Poker combinations with a partially wild joker
May 15th, 2014 at 3:27:02 PM
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https://wizardofodds.com/games/pai-gow-poker/appendix/3/
Michael presents the number of combinations for each different hand in Pai Gow Poker given seven cards. What I'd like to know is how to arrive at those numbers, so I can find out the number of combinations for different numbers of cards in a hand (eight-card hands, six-card hands, ten-card hands, etc).
I'm assuming it involved Combinations functions, but I'm not getting the numbers to quite add up. For example, by calculating the combinations for quads as
[combin(5,4)*combin(48,3)]+[combin(4,4)*combin(49,3)*12]
(Choose 4 aces from 5, choose 3 non-aces from 48, choose any set of quads, choose any three other cards)
I get 307,568 combinations, instead of the 307,472 combinations I should be getting.
Any help is appreciated.
PS: I know combin(x,x) equals 1 and is redundant. I just include that for my own clarity.
Michael presents the number of combinations for each different hand in Pai Gow Poker given seven cards. What I'd like to know is how to arrive at those numbers, so I can find out the number of combinations for different numbers of cards in a hand (eight-card hands, six-card hands, ten-card hands, etc).
I'm assuming it involved Combinations functions, but I'm not getting the numbers to quite add up. For example, by calculating the combinations for quads as
[combin(5,4)*combin(48,3)]+[combin(4,4)*combin(49,3)*12]
(Choose 4 aces from 5, choose 3 non-aces from 48, choose any set of quads, choose any three other cards)
I get 307,568 combinations, instead of the 307,472 combinations I should be getting.
Any help is appreciated.
PS: I know combin(x,x) equals 1 and is redundant. I just include that for my own clarity.
Casinos are not your friends, they want your money. But so does Disneyland.
And there is no chance in hell that you will go to Disneyland and come back with more money than you went with.
- AxelWolf and Mickeycrimm
May 15th, 2014 at 4:27:15 PM
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Sometimes you will have a Straight Flush or Royal Flush, which both beat Four of a Kind and should be subtracted from the Four of a Kind result.
For example, A♥ A♦ A♣ 2♣ 3♣ 4♣ Joker.
The possible ranks to complete the SF are 234, 235, 245, and 345 in the same suit as one of the three natural aces.
So from the Four Aces calculation, subtract:
combin(4,3) * combin(1,1) * combin(3,1) * combin(4,1) = 48
(natural aces) * (joker) * (SF suit) * (ways to complete the SF)
Likewise on the other end for a Royal Flush, you might have A♥ A♦ A♣ K♣ Q♣ J♣ Joker.
The possible rank combinations for a Royal Flush are TJQ, TJK, TQK, JQK. The calculation is the same, another 48 combinations to subtract.
307,568 - 48 - 48 = 307,472
For example, A♥ A♦ A♣ 2♣ 3♣ 4♣ Joker.
The possible ranks to complete the SF are 234, 235, 245, and 345 in the same suit as one of the three natural aces.
So from the Four Aces calculation, subtract:
combin(4,3) * combin(1,1) * combin(3,1) * combin(4,1) = 48
(natural aces) * (joker) * (SF suit) * (ways to complete the SF)
Likewise on the other end for a Royal Flush, you might have A♥ A♦ A♣ K♣ Q♣ J♣ Joker.
The possible rank combinations for a Royal Flush are TJQ, TJK, TQK, JQK. The calculation is the same, another 48 combinations to subtract.
307,568 - 48 - 48 = 307,472
