Perfect Pai Gow hand

Just wondering - what pai Gow hand will win more then loose or tie? A straight or flush is a great 5 card hand but would loose the 2 card hand. I saw this in Atlantic city where a player had a straight Ace high but had 2-8 on his 2 card hand and thus a push. It seem to me that 2 small pairs win more then 50% of the time. Thoughts?

AAAA*KK
In other words, five Aces and a pair of Kings..unbeatable hand.

AKQ*10 AA
Royal flush with a pair of aces

Quote: jdominik

Just wondering - what pai Gow hand will win more then loose or tie? A straight or flush is a great 5 card hand but would loose the 2 card hand. I saw this in Atlantic city where a player had a straight Ace high but had 2-8 on his 2 card hand and thus a push. It seem to me that 2 small pairs win more then 50% of the time. Thoughts?

Generally an average high hand is a pair of 9's. An average low hand is KQ

IIRC, I read somewhere that Sanford Wong's book says the magic hand is a pair of jacks with an A/8 up.

I'm not really sure what the OP's question is, but I stumbled across a conundrum while thinking about it.

I assert that it is impossible to devise a hand which has the same probability of winning, losing and tying. This does not seem intuitively obvious to me, but it seems to be the way the math works.

If the probability of the front hand winning is x, and the probability of the back hand winning is y, then the overall probabilities are:

win: x*y
lose: (1-x)*(1-y)
tie: (1-x)*y + (1-y)*x or 1 - x*y - (1-x)*(1-y)

For win=lose=tie=0.333, I think I determine that x and y must be imaginary. Seems weird to me.

Quote: PapaChubby

... tie: (1-x)*y + (1-y)*x or 1 - x*y - (1-x)*(1-y) ...

I think you made an algebra/multiplication error on that line. But the solution does appear to involve complex values of x and y.

Quote: PGBuster

IIRC, I read somewhere that Sanford Wong's book says the magic hand is a pair of jacks with an A/8 up.

PG - that's the "median" hand or average hand. The dead Center hand.

Quote: clarkacal

AKQ*10 AA
Royal flush with a pair of aces

Correct, best hand, Royal with Aces on top.

The AAAA*/KK hand could indeed push: dealer could have three pairs 66552/KK, and win the two-side copy KK (Dealer wins copies on each side.)

Quote: clarkacal

AKQ*10 AA
Royal flush with a pair of aces

This is unbeatable, but wouldn't it push against the dealer's suited AKQJ10 XY, since the dealer's copied royal flush would beat the player's royal flush?

You're right - there are only four aces in use with that hand, so there's one left...but it's a much smaller probably with the Royal than with two kings out on AAAA*/KK. Three pairs, or quads, or a full house with Kings up is much more common than a Royal with Aces against another royal.

There is no hand that is a 100% win against the dealer.....depressing thought.....but the royal with aces comes closest at 99.999% or so.
Still the best possible hand.

AAAAK / KK



Edit:

I momentarily forgot about a straight flush beating quads. Sigh....

Quote: DJTeddyBear

AAAAK / KK

I think you meant AAAAA/KK, using the joker. With two kings out on that hand, the KK top can copy and push on that side against a banker or dealer, against you via his full house with K's as the pair top, Quads with a pair of Kings top, three pairs with kings as top, and a straight or flush with a pair of kings on top in the dealer's or banker's hand.

With a Royal with Aces top, AKQJ10/AA, there are also two aces out - to also copy for the push and deny the win (against a a dealer or banker with: )
- a full house with A's as the pair top,
- Quads with a pair of Aces top,
- three pairs with Aces as top pair, and
- a straight or flush with a pair of Aces top.

On this basis, the top two-card side copying is much more common with AAAAA/KK than a Royal with AA on top, in spite of both cases having ONLY the two highest cards availalble to form a two-side pair that copies.

This is why:

With a Royal and aces top, FIVE ranks of cards are now very depleted (K through 10 are depleted by one card in the remaining hands, with the Aces severely depleted by four cards), making Quads, Full houses, three pairs, and straights and flushes more difficult to form with the top two remaining aces for top on the opposing hand.
So your dealer/banker opponent is less likely to form a hand that can put up the remaining Aces on the top of his hand from a split full house with Aces on top, Quads with Aces on top, Three pairs with Aces on top, etc...to push you, denying you YOUR win.

But with you holding AAAAA/KK, - then ALL of the ranks of Queens through 2's are now concentrated into the remaining hands, - including your opponent the dealer or banker - making all lower quads, straights, flushes, three pairs, and full houses more common - and allowing the two remaining kings to go up with his hand, and to copy and push you - not you win - on your monster hand against the banker or dealer.

So - you're better off Holding a Royal with Aces top, than Five aces with Kings top, in securing a win against the dealer or banker.

By a trillionth of a percentage.

I KNOW a Royal with a pair of aces is about four times more common than holding Five aces with a pair of kings to copy the high pair on top.

But holding five aces with a pair of Kings on top is much more likely to copy the pair of kings on top.

We need Mike the Wizard to resolve this theoretical Pai Gow Question.

I say AKQJ10/AA beats AAAAA/KK - as a "more guaranteed winner" against a banker or dealer...

By 99.9999999% versus 99.9999998%.

Edit:Natural Royal. Wild Royal with aces = game over. You ain't losing.

Quote: Paigowdan

There is no hand that is a 100% win against the dealer.....depressing thought.....but the royal with aces comes closest at 99.999% or so.
Still the best possible hand.

Depressing, or an opportunity for a bad push side bet?

But do I have an evil mind ;)

Quote: Paigowdan

Quote: DJTeddyBear

AAAAK / KK

I think you meant AAAAA/KK, using the joker.

No. I meant what I said.

With AAAAK/KK there is NO WAY for the dealer to beat or copy the two-card hand, and only a straight flush will beat the quads.

Now, as a Pai Gow dealer, you've seen a LOT of hands, so I ask you: What do you see more of, in ANY hand (player OR dealer/banker): A straight flush, or a top containing AA or KK? I suspect it's the AA or KK top.

No opportunity here.
I came up with a bad beat bet for Pai Gow poker, losing on trips or better (or two pairs in some versions).

Really worked it out:
1. Good payout table.
2. Math done.
3. Already have one hit new game out.
4. Product Description Guide.
5. Power Point presentation
6. Have a provisional on it, to expire next month.

ALL FOUR Game distributors I've shown it to said, "Dan, sorry man, your Bad Beat Pai Gow side-thing really sucks, really man...it's not happening....so show me some other games....NEXT!"

Ugg! I felt hurt. I really thought it kicked the Pai Gow insurance bet in the ass. It didn't.

So I showed a few other games. Like they said..."Next!"
Signed one new game,
Got interest on DJTeddybear's "Poker For Roulette" pitching it on his behalf - (He doesn't live in Las Vegas)...Still waiting for a Yeah/Nay
Got a call back to re-pitch another interesting and easy poker game for a guy in Texas (it's casino version he had me do with him)
All others were summarily dismissed.

Walked back to my car, drove home....

All I have to contribute is a personal hand from a couple nights ago in Louisiana... I was dealt 5-5-5-5-4 and 8-8. I didn't lose :)

I have to agree there is no unbeatable hand when not banking. I drank a cup of coffee trying to find one to impress you guys, but there doesn't seem to be any. So, let's look at the two types of hands being discussed:

(1) AAAAW-KK (W=wild)

As was said, this can copy against another KK in the low. The number of combinations with two kings is combin(44,5) = 1,086,008. The number of total combinations of 7 cards out of the 46 left is 53,524,680. So the probability of two kings is surprisingly high at 1,086,008/53,524,680 = 2.03%. With many of those combinations the dealer will not play the KK in the low. According to my Pai Gow Poker Appendix 1, the dealer will have a pair of aces or less in the high 74.93% of the time. That means he will make a two pair or better, allowing him to play the kings in the low, 25.07% of the time. So the probability the player will get two kings, and be able to play them in the low, is 2.03% × 25.07% = 0.51%, or 1 in 196.

(2) AWQJT (suited) -AA

Here I'm using the wild to substitute for a king to make a royal, but it could also have substituted for the Q, J, or T. I don't want it to substitute for an A, because then there would be two aces left in the deck for the dealer to tie the low. This way, the dealer can only tie the high with another royal. What are the odds of that? There are three suits left for the royal, and the other two cards could be anything. So the number of combinations that push is 3*combin(41,2) = 2,460. I get the 41 from 46 cards left in the deck after the player removed 7 from the original 43, and then subtract 5 more for the 5 cards in the dealer's royal. We still have combin(46,7) = 53,524,680 in the denominator. So the odds of a tie with wild royal - AA is 2,460/53,524,680 = 0.004596%, or 1 in 21,758.

Before some perfectionist writes to me, there may be some bizarre situations where the dealer doesn't play the hand the way I intended. I'm not looking to get an exact probability of each situation, but to substantiate why I think wild royal - AA is the best hand you can get in pai gow poker when not banking.

I'm ammending my statement above.

I say the best hand is AAA*K/KK - slightly better than AAAAK/KK as this produces the same quad Aces with a pair of Kings that I was talking about, but removes the wild from the dealer's possible cards, while leaving the dealer no way to get AA or KK top.

That means the dealer must have a straight flush to beat my quads in order for me to push.

But, depending on the 7 cards he has, there is a possibility that the house way may dictate playing a straight or flush, if it gives him a pair on top. For example, if the dealer had Jd Td 9d 8d 7d 2d Jh, wouldn't the house way play it as the flush with JJ?

Quote: DJTeddyBear

No. I meant what I said.

With AAAAK/KK there is NO WAY for the dealer to beat or copy the two-card hand, and only a straight flush will beat the quads.

Now, as a Pai Gow dealer, you've seen a LOT of hands, so I ask you: What do you see more of, in ANY hand (player OR dealer/banker): A straight flush, or a top containing AA or KK? I suspect it's the AA or KK top.

Yes, Dave, it is. But not by much. We see somewhat more AA tops than quads or straight flushes, and they're usually on three pair and full house hands played split.

But, we're talking here about a "lock down win" on the combined hand - for the very best Pai Gow win hand. And that's just AAAAA/KK or a Royal with AA up. The question is which one is it, really.

Dave,
AAAAK/KK just isn't it, because a SF will push it more often on the five-card side - as rare as it is, and regardless of it always losing the top - than AAAAA/KK and a Royal with AA copying the top - but winning the bottom.

Edit: Because both of these hands always beat a "mere" straight flush - yet will copy very often, comparatively.
BUT...the question arises - and still remaining - because the AAAAK/KK hand will never push on top or bottom - or EVER lose - and so will NEVER be denied a win by copy, - will the removal of the high copy percentage on the top side make this hand strangely best winning hand?? When you consider that the straight flush hand facing it is very rare - and the copies-push of it non-existant - is this a best hand in PGP?

Quote: Wizard

I have to agree there is no unbeatable hand when not banking. I drank a cup of coffee trying to find one to impress you guys, but there doesn't seem to be any. So, let's look at the two types of hands being discussed:

(1) AAAAW-KK (W=wild)

As was said, this can copy against another KK in the low. The number of combinations with two kings is combin(44,5) = 1,086,008. The number of total combinations of 7 cards out of the 46 left is 53,524,680. So the probability of two kings is surprisingly high at 1,086,008/53,524,680 = 2.03%. With many of those combinations the dealer will not play the KK in the low. According to my Pai Gow Poker Appendix 1, the dealer will have a pair of aces or less in the high 74.93% of the time. That means he will make a two pair or better, allowing him to play the kings in the low, 25.07% of the time. So the probability the player will get two kings, and be able to play them in the low, is 2.03% × 25.07% = 0.51%, or 1 in 196.

(2) AWQJT (suited) -AA

Here I'm using the wild to substitute for a king to make a royal, but it could also have substituted for the Q, J, or T. I don't want it to substitute for an A, because then there would be two aces left in the deck for the dealer to tie the low. This way, the dealer can only tie the high with another royal. What are the odds of that? There are three suits left for the royal, and the other two cards could be anything. So the number of combinations that push is 3*combin(41,2) = 2,460. I get the 41 from 46 cards left in the deck after the player removed 7 from the original 43, and then subtract 5 more for the 5 cards in the dealer's royal. We still have combin(46,7) = 53,524,680 in the denominator. So the odds of a tie with wild royal - AA is 2,460/53,524,680 = 0.004596%, or 1 in 21,758.

Before some perfectionist writes to me, there may be some bizarre situations where the dealer doesn't play the hand the way I intended. I'm not looking to get an exact probability of each situation, but to substantiate why I think wild royal - AA is the best hand you can get in pai gow poker when not banking.

Not a perfectionist... but... any royal with a sixth in that suit plays the ace up front. Any royal with a pair, say of the queens, plays queens with ace king up front. Any royal with a 9 plays a straight with the ace up front. so if you have a royal any extra 10, jack, queen, king, or 9, or any of the low cards in your suit, cause you not to play your royal. Probably 50% of the time.
I am also not sure about how you figured out you can put kings in the low 25% of the time you get them. The only 2 pair that allows that is aces and kings, and I don't think the other possibilities ((flush, straight, 3 of a kind, 3 pair) when the other 2 cards are defined as kings would even come close to 25%. Time to recalculate, or am I missing something here?

Something else...
So..the best Pai Gow hand is now the WILD Royal with AA up - because the AA can now never copy with FOUR aces out in the winning hand.

Dave Miller might have presented the best natural hand in PGP as AAAAK/KK, because it copies less often as it NEVER copies, - and can only lose to a S.F., which is hugely rare - in comparison to how often the so-called monster hands DO push and copy with AA and KK top.

Quote: Wizard

This way, the dealer can only tie the high with another royal. What are the odds of that? There are three suits left for the royal, and the other two cards could be anything.

I don't play PGP, but for AWQJT (suited) - AA, isn't there only one suit left for the dealer Royal? That of the one remaining A?

Quote: Paigowdan

Dave Miller might have presented the best natural hand in PGP as AAAAK/KK, because it copies less often as it NEVER copies, - and can only lose to a S.F., which is hugely rare - in comparison to how often the so-called monster hands DO push and copy with AA and KK top.

Thanks. Mind you, I'm not sure, and will bow to whoever wants to run the numbers, but this is my gut feeling.

But who cares about it being natural? Replace one Ace with the Joker to further reduce the dealer's chances.

Quote: DJTeddyBear

I'm ammending my statement above.

I say the best hand is AAA*K/KK - slightly better than AAAAK/KK as this produces the same quad Aces with a pair of Kings that I was talking about, but removes the wild from the dealer's possible cards, while leaving the dealer no way to get AA or KK top.

That means the dealer must have a straight flush to beat my quads in order for me to push.

But, depending on the 7 cards he has, there is a possibility that the house way may dictate playing a straight or flush, if it gives him a pair on top. For example, if the dealer had Jd Td 9d 8d 7d 2d Jh, wouldn't the house way play it as the flush with JJ?

The dealer could beat that with:

1. A2345 in the suit of the one ace left in the deck.
2. 23456 up to 89TJQ in four different suits. That is 7 different spans.
3. 9TJQK in the suit of the one king left in the deck.

So there are 1 + 4*7 + 1 = 30 straight flushes that can beat four aces. With each one there are combin(41,2)=820 combinations for the low cards. This is not exact, but close enough to show it isn't close to the best hand.

So there are 30×820 = 24,600 ways the dealer could beat the high. Compare that to 2460 for the royal-AA already discussed.

Quote: Wizard

The dealer could beat that with:

1. A2345 in the suit of the one ace left in the deck.
2. 23456 up to 89TJQ in four different suits. That is 7 different spans.
3. 9TJQK in the suit of the one king left in the deck.

So there are 1 + 4*7 + 1 = 30 straight flushes that can beat four aces. With each one there are combin(41,2)=820 combinations for the low cards. This is not exact, but close enough to show it isn't close to the best hand.

So there are 30×820 = 24,600 ways the dealer could beat the high. Compare that to 2460 for the royal-AA already discussed.

With ties going to the dealer, isn't it true to say that there's no such thing as a 100% winner, only a sure not-loser? I mean, AKQJTs/AA (wild or otherwise) ties with 2233K/A* or AKQJTs/KK (wild or otherwise) right?

But on the flip side, is there an automatic 100% loser? 95432/87 draws with AKQJTs/7x, etc. Is there such a thing as a 100% loser, or is there a sure not-winner outside of 95432/87?

There is no auto loser. The 'worst' hand of 2 3 4 5 7 8 9 has (7 8) up top. Dealers will play lower than that with many flushes or straights and even some 4 of a kinds. And some 3 of a kinds (7 7 7 5 4 3 2). I won a hand with a 6 3 up top and a flush against a dealer's straight with 5 something. So you never know.
Als0- 2 3 4 6 7 8 9 can never win if dealer wins copy hands.

Quote: Wizard

The dealer could beat that with:

1. A2345 in the suit of the one ace left in the deck.
2. 23456 up to 89TJQ in four different suits. That is 7 different spans.
3. 9TJQK in the suit of the one king left in the deck.

So there are 1 + 4*7 + 1 = 30 straight flushes that can beat four aces. With each one there are combin(41,2)=820 combinations for the low cards. This is not exact, but close enough to show it isn't close to the best hand.

So there are 30×820 = 24,600 ways the dealer could beat the high. Compare that to 2460 for the royal-AA already discussed.

But now removed are the ways that the dealer could beat - via copies - the KK pair top (on AAAAA/KK) and AA (on natural Royal with AA) - in comparison with the player holding three kings, in player's AAAAK/KK, as the straight flush beat on the 5-card side is less common than the high pair two-card side copy loss?

We were also looking at copies losses on the two-card side denying the player win.

With countless more copy-loses now gone on the two-card side, does the dealer's rare straight flush high-side win make that hand push (deny a win) more often with the player's certain win now on the two-card side?

The problem with AAAAA/KK and natural AKQJ10/AA copies is that is that player's hand now "player pushes" more often than Dave's AAAAK/KK losing to a S.F. - a never-losing player hand?

With this, is the best winning hand ONLY a Wild Royal with AA up? : AKQ*10/AA?

Quote: Paigowdan

But now removed are the ways that the dealer could beat - via copies - the KK pair top - with the player holding three kings, in player's AAAAK/KK

Right.

I accept the Wiz' math that there are 24,600 ways to beat AAAAK.

But, with the other hands beign discussed, (i.e. AAAA*/KK and Royal/AA) how many ways are there to copy either two-card hand? I'm thinking that there's more than 24,600.

Quote: Paigowdan

The problem with AAAAA/KK and natural AKQJ10/AA copies is that is that player's hand now "player pushes" more often than Dave's AAAAK/KK losing to a S.F. - a never-losing player hand?

Exactly.

But, again, I defer to someone who actually does the math.

And I remind you that AAA*K/KK is even stronger since it removes a lot of the dealer's combinations.

I had asked this above but it was my first post in the forums, so maybe went unnoticed. I only bring it up again since this question made the latest "Ask the Wizard" post on WOO.

Quote: Wizard

(2) AWQJT (suited) -AA

Here I'm using the wild to substitute for a king to make a royal, but it could also have substituted for the Q, J, or T. I don't want it to substitute for an A, because then there would be two aces left in the deck for the dealer to tie the low. This way, the dealer can only tie the high with another royal. What are the odds of that? There are three suits left for the royal, and the other two cards could be anything. So the number of combinations that push is 3*combin(41,2) = 2,460. I get the 41 from 46 cards left in the deck after the player removed 7 from the original 43, and then subtract 5 more for the 5 cards in the dealer's royal. We still have combin(46,7) = 53,524,680 in the denominator. So the odds of a tie with wild royal - AA is 2,460/53,524,680 = 0.004596%, or 1 in 21,758.

Before some perfectionist writes to me, there may be some bizarre situations where the dealer doesn't play the hand the way I intended. I'm not looking to get an exact probability of each situation, but to substantiate why I think wild royal - AA is the best hand you can get in pai gow poker when not banking.

As a newbie, I don't presume to be the "perfectionist" to second guess the Wiz. I am just trying to understand the numbers offered as well as the game PGP. So in the case described above, AWQJT - AA, The Wiz says there are three suits left for the dealer's royal. I would have thought there is only one left, that of the remaining Ace? The AA low hand doesn't somehow play as "community cards" like the board in Hold'em does it. I thought in PGP everybody played their own 7 cards?

If so, wouldn't the odds of tie with the wild royal be even lower by factor of 3?

I have not followed this thread and have not read all the posts. But if I had to guess the best hand you could have would be AKQJ*/AA.
With you having 3 aces there is only one suit left for the dealer to copy you on the high hand with a "natural royal" since you have the joker. And of course you cant lose or tie on the low hand because you have 3 of the A's. But thats just my 2 cents.

Quote: chevy

I had asked this above but it was my first post in the forums, so maybe went unnoticed. I only bring it up again since this question made the latest "Ask the Wizard" post on WOO.




As a newbie, I don't presume to be the "perfectionist" to second guess the Wiz. I am just trying to understand the numbers offered as well as the game PGP. So in the case described above, AWQJT - AA, The Wiz says there are three suits left for the dealer's royal. I would have thought there is only one left, that of the remaining Ace? The AA low hand doesn't somehow play as "community cards" like the board in Hold'em does it. I thought in PGP everybody played their own 7 cards?

If so, wouldn't the odds of tie with the wild royal be even lower by factor of 3?

You're right. 3 of the 4 suits are definitely eliminated since you're holding their aces in this case. Only one suit remains for the dealer to get the royal in and copy your 5 card hand.

Quote: ChesterDog

This is unbeatable, but wouldn't it push against the dealer's suited AKQJ10 XY, since the dealer's copied royal flush would beat the player's royal flush?

Yes, it will push the Dealer for the Ace-remaining Royal.
and AAAA* + KK will push Dealer 2233x + KK.

"It cannot lose" is different than "it always wins".

How about the best Pai Gow tiles hand? Gee Joon/Teen & Gee Joon/Day cannot lose! Imagine that two hands that cannot lose in one game, compared to Pai Gow Poker where a push is always possible!