Discount Gambling

Pai-Gow Progressive @ Barona

Posted in pai-gow poker, progressive sidebets by stephenhow on September 17, 2011

The Pai-Gow progressive jackpot often gets very large at Barona, so here’s the breakdown, if you feel like playing it sometimes:

Hand Payout Probability Return
5 Aces 100% 7.3179 x 10-6 jackpot/136,651
5-of-a-Kind 1000-for-1 8.7815 x 10-5 0.087815
Royal Flush 500-for-1 1.7563 x 10-4 0.087815
Bad Hand* 250-for-1 1.0082 x 10-4 0.025204
Straight Flush 100-for-1 1.4503 x 10-3 0.145034
4-of-a-Kind 20-for-1 7.2935 x 10-3 0.145870
Magic Card** 20-for-1 5.0798 x 10-3 0.101597
all others 0-for-1 ~ 98.6% 0
total 0.593335 +

So the break-even point for the jackpot is $55,571. Last month, someone hit it for $125k. That means the EV was 1.51, while it lasted.

*Bad Hand = 9-high, no 6, no flush
**Magic Card frequency assumed @ 1/26 (from DEQ, the game publisher)

Joker/Ace Collusion Analysis For Pai-Gow Poker

Posted in collusion, pai-gow poker by stephenhow on June 5, 2011

Experienced Pai-Gow players (i.e., those who play every day) often tell each other where the Joker and Aces are. At a full table, 6 players hold 42 cards, so if no one says “joker busy”, then the dealer probably has it (63.6%). Also, if the players don’t hold many Aces, then the dealer is going to have a stronger-than-usual hand. I’ve often wondered how practical and powerful a Pai-Gow collusion strategy would be, so I worked out an analysis of the problem.

Immediately, I saw that Ace/Joker info would help mostly for deciding whether to play “pair-pair” (small pair in front, big pair in back) or “two pair behind” (kickers in front, two pair in back). The two pair hands (or other “break/keep” decisions) occur about 20% of the time, frequently enough for a collusion strategy to have some promise. For these two pair decisions, I thought it’d be important to see how the Ace/Joker count affects the dealer front pair %, and the dealer two pair or better behind %. This seemed natural, because I figured if the players held 3 Aces, then the chance of a single dealer Ace was pretty high, which he’d probably play up front (the “3 Ace Effect”).

I ran the analysis (CA rules where the Joker is completely wild), and was amazed to see the following results!

Probabilities of dealer hands as a function of known Ace/Jack locations (6 players).

Sure enough, the expected results pop right out of the graphs. On the left side are hands where the Joker is “busy”, i.e., one of the players holds it, so the dealer cannot have it. On the right the Joker is not busy, and is probably in the dealer’s hand. The top graphs represent the strength of the dealer front hand vs. the known Ace count, and the bottom graphs show the strength of the dealer back hand vs. the Ace count.

As expected, the dealer hand is strong when the Joker is not busy. Also, the dealer front hand is weakest when the players hold 3 Aces. As I pointed out earlier, the “3 Ace effect” results from the high probability that the dealer has the remaining Ace, and will play it up front. This is the “sweet spot”. So, we see the weakest dealer hand happens when the Joker is Busy and the players hold exactly 3 Aces (12.5% front pair or better) and the strongest dealer hand happens when the Joker is not busy, and the players hold no Aces (80% front pair or better). This is a huge, huge difference, and suggests a big difference in the right way to play two pairs for these two cases.

So, by sharing Ace/Joker information at a full table, you learn when the dealer has either a very weak hand (Joker Busy, 3 Aces seen; most common case, 34.3% of the time), or a very strong hand (Joker not Busy, and 2 or less Aces seen; rare 3% of the time). You should alter your 2 pair decisions to take advantage of this information, as shown below.


The following examples show cases where Joker/Ace info would save you around 20% of your bet, on average.

Strong Dealer Hand

Let’s say you have two pairs, Jacks and 5’s, and no kickers: Jd Jh 5s 5d 8s 7h 3c. Normally, you’d play this pair-pair, with a pair of 5’s up front, and a pair of Jacks behind. No one in the world would play 8-high in front, and two pairs (Jacks and 5’s behind). But, in the rare (3%) case where the Joker is hiding, and the players have 0, 1, or 2 Aces, then you’re actually better off playing two pair behind!

Jacks and 5’s, Joker Hiding, 8-high Kicker
Aces Seen EV(Pair-Pair) EV(2 Pair Behind) Decision
0 -0.72 -0.52 play 8-high front, J’s and 5’s behind
1 -0.57 -0.36 play 8-high front, J’s and 5’s behind
2 -0.34 -0.25 play 8-high front, J’s and 5’s behind
3 -0.06 -0.24 play 5’s in front, J’s behind
4 -0.11 -0.26 play 5’s in front, J’s behind

Weakest Dealer Hand

Now, say you had the J’s and 5’s again, but this time you have an Ace kicker. This is a decision point that people may think about. Most house-way strategies will play two pair behind. You should definitely play two pair behind when the Joker is hiding. However, when the dealer hand is extremely weak because the Joker is busy and exactly 3 Aces are seen (“3 Ace Effect”), ignore your Ace, and play pair-pair. This is normally considered aggressive, but against the weakest dealer hand (the most common case, 34.3% of the time), you should go for it. The “3 Ace Effect” creates the “sweet spot” that minimizes the expected dealer front hand strength.

Jacks and 5’s, Joker Busy, Ace Kicker
Aces Seen EV(Pair-Pair) EV(2 Pair Behind) Decision
1 -0.12 +0.17 play Ace front, J’s and 5’s behind
2 +0.18 +0.30 play Ace front, J’s and 5’s behind
3 +0.52 +0.32 play 5’s in front, J’s behind
4 +0.41 +0.55 play Ace front, J’s and 5’s behind

How To Exploit Ace/Joker Info

It’s kind of pointless to work out the extreme details of the optimal Pai-Gow strategy given Joker/Ace info. Most people won’t remember the details, or even the broad strokes, given the gory details. However, it’s pretty easy to boil it all down to a simple collusion strategy. The players should first find out if the Joker is “busy” or not. If players have not seen the Joker, then they should play conservatively and favor two pair behind (like house way). If the Joker is hiding, check if there are only 0 or 1 Aces out. If so, the dealer has a very strong hand, so play extremely conservatively. For example, I would play no front and Kings and 6’s behind against the very strongest dealer hand. But the strongest dealer hand is rare 0.4%), so it’s not worth checking for, unless you have a huge bet out there. The Joker is busy 80% of the time. When the Joker is busy, and you have a possible two pair decision, find out if 3 Aces are seen, and thus the dealer has the weakest possible hand (the most common case, 34.3% of the time). If so, play two pair aggressively according to the below table.

Here’s a summary of the practical 6-player collusion strategy:

  • If Joker is Busy
    • If exactly 3 Aces seen, the dealer has the weakest possible hand (most common), so play aggressively.
    • Else play normal.
  • Else, Joker is Hiding
    • If 0 or 1 Aces seen (extremely rare), dealer has the strongest possible hand, play super conservatively
    • Else, play conservatively
Minimum Front for Two Pair Behind (Aggressive, only use against the weakest dealer hand.)
22 33 44 55 66 77 88 99 TT JJ QQ KK
AA p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p
KK p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p
QQ p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p
JJ AK AK pp AK AK AK p/p p/p p/p
77 A4 A4 A8 AT AJ
66 A4 A5 A5 A7
55 A4 A7 A8
44 KJ KQ
33 K9

where “p/p” means always play pair-pair. Note that this table more aggressively plays pair-pair, because of the stronger-than-normal minimum front hand required to play two pair behind. Most tables require only an Ace for the larger pairs, and only a Jack or Queen for the lower pairs. However, the above table requires not only an Ace, but often AK, AQ, or AJ. Even the smallest pairs require K9 to play two pair behind. This is an aggressive table to play against a weak dealer hand that likely (87.5%) will not have a front pair.

Compare the above aggressive table to the more conservative strategy below, where the dealer hand is moderately strong (Joker is Hiding, and all 4 Aces are held by the players). Notice front hand requirements are much lower than the aggressive strategy. This means you end up playing “two pair behind” much more often. This is as you would expect against a stronger dealer hand. I don’t provide two pair tables for all Joker/Ace combinations, but provide these two to show the effect of expected dealer hand strength on how you play two pairs.

Minimum Front for Two Pair Behind (Conservative, used when Joker Hiding and 4 Aces Seen)
22 33 44 55 66 77 88 99 TT JJ QQ KK
AA p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p
KK Ax Ax p/p p/p p/p p/p p/p p/p p/p p/p p/p
QQ Kx Kx Ax Ax Ax p/p p/p p/p p/p p/p
JJ 2p Qx Qx Kx Kx Ax Ax Ax Ax
TT 2p 2p 2p Qx Kx Kx Kx Ax
99 2p 2p 2p 2p Jx Qx Kx
88 2p 2p 2p 2p Jx Jx
77 2p 2p 2p 2p 2p
66 2p 2p 2p 2p
55 2p 2p 2p
44 2p 2p
33 2p

where “2p” means always play two pair behind.


Frequencies of Dealer Hand Strengths
Description Strength Frequency
Joker Hiding, 0 Aces Seen Strongest 0.02%
Joker Hiding, 1 Aces Seen 0.4%
Joker Hiding, 2 Aces Seen 3.0%
Joker Hiding, 4 Aces Seen 8.6%
Joker Hiding, 3 Aces Seen 8.8%
Joker Busy, 0 Aces Seen 0.1%
Joker Busy, 1 Aces Seen 2.0%
Joker Busy, 2 Aces Seen 13.2%
Joker Busy, 4 Aces Seen 29.6%
Joker Busy, 3 Aces Seen Weakest 34.3%
Total 100%