Pai Gow Poker Dealer Probability calculations (Trump Plaza House Way)

Hello Wizard
I am writing a Pai Gow calculator and have started by implementing the Trump Plaza house way. So far I have successfully reproduced the results you publish on your Wizard of Odds site. After 100 million simulations;

Outcome	Frequency	Probability
Dealer Wins Both	29914654	29.91%
Tie	41473073	41.47%
Player Wins Both	28612273	28.61%

This gives a House Edge (as a player) of --> 2.73%.

However, when I calculate the 2 and 5 card power rating tables, my results are not exactly the same as yours. I can reproduce all but the following in the 5 card table:

Hand	Wizard's Results	My Error
A,A	14832140	-155760
AAA	635504	-848
Straight, 6 high	1407470	-139146
Straight, 7 high	1157246	-614
Straight, 8 high	1093070	-1058
Straight, 9 high	1082354	-1862
Straight, 10 high	1073954	-2666
Straight, J high	1064390	-3470
Straight, Q high	1055156	-308
Straight, K high	1053224	-5168
Straight, A-5	965276	192124
Straight 10-A	1025276	45976
Flush, Q-7 high	1326412	2700
Flush, K high	919220	1080
Flush, A high	2572158	48950
Straight flush	75914	20070

I am guessing that our rules for Straight/Flushes do not match. The Trump Plaza House Way for Straights/Flushes is listed as follows:

• Straights, flushes, straight flushes, and royal flush:
• With no pair: When choosing whether to play a straight, flush, or straight flush play the category which will allow the highest two cards in front.
• With 6th or 7th card: Play the lower straight or flush in the back to place the highest cards in front.
• With one pair: Play pair in the front only if a straight, flush, or straight can be played in the back.
• With two pair: Use two pair rule.
• With three pair: Use three pair rule.
• With three of a kind: Play pair in front.
• With full house: Use full house rule.

There is no mention of what to do when you also have a four of a kind, but I have assumed that you play the four of a kind rule. Also, with three of a kind I have assumed that you only "play pair in front" if the high hand rank remains higher than a pair. Outside of the "regardless rules" (2 pair, 3 pair, full house, 4 of a kind) it would seem that flushes and straights are pretty straight forward; play the straight/flush that gives the highest low hand possible. However, since I am not reproducing your results for straights and flushes, my deduction is likely incorrect.

Q: Are there any "unwritten rules" that I should have taken into account when playing straights and flushes that would help me match your results?

Here are some things to think about:

1) You cannot have a ST/FL/SF/RF with four of a kind
2) You cannot have a ST/FL/SF/RF with three pair or a full house either
3) Having a ST/FL/SF/RF with 3 of a kind will always let you play a ST/FL/SF/RF in the high hand and a pair in the low hand.

Is it possible that you are not correctly ranking the Joker in straights and flushes? I see you have a lot of error with A2345 straights. In Vegas they typically treat A2345 as the second-highest straight, whereas basically everywhere else the normal poker ranking is used (A2345 is the lowest straight). Given that the Trump Plaza is in Atlantic City, I think the Wizard followed normal poker ranking rules for A2345, treating it as the lowest straight. This has an impact on what rank you give the joker in a straight where the four natural cards are 2-3-4-5. (For Vegas rules the Joker should be an Ace; elsewhere it should be a 6.) Also, make sure you are giving the Joker the higher rank with four consecutive cards such as 7-8-9-10 (the Joker should be a Jack, not a 6); except for J-Q-K-A, where it should be the lowest value. And of course if there is a gap, it must assume the missing rank (the 3 in A-2-4-5, the 8 in 5-6-7-9, etc).

With Flushes, the joker should assume the highest rank not already present in the flush (*see note), even if it takes the rank of a card you are playing in your low hand. For example, Joker-A(h)-K(h)-K(s)-J(h)-8(h)-7(h): Play the pair of kings in the low hand, and the A-Joker-J-8-7 flush in the high hand, with the Joker assuming the rank of the King of Hearts even though you have the King of Hearts but are playing it in the low hand.

As far as unwritten rules, it is safe to assume that if one of the 21 splits results in a low hand of equal or greater rank than the other 20 low hands, and a high hand of equal or greater rank than the other 20 high hands, then that is the split the house (or player) will make. Some ST/FL/SF/RF with 3 of a kind hands meet this criteria, such as 2-2-2-3-4-5-6; there's only one way to play (23456/22). Joker-Ace-Ace-K-Q-J-9 does not meet the "one way" criteria since your choices are essentially KQJT9/AA or AKQJT/A9. The KQJT9/AA would likely end up being the play made, but you can't rule out the other way since it makes the best high hand. The Flush example I gave above also meets the "only one way" criteria.

*Note: Some people claim that the Joker can act as a second Ace in a Flush. That might have been the original intent of the game designer, but realistically, no casino would let you get away with that: if the dealer had a natural A-K-Q-8-7 flush in diamonds, and you had Joker-A-K-8-7 in hearts, it would be treated as a copy.

I have treated the A2345 straight as the second highest. This is likely the cause for much of the errors; i will investigate and report back.
I have also allowed the joker to be a second ace in a flush.

In response to your comments 1), 2) and 3).
1) Ah Ad Ac Joker Kd Qd Jd. Is that not a 4 of a kind with a Royal Flush?
2) Ah joker Kh Kd Qh Qd Jh. Is that not a 3 of a kind with a Royal Flush?
3) Joker Ad Ac 7d 4c 3c 2c. If you play 2 aces in the low hand, you break the flush. Correct?

Thanks for your response!

Very good response, my points were all incorrect and yours were correct. Hopefully the info about the A2345 straight will prove useful to you.

It is my understanding that the Wheel being second best is not a Vegas thing but a Pai Gow thing.

That's the rank everywhere I've played.

It was explained this way: First you evaluate the type of hand. Straight. THEN you look at the cards INDIVIDUALLY. Therefore, a straight whose highest card is an ace beats the king high straight, but is second to Broadway because of the king vs five kickers.

I dealt Pai Gow Poker in Canada for many years and the A-5 straight was always 2nd highest after Broadway.
None of the other table poker games used that ranking.

I guess the question is what rank does A2345 have at Trump Plaza?

This is the best I can do for now:

Hand	Wizard's Results	My Error
A,A	14832140	-64014
AAA	635504	-848
Straight, 6 high	1407470	27178
Straight, 7 high	1157246	-614
Straight, 8 high	1093070	-1058
Straight, 9 high	1082354	-1862
Straight, 10 high	1073954	-2666
Straight, J high	1064390	-3470
Straight, Q high	1055156	-308
Straight, K high	1053224	-5168
Straight, A-5	965276	26286
Straight 10-A	1025276	9040
Flush, Q-7 high	1326412	2700
Flush, K high	919220	1080
Flush, A high	2572158	-4888
Straight flush	75914	18612

I'm a bit puzzled...

The Straight Flush entry is off by almost 25%; that's 1 in 4 plays. When I run my program under the debugger I should be able to see any misplays, but I don't. Anyone have any ideas?

Quote: canmathguy

This is the best I can do for now:

Hand	Wizard's Results	My Error
A,A	14832140	-64014
AAA	635504	-848
Straight, 6 high	1407470	27178
Straight, 7 high	1157246	-614
Straight, 8 high	1093070	-1058
Straight, 9 high	1082354	-1862
Straight, 10 high	1073954	-2666
Straight, J high	1064390	-3470
Straight, Q high	1055156	-308
Straight, K high	1053224	-5168
Straight, A-5	965276	26286
Straight 10-A	1025276	9040
Flush, Q-7 high	1326412	2700
Flush, K high	919220	1080
Flush, A high	2572158	-4888
Straight flush	75914	18612

I'm a bit puzzled...

Well, the thing all of those have in common is that they all use the joker, with it defaulting to the ace. So I would guess the joker's being processed incorrectly . Your right-hand column cancels itself out, so I would say the program is ranking the hands wrong internally. I'm not sure why FH/4OAK/5A aren't on your list, though; how are your AA and AAA incorrect, when those are components of those 3 hands? I think possibly you're not truncating correctly for the wraparound effect beyond the joker/ace in either direction, and the joker is being over-weighted in making straights that aren't really there, like somehow being counted to fill 2 gaps.

Probably no news to you on any of that, and it's all I have for you as a non-math paigow player. I did not find the right combination of subsets that cancelled each other out, which might help pinpoint the mis-ranking error if it's there (2 small errors intermixed that, if you found the right combination of + and -, would indicate where the ranking/weighting errors are).

Quote: DJTeddyBear

It is my understanding that the Wheel being second best is not a Vegas thing but a Pai Gow thing.

That's the rank everywhere I've played.

It was explained this way: First you evaluate the type of hand. Straight. THEN you look at the cards INDIVIDUALLY. Therefore, a straight whose highest card is an ace beats the king high straight, but is second to Broadway because of the king vs five kickers.

Reno A-5 straight is ranked lowest.

Reno is a bit in its own world.
Most areas - just about all - the wheel is 2nd highest in Pai Gow poker.
Interestingly, some crap tables in Reno pay 3:1 field on the 2 instead of the 12.

Quote: canmathguy

The Straight Flush entry is off by almost 25%; that's 1 in 4 plays. When I run my program under the debugger I should be able to see any misplays, but I don't. Anyone have any ideas?

Try the following hands:
Joker, Ac,2c,3h,4h,5s,6d
Joker, Ac,2c,3h,4h,6s,6d
Joker, Ac,2c,3h,4h,5s,5d
Joker, Ac,2c,3h,4h,5s,7d
Joker, Ac,2c,3h,4h,4s,5d

Quote: canmathguy

I have treated the A2345 straight as the second highest. This is likely the cause for much of the errors; i will investigate and report back.
I have also allowed the joker to be a second ace in a flush.

In response to your comments 1), 2) and 3).
1) Ah Ad Ac Joker Kd Qd Jd. Is that not a 4 of a kind with a Royal Flush?
2) Ah joker Kh Kd Qh Qd Jh. Is that not a 3 of a kind with a Royal Flush?
3) Joker Ad Ac 7d 4c 3c 2c. If you play 2 aces in the low hand, you break the flush. Correct?

Thanks for your response!

I think your comment about the joker being the second ace in a flush might be the issue in at least some of the problems; it's treated as the highest non-represented card in a flush in the actual game if it's necessary to complete the flush, as an ace if it's not (for win/loss, head to head with the dealer, it's as played; for bonus ranking values, it's irrelevant, because bonuses are paid win or lose) So if a hand contains 5 hearts, joker, ace of spades, the value is flush plus pair; but your program might be using the joker as an ace of hearts for the highest ranking flush, leaving ace of spades/littlest heart to be played up top.

Same thing with a straight; if the joker is not needed to complete a straight, it is given the Ace value and played up top. Exception on both these is when a pair can be played out of the sequence and the joker substituted for either straight or flush value; in that case, the heart and other card of the same rank would come out of the hand for the top, and the joker complete the straight or flush.

However, sometimes in either it has the value of an Ace as well as participating in the flush or straight. And this issue would carry over to straight flushes; they can happen without using the Joker, and also include it; any natural SF up to the 7 card royal would be improved by one rank by putting the joker on the top (wheel bottom) of it, but the hand ranks stronger with the lower ranked SF and an ace to put up top, unless the 7th card makes a pair. If your program is putting the joker on the higher ranking end, rather than playing it as an ace in the 2 card hand in these cases, it could explain the rest of your puzzling results beyond using the joker as a 2nd ace in a flush.

Quote: beachbumbabs

I think your comment about the joker being the second ace in a flush might be the issue in at least some of the problems; it's treated as the highest non-represented card in a flush in the actual game if it's necessary to complete the flush, as an ace if it's not (for win/loss, head to head with the dealer, it's as played; for bonus ranking values, it's irrelevant, because bonuses are paid win or lose) So if a hand contains 5 hearts, joker, ace of spades, the value is flush plus pair; but your program might be using the joker as an ace of hearts for the highest ranking flush, leaving ace of spades/littlest heart to be played up top.

Same thing with a straight; if the joker is not needed to complete a straight, it is given the Ace value and played up top. Exception on both these is when a pair can be played out of the sequence and the joker substituted for either straight or flush value; in that case, the heart and other card of the same rank would come out of the hand for the top, and the joker complete the straight or flush.

However, sometimes in either it has the value of an Ace as well as participating in the flush or straight. And this issue would carry over to straight flushes; they can happen without using the Joker, and also include it; any natural SF up to the 7 card royal would be improved by one rank by putting the joker on the top (wheel bottom) of it, but the hand ranks stronger with the lower ranked SF and an ace to put up top, unless the 7th card makes a pair. If your program is putting the joker on the higher ranking end, rather than playing it as an ace in the 2 card hand in these cases, it could explain the rest of your puzzling results beyond using the joker as a 2nd ace in a flush.

Sorry, this is a mishmash but I'm going to leave it be. In short, I think you have to consider the 7 card hand and strategy in how you best hold it, not just the best 5 card hand that can be made for the bonus. I'm not sure which you're evaluating, because they would be different between whether you're evaluating the game for the base pay or for a bonus pay occurence.