Player starts with four cards and decides whether to fold or to raise up to four times the initial wager. If the player doesn't fold, the player gets three more cards to make the pai gow hand. The dealer gets eight cards from which to make the seven-card pai gow hand.

One twist is that if the hand is a push in the traditional pai gow poker sense, the initial wager wins, while the raise pushes.

It looks like it might be a hard game to analyze.

More details from Galaxy Gaming:

https://www.galaxygaming.com/products/supreme-pai-gow

1. Even calculating what dealer's probabilities are for an 8 card hand from a fresh deck is challenging. With dealer having 8 cards, dealer will get many more 5 card straights and flushes which can be played behind, with 3 cards from which to make the two-card hand.

2. The House way for arranging the dealer's hands may be different because of the rule that a push means that the player wins the Ante Bet.

3. Then the player's strategy for decisions after receiving four cards is critical. Essentially player can fold, Raise 1x or Raise 4x

- What 4 card holdings would be good enough to justify a 4X raise? (Having the Joker would be a nice start.)

- How bad do 4 card holdings have to be to fold them?

4. Also, player strategy as to how arrange 7 card hands may be significantly different than conventional PGP, especially for 7 card hands that have two pair or higher. Also, the correct way to arrange the player's hand may occasionally depend on whether the player has raised 1x or 4x.

Game rules from Washington gaming

If one of the player’s hands is higher than the dealer’s respective hand and the other hand

is lower than or ties the dealer’s respective hand, the Ante wager wins and is paid 1 to 1

and the Raise wager pushes.

the underline is my editing.

Basically, if you win one of the two hands and lose/tie the other hand you will have a net win of one unit. In conventional PGP, the player pushes with the dealer 36% of the time; that win/lose outcome will occur less frequently in Supreme PGP but it's clear that this is a major driver in strategy.

As the player arranges a hand such as "Two Pair +3 singletons" or "FullHouse + 2 singletons" there will be a significant incentive to go all-out to win at least one of the two hands.

(Maybe I should have worded it differently.)Quote:...if the hand is a push in the traditional pai gow poker sense, the initial wager wins, while the raise pushes.

https://alphalackey.github.io/superpgdemogv2/

Dealer Has AQ754433 and arranges it:

33

44-AQ7

My working assumption is that the House Way is to always split Two Pair, it's an easy rule for Dealer to remember.

Is "Ultimate" falling out of favor?

Quote:gordonm888In the free demo, I noticed this hand:

Dealer Has AQ754433 and arranges it:

33

44-AQ7

My working assumption is that the House Way is to always split Two Pair, it's an easy rule for Dealer to remember.

link to original post

This is from the video:

"The house way here is any time we have two pair we're going to split the two pair unless you have an ace"

Also, on the demo I have seen that A5432 straights beats a K-High straight! Same loopy rule as conventional PGP.

T7

AAA-52

So, 3oak AAA does not appear to get split.

Quote:charliepatrickI wonder how it picks the best 7-cards. In this hand, which I was destined to lose, it seemed to prefer AQ-straight to TT AAxxx

link to original post

That is indeed a very curious House Way. They video did say, in a somewhat garbled way, that two pair is always split except when dealer has an Ace (to play on top). In this case, playing the AQ in front allows dealer to have a 5-card hand that is higher than AA-QQ-x.

I have been assuming that this game uses a much simpler strategy than conventional PGP for its House Way, as a selling point - to reduce Dealer training and mistakes. But, I have eyes wide open.

I wonder if the game analysts assumed that the player's basic strategy is to NEVER split two pair when the two pair are, say, TT-xx or lower - so as to try for a push and get the ante win. In that case the dealer's counter-strategy may be to always play A-X in front if you can get 2-pair or higher in the 5 card hand. Sort of a game theory thing.

Joker pays 100-1. I think the hand has to be a full house with a pair (with no aces) for it to be a joker. Could probably make that 500-1 no problem.

Quote:FinsRuleIsn’t this just bad programming?

link to original post

Could be. But we have so little necessary info on this game that it seems better to not ignore anything.

Quote:FinsRuleI just looked at the 8th dealer card bonus.

Joker pays 100-1. I think the hand has to be a full house with a pair (with no aces) for it to be a joker. Could probably make that 500-1 no problem.

link to original post

Ace high straight with a pair.

Royal flush with a pair.

You could discard the joker and still have the strongest possible hand.

Quote:JoeTheDragonthere is an free demo at

https://alphalackey.github.io/superpgdemogv2/

link to original post

Hey, great to see people still playing my little demos :) I did the math analysis on this game a long time ago. I still have the source code and the house way is meant to be simplified as much as possible. It's based on the 'rank pattern' of the eight cards (eight singletons, six singletons and one pair, and so forth).

For those curious, when the pattern is 2-2-1-1-1-1 (two pairs and four singletons), the following rule is used:

* If the dealer can play a straight or better with Ax or better, play the best top that still gives a straight or better.

* Otherwise, split the two pair

That's why the hand you saw, it kept the Ax with the straight but split the 5522

And basically, the design decision was to err or the side of simplicity in the house way, then adjust the paytables as needed to get a house edge.

Quote:SOOPOOQuote:FinsRuleI just looked at the 8th dealer card bonus.

Joker pays 100-1. I think the hand has to be a full house with a pair (with no aces) for it to be a joker. Could probably make that 500-1 no problem.

link to original post

Ace high straight with a pair.

Royal flush with a pair.

You could discard the joker and still have the strongest possible hand.

link to original post

I figured they would keep the joker in there to avoid the big payout.

Quote:CharlesMousseauQuote:JoeTheDragonthere is an free demo at

https://alphalackey.github.io/superpgdemogv2/

link to original post

Hey, great to see people still playing my little demos :) I did the math analysis on this game a long time ago. I still have the source code and the house way is meant to be simplified as much as possible. It's based on the 'rank pattern' of the eight cards (eight singletons, six singletons and one pair, and so forth).

For those curious, when the pattern is 2-2-1-1-1-1 (two pairs and four singletons), the following rule is used:

* If the dealer can play a straight or better with Ax or better, play the best top that still gives a straight or better.

* Otherwise, split the two pair

That's why the hand you saw, it kept the Ax with the straight but split the 5522

And basically, the design decision was to err or the side of simplicity in the house way, then adjust the paytables as needed to get a house edge.

link to original post

Charles,

Thanks for checking in. May I ask a few more questions about the Super Pai Gow House Way?

1. Dealer has 3-2-1-1-1. Does this always go Trips to the bottom and Pair to the top? If there is an A-x that can be played on top, does the dealer play full house on the bottom?

2. 4-1-1-1-1 Is Quads always broken up into two pair? Even with Quad 2s? What if you have the choice of making a straight or flush by breaking up the quads and playing a pair on top?

3. 4-3-1 If the rank of the 4oak is higher than the rank of the 3oak, does dealer ever split up the quads to play the highest possible top hand, and settle for playing a Boat on the bottom? I.e., if dealer has AAAA-222-8 is it:

4. 3-2-2-1 If the rank of trips is higher than either of the pairs, do you ever break up trips to play it as a high pair on top?

5. 3-1-1-1-1; If you have, say, AAA-J-6-4-3-2, would dealer break up the trip aces to play it as AJ|AA-643 or is it J6|AAA-43?

6. Is PGP House Way Rule "Always maximize the top hand" apply when choosing between playing a straight vs flush vs Straight flush on the bottom?

7. Just for fun, how would House Way arrange this 8 card dealer hand?

KsKdKc-WAs-QsQh-Js Is it Royal on bottom and KK pair on top? Or KKK-QQ on bottom and AA on top?

Thanks.

The rules you described are correct: players start with four cards and can either fold or raise up to four times the initial wager. They then receive three additional cards to make their best five-card hand, following the standard Pai Gow Poker ranking rules. The dealer receives eight cards and must make their best seven-card hand.

As you mentioned, one interesting twist in this game is that if the result is a push. the initial wager wins, while the raise pushes. This means that even if the player doesn't win the hand, they still have a chance to recover their raise bet.

Analyzing the game can be challenging because of the complex rules and the presence of the progressive jackpot. But, there are resources available that can help you learn more about the game and its strategies. For example, you can find books and online articles that provide detailed explanations of the game and its optimal strategies, as well as videos and tutorials that demonstrate how to play and win at Pai Gow Poker Progressive.

So, whether or not the game is "hard" to analyze depends on your level of experience and expertise in gaming analysis. If you have a strong background in probability and statistics, as well as experience with card games, you may find the challenge of analyzing Pai Gow Poker Progressive enjoyable and rewarding. On the other hand, if you're new to these topics, it may be helpful to start by learning more about basic Pai Gow Poker strategies and gradually work your way up to the progressive variant.

Indeed, the player's 2 pair strategy for Supreme Pai Gow Poker is going to depend somewhat on whether player has raised 1x or 4x.

When you have raised 1x (which is most of the time) the possible outcomes are:

Lose both hands: -2

Win one hand: +1

Win Both Hands +2

So player is highly incentivized to keep many two pair combos in the back hand, so as to maximize chances of winning at least one hand. I think this will prove to be a strong factor when your highest pair is TT or lower and your lowest pair is 55 or lower, regardless of what cards you have remaining to play in the front hand. I haven't finished the math yet, I still have a long, long way to go.

When you have raised 4x (which is) the possible outcomes are:

Lose both hands: -5

Win one hand: +1

Wing Both Hands +5

Its subtle, but there's slightly more incentive in those payouts to go for a scoop (win both hands) when you have raised 4x.

Raise 4X with:

1. Quads

2. Trips

3. Two Pair

4. Joker-A-xx or A-A-xx

5. K-K-A-x (or K-K-Joker-x)

6. Q-Q-Joker-x

7. Q-Q-A-K

Otherwise raise 1x.

Note Fold with: 8-7-3-2 when rainbow or with no more than 2 suited cards.

Comments? Ideas or advice? (Given that I have not yet worked the math.)

So the figures below are definitely wrong and underestimate the chances of the dealer making a pair, but it's a simple approach.

Combos | Pattern | Only Pair | Second pair | Best of four | Best of three | Best of two | High Hand | Notes |
---|---|---|---|---|---|---|---|---|

78 | 44 | 78 | Better of the Quads | |||||

27456 | 431 | 27456 | Plays quads OR FH | |||||

30888 | 422 | 30888 | Better of the Pairs | |||||

823680 | 4211 | 823680 | Always play Quads/Pair | |||||

1647360 | 41111 | 1647360 | * Doesn't split AAAA | |||||

82368 | 332 | 82368 | Plays FH plus best Pair | |||||

1098240 | 3311 | 1098240 | Uses better Trips | |||||

4942080 | 3221 | 4942080 | Better of the Pairs | |||||

39536640 | 32111 | 39536640 | * Uses the pair | |||||

42172416 | 311111 | 42172416 | * Doesn't split AAA | |||||

926640 | 2222 | 926640 | Best of the Pairs | |||||

44478720 | 22211 | 44478720 | Best of the Pairs | |||||

237219840 | 221111 | 237219840 | * Always splits | |||||

295206912 | 2111111 | 295206912 | * High Hand | |||||

84344832 | 11111111 | 84344832 | High Hand | |||||

752538150 | non-joker | 40360320 | 237219840 | 926640 | 44561088 | 6098742 | 423371520 | * Ignores making Str/Pair etc |

14611356 | P (A) | 3104640 | 285120 | 10283328 | 938268 | 1.942% | ||

15789279 | P (K) | 3104640 | 3041280 | 213840 | 8569440 | 860079 | 2.098% | |

17135970 | P (Q) | 3104640 | 6082560 | 155520 | 7011360 | 781890 | 2.277% | |

18650133 | P (J) | 3104640 | 9123840 | 108864 | 5609088 | 703701 | 2.478% | |

20330472 | P (T) | 3104640 | 12165120 | 72576 | 4362624 | 625512 | 2.702% | |

22175691 | P (9) | 3104640 | 15206400 | 45360 | 3271968 | 547323 | 2.947% | |

24184494 | P (8) | 3104640 | 18247680 | 25920 | 2337120 | 469134 | 3.214% | |

26355585 | P (7) | 3104640 | 21288960 | 12960 | 1558080 | 390945 | 3.502% | |

28687668 | P (6) | 3104640 | 24330240 | 5184 | 934848 | 312756 | 3.812% | |

31179447 | P (5) | 3104640 | 27371520 | 1296 | 467424 | 234567 | 4.143% | |

33829626 | P (4) | 3104640 | 30412800 | 155808 | 156378 | 4.495% | ||

36636909 | P (3) | 3104640 | 33454080 | 78189 | 4.868% | |||

39600000 | P (2) | 3104640 | 36495360 | 5.262% | ||||

423371520 | ZIP | 423371520 | 56.259% |

I sense if I can work out how to do the house way I might just run a simulation and see the hands that the dealer gets and how often the player got a double win/single win/loss, tallying it against the starting hands (or set the starting hands).

I have worked through all the combinations for natural 2111111, that is (1 pair+ six singletons; no joker). For a fresh 53-card deck these represent about 33.306933% of all hands. I have made the probabilities of all these dependent on the composition of the deck so that I can look at actual player hands vs the dealers hands.

But for this post I will quote only values for a fresh deck.

For (1 pair+ six singletons; no joker) from a fresh 53 card deck, these are the probabilities of occurrence of the bottom/back 5-card hands:

Hand | Prob |
---|---|

SF | 0.000458445 |

Flush | 0.082509125 |

Straight | 0.116212767 |

AA | 0.063641803 |

KK | 0.063489367 |

0.062346101 | |

JJ | 0.061279053 |

TT | 0.060135787 |

99 | 0.060516876 |

88 | 0.060516876 |

77 | 0.060669311 |

66 | 0.060516876 |

55 | 0.060212005 |

44 | 0.061355271 |

33 | 0.062498537 |

22 | 0.063641803 |

Hand | Prob |
---|---|

AA | 0.005430848 |

KK | 0.004891837 |

0.004319899 | |

JJ | 0.003818691 |

TT | 0.003399189 |

99 | 0.003061392 |

88 | 0.003251631 |

77 | 0.003659548 |

66 | 0.004399775 |

55 | 0.003027856 |

44 | 0.003256509 |

33 | 0.003408945 |

22 | 0.003408945 |

AK | 0.165719534 |

AQ | 0.108528556 |

AJ | 0.066517711 |

AT | 0.037469001 |

A2-A9 | 0.040553807 |

KQ | 0.104190853 |

KJ | 0.06627479 |

KT | 0.03815057 |

K2-K9 | 0.042437177 |

QJ | 0.061981232 |

QT | 0.037008402 |

Q9 | 0.01958698 |

Q2-Q8 | 0.021711443 |

JT | 0.034341757 |

J2-J9 | 0.018831022 |

T9 | 0.037883991 |

T2-T8 | 0.019406741 |

98 | 0.007883352 |

92-97 | 0.008896255 |

<92 | 0.013291762 |

Some of these probabilities look a bit squirrely in the 3rd or 4th digit (such as the probability of 77 pair + 3 singletons in the bottom hand), so I may have some errors. This is a work in progress.

Notice that about 20% of these 8-card hands arrange as a straight, flush or SF in the back 5-card hand! However, many of those hands have fairly weak front/top 2-card hands, and thus are readily beatable for a payout of +1.

A hand that comes close to breaking-even against this cohort of hands is QQ-975 | A-T.

I have started work on evaluating 2111111 hands that have a joker (but no Ace) as one of the kickers. Obviously, such hands are expected to have an even higher incidence of straights and flushes.

I have only looked at the hands which are 2111111 but do agree with your AA figure!!

(Sorry it's the other way round since I list them by their rank for i=1 upwards, yes I could change the code!)

As you say work in progess.

Edit: I have not yet looked at where the dealer splits the pair to make a 5-card hand, so definitely work in progress!!

(87) | 506 520 | 0.171 581% |

(97) | 675 360 | 0.228 775% |

(98) | 2 026 080 | 0.686 325% |

(T7) | 675 360 | 0.228 775% |

(T8) | 2 363 760 | 0.800 713% |

(T9) | 5 234 040 | 1.773 007% |

(J7) | 675 360 | 0.228 775% |

(J8) | 2 363 760 | 0.800 713% |

(J9) | 5 740 560 | 1.944 589% |

(JT) | 10 974 600 | 3.717 596% |

(Q7) | 675 360 | 0.228 775% |

(Q8) | 2 363 760 | 0.800 713% |

(Q9) | 5 740 560 | 1.944 589% |

(QT) | 11 649 960 | 3.946 371% |

(QJ) | 20 260 800 | 6.863 254% |

(K7) | 675 360 | 0.228 775% |

(K8) | 2 363 760 | 0.800 713% |

(K9) | 5 740 560 | 1.944 589% |

(KT) | 11 649 960 | 3.946 371% |

(KJ) | 21 105 000 | 7.149 223% |

(KQ) | 34 274 520 | 11.610 338% |

(A7) | 506 520 | 0.171 581% |

(A8) | 2 194 920 | 0.743 519% |

(A9) | 5 571 720 | 1.887 395% |

(AT) | 11 481 120 | 3.889 177% |

(AJ) | 20 936 160 | 7.092 029% |

(AQ) | 35 118 720 | 11.896 307% |

(AK) | 54 197 640 | 18.359 204% |

P(2) | 1 603 224 | 0.543 085% |

P(3) | 1 458 504 | 0.494 062% |

P(4) | 1 313 784 | 0.445 038% |

P(5) | 1 169 064 | 0.396 015% |

P(6) | 1 193 184 | 0.404 186% |

P(7) | 1 193 184 | 0.404 186% |

P(8) | 1 193 184 | 0.404 186% |

P(9) | 1 193 184 | 0.404 186% |

P(T) | 1 169 064 | 0.396 015% |

P(J) | 1 313 784 | 0.445 038% |

P(Q) | 1 458 504 | 0.494 062% |

P(K) | 1 603 224 | 0.543 085% |

P(A) | 1 603 224 | 0.543 085% |

Quote:CharlesMousseauQuote:JoeTheDragonthere is an free demo at

https://alphalackey.github.io/superpgdemogv2/

link to original post

Hey, great to see people still playing my little demos :)

link to original post

I just played your demo — beautifully presented. Thank you for your work on this!

Quote:charliepatrickI've managed to crank the 2111111 hands (ignoring any jokers, and assuming a fresh 52-card deck (since we can ignore the joker in it). I would expect the same numbers for AA KK and 22 since there are two ways to get a straight that have been eliminated, similarly for 33 and QQ, 44 and JJ, I'm not sure why 55 and TT are different from 66-99, perhaps it's the chance of a six-card straight. By symmetry any hand featuring KK abcdef will be matched by a corresponding hand 22 uvwxyz where if a=Ace, u=Two etc. Thus the chances of straights etc, hence the lo hand being a pair, should be the same.

I have only looked at the hands which are 2111111 but do agree with your AA figure!!

(Sorry it's the other way round since I list them by their rank for i=1 upwards, yes I could change the code!)

As you say work in progess.

Edit: I have not yet looked at where the dealer splits the pair to make a 5-card hand, so definitely work in progress!!

-snip-

link to original post

1. You said: . I would expect the same numbers for AA KK and 22 since there are two ways to get a straight that have been eliminated.

This is wrong. You are examining hands with 7 ranks, thus there are 6 and 7 card straights and 6 and 7 card flushes. When you have these types of hands, the dealer standardly uses the lowest possible 5-card straight or flush so as to allow the higher cards in the (6,7-card) straights or flushes to be used in the front hand. This causes differences in the amount of 5-card straights and flushes that contain 2 vs King vs Ace.

So, the rules of thumb you are quoting apply to 5-card hands, but are incorrect for 7card and 8 card hands.

There are additional subtleties when dealing with Supreme Pai Gow Poker with its 8-card dealer hand that includes designating a card as a discard.

The numbers I posted were the results of an analysis that addressed all of those issues correctly.

2. The entries in your table you posted are incorrect because you have omitted approximately 15.5% of the hands. You may learn this by studying the results that I posted earlier, because that is the fraction of hands where it was necessary to split the pair to make a straight or flush.

This arising of straights/flushes that split the pairs is important -particularly if you are attempting to divine a player strategy on how to arrange 2 pair hands - because these straights/flushes tend to strengthen the dealer's back hands and weaken the dealer's front hand because they create many front hands that are "best 2 of 3" hi card hands.

Again, the methodology employed in my analysis correctly incorporates straights and flushes that split the pair, including 6,7-card straights and flushes with pairs. My table should be preferred to yours.

3. You appear to be adopting a "power rankings" strategy to assist you in developing a player strategy as to when to split two pair versus when to keep two pair together. I doubt you can develop anything but a very rough strategy by this method; that is a polite way of saying that any strategy you develop is likely to have many details that are incorrect. This is because you will be ignoring the effect of the player's cards on the composition of the deck and that's 15.1% of the fresh deck. How accurate can your analyses be when they ignore that 15.1% of the deck is gone?

Example: Consider TT-55-Q762. Placing TT in the back hand will create a hand that is much weaker than you would expect from a fresh deck analysis because the presence of 55-762 in the players hand will greatly reduce the probability of pairs that TT can beat. Similarly, playing 55 in the front will be more powerful because the removal of TT-Q762 from the deck will eliminate many pairs that will beat 55. Also, the dealer's probability of making a straight will be greatly reduced because every possible 5-card straight must contain a 10 or a 5 and this player hand has removed two 10s and two 5s from the deck.

Why not just write a simulator, and test player 2-pair hands to see which arrangements have higher EV? For TT-55-A762, test TTA76|55 and TT556|A7 and see which is superior? That is likely to be far superior to a Power Rankings approach based on fresh deck analyses.

This post is solely looking further into when the Dealer can place the pair into the lo hand because the six singletons can make a 5-card hand. This gives the counts for the pairs in the Lo Hand.

Whichever set of six cards there are, it will always be the same chances of creating a flush (although some may be straights).

The chances are (a) All six in same suit ABCDEF (4 options) (b) The top five are in the same suit (hence the sixth card isn't) ABCDEf (4 options for the flush suit, and 3 options for the odd card) (c) similarly for ABCDeF ABCdEF ABcDEF AbCDEF aBCDEF.

Depending on the pair there will be a different number of sets-of-six that can form a straight. For instance with an Ace you can make a straight with KQJT9x (7) QJT98x (6) xJT987 (1) JT987x (5) xT9876 (2) T9876x (4) x98765 (3) 98765x (3) x87654 (4) 87654x (2) x76543 (5) 76543x (1) x65432 (6) = 49 (seven ways to make KQJT9 and six of the others) Only AKQJT and A5432 have been excluded.

With a middle card this rules out more of the straights, so with these cards the ways for the six cards to make a straight is less. Therefore with these cards the chances of the Pair going into the Lo Hand reduces.

Starting with the 2 and working up to K and A, the ways to make six-card sets and how many of those form a straight.

Combos: 924 Straight Combos: 49

Combos: 924 Straight Combos: 43

Combos: 924 Straight Combos: 37

Combos: 924 Straight Combos: 31

Combos: 924 Straight Combos: 32

Combos: 924 Straight Combos: 32

Combos: 924 Straight Combos: 32

Combos: 924 Straight Combos: 32

Combos: 924 Straight Combos: 31

Combos: 924 Straight Combos: 37

Combos: 924 Straight Combos: 43

Combos: 924 Straight Combos: 49

Combos: 924 Straight Combos: 49

I sense from your post I'm overlooking something and perhaps not seeing something that is obvious - thanks.

I agree with your points above.about pairs in the middle ranks being split more frequently by straights.

*******************************************

Reviewing your calculations.

I agree that when you have 7 ranks, that any given rank will appear in 924 combos.

I wonder if you are calculating straight combos that do not involve the pair? You mention 'six card sets than can form a straight)' but it is really a 7 rank problem.

7 ranks have 1716 combos. Ignoring suits, the set of 2111111 hands can be represented as 7 versions of these 1716 combos, in which the highest rank is paired, the 2nd highest rank is paired, etc.

Of the 1716 combos :

1501 have no straights (of 5 or more cards)

159 have 5 card straights

48 have 6 card straights

8 have 7 card straights. So, 215 straight hands out of 1716 = 12.519 %

So, this is a higher fraction of hands with straights than you are calculating.

Yes at the moment I am only looking at 2111111 hands which (a) do not have a joker (b) where the pair is put into the low hand, thus the other six create a 5-card hand (c) ignoring (at this stage) the possibility of breaking the pair to form a 5-card hand.Quote:gordonm888... I wonder if you are calculating straight combos that do not involve the pair? You mention 'six card sets than can form a straight)' but it is really a 7 rank problem...

I haven't yet worked out a reasonable way to code this since the objective is to find any 5-card hand which gives the best Lo Hand. I sense I'll be needing to create a sub-routine to analyse the various hands, probably picking two cards for the LoHand and seeing what the other six cards can form. This logic will also be useful when getting to the 8x1 hands. (Perhaps I should have started this way!)

Thus at the moment the only figures which I might have calcualted correctly are where the pair makes it into the low hand and there wasn't a joker involved.

************************************************************

The 3oak hand, 311111, has the interesting property that any flush or straight will be played in the back hand with the trips split to play a pair in the front hand. So, always a pair in the 2-card hand for 311111 hands with a straight/flush.

Pai Gow(cards) is already the perfect "I just wanna sit and not think too hard, drink a couple mixed drinks, flat bet and maybe spike a 5+ hand straight flush occasionally" type game. Maybe you get lucky and spike the rare 7 card straight flush for a large payday once in your life at some point.

In doing this I used the following strategy for arranging the player cards:

- 2 Pair Hands: Always split the pairs, except 3322 to 7722, 4433, 5533 and (8822 with an Ace or Joker kicker), (9922 with an AK kicker).

Quads: Keep 2222 to 7777 and 8888-AKxx together in the back hand, otherwise Split 8888 to AAAA|AAAJoker to put a pair in the front hand.

Decisions on arranging 2PR + Str/Flush and 1PR + Str/Flush were made on a case by case basis keeping in mind that winning at least one hand (2-crad or 5-card) is 3x more important than winning the second hand.

On hands where a Joker is played in the two-card hand, it is considered in the table below to be equivalent to an Ace. Thus, if Joker-K is played in the two-card hand, it shows up in the table below as AK.

Player Hand | Win | Loss | Win Fraction |
---|---|---|---|

AA | 11 | 1.000 | |

KK | 8 | 2 | 0.800 |

16 | 2 | 0.889 | |

JJ | 17 | 1.000 | |

TT | 10 | 1 | 0.909 |

99 | 14 | 4 | 0.778 |

88 | 15 | 1 | 0.938 |

77 | 11 | 4 | 0.733 |

66 | 19 | 9 | 0.679 |

55 | 16 | 9 | 0.640 |

44 | 20 | 11 | 0.645 |

33 | 12 | 12 | 0.500 |

22 | 13 | 14 | 0.481 |

AK | 30 | 37 | 0.448 |

AQ | 27 | 36 | 0.429 |

AJ | 17 | 18 | 0.486 |

AT | 13 | 11 | 0.542 |

A9 | 13 | 10 | 0.565 |

A8 | 5 | 6 | 0.455 |

A7 | 5 | 3 | 0.625 |

A6 | 4 | 0.0 | |

A5 | 1 | 1.000 | |

A4 | 1 | 2 | 0.333 |

A3 | 1 | 2 | 0.333 |

A2 | 1 | 0.000 | |

KQ | 17 | 46 | 0.270 |

KJ | 8 | 26 | 0.235 |

KT | 8 | 25 | 0.242 |

K9 | 12 | 0.0 | |

K8 | 1 | 9 | 0.100 |

K7 | 3 | 1 | 0.750 |

K6 | 3 | 0.0 | |

K5 | |||

K4 | 1 | 2 | 0.333 |

K3 | 2 | 0.0 | |

K2 | 1 | 0.0 | |

QJ | 9 | 37 | 0.196 |

QT | 1 | 37 | 0.026 |

Q9 | 1 | 25 | 0.038 |

Q8 | 1 | 15 | 0.063 |

Q7 | 1 | 6 | 0.143 |

Q6 | 3 | 0.0 | |

Q5 | |||

Q4 | |||

Q3 | 1 | 0.0 | |

Q2 | 4 | 0.0 | |

JT | 3 | 39 | 0.071 |

J9 | 21 | 0.0 | |

J8 | 10 | 0.0 | |

J7 | 1 | 9 | 0.100 |

J6 | 2 | 0.0 | |

J5 | 1 | 1 | 0.500 |

J4 | 1 | 0.0 | |

J3 | 1 | 0.0 | |

J2 | |||

T9 | 21 | 0.0 | |

T8 | 11 | 0.0 | |

T7 | 10 | 0.0 | |

T6 | 3 | 0.0 | |

T5 | |||

T4 | 1 | 0.0 | |

T3 | |||

T2 | 1 | 0.0 | |

98 | 17 | 0.0 | |

97 | 8 | 0.0 | |

96 | 4 | 0.0 | |

95 | 1 | 0.0 | |

94 | 1 | 0.0 | |

93 | 1 | 0.0 | |

92 | 1 | 0.0 | |

87 | 6 | 0.0 | |

86 | 4 | 0.0 | |

85 | 2 | 0.0 | |

84 | 1 | 0.0 | |

83 | 1 | 0.0 | |

82 | 1 | 0.0 | |

76 | 2 | 0.0 | |

75 | 2 | 0.0 | |

74 | |||

73 | 2 | 0.0 | |

72 | 2 | 0.0 | |

65 | 2 | 0.0 | |

64 | 2 | 0.0 | |

63 | 1 | 0.0 | |

62 | |||

54 | 2 | 0.0 | |

53 | 1 | 0.0 | |

52 | 2 | 0.0 | |

43 | |||

42 | 2 | 0.0 | |

32 | 1 | 0.0 | |

Total | 351 | 654 | 0.349 |

There is a lot of scatter in the win fractions because this is only 1000 cards. But clearly, there is very little value/equity in playing a two-card hand lower than JQ. A hand like AK appears to lose more often than it wins, indeed even a 22 pair may lose slightly more often than it wins.

If you have a hand like T98765-2, it may turn out to be better to play it as T9876|52 than 98765|T2, because T2 has near-zero worth, but Dealer straights are so common that a Ten-High Straight may be noticeably better than a 9-High Straight. Not sure until the numbers can be run.

My approach to this was to look at how many ways there are to make eight different ranks, and then determine what is the best straight one might make. If we call the cards A B C D E F G H then, in order what it leaves for the low hand, the straights are D-H, C-G (both leave AB for the lo), then B-F (leaving AG), then AEFGH (5432A) leaving BC, finally ABCDE leaving FG. Determine for each set of eight what is the best lo hand, if any, straight you can make.

For any set of eight different ranks, one can work out which flushes are possible (at this stage ignoring which ones might be straights). Using a binary of eight 0s or 1s, one can work this out for all the possibilities. e.g. 01110110 means you can make a flush leaving AE as the lo hand. Where a straight isn't possible then all these flushes possibilities will create various lo hands. Where a straight is possible, you make the fushes that make better low hands, until you reach the low hand which the straight makes.

Since there are only seven kinds of straight-making hands, you can work out the pattern of totals for these, and then apply them appropriately to each set of eight ranks.

I haven't double-checked this but this is what I got for the "11111111" pattern. I'm hoping to use similar logic for the other patterns, and then look at Jokers.

Hand | |
---|---|

AK | 2 877 312 |

AQ | 2 036 880 |

AJ | 1 368 864 |

AT | 854 992 |

A9 | 484 240 |

A8 | 250 720 |

A7 | 462 720 |

A6 | 614 352 |

A5 | 650 880 |

A4 | 560 928 |

A3 | 344 064 |

KQ | 13 152 180 |

KJ | 8 167 032 |

KT | 3 686 092 |

K9 | 1 391 356 |

K8 | 402 856 |

K7 | 128 880 |

K6 | 349 296 |

K5 | 446 904 |

K4 | 424 728 |

K3 | 275 856 |

QJ | 12 630 432 |

QT | 6 519 892 |

Q9 | 2 303 476 |

Q8 | 560 716 |

Q7 | 118 260 |

Q6 | 91 512 |

Q5 | 253 296 |

Q4 | 295 476 |

Q3 | 211 140 |

JT | 7 725 604 |

J9 | 3 221 140 |

J8 | 723 148 |

J7 | 109 116 |

J6 | 87 912 |

J5 | 62 928 |

J4 | 168 420 |

J3 | 147 540 |

T9 | 3 207 568 |

T8 | 887 704 |

T7 | 101 412 |

T6 | 83 160 |

T5 | 61 956 |

T4 | 42 480 |

T3 | 84 516 |

98 | 702 844 |

97 | 94 068 |

96 | 78 300 |

95 | 59 400 |

94 | 42 336 |

93 | 21 708 |

87 | 378 948 |

86 | 306 852 |

85 | 231 624 |

84 | 158 232 |

83 | 80 196 |

76 | 534 756 |

75 | 403 092 |

74 | 273 264 |

73 | 138 792 |

65 | 572 724 |

64 | 386 352 |

63 | 195 336 |

54 | 496 416 |

53 | 250 368 |

43 | 309 288 |