# Discount Gambling

## 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.

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.

## Panda-8 Bonus Bet @ EZ Baccarat

Posted in +EV, baccarat, panda-8 by stephenhow on October 9, 2011

Read all my posts on the Panda-8 sidebet.

There’s another bonus bet on the EZ Baccarat table, called Panda-8, that pays 25-1 for a 3-card player 8 win. Using the same methods as the prior posts on countable baccarat side bets, I came up with the following card count values:

Card Count Value
Ace 1
Deuce 1
Trey -2
Four -2
Five -2
Six -1
Seven -1
Eight -2
Nine 4
Ten/Face 1

The player should bet the Panda-8 when the true count is ≥ 11. (The plot below shows the EV curve crosses the x-axis at a true count of 10.5):

Use the following running count (RC) thresholds when betting the Panda-8 side bet:

Hand # RCMin
55 31
56 29
57 28
58 27
59 26
60 25
61 24
62 23
64 21
65 20
66 19
67 18
68 17
69 16
70 15
71 14
72 13
73 12
74 11
75 10
76 9
77 8
78 6
79 5
80 4
81 3
82 2
83 1
84 0

Using the above RCmin thresholds, simulations showed a profit of 22% of a fixed bet per shoe, at an average 3.8 bets/shoe.

## Player Dragon Bonus Tracking Sheet

Posted in +EV, baccarat, dragon-bonus by stephenhow on October 5, 2011

I worked out a baccarat tracking sheet for counting the player dragon bonus available at my local Barona Casino. The tracking sheet simulates at +0.112 EV/shoe, gained from an average 3.9 bets/shoe. That means the count gets good enough to bet the player dragon about 3.9 times per shoe (4.8% of the time), and you’ll profit a total of 11.2% of a fixed bet, per shoe. So, if you’re betting the max \$300 per dragon, you’ll make about \$33.60/shoe, on average.

That’s not bad, since you’re allowed (encouraged, actually) to use tracking sheets while sitting at the Rapid Baccarat terminals. The player dragon has payouts ranging from 30-to-1 down to 1-to-1 and pushes, so you’ll win something about 30% of the time. This is much better than waiting for a 40-to-1 payout on an infrequent \$100 dragon-7 bet.

The tracking sheet below helps you maintain the running count (RC) for each hand of the shoe, and lists the required minimum RC for betting the player dragon on any given hand:

You’ll see the minimum betting RCs decrease as more hands are played. Typically, you’ll probably only be able to bet the dragon for the last 8 hands of the shoe, if at all. Welcome to the world of baccarat.

### Instructions

Use the following count values for each card:

Player Dragon Count Values
Card Count Value
Ace 1
Deuce 2
Trey 2
Four 1
Five 1
Six 0 (ignore)
Seven -1
Eight -1
Nine -1
Face/Ten -1

The dealer pulls the first card out of the shoe, and turns it face up. Start the running count with the count value of the card. The following unseen burn cards do not affect the RC or the bet thresholds.

Make sure you only use this sheet for betting the player dragon bet (1-1 for natural player win, push for natural player tie, else 30-1 for player win by 9 points, 10-1 for win by 8 points, 6-1 for win by 7 points, 4-1 for win by 6 points, 2-1 for win by 5 points, 1-1 for win by 4 points, lose all others).

For each hand dealt, add up the count values of each card to get the count for the hand. Notice that 2’s and 3’s are +2, then “high cards” (7,8,9,10) are -1, and “low cards” (1,4,5) are +1. Write down the count for the hand in the sub-box, and add it to the running count (RC). Write the new RC value in the box.

If the new RC is greater or equal to the number printed in the next hand’s box, then bet the dragon. That’s it.

### Examples

You’re at the start of the shoe. The dealer pulls out a 7, and burns seven cards. Start the running count at -1.

The first hand dealt is player (10,9) and banker (4,5). There are no 2’s or 3’s in the hand. The two high cards cancel out the two low cards, so the count for the hand is 0. The running count remains at -1, and write it in the box.

The next hand dealt is player (1,2) and banker (8,10). The count value of the hand is 1+2-1-1 = +1. Write the new RC of 0 into the box. The RC is less than 56, so don’t bet the player dragon the next hand.

The next hand dealt is player (3,1,5) and banker (4,10,5). The count value of the hand is 2+1+1+1-1+1 = +5. Update the running count by +5, and write the new RC of 5 in the box. The RC is less than 55, so don’t bet the player dragon the next hand.

## Baccarat Dragon-7 Tracking Sheet

Posted in +EV, baccarat, dragon-7 by stephenhow on October 4, 2011

Update: See the tracking sheet in action with my Dragon-7 shoe-by-shoe simulator. Also, check out the much easier-to-use unbalanced Dragon-7 count.

I tested out Eliot Jacobson’s true count system for the 40-to-1 baccarat dragon-7 bet (banker wins with 3 card 7), and got excellent results with the following tracking sheet:

The sheet helps you track the running count (RC) for each hand in the shoe. You just compare the running count (RC) to the minimum betting threshold for the next hand. When the RC is greater or equal to the number printed in next box, bet the dragon.

The tracking sheet simulates at an average profit of about \$53 per shoe, for \$100 dragon bets. On average, you’ll make about 5.2 dragon bets per shoe.

I put together this tracking sheet because I know no one is going to read the WoO post and implement the true count correctly at the table. Half the people around the baccarat table write down something complicated every hand, so this will be my craziness.

### Instructions

Use the following count values for each card:

Dragon-7 Count Values
Card Count Value
Ace 0 (ignore)
Deuce 0 (ignore)
Trey 0 (ignore)
Four -1
Five -1
Six -1
Seven -1
Eight 2
Nine 2
Face/Ten 0 (ignore)

The dealer pulls the first card out of the shoe, and turns it face up. Start the running count with the count value of the card. The following unseen burn cards do not affect the RC or the bet thresholds.

For each hand dealt, add up the count values of each card to get the count for the hand. Notice that 8’s and 9’s are +2, and 4-thru-7’s are -1. Ignore all other cards. Write down the count for the hand in the sub-box, and add it to the running count (RC). Write the new RC value in the box.

If the new RC is greater or equal to the number printed in the next hand’s box, then bet the dragon. That’s it.

### Examples

You’re at the start of the shoe. The dealer pulls out a 7, and burns seven cards. Start the running count at -1.

The first hand dealt is player (10,9) and banker (4,5). The count value of this hand is 2-1-1 = 0. The RC doesn’t change, so write the RC of -1 in the box. The RC of -1 is less than the minimum 40 required to bet the dragon on the next hand.

The next hand dealt is player (1,2) and banker (8,10). The count value of the hand is +2. Write the new running count of +1 in the box. Again, the RC of +1 is too low to bet the dragon on the next hand (need at least 40).

Keep filling in the hand boxes, from left to right, and down the page. There are about 80.9 hands/shoe on average. There’s room for 88 hands, which occurs very infrequently. You can track two shoes per sheet. You’ll see as you get deeper into the shoe, the minimum RC for betting the dragon decreases. This follows from true count = RC/(decks remaining).

## +EV Rapid Baccarat Dragon @ Barona Casino

Posted in +EV, baccarat, dragon-bonus by stephenhow on October 3, 2011

OMG, did you see the recent WoO post on counting the baccarat dragon bet?! That bet gets so +EV that a simple counting scheme yields an average +4% edge over 7.5% of the time. Betting \$100 dragons yields an average \$25 profit per shoe with the simple count.

Update: use my simple dragon tracking sheet that simulates at an \$53 profit/shoe (betting \$100 dragons 6.4% of the time).

I had never looked into counting for the dragon, because I assumed the 7.611% house edge was too big to overcome. However, given the 40-to-1 payout odds, a 0.25% (quarter percent) change in the dragon frequency yields a 10% change in the bet EV. And it works out that the effect-of-removal (EOR), i.e. the sensitivity, of a card to the dragon-7 EV is rather large.

So I looked into other countable dragon opportunities at my nearby Barona Casino. They have a Rapid Baccarat bank of player consoles, with the following dragon bet:

Rapid Baccarat Dragon Bet (on Player Hand)
Hand Payout Frequency Return
Non-natural winner by 9 points 30-to-1 0.003683 0.110492
Non-natural winner by 8 points 10-to-1 0.006822 0.068217
Non-natural winner by 7 points 6-to-1 0.017924 0.107543
Non-natural winner by 6 points 4-to-1 0.028257 0.113027
Non-natural winner by 5 points 2-to-1 0.033244 0.066489
Non-natural winner by 4 points 1-to-1 0.037368 0.037368
Natural winner 1-to-1 0.162589 0.162589
Natural tie push 0.017871 0.000000
All Others -1 0.692242 -0.692242
Total 1.0000 -0.026517

The sensitivity of the player dragon bet to any given card is listed in the following EOR table (8 deck shoe):

Effect-of-Removal per card for Player Dragon EV
Card EOR
Face/Ten -0.000530
Ace +0.000777
Deuce +0.001160
Trey +0.001273
Four +0.000470
Five +0.000172
Six +0.000053
Seven -0.000563
Eight -0.000619
Nine -0.000604

The result is that the player dragon bet on the Rapid Baccarat machines at Barona is countable (you can calculate when the bet is +EV). Here’s a baccarat tracking sheet that tells you when to make the dragon bet. Below are the theoretical distributions of the dragon-7 and the player dragon EVs after 7 decks are dealt.

The graph shows that the dragon-7 (banker wins with a 3 card 7) often gets very advantageous at the 7th-deck point of the shoe. This graph shows the actual EV of the bets, independent of counts. You’ll need a mobile app to track the shoe exactly to get this calculation. In the next posts, I’ll compare the return using a simple count to these ideal returns.

The graph of the Rapid Baccarat Player Dragon shows a much narrower distribution, which still yields a player advantage. The player dragon compares favorably to the dragon-7, in a few important aspects. First, you can sit around at a Rapid Baccarat terminal all day, without any floor person even noticing you. You can use your mobile device at a Rapid Baccarat terminal. Secondly, the variance of the player dragon is much, much less than the dragon-7, because you hit something 30% of the time, more than 10x as often as the dragon-7’s infrequent 2.5% hit rate. The Kelly criterion will allow you to bet much more on the player dragon than the dragon-7 for any given bankroll. (Think about it, you’re still betting on a 40-to-1 longshot on the dragon-7. On the other hand, the player dragon pays even money for a natural winner.)

Would I sit around all day at a Rapid Baccarat terminal waiting for the count to go +EV on the player dragon? While listening to my podcasts and music on my iPhone? Probably not, unless the numbers work out amazing well. I’d need the shoe to be +EV around 10% of the time. Otherwise, I’d rather play the fun -EV carnival games.