offsuit, in order $37.50 3.96E-04 5.98%
$0.25 2,3,4 suited, any order $75.00 1.41E-04 4.24%
$0.25 2,3,4 suited, in order $500.00 2.82E-05 5.64%
$0.25 2,3,4,5 offsuit, any order $25.00 7.44E-04 7.52%
$0.25 2,3,4,5 offsuit, in order $250.00 3.24E-05 3.24%
$0.25 2,3,4,5 suited, any order $1,000.00 1.26E-05 5.03%
$0.25 2,3,4,5 suited, in order $10,000.00 5.47E-07 2.19%
$0.25 2,3,4,5,7 offsuit, any order $50.00 3.30E-04 6.63%
$0.25 2,3,4,5,7 offsuit, in order $5,000.00 2.77E-06 5.54%
$0.25 2,3,4,5,7 suited, any order $20,000.00 1.29E-06 10.34%
$0.25 2,3,4,5,7 suited, in order $75,000.00 1.09E-08 0.33%
           
           
total         64.78%
           
           
decks 6        

Perfect Pairs @ Barona Casino, CA

Posted in blackjack sidebets by stephenhow on March 7, 2009

Barona Casino is offering a new “promotion” called Perfect Pair blackjack.  The various taglines are “The Best Blackjack in the U.S. is getting even better” and “Your odds have never been better”.
What it is, of course, is just another “bonus” sidebet that some players like, and the house loves.  A quick analysis of the odds reveals:

Hand Frequency Probability Payout Return
suited pair 780 1.6077% 30 0.482315
colored pair 936 1.9293% 10 0.192926
other pair 1872 3.8585% 5 0.192926
nothing 44928 92.6045% -1 -0.926045
total 48516 100% -5.7878%

The house edge on this game is 5.8%. That’s pretty bad, and most people bet accordingly.
baronaperfectpairs

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Buster Blackjack @ Sycuan, CA

Posted in blackjack sidebets by stephenhow on February 18, 2009

At my nearby Sycuan Casino, they offer a sidebet on some of their games called “Buster Blackjack”. It’s an optional “bonus” bet the player can place, up to the amount of his blackjack bet, and limited to $100. It’s a bet that the dealer will bust, and its paytable is based on the number of cards the dealer busts with.
I wrote a little recursive program to calculate the distribution of dealer busts for a 6-decks shoe, and worked in their paytable.

Dealer Bust Cards Payout Probability Return
3 2:1 1.73032e-01 0.346064
4 2:1 8.93918e-02 0.178784
5 4:1 2.04726e-02 0.081890
6 12:1 2.63760e-03 0.031651
7 50:1 2.14444e-04 0.010722
8 250:1 1.15280e-05 0.002882
no bust -1 0.714240 -0.714240
total 1.000000 -0.062247

So the bet has a typical bonus bet edge of 6.2%.  It’s a lot better than I thought it’d be.

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