Poker Probabilities for 6-10 Cards

Quote: miplet

I have no clue either.

O.K. I guess you enjoy the challenge of doing the analysis. I can live with that.

Quote: miplet

Someone already did an analysys using 5 of 8-10 cards here.

That analysis was my suggestion back on page 1 of this thread. Guess I missed the linked thread first time around. Or ignored it because I don't play poker.

Thanks for your response and explanations.

Thanks Miplet, I agree with your 8-card numbers. Here are my tables for 9 and 10, if you have the inclination.

9 Cards

Hand	Combinations
Royal flush	4
Straight flush	20
Four of a kind	22,256,520
Full house	424,213,504
Flush	2,836
Straight	1,572,840
Three of a kind	196,804,608
Two pair	1,836,229,824
Pair	1,012,137,984
Garbage	185,857,260
Total	3,679,075,400

10 Cards

Hand	Combinations
Royal flush	4
Straight flush	16
Four of a kind	159,455,868
Full house	2,977,017,472
Flush	1,124
Straight	5,242,860
Three of a kind	674,758,656
Two pair	9,178,554,528
Pair	2,530,344,960
Garbage	294,648,732
Total	15,820,024,220

Quote: Doc

And now, even seven cards?That is indeed what the initial post seems to indicate. I'm just curious why anyone would ask that question and why it would warrant this much work on the part of the Wizard and miplet. Doesn't there exist some limit on how esoteric an issue will be exhaustively investigated by a geek (not counting Trekkies)? I thought that suggested there might actually be a real-world use for this info. Perhaps that notion reflects a personal bias drawn from a career in applied rather than basic research.

I don't know why people ask for this stuff either, but they do. It also makes for a good exercise in combinatorial mathematics.

I get the same numbers. :+)

Quote: miplet

I get the same numbers. :+)

Yeah! Thanks again. I put all this up on my poker probabilities page, with a nod to you for your help. I also added tables at the top for 6-card to 10-card stud, where you make the best 5-card poker hand. I did those by brute force looping. The 10-card table took several hours.

Quote: Ibeatyouraces

How would a full house be composed out of 6 cards? Also, how about a new combo , three of a kind-three of a kind.

The Full House minimally would be XXXYYZ with 164,736 ways. An XXXYYY would rightly be a separate winning hand, called "Triplets" Or "Two-Set" with 1,248 ways. Also there would now be a "Paired Four-of-a-Kind" (XXXXYY) 936 different ways vs "Unpaired" (XXXXYZ) the other 13,728; and "Three-Pair" (XXYYZZ) any of 61,776 ways over a standard "Two-Pair" 2,471,040 ways.

Quote: Wizard

makes for a good exercise in combinatorial mathematics.

Yep.

Quote: miplet

Someone already did an analysys using 5 of 8-10 cards here.

Those calculations assume a five-card winning sequence. For pair-plus hands the numbers should be the same as ours here, but for non-pair winners (royals, straight flushes, flushes, and straights) the numbers will be very different. For example, with a six-card hand, DB shows (4 * 47) = 188 royal flushes possible, and with a seven-card hand 4 * combin(47, 2) = 4,324 royals, because of the difference between DB's and Wizard's respective definitions of a Royal Flush sequence. It's harder with flushes and straights to confirm the numbers because of the "hidden" double-wins that don't actually count as a second, simultaneous flush on the same hand. Hence for a six-card hand, one would naively expect 36 * 47 = 1,692 to be the number of non-royal straight flushes scorable; there are only 36 * 46 = 1,656 because exactly one of the 52 - 5 = 47 remaining cards will cause the hand to contain two straight flushes, with only the higher of the two being scored (say the wheel flush - As, 2s, 3s, 4s, 5s - with 6s added doesn't add a second tally to the straight flush count). It's even harder for me to come up with DB's 205,792 flushes for six-cards.