Poker Probabilities for 6-10 Cards
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| Hand | Combinations |
|---|---|
| Royal flush | 4 |
| Straight flush | 32 |
| Four of a kind | 14,664 |
| Full house | 165,984 |
| Flush | 6,828 |
| Straight | 36,828 |
| Three of a kind | 732,160 |
| Two pair | 2,532,816 |
| Pair | 9,884,160 |
| Garbage | 6,985,044 |
| Total | 20,358,520 |
Quote: DocI'm not a poker player, so maybe I don't understand. Or maybe I just shouldn't comment. However, it seems it would be more useful to know the probability of the various 5-card hands given that you have 6 or more cards to work with, such as the 7 cards of Hold-em. How often do you have to use more than five cards in one hand (not split into two hands)?
Point taken. I may do that next, but the math will be very messy.
Well, draw poker would give you 8, not all available at the same time (or maybe even 10, depending on the local rules.) As I said, I'm not a poker player, so I don't know all the variations that might be out there. Apparently, the Wizard is considering hands for a very different kind of game that I don't know about.Quote: DeMangoWe play with 5 or 7 no?
Quote: WizardSomebody suggested that I add a table to WoO for poker probabilities with 6-10 cards. Before I go live with that, if anybody would like to check my work, please do! To start, here is what I get for six cards. Straights, flushes, straight flushes, and royal flushes must make use of all cards. For example 89TJQK would be a straight, but 89TJQA would not. Once somebody agrees with me on this, and hopefully somebody will, we'll move onto 7 cards.
Must.... go... to.... bed...... Don't even think about it.
I'll check after work tonight.
On another topic, I have an easy paying project. It has to do with a baccarat side bet. Send me a message via the contact form if you're interested.
Quote: WizardThanks! I'm away from my computer with the 7-card numbers, but I'll post those by Monday morning at the latest. If you wish to work ahead, by all means, don't let me stop you.
You're welcome. I should have 7 card numbers about this time tomarrow.
Quote:On another topic, I have an easy paying project. It has to do with a baccarat side bet. Send me a message via the contact form if you're interested.
Message sent.
| Hand | Combinations |
|---|---|
| Royal flush | 4 |
| Straight flush | 28 |
| Four of a kind | 22,4848 |
| Full house | 3,473,184 |
| Flush | 6,832 |
| Straight | 131,040 |
| Three of a kind | 6,589,440 |
| Two pair | 32,123,520 |
| Pair | 63,258,624 |
| Garbage | 27,977,040 |
| Total | 133,784,560 |
| Hand | Combinations |
|---|---|
| Royal flush | 4 |
| Straight flush | 24 |
| Four of a kind | 2,529,462 |
| Full house | 45,659,328 |
| Flush | 5,120 |
| Straight | 458,724 |
| Three of a kind | 42,172,416 |
| Two pair | 282,625,200 |
| Pair | 295,206,912 |
| Garbage | 83,880,960 |
| Total | 752,538,150 |
As I mess with the other 8-card hands, it gets pretty messy for three of a kind and two pair. I think going through 7 cards the most I should bother you for. For the rest, I'll probably write a looping program.
Quote: IbeatyouracesHow would a full house be composed out of 6 cards? Also, how about a new combo , three of a kind-three of a kind.
A 6-card full house could be something like KKKQQQ or KKKQQJ. It is the best 5-card hand you can make using 6 cards.
Quote: WizardThanks Miplet for the 7-card and 8-card numbers. I agree with the 7-card numbers. However, beg to differ on some of the 8-card numbers. Let's look at a four of a kind. I think you are counting hands like Ah As Ac Ad Kd Qd Jd Td as a four of a kind, when it should be a royal flush.
Quote: Wizard
Straights, flushes, straight flushes, and royal flushes must make use of all cards. For example 89TJQK would be a straight, but 89TJQA would not.
These seem contradictory. For 7 cards I have As Ac Ad Kd Qd Jd Td as a three of a kind based on the second quote, but by the first quote it should be a royal.
Quote: mipletThese seem contradictory. For 7 cards I have As Ac Ad Kd Qd Jd Td as a three of a kind based on the second quote, but by the first quote it should be a royal.
You're absolutely right. Let me redo my math and get back to you.
And now, even seven cards?Quote: DocActually, I'm still confused on the purpose of this whole topic. Can someone tell me what poker game it is that requires a player to use six, eight, ten cards in a hand?
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.Quote: IbeatyouracesI dont think there is one. I think someone maybe asked about the probabilities of 6-10 card hands.
O.K. I guess you enjoy the challenge of doing the analysis. I can live with that.Quote: mipletI have no clue either.
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.Quote: mipletSomeone already did an analysys using 5 of 8-10 cards here.
Thanks for your response and explanations.
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: DocAnd 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.
Quote: mipletI 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: IbeatyouracesHow 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: Wizardmakes for a good exercise in combinatorial mathematics.
Yep.
Quote: mipletSomeone 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.
