5 card Pai Gow Poker?

Local here and rarely in Harrah’s. I was walking through last week and saw a “5 Card Pai Gow” table. There were no chips in the rack, so I have no idea if the game is on its way out or on its way in. Does anybody have any info on this?

I suspect it is CGD's game 5-Card Pai Gow, I believe it is on the way in for a field trial.
https://wizardofodds.com/video/5-card-pai-gow/

In the 5 card pai gow game, what are the hand rankings for the 3 card hand?

Specifically if one hand is 3oak and the other hand is a 3-card flush, which hand wins?

If one hand is a 3-card flush and the other hand is a 3-card straight, who wins?

Yes. Curious on this also.

Played a little with a friend after seeing this thread and very interested in ranking, rules etc

I am pretty sure the 3-card poker hand follows typical 3-card poker hand rankings:
Straight Flush
Trips
Straight
Flush
Pair
High Card

I found a demo of their game online, and my higher flush lost to a straight, but the other combinations would be harder to confirm.
https://casinogamingdevelopment.herokuapp.com/paigow

Quote: zbrownson

I am pretty sure the 3-card poker hand follows typical 3-card poker hand rankings:
Straight Flush
Trips
Straight
Flush
Pair
High Card

I found a demo of their game online, and my higher flush lost to a straight, but the other combinations would be harder to confirm.
https://casinogamingdevelopment.herokuapp.com/paigow
link to original post

Thanks. I suspected this,but obviously felt a need to check.

Thanks for the link to the demo game. Very interesting Dealer plays this hand J-J-9-7-4 as J-9-4| J-7

and Q-Q-8-3-2 as Q-8-2 | Q-3

This game seems to have a very high frequency of pushes. I suspect it's greater than 50%. You automatically push on dealer hand when his two card hand is 10 high or lower. That happens a lot. Plus, of course it's quite frequent that Dealer and Player will split, i.e., each wins one of the two hands.

The worst hand dealer can have is JJ-532 (no 3-card flush). It will qualify and dealer will arrange it as J32 | J5.

^ Thanks for clarifying that - I read is as "A 10" high rather than a 10 high so had been totally confused by some of its decisions and gave in playing! That also explains why it would split large pairs.
btw is there any idea of House Way, otherwise I might find myself doing some fun-time coding sometime. I can imagine with AKQ42 it might be best to play A42 and KQ.

Quote: charliepatrick

^ Thanks for clarifying that - I read is as "A 10" high rather than a 10 high so had been totally confused by some of its decisions and gave in playing! That also explains why it would split large pairs.
btw is there any idea of House Way, otherwise I might find myself doing some fun-time coding sometime. I can imagine with AKQ42 it might be best to play A42 and KQ.
link to original post

From the video, house way is to maximize the two card hand while "not fouling" the three card hand. I interpret not fouling as not having a three card hand that is lower than the two card hand. There are definitely some very non-optimum moves implied by this house way. House way is meant to be simple for the dealers to learn and to minimize mistakes on.

AsQsJd9c3s is arranged as As6c3s- QsJd and not as AsQs3s - Jd9c

AsKsQsTd9h is arranged as AT9-KQ, which makes sense since AsKsQs-Td9h would be an automatic push.

Quote: gordonm888

This game seems to have a very high frequency of pushes. I suspect it's greater than 50%. You automatically push on dealer hand when his two card hand is 10 high or lower. That happens a lot. Plus, of course it's quite frequent that Dealer and Player will split, i.e., each wins one of the two hands.

The worst hand dealer can have is JJ-532 (no 3-card flush). It will qualify and dealer will arrange it as J32 | J5.
link to original post

I believe it would be J52 / J3. The bottom hand still must be stronger than the top.

Quote: linksjunkie

Quote: gordonm888

This game seems to have a very high frequency of pushes. I suspect it's greater than 50%. You automatically push on dealer hand when his two card hand is 10 high or lower. That happens a lot. Plus, of course it's quite frequent that Dealer and Player will split, i.e., each wins one of the two hands.

The worst hand dealer can have is JJ-532 (no 3-card flush). It will qualify and dealer will arrange it as J32 | J5.
link to original post

I believe it would be J52 / J3. The bottom hand still must be stronger than the top.
link to original post

You are absolutely right. My error.

I suspect this might be House Way when making the "Best Lo", if so it should be possible to work out how often the dealer qualifies.
House Way assuming always makes Best Lo
(i) Quads : always split to put Pair in Lo hand
(ii) Full House : always put Pair into the Lo hand and Trips in Hi hand (AAA/KK KKK/AA)
(iii) Trips
- (a) if Pair in Lo leaves a 3-card hand then do that e.g. (JTTT9)
- (b) else put two highest ranked cards in Lo and add remaining card to the pair
e.g. AAAQJ AAJ/AQ, AQQQJ QQJ/AQ, AQJJJ JJJ/AQ
(iv) Two Pairs : Put lower Pair in Lo hand
(v) One Pair : similar idea to Trips
- (a) if Pair in Lo leaves a 3-card hand then do that (TT654)
- (b) if the two highest ranked singletons are not part of the pair, put them in Lo hand (AQTT6)
- (c) if the highest ranked singleton is not part of the pair, the two highest ranked singletons go into Lo hand (K9932 992/K3)
- (d) if the Pair uses the highest ranked cards, then split the pair ensuring Hi>Lo (JJ532 J52/J3)
(vi) Five unmatched cards
- (a) if three of 2nd thru 5th cards can form a 3-card hand, create the best Lo that keeps a 3-card hand in Hi (AQJT8 QJT/A8; AsQhJdTd6d JT6d/AQ; etc.)
- (b) play 1st,4th and 5th in Hi, and 2nd and 3rd in Lo (AKQT9 AT9/KQ)

I've only had a quick look so haven't checked any figures, but I get the dealer makes 10-high or lower only 592476 out of 2598960 hands, which is about 22.8%. I can imagine there are quite a few normal pushes. I've also realised if you can't make Lo J-high or better then you might as well make the best hi hand (e.g. 76532). (I'm assuming a slightly different game where you don't know the Dealer's cards.)

Quote: gordonm888

This game seems to have a very high frequency of pushes. I suspect it's greater than 50%. You automatically push on dealer hand when his two card hand is 10 high or lower. That happens a lot. Plus, of course it's quite frequent that Dealer and Player will split, i.e., each wins one of the two hands.

The worst hand dealer can have is JJ-532 (no 3-card flush). It will qualify and dealer will arrange it as J32 | J5.
link to original post

Are you sure? The dealer is allowed to make his low hand higher than his high hand? J5 is higher than J32. If you do that in ‘regular’ 7 card Pai Gow that’s a foul hand.

The high hand still has to be higher than the low hand.. But the dealer will split high pairs to put one in the low hand. The example given was JJ532 but it has to be J52/J3.

Quote: SOOPOO

Quote: gordonm888

This game seems to have a very high frequency of pushes. I suspect it's greater than 50%. You automatically push on dealer hand when his two card hand is 10 high or lower. That happens a lot. Plus, of course it's quite frequent that Dealer and Player will split, i.e., each wins one of the two hands.

The worst hand dealer can have is JJ-532 (no 3-card flush). It will qualify and dealer will arrange it as J32 | J5.
link to original post

Are you sure? The dealer is allowed to make his low hand higher than his high hand? J5 is higher than J32. If you do that in ‘regular’ 7 card Pai Gow that’s a foul hand.
link to original post

If you scroll up, you will see that someone else caught this error earlier, and I instantly admitted that this was a terrible, bone-headed error on my part.

Quote: charliepatrick

I suspect this might be House Way when making the "Best Lo", if so it should be possible to work out how often the dealer qualifies.
House Way assuming always makes Best Lo
(i) Quads : always split to put Pair in Lo hand
(ii) Full House : always put Pair into the Lo hand and Trips in Hi hand (AAA/KK KKK/AA)
(iii) Trips
- (a) if Pair in Lo leaves a 3-card hand then do that e.g. (JTTT9)
- (b) else put two highest ranked cards in Lo and add remaining card to the pair
e.g. AAAQJ AAJ/AQ, AQQQJ QQJ/AQ, AQJJJ JJJ/AQ
(iv) Two Pairs : Put lower Pair in Lo hand
(v) One Pair : similar idea to Trips
- (a) if Pair in Lo leaves a 3-card hand then do that (TT654)
- (b) if the two highest ranked singletons are not part of the pair, put them in Lo hand (AQTT6)
- (c) if the highest ranked singleton is not part of the pair, the two highest ranked singletons go into Lo hand (K9932 992/K3)
- (d) if the Pair uses the highest ranked cards, then split the pair ensuring Hi>Lo (JJ532 J52/J3)
(vi) Five unmatched cards
- (a) if three of 2nd thru 5th cards can form a 3-card hand, create the best Lo that keeps a 3-card hand in Hi (AQJT8 QJT/A8; AsQhJdTd6d JT6d/AQ; etc.)
- (b) play 1st,4th and 5th in Hi, and 2nd and 3rd in Lo (AKQT9 AT9/KQ)
link to original post

This statement is incorrect: " if three of 2nd thru 5th cards can form a 3-card hand, create the best Lo that keeps a 3-card hand in Hi (AQJT8 QJT/A8; AsQhJdTd6d JT6d/AQ; etc.)"

Dealer must go all out to make the best low, even when it busts up a flush or straight in the bottom (high) hand.

Example from demo game: AsJd5s4d2s is is played as As4d2s : Jd5s rather than As5s2s : Jd4d. Dealer breaks up his 3-card flush just to obtain a J5 rather than a J4. This kind of non-optimal play by the dealer is apparently part of the advantage given to player.

Quote: gordonm888

Quote: SOOPOO

Quote: gordonm888

This game seems to have a very high frequency of pushes. I suspect it's greater than 50%. You automatically push on dealer hand when his two card hand is 10 high or lower. That happens a lot. Plus, of course it's quite frequent that Dealer and Player will split, i.e., each wins one of the two hands.

The worst hand dealer can have is JJ-532 (no 3-card flush). It will qualify and dealer will arrange it as J32 | J5.
link to original post

Are you sure? The dealer is allowed to make his low hand higher than his high hand? J5 is higher than J32. If you do that in ‘regular’ 7 card Pai Gow that’s a foul hand.
link to original post

If you scroll up, you will see that someone else caught this error earlier, and I instantly admitted that this was a terrible, bone-headed error on my part.
link to original post

I think it’s a tiny, inconsequential error on your part. But this is WoV, where quibbling over minutia rules!

^ What I was trying to say is with five different cards, say AKxxx, then usually the Ace goes in the Hi and King to Lo. However if you can make a straight or flush of StF without using the Ace, I.e. with 2nd-5th cards, then you can put Ace into Lo hand.

This is only a close approximation as it assumes the Dealer never folds and the cards of the Player doesn't affect the Dealer's and the way I've taken averages.
I guess what I could do is, rather like peeking at Blackjack, look at the hands where the Dealer qualifies.
What seems interesting is with AAAAx you would play AAA/Ax not AA/AA (obviously Dealer can't make any Ax) - this may be different if the Dealer folds some hands.

	Hi Hand	Lo Hand
StFlushes	99.697%
Trips	99.000%
Straights	93.536%
Flushes	81.238%
P(A)	72.413%	99.790%
P(K)	71.359%	99.342%
P(Q)	69.676%	98.843%
P(J)	67.492%	98.296%
P(T)	64.926%	97.693%
P(9)	62.081%	97.034%
P(8)	59.049%	96.319%
P(7)	55.907%	95.548%
P(6)	52.722%	94.721%
P(5)	49.546%	93.838%
P(4)	46.420%	92.900%
P(3)	43.372%	91.905%
P(2)	40.403%	90.850%
AK	37.852%	89.215%
AQ	37.154%	87.002%
AJ	36.589%	84.808%
AT	35.970%	82.723%
A9	34.919%	80.755%
A8	33.419%	78.912%
A7	31.501%	77.219%
A6	29.285%	75.700%
A5	26.980%	74.380%
A4	24.878%	73.287%
A3		72.492%
A2		71.913%
K-high	21.219%	70.195%
		67.289%
		64.544%
		62.139%
		60.052%
		58.259%
		56.754%
		55.531%
		54.584%
		53.868%
		53.343%
Q-high	11.979%	51.550%
		48.403%
		45.530%
		43.166%
		41.256%
		39.748%
		38.605%
		37.789%
		37.207%
		36.791%
J-high	6.247%	35.097%
		32.070%
		29.399%
		27.342%
		25.817%
		24.738%
		24.037%
		23.576%
		23.254%
T-high	2.930%	21.784%
		19.147%
		16.916%
		15.345%
		14.317%
		13.719%
		13.365%
		13.124%
9-high	1.184%	11.962%
		9.892%
		8.249%
		7.246%
		6.740%
		6.481%
		6.311%
8-high	0.384%	5.493%
		4.077%
		3.075%
		2.635%
		2.464%
		2.354%
7-high	0.090%	1.873%
		1.104%
		0.708%
		0.600%
		0.544%
6-high	0.014%	0.348%
		0.128%
		0.063%
5-high		0.030%
		0.010%

Here are the results for player of 25 hands on the demo game:

Win: 4
LOSE: 5
DNQ;Push: 5
Push (dealer qualifies): 11

So, 16 of 25 hands ended in push. I realize the sample size is small; but one can still be informed with reasonable confidence by small samples when the results are extreme.

EDIT: Results for 50 trials

Win: 7
Lose: 9
DNQ PUSH: 13
Qualify Push: 21

Charlie, thanks for those calcs. I continue to be amazed at how fast you can do certain kinds of calcs.

Obviously, both high and low hands that are 10 high or lower never win, because of the DNQ rule for dealer. But we can use your results to make some back-of-the-envelope adjustments and gain some insights.

fyi If one does consider the DNQ, obviously the Lo hands distribution will be the same, but this table excludes High hands where the Low hand didn't qualify. For instance 66652 will be played as 662/65 and still not qualify.
DNQ is all hands which do not qualify. Hands such as J-high assume that half beat Dealer's J-high, and half lose.
As always I've haven't had a chance to check these figures nor compare them with a simulation. So please don't take them as gospel!!

DNQ	601 116	23.129%
J-High	27 720	23.662%
Q-high	95 700	26.037%
K-high	193 440	31.599%
A-high	311 940	41.322%
P(2)	57 648	48.432%
P(3)	59 388	50.684%
P(4)	60 872	52.998%
P(5)	62 664	55.374%
P(6)	64 456	57.820%
P(7)	66 248	60.334%
P(8)	68 040	62.918%
P(9)	70 192	65.577%
P(T)	72 292	68.318%
P(J)	65 716	70.974%
P(Q)	54 804	73.292%
P(K)	40 240	75.121%
P(A)	22 384	76.325%
Flush	305 072	82.625%
Straight	267 852	93.647%
Trips	13 988	99.070%
StFlushes	17 188	99.669%

I have completed a 5-card Pai Gow 'hand analyzer' in which a given 5-card player hand is compared to every possible dealer hand to calculate its Expected Value (EV). The analysis is composition dependent, i.e., the probability of dealer hands is affected by the cards in the player's hand. And because the dealer's hand is face up; all legal arrangements of the player's hand are evaluated against each dealer hand to find the optimal arrangement.

Thus if player has AAJJ5 and the dealer has AAKK8, then the player's hand is arranged as AAK | K8 so that the player can at least win the high hand and gain a push.

And, of course when dealer has a DNQ the hand is an automatic push.

First let's look at hands where the player has a low hand that is 10-high or lower and thus cannot possibly win against a dealer hand that has qualified.

Descriptor	Player Hand	EV
Molten poop	T9742	-0.777807331
High Card + poop	J9742	-0.743944186
High Card + poop	Q9742	-0.716324443
High Card + poop	K9742	-0.643356744
High Card + poop	A9742	-0.514189287
Low Pair + poop	33-T95	-0.460947274
Low Pair + poop	66-T95	-0.387282676
Low Pair + poop	TT-965	-0.320718099
Low flush + poop	5s3s2s-Td9c	-0.278969372
Low flush + poop	Ts9s5s-3d2c	-0.22927574
Low straight + poop	432-T9	-0.116017651
Low straight + poop	T98-32	-0.041581836
Low SF + poop	Ts9s8s-3d2c	-0.002642217

Now let's look at some hands with multiple high cards, including hands with high pairs (JJ to AA). The high pair hands benefit from being able to be arranged as "the pair in the bottom hand" or "splitting the pair between the bottom and top hands", depending upon what is seen in the dealer's hand.

Descriptor	Player Hand	EV
2 High Cards	AKT86	-0.088133231
2 High Cards	AQT86	-0.262343548
2 High Cards	AJ986	-0.439651772
2 High Cards	KQ986	-0.414233552
2 High Cards	KJ986	-0.568585843
2 High Cards	QJ986	-0.633105358
JJ Pair	JJ-976	-0.289514772
JJ Pair+High Card	JJ-Q76	-0.182833216
JJ Pair+High Card	JJ-K76	-0.003931708
JJ Pair+High Card	JJ-A76	0.198720418
QQ Pair	QQ-T76	-0.215095907
QQ Pair + High Card	QQ-J96	-0.184165081
KK Pair	KK-T96	-0.081276374
KK Pair + High Card	KK-J96	-0.071018469
KK Pair + High Card	KK-Q96	0.000420486
AA Pair	AA-T96	0.151505373
AA pair + High Card	AA-J96	0.141201182
AA pair + High Card	AA-Q96	0.17648029
AA pair + High Card	AA-K96	0.227085301

Notice that an AA-T96 has a higher EV than an AA-J96. This surprising result is due to the AA-T96 hands facing a lower frequency of Dealer DNQ than the AA-J96 hands. Having a 10 or 2 in your hand has a significant influence toward lowering the Dealer's DNQ frequency.

More in my next post (including 5-card hands with higher EVs) which will be a bit later.

Search for the High Hand
So, given this new analysis capability, let's ask: What is the best hand that player can get in 5-card Pai Gow Poker?

Well, that should be easy. Let's give this hand a Royal flush in the 3-card bottom hand, and a pair of aces in the 2-card top hand. Like this


Nothing can beat or tie the AA because there's only one Ace left in the deck. And the royal flush can't be beaten and it can only be tied by another royal. And another royal would be even rarer than usual because there's only one ace left in the desk. And also because the dealer's house way would split a royal as an ace in the bottom hand and a KQ in the top hand - unless the 4th and 5th cards are a KQ or a pair 22-JJ (remember that all the aces are now gone.) So, this hand is very close to unbeatable. Here we go with the calculation:


Huh??? Crap, this was not what I expected. The dealer must be DNQ at about 37%!!!!!

Well, I realize that AKQ-AA is chewing up 5 of the 16 high cards, and that the dealer often needs two high cards to qualify. So, I start to wonder if Ts9s8s | AsAd might be a better hand? Straight flushes are uncommon enough that a T98 straight flush might not be a very big penalty. And I'll line up the suits such that two suits still have 13 cards remaining. So, I do the calculation and get



Wow, More than a 10 point improvement. Wow. But, can I do even better? What if I boldly chuck the AA for two cards that are 10 or less, so that I have all 5 cards at 10 or less and thus all partially blocking the DNQ. So, here's what I got:

AsKsQs | AhAd___ EV = +0.62600664

Ts9s8s | AsAd___ EV = +0.729019211

9s9h9d | ThTd___ EV = +0.762306063

TsThTd | 9h9d___ EV = +0.764380461

Ts9s8s | ThTd___ EV = +0.770747729

So there it is: (Ts9s8s | TT) is the best Player hand in 5-card Pai Gow Poker! I've always thought of poker as being a game of high cards. But this makes sense: high pairs JJ-AA are quite rare in the 2-card low hand and straight flushes higher than T-high occur infrequently. Avoidance of the Dealer DNQ in 5-card Pai Gow is a very influential factor.

QUADS

Quads are a rare hand in 5-card Pai Gow Poker, but they are an interesting card category. The quads are usually split into two pair like this: 6666-9 --> 669 | 66. But the player may elect, after seeing the dealer's hand, to split them into trips and 2 singletons: 6666-9 --> 666 | 96. This arrangement might be used when facing a dealer hand such as QsTs3s-88 in order to convert a losing hand into a push.

And it turns out, to my surprise, that the usually irrelevant singleton kicker in a Quads hand can actually be quite significant because it affects the strength of the two card hand when playing Trips in the 3-card hand.

Here's a table of calculated EVs, where the columns headed by 6 to A denote the singleton kicker in the player's quad hand. I 've kept the entries to only 4 digits to make it easily readable.

Quads	6	T	J	Q	K	A
AAAA	0.5913	0.5987	0.5603	0.5598	0.5593
KKKK	0.4884	0.4961	0.4565	0.4555		0.4979
QQQQ	0.4232	0.4289	0.3887		0.4129	0.4259
JJJJ	0.3764	0.3789		0.3592	0.3684	0.3826
TTTT	0.4648		0.4640	0.4714	0.4808	0.4965
9999	0.4556	0.4664	0.4430	0.4506	0.4601	0.4793
8888	0.4256	0.4408	0.4163	0.4228	0.4354	0.4594
7777	0.4016	0.4189	0.3931	0.4022	0.4188	0.4487
6666		0.3922	0.3665	0.3786	0.4000	0.4368
5555	0.3591	0.3637	0.3384	0.3542	0.3811	0.4253
4444	0.3300	0.3334	0.3086	0.3288	0.3618	0.4138
3333	0.3079	0.3103	0.2863	0.3107	0.3500	0.4085
2222	0.2763	0.2786	0.2557	0.2847	0.3301	0.3965

In this table we also see, once again, the influence of having a low or high card in your hand on the DNQ frequency of the dealer's hand.