Perfect Pai Gow hand

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.

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.

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.

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?

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.

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.

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

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?

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.