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Old 08-23-2017, 04:16 PM   #1
whosnext
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Six Plus Hold-Em Hand Rankings

This is going to be a long and winding post regarding Six Plus Hold-Em.

My daughter and I were watching some of the recent Poker After Dark live stream in which several of the players were discussing Six Plus Hold-Em, a poker variant in which all of the 2's, 3's, 4's, and 5's are removed from the deck. Sixes are the lowest card (hence the name Six Plus Hold-Em). Of course, there are now 36 cards in the deck rather than the traditional 52.

The players mentioned that the hand rankings of Six Plus Hold-Em are slightly different from the traditional poker hand rankings. My daughter asked me why they would make any changes.

Of course, the traditional poker hand rankings are: royal flush, straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, one pair, high card. These hand rankings reflect how rare a certain type of hand is.

For example, when dealing five cards from a traditional 52-card deck there are fewer flushes than straights, and, thus, a flush is ranked higher than a straight in traditional poker hand rankings. Note that the frequencies change slightly in 7-card hands vs. 5-card hands (these respective frequencies can be easily found on the internet), a point we will come back to shortly.

So the first thing my daughter and I did was to tabulate all 5-card Six Plus Hold-Em hands, using the hand ranking that the players mentioned on the live stream. Of course, a hand is categorized in the highest category it achieves, something that will become relevant when we turn our attention to 7-card hands below.

5-Card Six Plus Hand Frequencies

Category 1: Royal Flush
Clearly there are 4 possible royal flushes (one in each suit).

Category 2: Straight Flush (excluding royal flush)
High card can be K,Q,J,T,9 (note that 9876A is considered a straight in Six Plus poker), so that there are 4*5 = 20 straight flushes.

Category 3: Four of a Kind
9 ranks, other card can be any of (36-4) cards, so 9*32 = 288

Category 4: Flush (not royal or straight flush)
4*[C(9,5)-6] = 4*(126-6) = 4*120 = 480

Category 5: Full House
C(9,2)*2*C(4,3)*C(4,2) = 1,728

Category 6: Three of a Kind (excluding full house)
C(9,1)*C(4,3)*C(8,2)*4*4 = 16,128

Category 7: Straight (not royal or straight flush)
6*(4^5) - (4+20) = 6,120

Category 8: Two Pair (not full house)
C(9,2)*C(4,2)*C(4,2)*(36-8) = 36,288

Category 9: One Pair (not two pair, etc.)
C(9,1)*C(4,2)*C(8,3)*(4^3) = 193,536

Category 10: High Card (no straight, flush, straight-flush, or royal-flush)
C(9,5)*(4^5) - (4+20+480+6120) = 122,400

TOTAL NUMBER OF POSSIBLE HANDS = C(36,5) = 376,992

So, based on the above, it does look like it makes sense to move Flush ahead of Full House in the Six Plus poker hand rankings, but moving Three of a Kind ahead of Straight looks a little funny.

7-Card Six Plus Hand Frequencies

Since the game the players were discussing was Six Plus Hold-Em, we need to look at the frequencies of 7-card hands rather than, or in addition to, 5-card frequencies.

Here are the detailed derivations that my daughter and I slogged through (undoubtedly there are easier or more condensed ways to do this).

Category 1: Royal Flush
4*C((36-5),2) = 1,860

Category 2: Straight Flush (excluding royal flush)
4*5*C((36-6),2) = 8,700

Category 3: Four of a Kind
9*C((36-4),3) = 44,640

Category 4: Flush (not royal or straight flush)
Seven cards are in same suit = 4*C(9,7) = 144
Six cards are in same suit = 4*C(9,6)*C(27,1) = 9,072
Five cards are in same suit = 4*C(9,5)*C(27,2) = 176,904
So we have (144+9072+176904) - (1860+8700) = 175,560

Category 5: Full House
Ranks are [3,3,1] = C(9,2)*C(4,3)*C(4,3)*C(7,1)*C(4,1) = 16,128
Ranks are [3,2,2] = C(9,3)*3*C(4,3)*C(4,2)*C(4,2) = 36,288
Ranks are [3,2,1,1] = C(9,2)*2*C(4,3)*C(4,2)*C(7,2)*C(4,1)*C(4,1) = 580,608
Total = 633,024

Category 6: Three of a Kind (no full house, flush, straight-flush, or royal flush)
Ranks need to be [3,1,1,1,1] = C(9,1)*C(4,3)*C(8,4)*(4^4) - 4*C(9,5)*5*C(3,2) = 637,560

Category 7: Straight (no three of a kind, flush, straight-flush, or royal flush)
Ranks are [2,2,1,1,1] = 6*C(5,2)*C(4,2)*C(4,2)*C(4,1)*C(4,1)*C(4,1) = 138,240

Ranks are [2,1,1,1,1,1] & pair is part of straight:
Ace high straight = 5*C(4,2)*(4^4)*C(4,1)*4 = 122,880
King-Nine high straight = 5*5*C(4,2)*(4^4)*C(3,1)*4 = 460,800

Ranks are [2,1,1,1,1,1] & pair is not part of straight:
Ace high straight = (4^5)*C(4,1)*C(4,2) = 24,576
King-Nine high straight = 5*(4^5)*C(3,1)*C(4,2) = 92,160

(Subtotal of ranks are [2,1,1,1,1,1] = 700,416 (we will use this below))

Ranks are [1,1,1,1,1,1,1]:
Ace high straight = (4^5)*C(4,2)*(4^2) = 98,304
King-Nine high straight = 5*(4^5)*C(3,2)*(4^2) = 245,760

(Subtotal of ranks are [1,1,1,1,1,1,1] = 344,064)

Subtotal of all straights = 1,182,720

Now we need to subtract off straights that are also flushes:

Ranks are [2,2,1,1,1] = 4*6*C(5,2)*C(3,1)*C(3,1) = 2,160

(Subtotal of straights & flushes with ranks [2,2,1,1,1] = 2,160)

Ranks are [2,1,1,1,1,1] with pair part of straight:
Ace high straight with 6 suited cards, 5 in straight = 4*5*3*4 = 240
Ace high straight with 5 suited cards, 5 in straight = 4*5*3*4*3 = 720
Ace high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*3*4*3*4 = 2,880
Ace high straight with 5 suited cards, 4 in straight, pair not in suit = 4*5*C(3,2)*4 = 240

King-Nine high straight with 6 suited cards, 5 in straight = 4*5*5*3*3 = 900
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*5*3*3*3 = 2,700
King-Nine high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*5*3*4*3*3 = 10,800
King-Nine high straight with 5 suited cards, 4 in straight, pair not in suit = 4*5*5*C(3,2)*3 = 900

Ranks are [2,1,1,1,1,1] with pair not part of straight:
Ace high straight with 6 suited cards, 5 in straight = 4*4*1*3 = 48
Ace high straight with 5 suited cards, 5 in straight = 4*4*C(3,2) = 48
Ace high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*3*4*1*3 = 720
Ace high straight with 5 suited cards, 4 in straight, pair not in suit = 0 (impossible)

King-Nine high straight with 6 suited cards, 5 in straight = 4*5*3*1*3 = 180
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*3*C(3,2) = 180
King-Nine high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*5*3*3*1*3 = 2,700
King-Nine high straight with 5 suited cards, 4 in straight, pair not in suit = 0 (impossible)

(Subtotal of straights & flushes with ranks [2,1,1,1,1,1] = 23,256)

Ranks are [1,1,1,1,1,1,1]:
Ace high straight with 7 suited cards, 5 in straight = 4*C(4,2) = 24
Ace high straight with 6 suited cards, 5 in straight = 4*C(4,2)*2*1*3 = 144
Ace high straight with 6 suited cards, 4 in straight = 4*5*3*C(4,2) = 360
Ace high straight with 5 suited cards, 5 in straight = 4*C(4,2)*3*3 = 216
Ace high straight with 5 suited cards, 4 in straight = 4*5*3*C(4,2)*2*1*3 = 2,160
Ace high straight with 5 suited cards, 3 in straight = 4*C(5,2)*3*3*C(4,2) = 2,160

King-Nine high straight with 7 suited cards, 5 in straight = 4*5*C(3,2) = 60
King-Nine high straight with 6 suited cards, 5 in straight = 4*5*C(3,2)*2*1*3 = 360
King-Nine high straight with 6 suited cards, 4 in straight = 4*5*5*3*C(3,2) = 900
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*C(3,2)*3*3 = 540
King-Nine high straight with 5 suited cards, 4 in straight = 4*5*5*3*C(3,2)*2*1*3 = 5,400
King-Nine high straight with 5 suited cards, 3 in straight = 4*5*C(5,2)*3*3*C(3,2) = 5,400

(Subtotal of straights & flushes with ranks [1,1,1,1,1,1,1] = 17,724)

Putting it all together, we have:
= 1,182,720 - (2,160+23,256+17,724)
= 1,139,580

Category 8: Two Pair (no full house, flush or straight)

Ranks are [2,2,2,1] = C(9,3)*C(4,2)*C(4,2)*C(4,2)*C(6,1)*C(4,1) = 435,456

Ranks are [2,2,1,1,1] = C(9,2)*C(4,2)*C(4,2)*C(7,3)*C(4,1)*C(4,1)*C(4,1) = 2,903,040

Now subtract off flushes:
Ranks are [2,2,1,1,1] = 4*C(9,5)*C(5,2)*C(3,1)*C(3,1) = 45,360

Now subtract off straights:
Ranks are [2,2,1,1,1] = 138,240 (from above)

Now add back in all hands with ranks [2,2,1,1,1] that are both straights and flushes:
We know this is 2,160 (from above)

Putting it all together, we have:
= (435,456 + 2,903,040) - (45,360 + 138,240) + 2,160
= 3,157,056

Category 9: One Pair (no straight or flush)
Ranks need to be [2,1,1,1,1,1] = C(9,1)*C(4,2)*C(8,5)*(4^5) = 3,096,576

Now subtract off flushes with ranks [2,1,1,1,1,1]:
Pair is not part of flush = 4*C(9,5)*C(4,1)*C(3,2) = 6,048
Pair is part of flush, other card is suited = 4*C(9,6)*6*C(3,1) = 6,048
Pair is part of flush, other card is not suited = 4*C(9,5)*5*C(3,1)*C(4,1)*C(3,1) = 90,720

(Subtotal of flushes with ranks [2,1,1,1,1,1] is 102,816)

Now subtract off straights with ranks [2,1,1,1,1,1]:
We know this is 700,416 (from above).

Now add back in hands with ranks [2,1,1,1,1,1,] that are both straights and flushes:
We know this is 23,256 (from above).

Putting it all together, we have:
= 3,096,576 - (102,816 + 700,416) + 23,256
= 2,316,600

Category 10: High Card (no straight or flush)
Ranks need to be [1,1,1,1,1,1,1] = C(9,7)*(4^7) = 589,824

Now subtract off flushes with ranks [1,1,1,1,1,1,1]:
Seven cards in suit = 4*C(9,7) = 144
Six cards in suit = 4*C(9,6)*3*3 = 3,024
Five cards in suit = 4*C(9,5)*C(4,2)*3*3 = 27,216

(Subtotal of flushes with ranks [1,1,1,1,1,1,1] is 30,384)

Now subtract off straights with ranks [1,1,1,1,1,1,1]:
Ace high straight = (4^5)*C(4,2)*(4^2) = 98,304
King-Nine high straight = 5*(4^5)*C(3,2)*(4^2) = 245,760

(Subtotal of straights with ranks [1,1,1,1,1,1,1] is 344,064)

Now add back hands with ranks [1,1,1,1,1,1,1] that are both straights and flushes:
We know this is 17,724 (from above)

Putting it all together, we have:
= 589,824 - (30,384 + 344,064) + 17,724
= 233,100

----------

Okay, let's create a table to display all of these 5-card and 7-card frequencies (I'll also add in columns for percentages to make comparisons easier):


Six Plus CategoryFive Card FreqFive Card PctSeven Card FreqSeven Card Pct
Royal Flush
4
0.00
1,860
0.02
Straight Flush
20
0.01
8,700
0.10
Four of a Kind
288
0.08
44,640
0.53
Flush
480
0.13
175,560
2.10
Full House
1,728
0.46
633,024
7.58
Three of a Kind
16,128
4.28
637,560
7.64
Straight
6,120
1.62
1,139,580
13.65
Two Pair
36,288
9.63
3,157,056
37.82
One Pair
193,536
51.34
2,316,600
27.75
High Card
122,400
32.47
233,100
2.79
TOTAL
376,992
100.00
8,347,680
100.00

Whether looking at the 5-card hand frequencies or the 7-card hand frequencies, we see that elevating Flush ahead of Full House makes perfectly good sense in Six Plus poker. Of course, by compressing the deck, flushes are harder to come by whereas full houses are easier to come by.

You will also see that when 7-card hands are considered, it makes perfectly good sense to elevate Three of a Kind ahead of Straights (three of a kind being more rare) in Six Plus Hold-Em. This was not apparent when we looked only at 5-card hands.

I hope that these derivations may provide some value for others.

Any comments are more than welcome.

Last edited by whosnext; 08-23-2017 at 06:15 PM. Reason: fixed typo
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