5 card Pai Gow Poker?
March 10th, 2025 at 6:09:56 PM
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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?
March 10th, 2025 at 6:17:47 PM
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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/
https://wizardofodds.com/video/5-card-pai-gow/
June 25th, 2025 at 2:44:24 PM
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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?
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?
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 25th, 2025 at 9:54:34 PM
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Yes. Curious on this also.
Played a little with a friend after seeing this thread and very interested in ranking, rules etc
Played a little with a friend after seeing this thread and very interested in ranking, rules etc
Son you ain’t paying attention
I’m cutting you but you ain’t bleeding
- Foghorn Leghorn
June 26th, 2025 at 11:22:32 AM
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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
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
June 26th, 2025 at 8:19:42 PM
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Quote: zbrownsonI 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
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 28th, 2025 at 12:16:11 PM
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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.
The worst hand dealer can have is JJ-532 (no 3-card flush). It will qualify and dealer will arrange it as J32 | J5.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 28th, 2025 at 12:28:07 PM
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^ 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.
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.
June 28th, 2025 at 6:23:58 PM
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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.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 29th, 2025 at 7:29:44 AM
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Quote: gordonm888This 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.
Son you ain’t paying attention
I’m cutting you but you ain’t bleeding
- Foghorn Leghorn
June 29th, 2025 at 7:55:26 AM
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Quote: linksjunkieQuote: gordonm888This 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.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
June 29th, 2025 at 11:28:00 AM
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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)
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)
June 30th, 2025 at 6:05:52 PM
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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.)
June 30th, 2025 at 7:01:24 PM
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Quote: gordonm888This 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.
June 30th, 2025 at 7:26:18 PM
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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.
July 1st, 2025 at 6:34:31 AM
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Quote: SOOPOOQuote: gordonm888This 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.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 1st, 2025 at 6:45:28 AM
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Quote: charliepatrickI 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.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 1st, 2025 at 6:49:12 AM
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Quote: gordonm888Quote: SOOPOOQuote: gordonm888This 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!
July 1st, 2025 at 11:09:53 AM
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^ 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.
July 1st, 2025 at 2:44:56 PM
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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.
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% |
July 1st, 2025 at 2:50:23 PM
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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
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
Last edited by: gordonm888 on Jul 1, 2025
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 1st, 2025 at 3:35:02 PM
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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.
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.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 2nd, 2025 at 3:47:34 PM
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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 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% |
July 5th, 2025 at 6:57:36 PM
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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.
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.
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.
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.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 6th, 2025 at 10:28:59 AM
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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:
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.
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
AsKsQs | AhAd
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:
AsKsQs | AhAd___ EV = +0.62600664
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
AsKsQs | AhAd___ EV = +0.62600664
Ts9s8s | AsAd___ EV = +0.729019211
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.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 9th, 2025 at 7:16:20 AM
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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.
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.
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.
6666-K --> 66K | 66 or 666| K6
KKKK-T--> KKT | KK or KKK | KT
KKKK-T--> KKT | KK or KKK | KT
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 |
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 12th, 2025 at 12:38:58 PM
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Sorry for the long post.
My first approach was to categorize hands by Quads, Full Houses etc. Eventually, since it didn't take that long, I just used brute force such as do I=1 to 48, do j=i+1 to 49... do m=l+1 to 52; and after finding a bug with Full Houses, they agreed.
Within this loop I look at all ten ways to set the Dealer's hands, setting lo1,lo2 hi1,hi2,hi3 mazimizing the low hand while keeping it valid.
Here's the tail end of my results for AKQs AhAd, showing how setting the Player's hand fares. (0) uses the best option having seen dealer's hand - in this case it's always best to set AA/AKQs! Note I have assumed the 62 ways of Dealer having Royal Flush, which leads to Tie/Win counts as a Tie rather than a Win. (It's easy to change the code!)
I've only been looking at this hand and have been trying for several days to get to your figure. However, having tried three different methods, I get a higher figure. I did play around and wondered whether you had considered a hand, such as Js 9h 7h 5d 3h where there is a 973 flush meaning the J5 qualifies, as this seemed to find quite a number of qualifying hands, typically where there is one-high card which can go into the Low hand due to a flush with three lower cards, and accounts for most, but not all, of the difference. (I guess there's similar logic with something like 66653 where 653 is a flush.)Quote: gordonm888...AsKsQs | AhAd___ EV = +0.62600664
My first approach was to categorize hands by Quads, Full Houses etc. Eventually, since it didn't take that long, I just used brute force such as do I=1 to 48, do j=i+1 to 49... do m=l+1 to 52; and after finding a bug with Full Houses, they agreed.
Within this loop I look at all ten ways to set the Dealer's hands, setting lo1,lo2 hi1,hi2,hi3 mazimizing the low hand while keeping it valid.
/* Evaluate Low Hand */
thislow=n2*drank[lo1]+n1*drank[lo2];
if (drank[lo1]==drank[lo2]) {thislow=pair+n1*drank[lo1];};
/* Evaluate High Hand */
thishigh=n2*drank[hi1]+n1*drank[hi2]+drank[hi3];
if (drank[hi1]==drank[hi2]) {thishigh=pair+n1*drank[hi1]+drank[hi3];};
if (drank[hi2]==drank[hi3]) {thishigh=pair+n1*drank[hi2]+drank[hi1];};
if ((drank[hi1]==drank[hi2])&&(drank[hi2]==drank[hi3])) {thishigh=trips+drank[hi3];};
if ((drank[hi1]==drank[hi2]+1)&&(drank[hi2]==drank[hi3]+1)) {thishigh=straight+drank[hi1];};
if ((drank[hi1]==13)&&(drank[hi2]==2)&&(drank[hi3]==1)) {thishigh=straight+1;};
if ((dsuit[hi1]==dsuit[hi2])&&(dsuit[hi2]==dsuit[hi3])) {thishigh+=flush;};
/* Get rid of this if it's invalid */
if (thislow>thishigh) {thislow=0; thishigh=0;};
/* Now see if this is the best low hand so far, no other checking needed */
if (thislow>bestlo) {bestlo=thislow; besthi=thishigh;};
Here's the tail end of my results for AKQs AhAd, showing how setting the Player's hand fares. (0) uses the best option having seen dealer's hand - in this case it's always best to set AA/AKQs! Note I have assumed the 62 ways of Dealer having Royal Flush, which leads to Tie/Win counts as a Tie rather than a Win. (It's easy to change the code!)
Hi: (5000013) 62
Lo: (40200) 156
Lo: (40300) 876
Lo: (50200) 804
Lo: (50300) 2040
Lo: (50400) 8788
Lo: (60100) 912
Lo: (60200) 1500
Lo: (60300) 3384
Lo: (60400) 14932
Lo: (60500) 20680
Lo: (70100) 1968
Lo: (70200) 3156
Lo: (70300) 4824
Lo: (70400) 15436
Lo: (70500) 29404
Lo: (70600) 34168
Lo: (80100) 3216
Lo: (80200) 5004
Lo: (80300) 7272
Lo: (80400) 16084
Lo: (80500) 28300
Lo: (80600) 43072
Lo: (80700) 46868
Lo: (90100) 4704
Lo: (90200) 7092
Lo: (90300) 9960
Lo: (90400) 17788
Lo: (90500) 27388
Lo: (90600) 39592
Lo: (90700) 53552
Lo: (90800) 56396
Lo: (100100) 6480
Lo: (100200) 9468
Lo: (100300) 12936
Lo: (100400) 19780
Lo: (100500) 27580
Lo: (100600) 36352
Lo: (100700) 46928
Lo: (100800) 58460
Lo: (100900) 60368
Lo: (110100) 6444
Lo: (110200) 8542
Lo: (110300) 11081
Lo: (110400) 15414
Lo: (110500) 20158
Lo: (110600) 25325
Lo: (110700) 30927
Lo: (110800) 37588
Lo: (110900) 44672
Lo: (111000) 46061
Lo: (120100) 7848
Lo: (120200) 10165
Lo: (120300) 12923
Lo: (120400) 16584
Lo: (120500) 20197
Lo: (120600) 23774
Lo: (120700) 27327
Lo: (120800) 31021
Lo: (120900) 35165
Lo: (121000) 39444
Lo: (121100) 30270
Lo: (130100) 3123
Lo: (130200) 2829
Lo: (130300) 3996
Lo: (130400) 4963
Lo: (130500) 5884
Lo: (130600) 6844
Lo: (130700) 7847
Lo: (130800) 8948
Lo: (130900) 10139
Lo: (131000) 11533
Lo: (131100) 9682
Lo: (131200) 10055
Lo: (1000100) 21888
Lo: (1000200) 20176
Lo: (1000300) 18541
Lo: (1000400) 17137
Lo: (1000500) 15733
Lo: (1000600) 14329
Lo: (1000700) 12925
Lo: (1000800) 11521
Lo: (1000900) 10194
Lo: (1001000) 8927
Lo: (1001100) 4187
Lo: (1001200) 3940
Totals : 1533939 HiH: 5000013 LoH: 50401
Hand: 0 W: 1024561 T: 62 L: 0 D: 509316 EV: 0.6679281249123987
Hand: 1 W: 573621 T: 366128 L: 84874 D: 509316 EV: 0.31862218771411377
Hand: 2 W: 567390 T: 368908 L: 88325 D: 509316 EV: 0.312310333070611
Hand: 3 W: 1004977 T: 19646 L: 0 D: 509316 EV: 0.6551609940160593
Hand: 4 W: 1004977 T: 19646 L: 0 D: 509316 EV: 0.6551609940160593
Hand: 5 W: 742103 T: 279079 L: 3441 D: 509316 EV: 0.48154587633536927
Hand: 6 W: 573621 T: 366128 L: 84874 D: 509316 EV: 0.31862218771411377
Hand: 7 W: 573621 T: 366128 L: 84874 D: 509316 EV: 0.31862218771411377
Hand: 8 W: 567390 T: 368908 L: 88325 D: 509316 EV: 0.312310333070611
Hand: 9 W: 567390 T: 368908 L: 88325 D: 509316 EV: 0.312310333070611
Hand: 10 W: 1024561 T: 62 L: 0 D: 509316 EV: 0.6679281249123987
Totals (d) : 1533939
July 12th, 2025 at 8:41:45 PM
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Charlie:
Thanks for checking me.
I certainly intended to rank the dealer no pair hands with a flush the way that you indicate. I'll check my spreadsheet and see if I made such an error.
Thanks for checking me.
I certainly intended to rank the dealer no pair hands with a flush the way that you indicate. I'll check my spreadsheet and see if I made such an error.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 13th, 2025 at 2:28:18 AM
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I certainly have had the most problems with five different ranks, hence it's quite possible I'm wrong!!
As you know with ABCDE, if there's no better solution, it will be turned into BC with ADE. So the calculations have initially to look at whether CDE can form a flush or straight, enabling AB to be the low hand. Then this is repeated for AC AD AE before accepting BC is the best low. I encountered some difficulty with flushes, for instance for (AC)BDE to be a flush (AB)CDE musn't have been, etc. Also where there's a straight, that mops up all the non-flushed hands as well.
I haven't yet tried a spreadsheet approach, so I might look at that.
As you know with ABCDE, if there's no better solution, it will be turned into BC with ADE. So the calculations have initially to look at whether CDE can form a flush or straight, enabling AB to be the low hand. Then this is repeated for AC AD AE before accepting BC is the best low. I encountered some difficulty with flushes, for instance for (AC)BDE to be a flush (AB)CDE musn't have been, etc. Also where there's a straight, that mops up all the non-flushed hands as well.
I haven't yet tried a spreadsheet approach, so I might look at that.
July 13th, 2025 at 5:32:56 AM
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For ABCDE hands (i.e., 5 ranks or no pairs), if A is a ten or lower I don't evaluate for straights or flushes, I just score the hand as a DNQ - as a push.
For a one pair hand in which no card is higher than a ten and for which the singletons do not form a straight nor flush, I also score it as a DNQ without arranging it into high and low hands.
Similarly, for a 3oak hand in which no card is higher than a ten and which does not contain a straight nor flush, I also score it as a DNQ without arranging it into high and low hands.
So its not straightforward to compare it to your numbers, I would have to bypass these calculational short-cuts and designate the low hand for those hand categories. But I might do that.
******************************************
I did a lot of error checking last night and found a number of errors that affect small numbers of hands, but the EV for (AKQ)s-AA is still a bit higher than 0.62.
For a one pair hand in which no card is higher than a ten and for which the singletons do not form a straight nor flush, I also score it as a DNQ without arranging it into high and low hands.
Similarly, for a 3oak hand in which no card is higher than a ten and which does not contain a straight nor flush, I also score it as a DNQ without arranging it into high and low hands.
So its not straightforward to compare it to your numbers, I would have to bypass these calculational short-cuts and designate the low hand for those hand categories. But I might do that.
******************************************
I did a lot of error checking last night and found a number of errors that affect small numbers of hands, but the EV for (AKQ)s-AA is still a bit higher than 0.62.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 13th, 2025 at 6:52:11 AM
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Here are my results for the hand AsKsQs-AA. These dealer hand categories are defined by the degree of pairing; the various flushes and straights are subcategories at a lower hierarchal level.
You'll notice that after correcting errors I am getting a somewhat lower EV; i.e. a somewhat higher DNQ fraction. The dealer hand categories "2 pair", "Boat" and "Quads" always qualify and thus (AKQ)s-AA hand has an EV =1 for those.
| Dealer Hand Category | Total Comb | Return | Player EV |
|---|---|---|---|
| No Pair | 756048 | 424344 | 0.561265951 |
| 1 Pair | 661500 | 422937 | 0.639360544 |
| 2 Pair | 77949 | 77949 | 1 |
| 3oaK | 35486 | 23472 | 0.66144395 |
| Boat | 2526 | 2526 | 1 |
| Quads | 430 | 430 | 1 |
| Total | 1533939 | 951658 | 0.620401463 |
You'll notice that after correcting errors I am getting a somewhat lower EV; i.e. a somewhat higher DNQ fraction. The dealer hand categories "2 pair", "Boat" and "Quads" always qualify and thus (AKQ)s-AA hand has an EV =1 for those.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 13th, 2025 at 8:10:47 AM
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I've created a spreadsheet only worrying about the Lo Hand for the No Pair hands, (I hope to move onto the other types later). btw I get 425064 hands where the Dealer qualifies, which is 720 more than yours. My guess is these need to be hands where the highest card ={AKQJ} but the second one is 10-2. For instance AT97x there are 12 ways to get T97 flush, four suits and three ways x<>suit. (97x 16 ways, T7x 12 ways, T9x 12 ways, A7x I don't care whether it's flushed or not) as the lo hand is T9.
In fact I've juist realised, if the two highest cards are in {AKQJ} the hand will qualify, if the two highest cards are in (10-2) then the hand can't qualify, so technically only the middle ones are worth worrying about!
So I looked into this approach, then organise the "middle ones" where there is a low or mid straight X8654, X9873, X6543. Then the rest can only qualify if there's a flush in the lowest four cards, and the ways to do this depend on whether the highest card is an Ace (only one around), K/Q (three around) J(four around).
With hands such as Axxxx Kxxxx Qxxxx Jxxxx there are 1,3,3,4 of the high card and 4*4*4*4 of the low card. The suit of the high card is irrelevant as it's needed for the Low hand, so the flushes are CDE (16 ways) BDE (12 ways) BCE (12 ways) BCD (12 ways) i.e. 52 of the 256 ways for each perm.
In fact I've juist realised, if the two highest cards are in {AKQJ} the hand will qualify, if the two highest cards are in (10-2) then the hand can't qualify, so technically only the middle ones are worth worrying about!
So I looked into this approach, then organise the "middle ones" where there is a low or mid straight X8654, X9873, X6543. Then the rest can only qualify if there's a flush in the lowest four cards, and the ways to do this depend on whether the highest card is an Ace (only one around), K/Q (three around) J(four around).
With hands such as Axxxx Kxxxx Qxxxx Jxxxx there are 1,3,3,4 of the high card and 4*4*4*4 of the low card. The suit of the high card is irrelevant as it's needed for the Low hand, so the flushes are CDE (16 ways) BDE (12 ways) BCE (12 ways) BCD (12 ways) i.e. 52 of the 256 ways for each perm.
| Qualify | DNQ | ||
| Two High | 272 208 | ||
| Lo Str | 59 136 | ||
| High Str | 42 240 | ||
| Ax | 23 040 | 4 680 | 18 360 |
| Kx | 69 120 | 14 040 | 55 080 |
| Qx | 69 120 | 14 040 | 55 080 |
| Jx | 92 160 | 18 720 | 73 440 |
| Two Low | 129 024 | ||
| Totals | 425 064 | 330 984 |
July 13th, 2025 at 11:06:33 AM
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Quote: charliepatrickI've created a spreadsheet only worrying about the Lo Hand for the No Pair hands, (I hope to move onto the other types later). btw I get 425064 hands where the Dealer qualifies, which is 720 more than yours. My guess is these need to be hands where the highest card ={AKQJ} but the second one is 10-2. For instance AT97x there are 12 ways to get T97 flush, four suits and three ways x<>suit. (97x 16 ways, T7x 12 ways, T9x 12 ways, A7x I don't care whether it's flushed or not) as the lo hand is T9.
In fact I've juist realised, if the two highest cards are in {AKQJ} the hand will qualify, if the two highest cards are in (10-2) then the hand can't qualify, so technically only the middle ones are worth worrying about!
So I looked into this approach, then organise the "middle ones" where there is a low or mid straight X8654, X9873, X6543. Then the rest can only qualify if there's a flush in the lowest four cards, and the ways to do this depend on whether the highest card is an Ace (only one around), K/Q (three around) J(four around).
With hands such as Axxxx Kxxxx Qxxxx Jxxxx there are 1,3,3,4 of the high card and 4*4*4*4 of the low card. The suit of the high card is irrelevant as it's needed for the Low hand, so the flushes are CDE (16 ways) BDE (12 ways) BCE (12 ways) BCD (12 ways) i.e. 52 of the 256 ways for each perm.
Qualify DNQ Two High Lo Str High Str Ax Kx Qx Jx Two Low Totals
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I'm struggling with your definitions, but I think this is what I get by interrogating my worksheet for ABCDE hands:
At least two high cards = 271760
Low straight (CDE) = 59392
High Straight (BCD) = 44,608
Those values for straights don't include 4 and 5 card straights, just 3 card straights.
However, there is double counting. Hands with 2 or more high cards that have a CDE or BCD straight are included in in both Two High and in one of the straight categories.
If I understand correctly what Ax and Kx mean (Ax = No pair, A high + 4 singletons that are 10 or less, no straights except for ABC (AKQ) and DEA (32A) straights which are included because they don't allow the Ace to be arranged in the 2-card hand), then I get:
Ax Qualify 4770 DNQ 18270
Kx Qualify 13770 DNQ 55350
Once again, I separately partition hands with 4 card and 5 cards straights (as well as 4 card and 5 card flushes), so that some of these discrepancies may be caused by different terminology or definitions.
For two low, I get: 129024. Which agrees!
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 13th, 2025 at 12:06:11 PM
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Here's the numbers for Two (or more) High.
Middle Hands which only have one of AKQJ. They only qualify if that card can go into the Low Hand.
Part 1 : see whether a straight can be made with the lowest four cards.
Once I've ignored the 2-High and 2-Low hands the "LoStr" is where CDE makes a straight (since the low hand can't be better than AB). "HiStr" is where BCD makes a straight (but CDE didn't). Note with middle hands, once either straight can be made, the hand qualifies since the Low Hand can use the highest card. Note that it is impossible for an A23 straight to be interesting as it would be ADE leaving BC in the low, which, for middle hands, doesn't qualify.
Part 2 : After removing 2-high, 2-low, Straight possible, this leaves hands of Axxxx Kxxxx Qxxxx or Jxxxx where x are all low cards, and in this case all four suits available. In hindsight it doesn't matter what the High card is since it needs to go into the Low Hand to qualify.
For each high-card (A,K,Q or J) and the other four ranks, there are 256 ways for the suits of the low cards. If any three of the low cards are the same suit, this creates a flush, allowing the High card to be used in the Low Hand.
For any given suit (y) and the other card's suit (x) there are the following five ways for a flush to occur
(i) yyyy - 4 ways (i.e. each suit)
(ii) xyyy - 12 ways (each suit for y, three other suits for x)
(iii) yxyy 12 ways
(iv) yyxy 12 ways
(v) yyyx 12 ways
4+12+12+12+12 = 52.
Therefore 52/256 of these hands can create a flush using the lowest four cards, so can qualify, and 204/256 won't qualify.
| A K Q J x | 1 | 3 | 3 | 4 | 36 | 1 296 | |
| A K Q x x | 1 | 3 | 3 | 36 | 32 | / 2 | 5 184 |
| A K J x x | 1 | 3 | 4 | 36 | 32 | / 2 | 6 912 |
| A Q J x x | 1 | 3 | 4 | 36 | 32 | / 2 | 6 912 |
| A K x x x | 1 | 3 | 36 | 32 | 28 | / 6 | 16 128 |
| A Q x x x | 1 | 3 | 36 | 32 | 28 | / 6 | 16 128 |
| A J x x x | 1 | 4 | 36 | 32 | 28 | / 6 | 21 504 |
| K Q J x x | 3 | 3 | 4 | 36 | 32 | / 2 | 20 736 |
| K Q x x x | 3 | 3 | 36 | 32 | 28 | / 6 | 48 384 |
| K J x x x | 3 | 4 | 36 | 32 | 28 | / 6 | 64 512 |
| Q J x x x | 3 | 4 | 36 | 32 | 28 | / 6 | 64 512 |
| Totals | 272 208 |
Middle Hands which only have one of AKQJ. They only qualify if that card can go into the Low Hand.
Part 1 : see whether a straight can be made with the lowest four cards.
Once I've ignored the 2-High and 2-Low hands the "LoStr" is where CDE makes a straight (since the low hand can't be better than AB). "HiStr" is where BCD makes a straight (but CDE didn't). Note with middle hands, once either straight can be made, the hand qualifies since the Low Hand can use the highest card. Note that it is impossible for an A23 straight to be interesting as it would be ADE leaving BC in the low, which, for middle hands, doesn't qualify.
Part 2 : After removing 2-high, 2-low, Straight possible, this leaves hands of Axxxx Kxxxx Qxxxx or Jxxxx where x are all low cards, and in this case all four suits available. In hindsight it doesn't matter what the High card is since it needs to go into the Low Hand to qualify.
For each high-card (A,K,Q or J) and the other four ranks, there are 256 ways for the suits of the low cards. If any three of the low cards are the same suit, this creates a flush, allowing the High card to be used in the Low Hand.
For any given suit (y) and the other card's suit (x) there are the following five ways for a flush to occur
(i) yyyy - 4 ways (i.e. each suit)
(ii) xyyy - 12 ways (each suit for y, three other suits for x)
(iii) yxyy 12 ways
(iv) yyxy 12 ways
(v) yyyx 12 ways
4+12+12+12+12 = 52.
Therefore 52/256 of these hands can create a flush using the lowest four cards, so can qualify, and 204/256 won't qualify.
July 13th, 2025 at 1:57:06 PM
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Using similar logic here are the Middle Hands which can make a straight. Note the hands consist of one of the eleven High cards and four cards 10-2. As an example the 432 straight can have each of those in four suits, then any card from 5 thru 10, then any of 11 High cards. When it comes for BCD forming a straight, E cannot be one less than D (as these have been counted, e.g J5432).
| 432 + High + (10-5) | 4 | 4 | 4 | 11 | 24 | 16 896 |
| 543 + High + (10-6) | 4 | 4 | 4 | 11 | 20 | 14 080 |
| 654 + High + (10-7) | 4 | 4 | 4 | 11 | 16 | 11 264 |
| 765 + High + (10-8) | 4 | 4 | 4 | 11 | 12 | 8 448 |
| 876 + High + (10-9) | 4 | 4 | 4 | 11 | 8 | 5 632 |
| 987 + High + T | 4 | 4 | 4 | 11 | 4 | 2 816 |
| Totals - Low straight | 59 136 | |||||
| 654+ High + 2 | 4 | 4 | 4 | 11 | 4 | 2 816 |
| 765+ High + (3-2) | 4 | 4 | 4 | 11 | 8 | 5 632 |
| 876+ High + (4-2) | 4 | 4 | 4 | 11 | 12 | 8 448 |
| 987+ High + (5-2) | 4 | 4 | 4 | 11 | 16 | 11 264 |
| T98+ High + (6-2) | 4 | 4 | 4 | 11 | 20 | 14 080 |
| Totals | 42 240 |
July 15th, 2025 at 10:56:26 AM
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Charlie,
I follow your logic.
A Long-winded Caveat
For 5 card hands with 3 card poker hands a factor, I divide and reorder the 1287 no pair hands into 11 different categories of straights:
1x "No straights"
2x 5 card straights: ABCDE; BCDEA (specifically A5432)
3 x 4 card straights: ABCD; BCDE; and CDEA (Ax432)
4 x 3 card straights: ABC; BCD; CDE; and DEA (Axx32)
1x Two 3-card straights, specifically AKQ32
I then take each of the 1287 ABCDE hands in the 11 straight bins and subdivide them (and their combinations) into 16 Flush hands:
1x No flush
1x 5-card flush
5x4 card flush, ABCD; ABCE; ABDE; ACDE; and BCDE
10x 3card flush: ABC; ABD; ABE; ACD; ACE; ADE; BCD; BCE; BDE; CDE
In order to efficiently calculate composition-dependent probabilities/combinations I array these 1287*16 hand categories not as a vertical list but as a matrix with Ranks in the row labels and flush categories in the column labels. And I will have one such matrix for combinations; a 2nd such matrix for bottom hand score, a 3rd matrix for top hand score; etc.
So with the ranks reordered and sub-divided into 11 straight categories, and with an over-all matrix of 1287 *16 hands for ABCDE hands its actually difficult for me to search for the probabilities of hands the way you define them. So the numbers I reported for comparison were not rigorous results that my spreadsheet was programmed to calculate. Example: For number of combinations of hands with 2 or more high cards I simply scanned the 11 sub matrixes containing a total of > 20000 hand categories and manually added up the number of combinations in the probability matrix based on the hand definitions. So, my initial suspicion is that, where I disagree with you it is because I made errors when manually scanning my matrices and compiling the combinations.
I've been busy, but I'll go back and work these comparisons by generating more rigorous results for comparison with your numbers and trying to track where our disagreement is arising from.
I follow your logic.
A Long-winded Caveat
For 5 card hands with 3 card poker hands a factor, I divide and reorder the 1287 no pair hands into 11 different categories of straights:
1x "No straights"
2x 5 card straights: ABCDE; BCDEA (specifically A5432)
3 x 4 card straights: ABCD; BCDE; and CDEA (Ax432)
4 x 3 card straights: ABC; BCD; CDE; and DEA (Axx32)
1x Two 3-card straights, specifically AKQ32
I then take each of the 1287 ABCDE hands in the 11 straight bins and subdivide them (and their combinations) into 16 Flush hands:
1x No flush
1x 5-card flush
5x4 card flush, ABCD; ABCE; ABDE; ACDE; and BCDE
10x 3card flush: ABC; ABD; ABE; ACD; ACE; ADE; BCD; BCE; BDE; CDE
In order to efficiently calculate composition-dependent probabilities/combinations I array these 1287*16 hand categories not as a vertical list but as a matrix with Ranks in the row labels and flush categories in the column labels. And I will have one such matrix for combinations; a 2nd such matrix for bottom hand score, a 3rd matrix for top hand score; etc.
So with the ranks reordered and sub-divided into 11 straight categories, and with an over-all matrix of 1287 *16 hands for ABCDE hands its actually difficult for me to search for the probabilities of hands the way you define them. So the numbers I reported for comparison were not rigorous results that my spreadsheet was programmed to calculate. Example: For number of combinations of hands with 2 or more high cards I simply scanned the 11 sub matrixes containing a total of > 20000 hand categories and manually added up the number of combinations in the probability matrix based on the hand definitions. So, my initial suspicion is that, where I disagree with you it is because I made errors when manually scanning my matrices and compiling the combinations.
I've been busy, but I'll go back and work these comparisons by generating more rigorous results for comparison with your numbers and trying to track where our disagreement is arising from.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 16th, 2025 at 5:04:18 AM
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As I already have the 1287 combinations for ABCDE I thought it would be easy to assigned them to the straight categories you have. However some of them overlap, so I found I was originally double-counting some of them, e.g. DEA needs to omit CDEA or AKQ32.
July 16th, 2025 at 5:36:56 AM
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Quote: charliepatrickUsing similar logic here are the Middle Hands which can make a straight. Note the hands consist of one of the eleven High cards and four cards 10-2. As an example the 432 straight can have each of those in four suits, then any card from 5 thru 10, then any of 11 High cards. When it comes for BCD forming a straight, E cannot be one less than D (as these have been counted, e.g J5432).
432 + High + (10-5) 543 + High + (10-6) 654 + High + (10-7) 765 + High + (10-8) 876 + High + (10-9) 987 + High + T Totals - Low straight 654+ High + 2 765+ High + (3-2) 876+ High + (4-2) 987+ High + (5-2) T98+ High + (6-2) Totals
link to original post
Okay, I have been able to confirm that my spreadsheet agrees with these numbers. For hands with one high card and a low straight, I get 59136. For hands with one high card and the middle straight I get 42240.
Part of my initial confusion was that these hands must have only one high card. I think I am now tabulating this correctly according to your methodology
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 16th, 2025 at 8:39:42 AM
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Quote: charliepatrickHere's the numbers for Two (or more) High.
A K Q J x A K Q x x A K J x x A Q J x x A K x x x A Q x x x A J x x x K Q J x x K Q x x x K J x x x Q J x x x Totals
link to original post
I have now confirmed my spreadsheet agrees with the table above.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 16th, 2025 at 8:48:34 AM
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Quote: charliepatrick
Middle Hands[/uwhich only have one of AKQJ. They only qualify if that card can go into the Low Hand.
Part 1 : see whether a straight can be made with the lowest four cards.
Once I've ignored the 2-High and 2-Low hands the "LoStr" is where CDE makes a straight (since the low hand can't be better than AB). "HiStr" is where BCD makes a straight (but CDE didn't). Note with middle hands, once either straight can be made, the hand qualifies since the Low Hand can use the highest card. Note that it is impossible for an A23 straight to be interesting as it would be ADE leaving BC in the low, which, for middle hands, doesn't qualify.
Part 2 : After removing 2-high, 2-low, Straight possible, this leaves hands of Axxxx Kxxxx Qxxxx or Jxxxx where x are all low cards, and in this case all four suits available. In hindsight it doesn't matter what the High card is since it needs to go into the Low Hand to qualify.
For each high-card (A,K,Q or J) and the other four ranks, there are 256 ways for the suits of the low cards. If any three of the low cards are the same suit, this creates a flush, allowing the High card to be used in the Low Hand.
For any given suit (y) and the other card's suit (x) there are the following five ways for a flush to occur
(i) yyyy - 4 ways (i.e. each suit)
(ii) xyyy - 12 ways (each suit for y, three other suits for x)
(iii) yxyy 12 ways
(iv) yyxy 12 ways
(v) yyyx 12 ways
4+12+12+12+12 = 52.
Therefore 52/256 of these hands can create a flush using the lowest four cards, so can qualify, and 204/256 won't qualify.
link to original post
My spreadsheet agrees with the 52/256 and 204/256. Remember that with a 13 the combinations 52 and 204 have to multiplied by 3 and with a 12 & 11 they need to be multiplied by 4.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 16th, 2025 at 9:07:43 AM
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^ I suspect we agree, although in my spreadhseet I use 13 is an Ace ... 10=J, 9="10" ... 1="2") 13 x1, 12 11 x3, 10-1 x 4. So as you say K7642 or Q7642 are three times more likely than A7642, and J7642 is four times. Similar logic applies to say AK742 except you can only make Kxx flushes in H,D and C, as Ks has gone, so has a different result than QJ742 where all four Jxx flushes can happen.
July 16th, 2025 at 10:00:42 AM
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I found a minor error; 4 ABCDE hands were miscategorized. My answers shifted in the 3rd or 4th digit to:
My No Pair calculations now agree precisely with yours. Your numbers are equivalent to an EV(NoPair) = 425064/756048 = 0.562218272, just as mine are.
My Total EV has changed from 0.6204 to 0.6209. However, still not in agreement with your 0.66.
I am reviewing my 2-pair worksheet and I'm going to be revising some of my calculations for hands with 3-card flushes. If I have an error, that may be where it is.
| Hand Category | Total Comb | Return | EV |
|---|---|---|---|
| No Pair | 756048 | 425064 | 0.562218272 |
| 1 Pair | 661500 | 422937 | 0.639360544 |
| 2 Pair | 77949 | 77949 | 1 |
| 3oaK | 35486 | 23472 | 0.66144395 |
| Boat | 2526 | 2526 | 1 |
| Quads | 430 | 430 | 1 |
| Total | 1533939 | 952378 | 0.620870843 |
My No Pair calculations now agree precisely with yours. Your numbers are equivalent to an EV(NoPair) = 425064/756048 = 0.562218272, just as mine are.
My Total EV has changed from 0.6204 to 0.6209. However, still not in agreement with your 0.66.
I am reviewing my 2-pair worksheet and I'm going to be revising some of my calculations for hands with 3-card flushes. If I have an error, that may be where it is.
Last edited by: gordonm888 on Jul 16, 2025
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 16th, 2025 at 12:28:54 PM
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Using your logic of needing a high card here's an analysis of definite qualifying hands and this leaves Small pair plus three small cards which might be flushes or straighta. In fact if the highest of the five cards is an AKQJ then the hand must qualify.
Thus you can see at least 467964 should qualify.
Thus you can see at least 467964 should qualify.
| KK AQJ | 3 | 1 | 3 | 4 | 36 | |
| KK AQx | 3 | 1 | 3 | 36 | 324 | |
| KK AJx | 3 | 1 | 4 | 36 | 432 | |
| KK Axx | 3 | 1 | 36 | 32 | / 2 | 1 728 |
| KK QJx | 3 | 3 | 4 | 36 | 1 296 | |
| KK Qxx | 3 | 3 | 36 | 32 | / 2 | 5 184 |
| KK Jxx | 3 | 4 | 36 | 32 | / 2 | 6 912 |
| KK xxx | 3 | 36 | 32 | 28 | / 6 | 16 128 |
| QQ AKJ | 3 | 1 | 3 | 4 | 36 | |
| QQ AKx | 3 | 1 | 3 | 36 | 324 | |
| QQ AJx | 3 | 1 | 4 | 36 | 432 | |
| QQ Axx | 3 | 1 | 36 | 32 | / 2 | 1 728 |
| QQ KJx | 3 | 3 | 4 | 36 | 1 296 | |
| QQ Kxx | 3 | 3 | 36 | 32 | / 2 | 5 184 |
| QQ Jxx | 3 | 4 | 36 | 32 | / 2 | 6 912 |
| QQ xxx | 3 | 36 | 32 | 28 | / 6 | 16 128 |
| JJ AKQ | 6 | 1 | 3 | 3 | 54 | |
| JJ AKx | 6 | 1 | 3 | 36 | 648 | |
| JJ AQx | 6 | 1 | 3 | 36 | 648 | |
| JJ Axx | 6 | 1 | 36 | 32 | / 2 | 3 456 |
| JJ KQx | 6 | 3 | 3 | 36 | 1 944 | |
| JJ Kxx | 6 | 3 | 36 | 32 | / 2 | 10 368 |
| JJ Qxx | 6 | 3 | 36 | 32 | / 2 | 10 368 |
| JJ xxx | 6 | 36 | 32 | 28 | / 6 | 32 256 |
| XX AKQ | 54 | 1 | 3 | 3 | 486 | |
| XX AKJ | 54 | 1 | 3 | 4 | 648 | |
| XX AKx | 54 | 1 | 3 | 32 | 5 184 | |
| XX AQJ | 54 | 1 | 3 | 4 | 648 | |
| XX AQx | 54 | 1 | 3 | 32 | 5 184 | |
| XX AJx | 54 | 1 | 4 | 32 | 6 912 | |
| XX Axx | 54 | 1 | 32 | 28 | / 2 | 24 192 |
| XX KQJ | 54 | 3 | 3 | 4 | 1 944 | |
| XX KQx | 54 | 3 | 3 | 32 | 15 552 | |
| XX KJx | 54 | 3 | 4 | 32 | 20 736 | |
| XX Kxx | 54 | 3 | 32 | 28 | / 2 | 72 576 |
| XX QJx | 54 | 3 | 4 | 32 | 20 736 | |
| XX Qxx | 54 | 3 | 32 | 28 | / 2 | 72 576 |
| XX Jxx | 54 | 4 | 32 | 28 | / 2 | 96 768 |
467 964 | ||||||
| Might qualify | ||||||
| XX xxx | 54 | 32 | 28 | 24 | / 6 | 193 536 |
| Totals | 661 500 |
July 16th, 2025 at 1:08:26 PM
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I'm within a gnat's breath of your Trips figure -
(i) Must qualify = 19358
(ii) The remaining are 252 ways with 4*4*4 suit options.
I get 21 of these are straights (e.g. trips 8 with T9/97/76 etc) = 21*64
Of the other 231, 12 of the 64 suit ways produce a flush {sxx,s,s} four suits and three ways to get xx suits of the trip = 2772.
So...
(i) 19358
(ii) 1344 (21x64)
(iii) 2772 (231x12)
= 23474
(i) Must qualify = 19358
(ii) The remaining are 252 ways with 4*4*4 suit options.
I get 21 of these are straights (e.g. trips 8 with T9/97/76 etc) = 21*64
Of the other 231, 12 of the 64 suit ways produce a flush {sxx,s,s} four suits and three ways to get xx suits of the trip = 2772.
So...
(i) 19358
(ii) 1344 (21x64)
(iii) 2772 (231x12)
= 23474
| KKK AQ | 1 | 1 | 3 | 3 | |
| KKK AJ | 1 | 1 | 4 | 4 | |
| KKK Ax | 1 | 1 | 36 | 36 | |
| KKK QJ | 1 | 3 | 4 | 12 | |
| KKK Qx | 1 | 3 | 36 | 108 | |
| KKK Jx | 1 | 4 | 36 | 144 | |
| KKK xx | 1 | 36 | 32 | / 2 | 576 |
| QQQ AK | 1 | 1 | 3 | 3 | |
| QQQ AJ | 1 | 1 | 4 | 4 | |
| QQQ Ax | 1 | 1 | 36 | 36 | |
| QQQ KJ | 1 | 3 | 4 | 12 | |
| QQQ Kx | 1 | 3 | 36 | 108 | |
| QQQ Jx | 1 | 4 | 36 | 144 | |
| QQQ xx | 1 | 36 | 32 | / 2 | 576 |
| JJJ AK | 4 | 1 | 3 | 12 | |
| JJJ AQ | 4 | 1 | 3 | 12 | |
| JJJ Ax | 4 | 1 | 36 | 144 | |
| JJJ KQ | 4 | 3 | 3 | 36 | |
| JJJ Kx | 4 | 3 | 36 | 432 | |
| JJJ Qx | 4 | 3 | 36 | 432 | |
| JJJ xx | 4 | 36 | 32 | / 2 | 2 304 |
| XXX AK | 36 | 1 | 3 | 108 | |
| XXX AQ | 36 | 1 | 3 | 108 | |
| XXX AJ | 36 | 1 | 4 | 144 | |
| XXX Ax | 36 | 1 | 32 | 1 152 | |
| XXX KQ | 36 | 3 | 3 | 324 | |
| XXX KJ | 36 | 3 | 4 | 432 | |
| XXX Kx | 36 | 3 | 32 | 3 456 | |
| XXX QJ | 36 | 3 | 4 | 432 | |
| XXX Qx | 36 | 3 | 32 | 3 456 | |
| XXX Jx | 36 | 4 | 32 | 4 608 | |
19 358 | |||||
| Might qualify | |||||
| XXX xx | 36 | 32 | 28 | / 2 | 16 128 |
| Totals | 35 486 |
July 18th, 2025 at 5:09:06 AM
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I found some errors in the 2-pair hands on my 5-card Pai Gow Poker spreadsheet and corrected them. I have recalculated the player hand AsKsQs-AA, which is an interesting hand because it wins against almost every Dealer qualifying hand but obviously pushes against every Dealer DNQ. It has been interesting to see how much removing 3 Aces, a King and Queen has increased the probability of the dealer not qualifying.
So with errors corrected, the calculated EV of this hand has gone from 0.6208 to 0.6252. Still, Dealer DNQ is still >0.37.
| Hand Category | Total Comb | Return | EV |
|---|---|---|---|
| No Pair | 756048 | 425064 | 0.562218272 |
| 1 Pair | 661500 | 422937 | 0.649414966 |
| 2 Pair | 77949 | 77949 | 1 |
| 3oaK | 35486 | 23472 | 0.66144395 |
| Boat | 2526 | 2526 | 1 |
| Quads | 430 | 430 | 1 |
| Total | 1533939 | 952378 | 0.625206739 |
So with errors corrected, the calculated EV of this hand has gone from 0.6208 to 0.6252. Still, Dealer DNQ is still >0.37.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 18th, 2025 at 2:21:53 PM
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I still get different numbers for One Pair. The easiest way to see this is to be DNQ both the Pair (rank) and the three singletons have to be 10 or smaller. Otherwise one can ensure a Jack or better is in the low hand. Some of these PPxxx will qualify if the xxx can form a straight or flush. At the end this leaves 193536-16128-11088 non-qualifiers = 166320.
Low pairs so all the cards are 10 or lower.
Thus there are 9*8*7/6 = 84 ways to pick the three singletons
Then there are 6 unused ranks, so six ways to pick the pair
The suits of each singleton can be any of 4 suits,
There are six ways for the suits of the pair.
Hence suits = 6*4*4*4 = 384
You can see 84*6*384 = 193536
16128 =7*2304 Of these 84 ways, 7 are straights (T98, 987, … 432)
11088 = 77*144 The other 77 ways, there are 144 of 2304 ways to make a flush .
Low pairs so all the cards are 10 or lower.
Thus there are 9*8*7/6 = 84 ways to pick the three singletons
Then there are 6 unused ranks, so six ways to pick the pair
The suits of each singleton can be any of 4 suits,
There are six ways for the suits of the pair.
Hence suits = 6*4*4*4 = 384
You can see 84*6*384 = 193536
16128 =7*2304 Of these 84 ways, 7 are straights (T98, 987, … 432)
11088 = 77*144 The other 77 ways, there are 144 of 2304 ways to make a flush .
Total | Qualify | DNQ | |
Quads | 430 | 430 | |
Full H | 2 526 | 2 526 | |
TwoPairs | 77 949 | 77 949 | |
No Pair | 756 048 | 425 064 | 330 984 |
Trips | 35 486 | 23 474 | 12 012 |
One Pair | 661 500 | 495 180 | 166 320 |
1 533 939 | 1 024 623 | 509 316 |
July 18th, 2025 at 7:24:58 PM
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Okay, let me start with Low 1-pair hands where all cards are 10 or lower.
I get Qualify = 30996; DNQ = 162540. You get DNQ = 166320? ,if I am reading your post correctly?
**********************************************************************************
For 1 pair hands, I found a truly grievous error in my hand scoring whenever the second high card was paired. I fixed it and I now get Qualify (and WIN) = 496008 where as you get Qualify = 495180.
In my spread sheet there are a small number of Push hands in 1-pair where the dealer qualifies with three singletons that are suited AKQ. I think that only amounts to 60 combinations.
In Trips , my spread sheet also has two hands that qualified but Player pushed - where the dealer hand is suited AKQ-KK and suited AKQ-QQ. I think that implies that our DNQ/Qualify number for Trips do indeed agree!
*****************************************************************************
Thank you so much for giving me these benchmarks. At least we are now much closer. Obviously, I have still have more debugging work to do on the one pair worksheet.
I get Qualify = 30996; DNQ = 162540. You get DNQ = 166320? ,if I am reading your post correctly?
**********************************************************************************
For 1 pair hands, I found a truly grievous error in my hand scoring whenever the second high card was paired. I fixed it and I now get Qualify (and WIN) = 496008 where as you get Qualify = 495180.
In my spread sheet there are a small number of Push hands in 1-pair where the dealer qualifies with three singletons that are suited AKQ. I think that only amounts to 60 combinations.
In Trips , my spread sheet also has two hands that qualified but Player pushed - where the dealer hand is suited AKQ-KK and suited AKQ-QQ. I think that implies that our DNQ/Qualify number for Trips do indeed agree!
*****************************************************************************
Thank you so much for giving me these benchmarks. At least we are now much closer. Obviously, I have still have more debugging work to do on the one pair worksheet.
Last edited by: gordonm888 on Jul 18, 2025
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 18th, 2025 at 9:44:42 PM
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I have continued to find a few errors in my 1-pair worksheet. When I make all the corrections I get for this calculation:
2-pair Qualify (and Win) = 494175
2-pair Qualify = 494235 (accounting for the 60 1-pair hands with a royal in the bottom hand that qualify and push)
charliepatrick's calculation for 1 pair qualifiers = 495180.
So, still a discrepancy. At this moment in time, I have no idea,
On low 1-pair hands where all cards are 10 or lower I am now getting DNQ = 166320, which is the same as your number.
2-pair Qualify (and Win) = 494175
2-pair Qualify = 494235 (accounting for the 60 1-pair hands with a royal in the bottom hand that qualify and push)
charliepatrick's calculation for 1 pair qualifiers = 495180.
So, still a discrepancy. At this moment in time, I have no idea,
On low 1-pair hands where all cards are 10 or lower I am now getting DNQ = 166320, which is the same as your number.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 20th, 2025 at 6:05:31 AM
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I'm glad we're gradually getting close. As you say the recent discussions have been whether the Dealer qualifies or not, but when working out the EV there are 62 of these which are a push: AcKcQcKhKd AcKcQcQhQd AcKcQc plus a pair of J-2 (10 pairs, 6 ways each).
July 20th, 2025 at 9:51:57 AM
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I now have fixed errors and my results are in complete agreement with Charlie's numbers for the players AsKsQs-AA hand. I had two errors in the portion of the spreadsheet that compare dealer and player's hands - I had been referring to the wrong cells to read the number of combinations for the hands in question.
Player hand =AsKsQs-AA
My results for the Return against Dealer's 1 pair hands is 495120. That is 60 combinations less than Charlie's result because I calculate that Dealer has hands with 60 combinations that qualify but TIE against the royal flush in the 3-card hand, thus changing the outcome of those wagers to a Push.
Many thanks to charliepatrick for his excellent work providing me with benchmark results for this case!
Player hand =AsKsQs-AA
| Hand Category | Total Comb | Return | EV |
|---|---|---|---|
| No Pair | 756048 | 425064 | 0.562218272 |
| 1 Pair | 661500 | 495120 | 0.748480726 |
| 2 Pair | 77949 | 77949 | 1 |
| 3oaK | 35486 | 23472 | 0.66144395 |
| Boat | 2526 | 2526 | 1 |
| Quads | 430 | 430 | 1 |
| Total | 1533939 | 1024561 | 0.667928125 |
My results for the Return against Dealer's 1 pair hands is 495120. That is 60 combinations less than Charlie's result because I calculate that Dealer has hands with 60 combinations that qualify but TIE against the royal flush in the 3-card hand, thus changing the outcome of those wagers to a Push.
Many thanks to charliepatrick for his excellent work providing me with benchmark results for this case!
Last edited by: gordonm888 on Jul 20, 2025
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
