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Old 12-19-2018, 09:04 AM
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Six-card poker: Hand rankings

There are a wide variety of poker games. In seven-card stud you get seven cards. In Texas Hold'em you get either 2 cards or 7 cards, depending on how you count them. In Big Jacoby you finish with six cards. But in all these variations your hand value is determined by your best five cards.

Why not score based on six-card hands? Is it because the list of hand rankings would get too cumbersome? It is the goal of this thread to rank the possible 6-card hands.

The following list is roughly in increasing probability of getting the hand with six random cards from a 52-card deck. (Tie-breaking is shown for 5-card straight-flush since the 6th card might extend the straight, etc.)
  • 6-card straight flush
  • 5-card straight flush
    • with a 6-straight
    • with a 6-flush
    • ** (6-straight and 5-flush "should" rank here)
    • with a pair
    • ordinary
  • 6-straight and 5-flush
  • Monster Boat: Quads and a Pair
  • Big Boat: Two Triplets
  • 5-straight and 5-flush
  • 6-flush
  • Four-of-a-kind
  • 6-straight
  • Three pair
  • 5-straight and pair
  • 5-flush and pair
  • 5-flush
  • Ordinary Boat: Triplets and a Pair
  • 5-straight
  • Triplets
  • Two pair
  • No pair
  • One pair

A 6-straight with 5 in same suit is actually much less likely than an ordinary straight flush but I've demoted it to reduce confusion. The '5-straight and pair' is only very slightly less likely than '5-flush and pair.' Would it be less confusing to reverse these ranks? With six cards, One Pair is more likely than No Pair but should the hands be ranked that way?

Comments? (If ranking the 6-card hands is too tedious, feel free to hijack and discuss 4-card poker instead.)
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Old 12-19-2018, 09:19 AM
OldGuy OldGuy is offline
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I'm not sure why 5 straights and 5 flushes are even a thing. You get no credit for a four flush (with or without a pair) in regular 5 card poker.
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Old 12-19-2018, 09:26 AM
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Okay. But note that, in a 6-card deal, opposite to a 5-card deal, 5-flush is harder to get than an ordinary full house. 6-Flushes and 6-Straights would be very rare.
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Old 12-19-2018, 10:40 AM
Snarky_Kong Snarky_Kong is offline
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Quote:
Originally Posted by septimus View Post
Okay. But note that, in a 6-card deal, opposite to a 5-card deal, 5-flush is harder to get than an ordinary full house. 6-Flushes and 6-Straights would be very rare.
I'm with OldGuy. In addition I wouldn't want flush+pair or straight+pair to be hands either, although I'm less convinced of that.
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Old 12-19-2018, 11:03 AM
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I think that it's inherent in the notion of "straight" and "flush" that they involve all of the cards. And yes, this makes those hands less likely, at the same time that the match-based hands become more likely: What of it? It's a variant; we expect things to be different.

And if you require all of the cards to be in a straight or flush, then that also eliminates the possibility of having a pair with either, so that's a non-factor.
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Old 12-19-2018, 11:51 AM
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4-flushes count in Cribbage. (And 3-straights for that matter.)
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Old 12-19-2018, 12:17 PM
Snarky_Kong Snarky_Kong is offline
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And if you require all of the cards to be in a straight or flush, then that also eliminates the possibility of having a pair with either, so that's a non-factor.
Of course.
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Old 12-19-2018, 01:35 PM
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Originally Posted by septimus View Post
...feel free to hijack and discuss 4-card poker instead....
I don't know if 4-card poker is a thing, but three-card brag definitely is. (Wherein a straight beats a flush, as expected.)
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Old 12-19-2018, 01:43 PM
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There's some sort of 3-card poker variant played here in rural Thailand but I've never even tried to learn the rules. The weird dice game called 'Hi-Lo' is already enough excitement for me.
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Old 12-19-2018, 06:19 PM
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OK, this report describes Primero, which I have never personally seen played, but you have to bet on a hand with four cards. From lowest to highest, you can make

numerus: 2 or 3 cards from the same suit
primero: one card from each suit
supremus: highest possible numerus (67A, since the cards go face cards - 2 - 3 - 4 - 5 - A - 6 - 7)
fluxus: flush
chorus: four of a kind

So no pairs or straights; high cards are worth more than low cards.

Last edited by DPRK; 12-19-2018 at 06:20 PM.
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Old 12-21-2018, 05:18 AM
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I may as well post some calculated percentages. 5-card poker usually isn't played with just 5 cards of course: by "7 cards" I show the chances in Texas Hold'em for a player who always stays for the final showdown. Similarly "8 cards" shows the chances for the possible 6-card best hands when dealt 8 altogether. (I've combined related hand types compared with the earlier list of hand rankings.)

.................5_cards...7_cards..........||..........6_cards...8_cards

No_Pair...........50.1%.....17.4%...........||...........32.5%......7.1%
One_Pair..........42.3%.....43.8%...........||...........47.8%.....31.4%
Two_Pair...........4.75%....23.5%...........||...........12.1%.....28.4%
Triplets...........2.11%.....4.83%..........||............3.60%.....5.12%
5-Straight.........0.39%.....4.62%..........||............1.60%.....6.57%
5-Flush............0.20%.....3.02%..........||............0.98%.....5.86%
Ord-Boat...........0.14%.....2.60%..........||............0.81%.....5.09%
Three-Pair...........--........--...........||............0.30%.....6.69%
6-Straight...........--........--...........||............0.18%.....2.51%
Four-of............0.024%....0.17%..........||............0.068%....0.22%
6-Flush..............--........--...........||............0.034%....0.67%
Two-Trips............--........--...........||............0.0061%...0.16%
Monster-Boat.........--........--...........||............0.0046%...0.12%
5-StraightFlush....0.002%....0.031%.........||............0.0089%...0.077%
6-StraightFlush......--........--...........||............0.0002%...0.0048%


Notes:
(a) As shown, 5-Straight Flushes are not hugely more common with 6 cards compared with 5. In the 8-card Hold'em game, they're less than thrice as likely as in regular Hold'em.
(b) Similarly the Ordinary Full Boat is less than twice as likely.
(c) The extra two cards lead to changes in the relative odds. A straight is almost as likely as Triplets in the 7-card column contrast this with 5-card. Of course they won't be so likely in real Hold'em players are much more likely to play a pocket pair than a 2-card possible straight!
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