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Old 08-27-2017, 03:45 PM   #1
whosnext
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Breakdown of 5-card Boards (straighty-flushy-paired boards)

The composition of the five community cards in NLHE has been the topic of conversation many times over the years.

Earlier this week I was watching the PokerStars Championships live stream from Barcelona when the commentators pointed out a "straighty-flushy-paired" board (an example would be Qh Jc Th Td 6h). Such a board is maximally "wet" in that just about anything is possible. Somebody could have a flush, somebody could have a straight, somebody could have a full-house or even quads.

I wondered how often such a board occurs. In truth, in the live stream session I watched there were quite a few of them. I wrote a simple program to tally such boards and below I will report the results of the program.

First, let's do some calculations in preparation (and as a check of the program's tallies). Note that everything in this post will essentially be assuming "random" boards ignoring any skewness due to card-removal effects from players playing high cards, pocket pairs, suited cards, etc. Or another way of saying the same thing, the methodology in this post will be simple counting of all possible boards and putting them into categories.


Breakdown by Rank Count

These are fairly well-known but I thought I would include these for completeness. The distribution of 5-card "hands" in terms of ranks (pairs, trips, etc.) is as follows:

No Pair: rank counts are [1,1,1,1,1]
C(13,5)*C(4,1)*C(4,1)*C(4,1)*C(4,1)*C(4,1) = 1,317,888

One Pair: rank counts are [2,1,1,1]
C(13,4)*4*C(4,2)*C(4,1)*C(4,1)*C(4,1) = 1,098,240

Two Pair: rank counts are [2,2,1]
C(13,3)*3*C(4,2)*C(4,2)*C(4,1) = 123,552

Three of a Kind: rank counts are [3,1,1]
C(13,3)*3*C(4,3)*C(4,1)*C(4,1) = 54,912

Full House: rank counts are [3,2]
C(13,2)*2*C(4,3)*C(4,2) = 3,744

Four of a Kind: rank counts are [4,1]
C(13,2)*2*C(4,4)*C(4,1) = 624

TOTAL: C(52,5) = 2,598,960


Breakdown by Suit Count

Here is the breakdown of 5-card "hands" by the maximum number of cards in any suit. This is key in NLHE to determine if (or how likely) somebody could have a flush. (Of course, if there is no suit with at least three board cards, a flush is impossible.)

Max in any Suit=1: 0 (impossible)

Max in any Suit=2: There are two cases to consider here.

Case 1: Suit counts are [2,2,1] = C(4,3)*3*C(13,2)*C(13,2)*C(13,1) = 949,104

Case 2: Suit counts are [2,1,1,1] = C(4,4)*4*C(13,2)*C(13,1)*C(13,1)*C(13,1) = 685,464

Subtotal = 949,104 + 685,464 = 1,634,568

Max in any Suit=3: There are two cases to consider here.

Case 1: Suit counts are [3,2] = C(4,2)*2*C(13,3)*C(13,2) = 267,696

Case 2: Suit counts are [3,1,1] = C(4,3)*3*C(13,3)*C(13,1)*C(13,1) = 580,008

Subtotal = 267,696 + 580,008 = 847,704

Max in any Suit=4: clearly suit counts are [4,1]
C(4,2)*2*C(13,4)*C(13,1) = 111,540

Max in any Suit=5: clearly suit counts are [5]
C(4,1)*C(13,5) = 5,148

TOTAL: C(52,5) = 2,598,960


Breakdown by Straight Count

This breakdown is a bit more complex. Here we will categorize 5-card "hands" by the maximum number of cards it possesses in any possible straight. Again, to repeat what was said above, this is key in NLHE to determine if (or how likely) somebody could have a straight. (Of course, if there does not exist any combination of three or more cards on board that could make a straight with any two additional cards, then a straight is impossible.)

Max in any Straight=5: There are 10 possible straights (A-T, K-9, ... , 5-A).
Each of the five cards can take any of the four suits. So this is:
= 10*(4^5) = 10,240

Max in any Straight=4: Here there are six cases to consider.

Case 1: exactly 4 cards in sequence and 5th card is not a pair
Must treat A-J and 4-A (2 of these) separately from K-T thru 5-2 (9 of these).
= 2*[(4^5)*8] + 9*[(4^5)*7] = 80,896

Case 2: exactly 4 cards in sequence and 5th card is a pair
= 11*[4*C(4,2)*(4^3)] = 16,896

Case 3: exactly 3 cards in sequence, 4th card is one away, 5th card is not a pair
Must treat AKQ and 32A (2 of these) separately from KQJ and 432 (2 of these) separately from QJT thru 543 (8 of these).
= 2*[(4^5)*8] + 2*[(4^5)*7] + 8*[(4^5)*(7+6)] = 137,216

Case 4: exactly 3 cards in sequence, 4th card is one away, 5th card is a pair
Must treat AKQ, KQJ, 432, 32A (4 of these) separately from QJT thru 543 (8 of these).
= 4*[4*C(4,2)*(4^3))] + 8*[2*4*C(4,2)*(4^3)] = 30,720

Case 5: two sets of two consecutive cards, 5th card is not a pair
Must treat AKJT and 542A (2 of these) separately from KQT9 thru 6532 (8 of these).
= 2*[(4^5)*7] + 8*[(4^5)*6] = 63,488

Case 6: two sets of two consecutive cards, 5th card is a pair
= 10*[4*C(4,2)*(4^3)] = 15,360

Subtotal = 344,576

Max in any Straight=3: I will return to this case below.

Max in any Straight=2: Here there are four cases to consider.

Case 1: Five distinct ranks (clearly ranks are [1,1,1,1,1])
There are 79 combos of five distinct ranks that qualify (AK984, AK983, ... , Q8732).

For each rank combo, clearly there are 4^5 possible suit combos. So the tally for this case is:

= 79*(4^5) = 79*1,024 = 80,896.

Case 2: Four distinct ranks (clearly ranks are [2,1,1,1])
There are 283 combos of four distinct ranks that qualify (AK98, AK97, ... , 8732). So the tally for this case is:

= 283*[4*C(4,2)*C(4,1)*C(4,1)*C(4,1)] = 283*1,536 = 434,688

Case 3: Three distinct ranks
There are 218 combos of three distinct ranks that qualify (AK9, AK8, ... , 732). There are two sub-cases to consider.

Case 3A: Ranks are [3,1,1]
= 218*[3*C(4,3)*C(4,1)*C(4,1)] = 218*192 = 41,856

Case 3B: Ranks are [2,2,1]
= 218*[3*C(4,2)*C(4,2)*C(4,1)] = 218*432 = 94,176

(Case 3 subtotal = 41,856 + 94,176 = 136,032)

Case 4: Two distinct ranks
There are 46 combos of two distinct ranks that qualify (AK, AQ, ... , 32). There are two sub-cases to consider.

Case 4A: Ranks are [4,1]
= 46*[2*C(4,4)*C(4,1)] = 46*8 = 368

Case 4B: Ranks are [3,2]
= 46*[2*C(4,3)*C(4,2)] = 46*48 = 2,208

(Case 4 subtotal = 368 + 2,208 = 2,576)

Subtotal: 80,896 + 434,688 + 136,032 + 2,576 = 654,192

Max in any Straight=1: Here there are four cases to consider.

Case 1: Five distinct ranks (clearly ranks are [1,1,1,1,1])
It is easy to see that this is 0 (impossible)

Case 2: Four distinct ranks (clearly ranks are [2,1,1,1])
It is easy to see that this is 0 (impossible)

Case 3: Three distinct ranks
There are four combos of three distinct ranks that qualify (K83, K82, K72, Q72). There are two sub-cases to consider.

Case 3A: Ranks are [3,1,1]
= 4*[3*C(4,3)*C(4,1)*C(4,1)] = 4*192 = 768

Case 3B: Ranks are [2,2,1]
= 4*[3*C(4,2)*C(4,2)*C(4,1)] = 4*432 = 1,728

(Case 3 subtotal = 768 + 1,728 = 2,496)

Case 4: Two distinct ranks
There are 32 combos of two distinct ranks that qualify (A9, A8, ... , 72). There are two sub-cases to consider.

Case 4A: Ranks are [4,1]
= 32*[2*C(4,4)*C(4,1)] = 32*8 = 256

Case 4B: Ranks are [3,2]
= 32*[2*C(4,3)*C(4,2)] = 32*48 = 1,536

(Case 4 subtotal = 256 + 1,536 = 1,792)

Subtotal: 0 + 0 + 2,496 + 1,792 = 4,288

Max in any Straight=3: This is a fairly complex case so I will simply tally this case by subtracting the other cases from the known grand total.

= 2,598,960 - (4,288 + 654,192 + 344,576 + 10,240)
= 1,585,664


Summary of Breakdowns

The tables below summarize the results we have derived so far.


Here is the breakdown by rank count.

Rank CategoryNumber
No Pair
1,317,888
One Pair
1,098,240
Two Pair
123,552
Three of a Kind
54,912
Full House
3,744
Four of a Kind
624


Here is the breakdown by suit count.

Max in any SuitNumber
One
0
Two
1,634,568
Three
847,704
Four
111,540
Five
5,148


Here is the breakdown by straight count.

Max in any StraightNumber
One
4,288
Two
654,192
Three
1,585,664
Four
344,576
Five
10,240


Detailed Breakdowns

Next let's turn our attention to the meat of our exercise, the detailed breakdowns of mixtures of the three "dimensions" we are considering. For example, how many boards have exactly one pair, exactly three to a suit, and exactly three to a straight?

To answer this question, I wrote a simple computer program to tally how many boards fall into each of our 3-way categories (such as the one mentioned above).

These tables are a bit unwieldly, but hopefully it is clear what information is being displayed.

Rank CategoryMax in any SuitMax in any StraightNumber
No Pair
1
1
0
No Pair
1
2
0
No Pair
1
3
0
No Pair
1
4
0
No Pair
1
5
0
No Pair
2
1
0
No Pair
2
2
47,400
No Pair
2
3
553,800
No Pair
2
4
165,000
No Pair
2
5
6,000
No Pair
3
1
0
No Pair
3
2
28,440
No Pair
3
3
332,280
No Pair
3
4
99,000
No Pair
3
5
3,600
No Pair
4
1
0
No Pair
4
2
4,740
No Pair
4
3
55,380
No Pair
4
4
16,500
No Pair
4
5
600
No Pair
5
1
0
No Pair
5
2
316
No Pair
5
3
3,692
No Pair
5
4
1,100
No Pair
5
5
40


Rank CategoryMax in any SuitMax in any StraightNumber
One Pair
1
1
0
One Pair
1
2
0
One Pair
1
3
0
One Pair
1
4
0
One Pair
1
5
0
One Pair
2
1
0
One Pair
2
2
285,264
One Pair
2
3
394,128
One Pair
2
4
41,328
One Pair
2
5
0
One Pair
3
1
0
One Pair
3
2
135,840
One Pair
3
3
187,680
One Pair
3
4
19,680
One Pair
3
5
0
One Pair
4
1
0
One Pair
4
2
13,584
One Pair
4
3
18,768
One Pair
4
4
1,968
One Pair
4
5
0
One Pair
5
1
0
One Pair
5
2
0
One Pair
5
3
0
One Pair
5
4
0
One Pair
5
5
0


Rank CategoryMax in any SuitMax in any StraightNumber
Two Pair
1
1
0
Two Pair
1
2
0
Two Pair
1
3
0
Two Pair
1
4
0
Two Pair
1
5
0
Two Pair
2
1
1,296
Two Pair
2
2
70,632
Two Pair
2
3
20,736
Two Pair
2
4
0
Two Pair
2
5
0
Two Pair
3
1
432
Two Pair
3
2
23,544
Two Pair
3
3
6,912
Two Pair
3
4
0
Two Pair
3
5
0
Two Pair
4
1
0
Two Pair
4
2
0
Two Pair
4
3
0
Two Pair
4
4
0
Two Pair
4
5
0
Two Pair
5
1
0
Two Pair
5
2
0
Two Pair
5
3
0
Two Pair
5
4
0
Two Pair
5
5
0


Rank CategoryMax in any SuitMax in any StraightNumber
Three of a Kind
1
1
0
Three of a Kind
1
2
0
Three of a Kind
1
3
0
Three of a Kind
1
4
0
Three of a Kind
1
5
0
Three of a Kind
2
1
624
Three of a Kind
2
2
34,008
Three of a Kind
2
3
9,984
Three of a Kind
2
4
0
Three of a Kind
2
5
0
Three of a Kind
3
1
144
Three of a Kind
3
2
7,848
Three of a Kind
3
3
2,304
Three of a Kind
3
4
0
Three of a Kind
3
5
0
Three of a Kind
4
1
0
Three of a Kind
4
2
0
Three of a Kind
4
3
0
Three of a Kind
4
4
0
Three of a Kind
4
5
0
Three of a Kind
5
1
0
Three of a Kind
5
2
0
Three of a Kind
5
3
0
Three of a Kind
5
4
0
Three of a Kind
5
5
0


Rank CategoryMax in any SuitMax in any StraightNumber
Full House
1
1
0
Full House
1
2
0
Full House
1
3
0
Full House
1
4
0
Full House
1
5
0
Full House
2
1
1,536
Full House
2
2
2,208
Full House
2
3
0
Full House
2
4
0
Full House
2
5
0
Full House
3
1
0
Full House
3
2
0
Full House
3
3
0
Full House
3
4
0
Full House
3
5
0
Full House
4
1
0
Full House
4
2
0
Full House
4
3
0
Full House
4
4
0
Full House
4
5
0
Full House
5
1
0
Full House
5
2
0
Full House
5
3
0
Full House
5
4
0
Full House
5
5
0


Rank CategoryMax in any SuitMax in any StraightNumber
Four of a Kind
1
1
0
Four of a Kind
1
2
0
Four of a Kind
1
3
0
Four of a Kind
1
4
0
Four of a Kind
1
5
0
Four of a Kind
2
1
256
Four of a Kind
2
2
368
Four of a Kind
2
3
0
Four of a Kind
2
4
0
Four of a Kind
2
5
0
Four of a Kind
3
1
0
Four of a Kind
3
2
0
Four of a Kind
3
3
0
Four of a Kind
3
4
0
Four of a Kind
3
5
0
Four of a Kind
4
1
0
Four of a Kind
4
2
0
Four of a Kind
4
3
0
Four of a Kind
4
4
0
Four of a Kind
4
5
0
Four of a Kind
5
1
0
Four of a Kind
5
2
0
Four of a Kind
5
3
0
Four of a Kind
5
4
0
Four of a Kind
5
5
0


Summary

The tables in the previous section have so much detail that the information we seek may be difficult to glean. So below I group the results into the most informative way I can think of.

For what follows, a "straighty" board is any board with at least three cards in any straight. And a "flushy" board is any board with at least three cards in any suit.

Here is a table that tallies the number of "straighty-flushy" boards by the board's rank category.

Rank CategoryStraighty-Flushy Boards
No Pair
512,192
One Pair
228,096
Two Pair
6,912
Three of a Kind
2,304
Full House
0
Four of a Kind
0

As you can see there are 228,096 boards that can rightly be called "straighty-flushy-paired" boards. The lion's share of this total, of course, comes from boards with exactly one pair, exactly three to a suit, and exactly three to a straight (there are 187,680 of these).

Using the slightly more generous tally, we see that the percentage of boards that are "straighty-flushy-paired" boards is:

= 228,096 / 2,598,960

= 8.8%.

(In case anyone is interested, 187,680 represents 7.2% of all boards.)

Thus, no matter how you define your terms, I would say that these super-wet "straighty-flushy-paired" boards are uncommon but not exceedingly rare.

Comments welcome.

Last edited by whosnext; 08-27-2017 at 03:57 PM.
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