Flop Analysis Part 7 – 290 Million Flops

Jan 12th, 2010 | Posted by spadebidder

If you haven’t yet read the first six parts of this series, it will be helpful to understand some of the data shown here.   Please click the Flop Analysis link at the top of the page to read the whole series.  

This post will show a series of scans of real NLHE flops and how they break down statistically.  I scanned 571 million NLHE hands and 290 million flops were dealt. Unfortunately this data is going to disappoint the Action Flop theorists. That said, the quantified removal effects are interesting and I think the data adds some worthwhile new material to the poker knowledge base.   As we have predicted, flops are not entirely random but are subject to certain specific card removal effects.   However, with the exception of Rank Bias the effects are so small as to not have much practical effect on the play of the game.

 

So before I get into the bias analysis and drill down to the parts-per-ten-million, and start calculating the standard deviations from the mean, first here is a simple higher-level chart rounded to five decimal places, or thousandths of a percent, just to show how close things are. You have to get finer-grained than this just to evaluate most of the card removal effects. This is from 98 million flops at full ring, using Dataset 1 (in more detail afterward).

[ Flop Combinations ]    Calculated    Actual    +/- per 100,000 flops

[ Pair & cnctr  rainbow]     1.412%    1.415%     3
[ Pair & cnctr 2-suited]     1.412%    1.413%     1
[ Pair & 1gap   rainbow]     1.303%    1.303%     0
[ Pair & 1gap  2-suited]     1.303%    1.307%     4
[ Pair & 2gap   rainbow]     1.195%    1.196%     1
[ Pair & 2gap  2-suited]     1.195%    1.196%     1
[ Pair & 3+gap  rainbow]     4.561%    4.574%    13
[ Pair & 3+gap 2-suited]     4.561%    4.573%    12
[ Triplets             ]     0.235%    0.235%     0
[ 3-Straight    rainbow]     1.303%    1.303%     0
[ 3-Straight   2-suited]     1.955%    1.954%    -1
[ 3-Straight-Flush     ]     0.217%    0.217%     0
[ Cnctr & 1gap  rainbow]     2.606%    2.603%    -3
[ Cnctr & 1gap monotone]     0.434%    0.435%     1
[ Cnctr & 1gap 2-suited]     3.909%    3.901%    -8
[ Cnctr & 2gap  rainbow]     2.389%    2.382%    -7
[ Cnctr & 2gap monotone]     0.398%    0.400%     2
[ Cnctr & 2gap 2-suited]     3.584%    3.577%    -7
[ Cnctr & 3+gp  rainbow]     7.710%    7.714%     4
[ Cnctr & 3+gp monotone]     1.285%    1.288%     3
[ Cnctr & 3+gp 2-suited]    11.566%   11.565%    -1
[ KA2 dbl-cnct  rainbow]     0.109%    0.107%    -2
[ KA2 dbl-cnct monotone]     0.018%    0.018%     0
[ KA2 dbl-cnct 2-suited]     0.163%    0.161%    -2
[ Dbl gutshot   rainbow]     1.195%    1.193%    -2
[ Dbl gutshot  monotone]     0.199%    0.199%     0
[ Dbl gutshot  2-suited]     1.792%    1.791%    -1
[ Other 1-gaps  rainbow]     8.036%    8.034%    -2
[ Other 1-gaps monotone]     1.339%    1.338%    -1
[ Other 1-gaps 2-suited]    12.054%   12.043%   -11
[ No cnt/gp/pr  rainbow]     7.710%    7.711%     1
[ No cnt/gp/pr monotone]     1.285%    1.288%     3
[ No cnt/gp/pr 2-suited]    11.566%   11.567%     1
[--check combinations--]   100.000%  100.000%     0  (w/o rounding)

I think it’s pretty striking that virtually every pattern (except pairs, which we predicted) is accurate to within a hundredth of a percent. That’s 1 flop off per 10,000 trials. And about half the flop types are accurate to the thousandth of a percent, or 1 flop off (or zero) per 100,000 trials. With a sample this large, that really shouldn’t be surprising mathematically, but it amazes me. Not because I thought there really would be such a thing as Action Flops. But because the Law of Large Numbers (AKA the long run) really does work in poker. It’s very cool to see that.

So that’s a high level view. Now we’ll drill a lot deeper and do the real analysis.

—————

I’m going to start out with a batch of 176 million full ring hands, and discuss the biases that appear in the flop distribution. A full ring table should generally have the largest card removal effects because it has the most folded hands before the flop. An exception to this is the Rank Bias, which we know is stronger at Heads-Up (see Part 2).

I’ll also show batches for other size games and from other sites for comparison, without repeating the explanations unless something unusual appears in the other batches.

I’ve marked the significant removal effects in red and commented on them below each section. I chose to highlight them only when two conditions are true: 

  1. The effect size is at least +/- 4 per 100,000 flops (once per 25,000).  I chose this somewhat arbitrarily based on looking at the range of actual results.
  2.   AND

  3. The significance of the effect is at least 2.33 standard deviations from the calculated mean.  This gives us 98% confidence that the effect is not random variance. When the significance is over 3 standard deviations, we have over 99.7% confidence, and over 4 SD gives 99.99% confidence.

 

Dataset 1 – Hands scanned: 176,262,777

                          Site B, 7 to 9 active players, NL Hold’em
                          Flops seen:   98,107,274   (55.7%)

FLOP TYPES FREQUENCY

---------------------------------------------------------------------
[ Flop Type ]            Calculated    Actual +-/100K +/-SDs  Example

[ Rainbow              ]  39.76471%  39.77033%     6   1.14  6h-9c-Qs
[ Monotone             ]   5.17647%   5.18326%     7   3.04  Jh-3h-Ah  (1)
[ Two-suited           ]  55.05882%  55.04641%   -12  -2.47  Qd-8d-Ah
[---check suit types---] 100.00000% 100.00000%

[ Paired flop          ]  16.94118%  16.97668%    36   9.38  3h-3s-4h  (2)
[ Triplet flop         ]   0.23529%   0.23541%     0   0.23  Th-Ts-Td
[ Unpaired flop        ]  82.82353%  82.78791%   -36  -9.35  Qd-8d-Ah
[--check match types---] 100.00000% 100.00000%

(1) The +3SD for monotone flops is significant, and means that players are favoring suited hands for seeing the flop slightly more often than their natural ratio of 24% of all starting hands. So we infer that slightly more than 24% of the hands seeing the flop are suited (or two opponents hold the same two suits, which has a similar effect). See Part 5 on Suit Bias. The extras came out of the two-suited here, and the remaining 1SD that shifted from two-suited to rainbow isn’t significant enough to mean anything.

(2) We predicted the Pair Bias at +50/100K flops and this batch came out at +36.   This bias is expected and normal and means players tend to see flops when holding paired starting hands.  See Part 3 on Pair Bias.

 

[ Flop Type ]            Calculated    Actual +-/100K +/-SDs  Example

[ Pair & connector     ]   2.82353%   2.82821%     5   2.80  3h-3s-4h  (2)
[ Pair & 1gap          ]   2.60633%   2.60975%     3   2.12  5c-3d-5h
[ Pair & 2gap          ]   2.38914%   2.39162%     2   1.61  7d-Tc-Th
[ Pair & 3+gap         ]   9.12217%   9.14710%    25   8.58  Js-2d-2c  (2)(3b)
[ Triplets             ]   0.23529%   0.23541%     0   0.23  Th-Ts-Td
[ 3-Straight           ]   3.47511%   3.47383%    -1  -0.69  Th-Qh-Jd
[ Connector & 1gap     ]   6.95023%   6.93927%   -11  -4.27  2c-4c-5h  (3)
[ Connector & 2gap     ]   6.37104%   6.35889%   -12  -4.93  7d-6s-Ts  (3)
[ Connector & 3+gap    ]  20.56109%  20.56613%     5   1.24  Tc-6h-5h
[ KA2 double connector ]   0.28959%   0.28572%    -4  -7.14  2d-Ad-Kh  (4)
[ Double gutshot       ]   3.18552%   3.18325%    -2  -1.28  2s-4d-6h
[ Other 1gaps          ]  21.42986%  21.41538%   -14  -3.50  Qd-8d-Ah  (5)
[ No cnct no 1gap no pr]  20.56109%  20.56545%     4   1.07  6h-9c-Qs
[-check connect types--] 100.00000% 100.00000%

(3)We predicted that “Connector & 1gap” and “Connector & 2gap” flops would come up short due to the natural rank bias inherent to that pattern.   (3b)We also predicted the 3gap types would be a little high because they use Twos 33% more often than Aces, and more Twos than Aces hit the flop due to the Rank Bias. See Part 4

(4) The KA2 flops are expected to be highly skewed by Rank Bias since it is a rank-specific flop that always needs 2/3 high cards, which we know come up light in the flop distribution.   See Part 2 on Rank Bias, and the second half of Part 4 for more details.

(5) As for the “Other 1gaps”, it’s a pretty rank-neutral pattern, but those extra pairs had to come from somewhere. Look at the combination section below and notice that the skew is all in the “Other 1-gaps 2-suited” flops and makes up almost all the shortage for two-suited hands (see the first section). I could break this one down in more detail since we’ve lumped over 20% of flops in there.  I’ll bookmark it for later.

 

The combinations below are all intersections of the broader types above, and refer to the same footnotes in the right margin.

[ Flop Combinations ]    Calculated    Actual +-/100K +/-SDs  Example

[ Pair & cnctr  rainbow]   1.41176%   1.41527%     4   2.94  7c-6s-6d  (2)
[ Pair & cnctr 2-suited]   1.41176%   1.41295%     1   0.99  3h-3s-4h
[ Pair & 1gap   rainbow]   1.30317%   1.30304%     0  -0.11  5c-3d-5h
[ Pair & 1gap  2-suited]   1.30317%   1.30670%     4   3.09  2d-2s-4d  (2)
[ Pair & 2gap   rainbow]   1.19457%   1.19605%     1   1.35  7d-Tc-Th
[ Pair & 2gap  2-suited]   1.19457%   1.19557%     1   0.91  5c-5s-2c
[ Pair & 3+gap  rainbow]   4.56109%   4.57406%    13   6.16  Js-2d-2c  (2)(3b)
[ Pair & 3+gap 2-suited]   4.56109%   4.57305%    12   5.68  Ts-6s-6h  (2)(3b)
[ Triplets             ]   0.23529%   0.23541%     0   0.23  Th-Ts-Td
[ 3-Straight    rainbow]   1.30317%   1.30271%     0  -0.40  6d-7s-8h
[ 3-Straight   2-suited]   1.95475%   1.95398%    -1  -0.55  Th-Qh-Jd
[ 3-Straight-Flush     ]   0.21719%   0.21715%     0  -0.10  Th-9h-Jh
[ Cnctr & 1gap  rainbow]   2.60633%   2.60256%    -4  -2.35  Th-8s-Jd
[ Cnctr & 1gap monotone]   0.43439%   0.43526%     1   1.31  As-Js-Qs
[ Cnctr & 1gap 2-suited]   3.90950%   3.90146%    -8  -4.11  2c-4c-5h  (3)
[ Cnctr & 2gap  rainbow]   2.38914%   2.38163%    -8  -4.87  2h-As-5c  (3)
[ Cnctr & 2gap monotone]   0.39819%   0.40030%     2   3.33  Js-8s-7s
[ Cnctr & 2gap 2-suited]   3.58371%   3.57695%    -7  -3.60  7d-6s-Ts  (1)(3)
[ Cnctr & 3+gp  rainbow]   7.71041%   7.71365%     3   1.21  8s-Ac-2d
[ Cnctr & 3+gp monotone]   1.28507%   1.28773%     3   2.34  4d-5d-Kd
[ Cnctr & 3+gp 2-suited]  11.56561%  11.56474%    -1  -0.27  Tc-6h-5h
[ KA2 dbl-cnct  rainbow]   0.10860%   0.10711%    -1  -4.47  As-Kd-2h
[ KA2 dbl-cnct monotone]   0.01810%   0.01783%     0  -1.96  Ac-2c-Kc
[ KA2 dbl-cnct 2-suited]   0.16290%   0.16077%    -2  -5.21  2d-Ad-Kh
[ Dbl gutshot   rainbow]   1.19457%   1.19323%    -1  -1.22  2s-4d-6h
[ Dbl gutshot  monotone]   0.19910%   0.19927%     0   0.38  Ah-Qh-Th
[ Dbl gutshot  2-suited]   1.79186%   1.79076%    -1  -0.82  Tc-Ah-Qh
[ Other 1-gaps  rainbow]   8.03620%   8.03427%    -2  -0.70  Qc-7d-As
[ Other 1-gaps monotone]   1.33937%   1.33814%    -1  -1.06  Jh-3h-Ah
[ Other 1-gaps 2-suited]  12.05430%  12.04296%   -11  -3.45  Qd-8d-Ah  (5)
[ No cnt/gp/pr  rainbow]   7.71041%   7.71134%     1   0.35  6h-9c-Qs
[ No cnt/gp/pr monotone]   1.28507%   1.28758%     3   2.21  6c-2c-Jc
[ No cnt/gp/pr 2-suited]  11.56561%  11.56652%     1   0.28  Kh-7s-4s
[--check combinations--] 100.00000% 100.00000%

 

These additional flops below are all rank-specific and so are heavily skewed by rank bias (making the SD measure useless). The percentages correspond closely to what would be expected as described in Part 2.  For example, the “any A or K” flops are about 0.4% under the calculation, which is about how much Aces and Kings combined come up short in the rank distribution. 

[More Interesting flops] Calculated    Actual +-/100K +/-SDs  Example

[ Any A or K or both   ]  40.07240%  39.67449%  -398     NA  Qd-8d-Ah
[ Triple-BWay unpaired ]   2.89593%   2.82533%   -71     NA  Kd-Qd-Jd
[ Triple-BWay paired   ]   2.17195%   2.12065%   -51     NA  Qs-Kd-Kh
[ Double-BWay unpaired ]  23.16742%  22.90982%  -258     NA  Qd-8d-Ah
[ Double-BWay paired   ]   4.34389%   4.29886%   -45     NA  7d-Tc-Th
[ 3 to a Wheel no pr   ]   2.89593%   2.92157%    26     NA  2c-4c-5h
[ 2 to a Wheel no pr   ]  23.16742%  23.26574%    98     NA  2h-Jd-5c

————————————-

 

And here’s how the rank distribution breaks down for the main types (same sample set).  These offsets are as compared to the calculated frequency in the second half of part 2.

rank-dist-types

 

So there’s no surprises there.  I’ve highlighted the triplet and paired flops to show how rank bias affects those vs. unpaired flops.   As you can see, the bias effect should be roughly doubled for paired flops and roughly tripled for triplet flops.

Now we’re just going to look at some more datasets.

————— 

 

Dataset 2- Hands scanned: 31,527,060

                         Site A,  7 to 9 active players, NL Hold’em
                         Flops seen:   18,357,792   (58.2%)

FLOP TYPES FREQUENCY

---------------------------------------------------------------------
[ Flop Type ]            Calculated    Actual +-/100K +/-SDs  Example

[ Rainbow              ]  39.76471%  39.75206%   -13  -1.11  8h-8c-4s
[ Monotone             ]   5.17647%   5.18843%    12   2.31  3s-Ks-6s
[ Two-suited           ]  55.05882%  55.05951%     1   0.06  Qh-5h-7s
[---check suit types---] 100.00000% 100.00000%   

[ Paired flop          ]  16.94118%  16.96161%    20   2.33  8h-8c-4s  (2)
[ Triplet flop         ]   0.23529%   0.23561%     0   0.28  Qc-Qs-Qd
[ Unpaired flop        ]  82.82353%  82.80278%   -21  -2.36  Qh-5h-7s
[--check match types---] 100.00000% 100.00000%   

[ Pair & connector     ]   2.82353%   2.83147%     8   2.05  9h-8s-8h
[ Pair & 1gap          ]   2.60633%   2.60543%    -1  -0.24  5s-5d-7c
[ Pair & 2gap          ]   2.38914%   2.38646%    -3  -0.75  Qc-9h-Qs
[ Pair & 3+gap         ]   9.12217%   9.13826%    16   2.39  8h-8c-4s  (2)(3b)
[ Triplets             ]   0.23529%   0.23561%     0   0.28  Qc-Qs-Qd
[ 3-Straight           ]   3.47511%   3.47858%     3   0.81  5s-7c-6d
[ Connector & 1gap     ]   6.95023%   6.95003%     0  -0.03  Ah-3h-Kh
[ Connector & 2gap     ]   6.37104%   6.36171%    -9  -1.64  Ts-Ad-Ks
[ Connector & 3+gap    ]  20.56109%  20.56818%     7   0.75  Jc-2s-3c
[ KA2 double connector ]   0.28959%   0.28646%    -3  -2.50  2h-Kh-Ac
[ Double gutshot       ]   3.18552%   3.17738%    -8  -1.99  4d-2h-6c
[ Other 1gaps          ]  21.42986%  21.42096%    -9  -0.93  Qh-5h-7s
[ No cnct no 1gap no pr]  20.56109%  20.55948%    -2  -0.17  2s-Ks-5c
[-check connect types--] 100.00000% 100.00000%   

[ Flop Combinations ]    Calculated    Actual +-/100K +/-SDs  Example

[ Pair & cnctr  rainbow]   1.41176%   1.41450%     3   0.99  4d-5s-5h
[ Pair & cnctr 2-suited]   1.41176%   1.41697%     5   1.89  9h-8s-8h
[ Pair & 1gap   rainbow]   1.30317%   1.30249%    -1  -0.26  5s-5d-7c
[ Pair & 1gap  2-suited]   1.30317%   1.30295%     0  -0.08  9h-Jh-9d
[ Pair & 2gap   rainbow]   1.19457%   1.19582%     1   0.49  Qc-9h-Qs
[ Pair & 2gap  2-suited]   1.19457%   1.19063%    -4  -1.55  Js-8s-Jh
[ Pair & 3+gap  rainbow]   4.56109%   4.56648%     5   1.11  8h-8c-4s
[ Pair & 3+gap 2-suited]   4.56109%   4.57178%    11   2.20  2h-2c-Jh
[ Triplets             ]   0.23529%   0.23561%     0   0.28  Qc-Qs-Qd
[ 3-Straight    rainbow]   1.30317%   1.30263%    -1  -0.20  5s-7c-6d
[ 3-Straight   2-suited]   1.95475%   1.95678%     2   0.63  Ac-2h-3h
[ 3-Straight-Flush     ]   0.21719%   0.21917%     2   1.81  3d-2d-4d
[ Cnctr & 1gap  rainbow]   2.60633%   2.61049%     4   1.12  9s-Qh-Jd
[ Cnctr & 1gap monotone]   0.43439%   0.43406%     0  -0.22  Ah-3h-Kh
[ Cnctr & 1gap 2-suited]   3.90950%   3.90549%    -4  -0.89  8h-Tc-7c
[ Cnctr & 2gap  rainbow]   2.38914%   2.38591%    -3  -0.91  8h-9d-5s
[ Cnctr & 2gap monotone]   0.39819%   0.39633%    -2  -1.26  7d-4d-8d
[ Cnctr & 2gap 2-suited]   3.58371%   3.57947%    -4  -0.98  Ts-Ad-Ks
[ Cnctr & 3+gp  rainbow]   7.71041%   7.70137%    -9  -1.45  As-8d-Kc
[ Cnctr & 3+gp monotone]   1.28507%   1.29014%     5   1.93  Ks-7s-Qs
[ Cnctr & 3+gp 2-suited]  11.56561%  11.57668%    11   1.48  Jc-2s-3c
[ KA2 dbl-cnct  rainbow]   0.10860%   0.10757%    -1  -1.34  As-Kc-2d
[ KA2 dbl-cnct monotone]   0.01810%   0.01799%     0  -0.36  2h-Kh-Ah
[ KA2 dbl-cnct 2-suited]   0.16290%   0.16091%    -2  -2.11  2h-Kh-Ac
[ Dbl gutshot   rainbow]   1.19457%   1.19266%    -2  -0.75  4d-2h-6c
[ Dbl gutshot  monotone]   0.19910%   0.19883%     0  -0.25  Qs-3s-As
[ Dbl gutshot  2-suited]   1.79186%   1.78588%    -6  -1.93  5d-Ad-3s
[ Other 1-gaps  rainbow]   8.03620%   8.03122%    -5  -0.78  5d-3c-8s
[ Other 1-gaps monotone]   1.33937%   1.34495%     6   2.08  Tc-8c-2c
[ Other 1-gaps 2-suited]  12.05430%  12.04479%   -10  -1.25  Qh-5h-7s
[ No cnt/gp/pr  rainbow]   7.71041%   7.70532%    -5  -0.82  Ks-4h-7c
[ No cnt/gp/pr monotone]   1.28507%   1.28697%     2   0.72  3s-Ks-6s
[ No cnt/gp/pr 2-suited]  11.56561%  11.56719%     2   0.21  2s-Ks-5c
[--check combinations--] 100.00000% 100.00000%   

[More Interesting flops] Calculated    Actual +-/100K +/-SDs  Example

[ Any A or K or both   ]  40.07240%  39.70195%  -370     NA  5s-3s-Kd
[ Triple-BWay unpaired ]   2.89593%   2.83013%   -66     NA  Qc-Kc-Ac
[ Triple-BWay paired   ]   2.17195%   2.12843%   -44     NA  Qh-Qc-Js
[ Double-BWay unpaired ]  23.16742%  22.91975%  -248     NA  As-8d-Kc
[ Double-BWay paired   ]   4.34389%   4.29206%   -52     NA  Qc-9h-Qs
[ 3 to a Wheel no pr   ]   2.89593%   2.92546%    30     NA  5d-Ad-3s
[ 2 to a Wheel no pr   ]  23.16742%  23.27601%   109     NA  5s-3s-Kd

Note that this dataset doesn’t include any games with stakes below $0.25BB, and the first set does. That probably explains why this second one is a little tighter, with smaller removal effect sizes, as there are fewer average players to the flop here. Nevertheless, the results are essentially the same and only differ by degree. It’s also a smaller sample, making the standard error smaller. Offsets under 2.33 SD are not highlighted.

—————

 

Dataset 3- Hands scanned: 40,177,601

                         Site B, Heads-Up NL Hold’em
                         Flops seen: 14,722,445 (36.6%)

FLOP TYPES FREQUENCY

---------------------------------------------------------------------
[ Flop Type ]            Calculated    Actual +-/100K +/-SDs  Example

[ Rainbow              ]  39.76471%  39.76897%     4   0.33  8h-Jd-Qc
[ Monotone             ]   5.17647%   5.18179%     5   0.92  6c-Kc-7c
[ Two-suited           ]  55.05882%  55.04924%   -10  -0.74  7h-2h-Tc
[---check suit types---] 100.00000% 100.00000%   

[ Paired flop          ]  16.94118%  16.93620%    -5  -0.51  8c-8d-2s
[ Triplet flop         ]   0.23529%   0.23779%     2   1.97  As-Ac-Ad
[ Unpaired flop        ]  82.82353%  82.82601%     2   0.25  6c-Kc-7c
[--check match types---] 100.00000% 100.00000%   

[ Pair & connector     ]   2.82353%   2.81728%    -6  -1.45  Js-Tc-Jc
[ Pair & 1gap          ]   2.60633%   2.59817%    -8  -1.97  Ac-Qs-Qh
[ Pair & 2gap          ]   2.38914%   2.38045%    -9  -2.18  2h-2d-5d
[ Pair & 3+gap         ]   9.12217%   9.14030%    18   2.42  8c-8d-2s  (3b)
[ Triplets             ]   0.23529%   0.23779%     2   1.97  As-Ac-Ad
[ 3-Straight           ]   3.47511%   3.47929%     4   0.88  Th-9s-Jc
[ Connector & 1gap     ]   6.95023%   6.93777%   -12  -1.88  Jc-Tc-8s
[ Connector & 2gap     ]   6.37104%   6.34518%   -26  -4.06  8h-Jd-Qc  (3)
[ Connector & 3+gap    ]  20.56109%  20.59322%    32   3.05  6c-Kc-7c  (3b)
[ KA2 double connector ]   0.28959%   0.28797%    -2  -1.16  2c-Kc-Ac
[ Double gutshot       ]   3.18552%   3.18129%    -4  -0.92  9s-Js-7c
[ Other 1gaps          ]  21.42986%  21.43207%     2   0.21  4d-Tc-Qc
[ No cnct no 1gap no pr]  20.56109%  20.56921%     8   0.77  7h-2h-Tc
[-check connect types--] 100.00000% 100.00000%   

[ Flop Combinations ]    Calculated    Actual +-/100K +/-SDs  Example

[ Pair & cnctr  rainbow]   1.41176%   1.40874%    -3  -0.98  Kc-Qs-Kh
[ Pair & cnctr 2-suited]   1.41176%   1.40854%    -3  -1.05  Js-Tc-Jc
[ Pair & 1gap   rainbow]   1.30317%   1.29747%    -6  -1.93  Ac-Qs-Qh
[ Pair & 1gap  2-suited]   1.30317%   1.30069%    -2  -0.84  2d-4h-2h
[ Pair & 2gap   rainbow]   1.19457%   1.18997%    -5  -1.62  Tc-Ks-Kh
[ Pair & 2gap  2-suited]   1.19457%   1.19048%    -4  -1.44  2h-2d-5d
[ Pair & 3+gap  rainbow]   4.56109%   4.56826%     7   1.32  8c-8d-2s
[ Pair & 3+gap 2-suited]   4.56109%   4.57205%    11   2.02  Ac-9c-As
[ Triplets             ]   0.23529%   0.23779%     2   1.97  As-Ac-Ad
[ 3-Straight    rainbow]   1.30317%   1.30743%     4   1.44  Th-9s-Jc
[ 3-Straight   2-suited]   1.95475%   1.95463%     0  -0.03  Jc-Tc-9s
[ 3-Straight-Flush     ]   0.21719%   0.21724%     0   0.04  Kc-Qc-Jc
[ Cnctr & 1gap  rainbow]   2.60633%   2.60441%    -2  -0.46  7s-9d-Tc
[ Cnctr & 1gap monotone]   0.43439%   0.43102%    -3  -1.97  3s-5s-6s
[ Cnctr & 1gap 2-suited]   3.90950%   3.90235%    -7  -1.42  Jc-Tc-8s
[ Cnctr & 2gap  rainbow]   2.38914%   2.37624%   -13  -3.24  8h-Jd-Qc  (3)
[ Cnctr & 2gap monotone]   0.39819%   0.39722%    -1  -0.59  6d-9d-Td
[ Cnctr & 2gap 2-suited]   3.58371%   3.57172%   -12  -2.47  Js-Th-As  (3)
[ Cnctr & 3+gp  rainbow]   7.71041%   7.71895%     9   1.23  5h-4c-Ks
[ Cnctr & 3+gp monotone]   1.28507%   1.28752%     2   0.83  6c-Kc-7c
[ Cnctr & 3+gp 2-suited]  11.56561%  11.58676%    21   2.54  Tc-Jh-2h  (3b)
[ KA2 dbl-cnct  rainbow]   0.10860%   0.10843%     0  -0.20  2h-Ks-Ac
[ KA2 dbl-cnct monotone]   0.01810%   0.01785%     0  -0.71  2c-Kc-Ac
[ KA2 dbl-cnct 2-suited]   0.16290%   0.16169%    -1  -1.15  Kh-2d-Ah
[ Dbl gutshot   rainbow]   1.19457%   1.19207%    -2  -0.88  2c-4d-6h
[ Dbl gutshot  monotone]   0.19910%   0.19903%     0  -0.06  3s-Qs-As
[ Dbl gutshot  2-suited]   1.79186%   1.79019%    -2  -0.48  9s-Js-7c
[ Other 1-gaps  rainbow]   8.03620%   8.04090%     5   0.66  Td-Qh-7s
[ Other 1-gaps monotone]   1.33937%   1.34402%     5   1.55  6h-Ah-Qh
[ Other 1-gaps 2-suited]  12.05430%  12.04714%    -7  -0.84  4d-Tc-Qc
[ No cnt/gp/pr  rainbow]   7.71041%   7.71832%     8   1.14  9h-Qc-6d
[ No cnt/gp/pr monotone]   1.28507%   1.28790%     3   0.96  3d-8d-Qd
[ No cnt/gp/pr 2-suited]  11.56561%  11.56300%    -3  -0.31  7h-2h-Tc
[--check combinations--] 100.00000% 100.00000%   

[More Interesting flops] Calculated    Actual +-/100K +/-SDs  Example

[ Any A or K or both   ]  40.07240%  39.41741%  -655     NA  6c-Kc-7c
[ Triple-BWay unpaired ]   2.89593%   2.79392%  -102     NA  Kc-Qc-Jc
[ Triple-BWay paired   ]   2.17195%   2.08573%   -86     NA  Ac-Qs-Qh
[ Double-BWay unpaired ]  23.16742%  22.78819%  -379     NA  4d-Tc-Qc
[ Double-BWay paired   ]   4.34389%   4.26587%   -78     NA  Ac-9c-As
[ 3 to a Wheel no pr   ]   2.89593%   2.97933%    83     NA  2s-5s-3c
[ 2 to a Wheel no pr   ]  23.16742%  23.45926%   292     NA  5h-4c-Ks

 The lack of a pair bias in this dataset is interesting, but it’s clear that heads-up play is less affected by card removal effects overall (except rank bias), as it should be. Rank bias is very clear in heads-up, as shown in Part 2. The Dataset below is for Heads-Up on another site, and it shows a bit more Pair Bias, perhaps by chance or perhaps due to differences in playing style.

—————

 

Dataset 4- Hands scanned: 20,485,290

                         Site A, Heads-Up NL Hold’em
                         Flops seen: 7,245,421 (35.4%)

FLOP TYPES FREQUENCY

---------------------------------------------------------------------
[ Flop Type ]            Calculated    Actual +-/100K +/-SDs  Example

[ Rainbow              ]  39.76471%  39.77496%    10   0.56  5h-4s-Td
[ Monotone             ]   5.17647%   5.17307%    -3  -0.41  Tc-3c-7c
[ Two-suited           ]  55.05882%  55.05197%    -7  -0.37  7d-Qd-Qh
[---check suit types---] 100.00000% 100.00000%   

[ Paired flop          ]  16.94118%  16.98538%    44   3.17  7d-Qd-Qh  (2)
[ Triplet flop         ]   0.23529%   0.23705%     2   0.97  3c-3s-3d
[ Unpaired flop        ]  82.82353%  82.77758%   -46  -3.28  5h-4s-Td
[--check match types---] 100.00000% 100.00000%   

[ Pair & connector     ]   2.82353%   2.83082%     7   1.19  4c-5c-5h
[ Pair & 1gap          ]   2.60633%   2.60997%     4   0.61  Tc-Qc-Qs
[ Pair & 2gap          ]   2.38914%   2.38687%    -2  -0.40  5s-2d-5c
[ Pair & 3+gap         ]   9.12217%   9.15771%    36   3.32  7d-Qd-Qh  (2)(3b)
[ Triplets             ]   0.23529%   0.23705%     2   0.97  3c-3s-3d
[ 3-Straight           ]   3.47511%   3.48489%    10   1.44  6d-7h-8s
[ Connector & 1gap     ]   6.95023%   6.94036%   -10  -1.04  2c-5c-4d
[ Connector & 2gap     ]   6.37104%   6.33800%   -33  -3.64  7d-4s-3c  (3)
[ Connector & 3+gap    ]  20.56109%  20.57396%    13   0.86  5h-4s-Td
[ KA2 double connector ]   0.28959%   0.28665%    -3  -1.47  Ks-2h-Ah
[ Double gutshot       ]   3.18552%   3.17349%   -12  -1.84  Ad-5d-3d
[ Other 1gaps          ]  21.42986%  21.42657%    -3  -0.22  Jd-4d-2s
[ No cnct no 1gap no pr]  20.56109%  20.55366%    -7  -0.49  8d-Qd-3s
[-check connect types--] 100.00000% 100.00000%   

[ Flop Combinations ]    Calculated    Actual +-/100K +/-SDs  Example

[ Pair & cnctr  rainbow]   1.41176%   1.41564%     4   0.88  8c-9s-9d
[ Pair & cnctr 2-suited]   1.41176%   1.41518%     3   0.78  4c-5c-5h
[ Pair & 1gap   rainbow]   1.30317%   1.31299%    10   2.33  5s-3d-3h  (2)
[ Pair & 1gap  2-suited]   1.30317%   1.29697%    -6  -1.47  Tc-Qc-Qs
[ Pair & 2gap   rainbow]   1.19457%   1.19503%     0   0.11  5s-2d-5c
[ Pair & 2gap  2-suited]   1.19457%   1.19184%    -3  -0.68  6s-9c-6c
[ Pair & 3+gap  rainbow]   4.56109%   4.58237%    21   2.75  3c-3h-7d  (2)(3b)
[ Pair & 3+gap 2-suited]   4.56109%   4.57534%    14   1.84  7d-Qd-Qh
[ Triplets             ]   0.23529%   0.23705%     2   0.97  3c-3s-3d
[ 3-Straight    rainbow]   1.30317%   1.30110%    -2  -0.49  6d-7h-8s
[ 3-Straight   2-suited]   1.95475%   1.96461%    10   1.92  Qd-Ah-Kh
[ 3-Straight-Flush     ]   0.21719%   0.21919%     2   1.15  9c-Tc-8c
[ Cnctr & 1gap  rainbow]   2.60633%   2.60640%     0   0.01  2c-5d-3s
[ Cnctr & 1gap monotone]   0.43439%   0.42971%    -5  -1.92  2c-5c-4c
[ Cnctr & 1gap 2-suited]   3.90950%   3.90425%    -5  -0.73  2c-5c-4d
[ Cnctr & 2gap  rainbow]   2.38914%   2.37945%   -10  -1.71  7d-4s-3c
[ Cnctr & 2gap monotone]   0.39819%   0.39782%     0  -0.16  As-Ts-Ks
[ Cnctr & 2gap 2-suited]   3.58371%   3.56073%   -23  -3.33  7s-6h-3s  (3)
[ Cnctr & 3+gp  rainbow]   7.71041%   7.71221%     2   0.18  5h-4s-Td
[ Cnctr & 3+gp monotone]   1.28507%   1.28438%    -1  -0.16  4d-9d-8d
[ Cnctr & 3+gp 2-suited]  11.56561%  11.57737%    12   0.99  As-6c-Ks
[ KA2 dbl-cnct  rainbow]   0.10860%   0.10694%    -2  -1.36  2d-Ac-Ks
[ KA2 dbl-cnct monotone]   0.01810%   0.01760%    -1  -1.00  Kc-Ac-2c
[ KA2 dbl-cnct 2-suited]   0.16290%   0.16212%    -1  -0.52  Ks-2h-Ah
[ Dbl gutshot   rainbow]   1.19457%   1.18758%    -7  -1.73  5s-3c-7d
[ Dbl gutshot  monotone]   0.19910%   0.19796%    -1  -0.69  Ad-5d-3d
[ Dbl gutshot  2-suited]   1.79186%   1.78796%    -4  -0.79  9s-Ks-Jh
[ Other 1-gaps  rainbow]   8.03620%   8.03775%     2   0.15  Qc-5s-3d
[ Other 1-gaps monotone]   1.33937%   1.34045%     1   0.25  Td-5d-Qd
[ Other 1-gaps 2-suited]  12.05430%  12.04837%    -6  -0.49  Jd-4d-2s
[ No cnt/gp/pr  rainbow]   7.71041%   7.70045%   -10  -1.00  9d-Ah-4s
[ No cnt/gp/pr monotone]   1.28507%   1.28597%     1   0.22  Tc-3c-7c
[ No cnt/gp/pr 2-suited]  11.56561%  11.56724%     2   0.14  8d-Qd-3s
[--check combinations--] 100.00000% 100.00000%   

[More Interesting flops] Calculated    Actual +-/100K +/-SDs  Example

[ Any A or K or both   ]  40.07240%  39.38196%  -690     NA  9d-Ah-4s
[ Triple-BWay unpaired ]   2.89593%   2.78772%  -108     NA  Ts-Js-Qs
[ Triple-BWay paired   ]   2.17195%   2.10159%   -70     NA  Tc-Qc-Qs
[ Double-BWay unpaired ]  23.16742%  22.78090%  -387     NA  As-6c-Ks
[ Double-BWay paired   ]   4.34389%   4.26523%   -79     NA  7d-Qd-Qh
[ 3 to a Wheel no pr   ]   2.89593%   2.98426%    88     NA  2c-5c-4d
[ 2 to a Wheel no pr   ]  23.16742%  23.45617%   289     NA  5h-4s-Td

 

—————

Now we look at some 6-max games.

 

Dataset 5- Hands scanned: 80,086,672

                         Site A, 6-max NL Hold’em
                         Flops seen: 39,621,987   (49.5%)

FLOP TYPES FREQUENCY

---------------------------------------------------------------------
[ Flop Type ]            Calculated    Actual +-/100K +/-SDs  Example

[ Rainbow              ]  39.76471%  39.74688%   -18  -2.29  9s-7h-7d
[ Monotone             ]   5.17647%   5.18260%     6   1.74  7d-3d-9d
[ Two-suited           ]  55.05882%  55.07051%    12   1.48  6h-Qh-Kd
[---check suit types---] 100.00000% 100.00000%   

[ Paired flop          ]  16.94118%  16.96057%    19   3.25  9s-7h-7d  (2)
[ Triplet flop         ]   0.23529%   0.23564%     0   0.45  7s-7h-7c
[ Unpaired flop        ]  82.82353%  82.80379%   -20  -3.30  6h-Qh-Kd
[--check match types---] 100.00000% 100.00000%   

[ Pair & connector     ]   2.82353%   2.82687%     3   1.27  8d-7s-7c
[ Pair & 1gap          ]   2.60633%   2.60520%    -1  -0.45  9s-7h-7d
[ Pair & 2gap          ]   2.38914%   2.39412%     5   2.05  9h-Qc-9c
[ Pair & 3+gap         ]   9.12217%   9.13438%    12   2.67  7c-Kc-7d  (2)(3b)
[ Triplets             ]   0.23529%   0.23564%     0   0.45  7s-7h-7c
[ 3-Straight           ]   3.47511%   3.48227%     7   2.46  2s-3h-As  (6)
[ Connector & 1gap     ]   6.95023%   6.94549%    -5  -1.17  5c-7s-8d
[ Connector & 2gap     ]   6.37104%   6.36024%   -11  -2.78  Tc-Jd-7s  (3)
[ Connector & 3+gap    ]  20.56109%  20.56163%     1   0.08  6h-Qh-Kd
[ KA2 double connector ]   0.28959%   0.28238%    -7  -8.45  Kh-As-2d  (4)
[ Double gutshot       ]   3.18552%   3.18453%    -1  -0.35  Jd-7d-9c
[ Other 1gaps          ]  21.42986%  21.41410%   -16  -2.42  Ts-Qc-5s  (5)
[ No cnct no 1gap no pr]  20.56109%  20.57316%    12   1.88  Ts-Kh-4c
[-check connect types--] 100.00000% 100.00000%   

[ Flop Combinations ]    Calculated    Actual +-/100K +/-SDs  Example

[ Pair & cnctr  rainbow]   1.41176%   1.41365%     2   1.01  8d-7s-7c
[ Pair & cnctr 2-suited]   1.41176%   1.41322%     1   0.78  Qc-Kd-Qd
[ Pair & 1gap   rainbow]   1.30317%   1.29960%    -4  -1.98  9s-7h-7d
[ Pair & 1gap  2-suited]   1.30317%   1.30560%     2   1.35  7h-5h-7c
[ Pair & 2gap   rainbow]   1.19457%   1.19455%     0  -0.01  Jd-Jc-8s
[ Pair & 2gap  2-suited]   1.19457%   1.19957%     5   2.90  9h-Qc-9c  (2)
[ Pair & 3+gap  rainbow]   4.56109%   4.56324%     2   0.65  Kh-4c-4s
[ Pair & 3+gap 2-suited]   4.56109%   4.57113%    10   3.03  7c-Kc-7d  (2)(3b)
[ Triplets             ]   0.23529%   0.23564%     0   0.45  7s-7h-7c
[ 3-Straight    rainbow]   1.30317%   1.30488%     2   0.95  Js-Td-Qc
[ 3-Straight   2-suited]   1.95475%   1.95956%     5   2.18  2s-3h-As
[ 3-Straight-Flush     ]   0.21719%   0.21783%     1   0.86  2c-3c-4c
[ Cnctr & 1gap  rainbow]   2.60633%   2.60725%     1   0.36  5c-7s-8d
[ Cnctr & 1gap monotone]   0.43439%   0.43282%    -2  -1.50  9s-Ts-Qs
[ Cnctr & 1gap 2-suited]   3.90950%   3.90542%    -4  -1.32  Ah-Kd-Jd
[ Cnctr & 2gap  rainbow]   2.38914%   2.38889%     0  -0.10  Tc-Jd-7s
[ Cnctr & 2gap monotone]   0.39819%   0.39755%    -1  -0.64  Ac-Tc-Jc
[ Cnctr & 2gap 2-suited]   3.58371%   3.57380%   -10  -3.36  Kd-4d-Ah  (3)
[ Cnctr & 3+gp  rainbow]   7.71041%   7.70633%    -4  -0.96  5h-4d-Jc
[ Cnctr & 3+gp monotone]   1.28507%   1.28660%     2   0.86  3c-Qc-4c
[ Cnctr & 3+gp 2-suited]  11.56561%  11.56869%     3   0.61  6h-Qh-Kd
[ KA2 dbl-cnct  rainbow]   0.10860%   0.10592%    -3  -5.12  Kh-As-2d
[ KA2 dbl-cnct monotone]   0.01810%   0.01781%     0  -1.36  2d-Kd-Ad
[ KA2 dbl-cnct 2-suited]   0.16290%   0.15865%    -4  -6.63  Ks-2s-Ah  (4)
[ Dbl gutshot   rainbow]   1.19457%   1.19405%    -1  -0.30  Tc-Ah-Qd
[ Dbl gutshot  monotone]   0.19910%   0.19983%     1   1.04  Tc-6c-8c
[ Dbl gutshot  2-suited]   1.79186%   1.79066%    -1  -0.57  Jd-7d-9c
[ Other 1-gaps  rainbow]   8.03620%   8.02472%   -11  -2.66  Kc-6h-8d  (5)
[ Other 1-gaps monotone]   1.33937%   1.34212%     3   1.51  7d-3d-9d
[ Other 1-gaps 2-suited]  12.05430%  12.04726%    -7  -1.36  Ts-Qc-5s
[ No cnt/gp/pr  rainbow]   7.71041%   7.70816%    -2  -0.53  Ts-Kh-4c
[ No cnt/gp/pr monotone]   1.28507%   1.28804%     3   1.66  6c-2c-9c
[ No cnt/gp/pr 2-suited]  11.56561%  11.57696%    11   2.23  8s-3h-Ks
[--check combinations--] 100.00000% 100.00000%   

[More Interesting flops] Calculated    Actual +-/100K +/-SDs  Example

[ Any A or K or both   ]  40.07240%  39.49718%  -575     NA  6h-Qh-Kd
[ Triple-BWay unpaired ]   2.89593%   2.80061%   -95     NA  Ks-As-Qs
[ Triple-BWay paired   ]   2.17195%   2.10092%   -71     NA  Ad-Qh-Qc
[ Double-BWay unpaired ]  23.16742%  22.79667%  -371     NA  6h-Qh-Kd
[ Double-BWay paired   ]   4.34389%   4.27921%   -65     NA  Jh-Jd-2d
[ 3 to a Wheel no pr   ]   2.89593%   2.93579%    40     NA  2s-3h-As
[ 2 to a Wheel no pr   ]  23.16742%  23.31399%   147     NA  Ah-Jc-5s

(6) In our part 6 predictions we predicted a surplus of 3-straights due to rank bias, but it didn’t show up as much in the full ring hands as in this 6-max sample.   See the second half of Part 4 for the reasons.

 

—————

This next chart shows the largest NLHE sample set used in this study so far.  It will be helpful to recall the nature of the standard deviation calculation here.  Since standard deviation is proportional to the square root of the sample size, if we double sample size and have some constant skew like a card removal effect, the SD offset should go up by the square root of 2, or about 1.4.  If we quadruple sample size,  SD offset should double (sqrt of 4).  Comparing to our other samples, that will be roughly true in the dataset below even after differences for game structure (6max or HU or full ring). The bigger SD offsets don’t necessarily mean the effect got bigger, they mean the sample size got bigger. You can compare the +/-100K numbers to see the relatively constant effect size (on most types). For example, “Other 1-gaps” above shows -16 (per 100K) and -2.42 (SD) and in the chart below it shows -16 (per 100K) and -4.26 (SD). So we have a constant effect size but the SD is bigger just because of sample size. That means the effect is more significant, not larger. Put another way, when an effect size remains proportionally constant with larger sample sizes, you have higher confidence that the effect is real, i.e. not random variance.

 

Dataset 6- Hands scanned: 222,163,484

                         Site B, 6-max NL Hold’em
                         Flops seen: 112,454,423 (50.6%)

FLOP TYPES FREQUENCY

---------------------------------------------------------------------
[ Flop Type ]            Calculated    Actual +-/100K +/-SDs  Example

[ Rainbow              ]  39.76471%  39.75900%    -6  -1.24  7h-6s-8c
[ Monotone             ]   5.17647%   5.19191%    15   7.39  Kh-7h-2h  (1)
[ Two-suited           ]  55.05882%  55.04909%   -10  -2.07  9s-4c-7s
[---check suit types---] 100.00000% 100.00000%   

[ Paired flop          ]  16.94118%  16.96469%    24   6.65  Jh-5h-Jd  (2)
[ Triplet flop         ]   0.23529%   0.23596%     1   1.46  Th-Td-Ts
[ Unpaired flop        ]  82.82353%  82.79935%   -24  -6.80  9s-4c-7s
[--check match types---] 100.00000% 100.00000%   

[ Pair & connector     ]   2.82353%   2.82790%     4   2.80  Ts-Td-9s  (2)
[ Pair & 1gap          ]   2.60633%   2.60907%     3   1.82  8d-8h-Tc
[ Pair & 2gap          ]   2.38914%   2.38850%    -1  -0.45  9c-9s-Qs
[ Pair & 3+gap         ]   9.12217%   9.13922%    17   6.28  Jh-5h-Jd  (2)(3b)
[ Triplets             ]   0.23529%   0.23596%     1   1.46  Th-Td-Ts
[ 3-Straight           ]   3.47511%   3.47412%    -1  -0.57  7h-6s-8c
[ Connector & 1gap     ]   6.95023%   6.93752%   -13  -5.30  6d-4c-7c  (3)
[ Connector & 2gap     ]   6.37104%   6.35572%   -15  -6.65  Jh-8c-Qc  (3)
[ Connector & 3+gap    ]  20.56109%  20.57457%    13   3.54  4s-5h-Ts  (3b)
[ KA2 double connector ]   0.28959%   0.28447%    -5 -10.12  As-2d-Kh  (4)
[ Double gutshot       ]   3.18552%   3.18191%    -4  -2.18  6h-4s-2h
[ Other 1gaps          ]  21.42986%  21.41339%   -16  -4.26  9s-4c-7s  (5)
[ No cnct no 1gap no pr]  20.56109%  20.57764%    17   4.34  9s-5d-2h  (7)
[-check connect types--] 100.00000% 100.00000%   

[ Flop Combinations ]    Calculated    Actual +-/100K +/-SDs  Example

[ Pair & cnctr  rainbow]   1.41176%   1.41427%     3   2.25  3c-4s-3h
[ Pair & cnctr 2-suited]   1.41176%   1.41363%     2   1.68  Ts-Td-9s
[ Pair & 1gap   rainbow]   1.30317%   1.30483%     2   1.56  8d-8h-Tc
[ Pair & 1gap  2-suited]   1.30317%   1.30424%     1   1.00  3h-5h-5s
[ Pair & 2gap   rainbow]   1.19457%   1.19430%     0  -0.27  9c-Qs-Qh
[ Pair & 2gap  2-suited]   1.19457%   1.19420%     0  -0.36  9c-9s-Qs
[ Pair & 3+gap  rainbow]   4.56109%   4.57138%    10   5.23  5c-9s-9d  (2)(3b)
[ Pair & 3+gap 2-suited]   4.56109%   4.56784%     7   3.43  Jh-5h-Jd  (2)(3b)
[ Triplets             ]   0.23529%   0.23596%     1   1.46  Th-Td-Ts
[ 3-Straight    rainbow]   1.30317%   1.30286%     0  -0.29  7h-6s-8c
[ 3-Straight   2-suited]   1.95475%   1.95329%    -1  -1.12  Qc-Kd-Ad
[ 3-Straight-Flush     ]   0.21719%   0.21797%     1   1.77  Qd-Ad-Kd
[ Cnctr & 1gap  rainbow]   2.60633%   2.60002%    -6  -4.20  2d-3s-5c  (3)
[ Cnctr & 1gap monotone]   0.43439%   0.43515%     1   1.22  5s-2s-3s
[ Cnctr & 1gap 2-suited]   3.90950%   3.90235%    -7  -3.91  6d-4c-7c  (3)
[ Cnctr & 2gap  rainbow]   2.38914%   2.38140%    -8  -5.38  6s-5c-2h  (3)
[ Cnctr & 2gap monotone]   0.39819%   0.39948%     1   2.17  7c-Tc-Jc
[ Cnctr & 2gap 2-suited]   3.58371%   3.57484%    -9  -5.06  Jh-8c-Qc  (3)
[ Cnctr & 3+gp  rainbow]   7.71041%   7.71300%     3   1.03  2d-Ts-9c
[ Cnctr & 3+gp monotone]   1.28507%   1.28927%     4   3.96  7c-8c-Kc
[ Cnctr & 3+gp 2-suited]  11.56561%  11.57231%     7   2.22  4s-5h-Ts
[ KA2 dbl-cnct  rainbow]   0.10860%   0.10672%    -2  -6.05  As-2d-Kh
[ KA2 dbl-cnct monotone]   0.01810%   0.01756%    -1  -4.25  Kc-Ac-2c
[ KA2 dbl-cnct 2-suited]   0.16290%   0.16019%    -3  -7.13  2d-Ah-Kh
[ Dbl gutshot   rainbow]   1.19457%   1.19277%    -2  -1.76  Th-Qs-8d
[ Dbl gutshot  monotone]   0.19910%   0.19921%     0   0.29  Qd-Td-8d
[ Dbl gutshot  2-suited]   1.79186%   1.78993%    -2  -1.54  6h-4s-2h
[ Other 1-gaps  rainbow]   8.03620%   8.02910%    -7  -2.77  4h-Ts-8d
[ Other 1-gaps monotone]   1.33937%   1.34314%     4   3.48  2h-7h-9h
[ Other 1-gaps 2-suited]  12.05430%  12.04114%   -13  -4.28  9s-4c-7s  (5)
[ No cnt/gp/pr  rainbow]   7.71041%   7.71238%     2   0.78  9s-5d-2h
[ No cnt/gp/pr monotone]   1.28507%   1.29013%     5   4.76  Kh-7h-2h
[ No cnt/gp/pr 2-suited]  11.56561%  11.57514%    10   3.16  As-4c-7c  (7)
[--check combinations--] 100.00000% 100.00000%   

[More Interesting flops] Calculated    Actual +-/100K +/-SDs  Example

[ Any A or K or both   ]  40.07240%  39.55853%  -514     NA  As-4c-7c
[ Triple-BWay unpaired ]   2.89593%   2.80487%   -91     NA  Qd-Ad-Kd
[ Triple-BWay paired   ]   2.17195%   2.10372%   -68     NA  Qh-Kd-Qd
[ Double-BWay unpaired ]  23.16742%  22.83191%  -336     NA  Ac-8c-Js
[ Double-BWay paired   ]   4.34389%   4.28280%   -61     NA  Jh-5h-Jd
[ 3 to a Wheel no pr   ]   2.89593%   2.93313%    37     NA  3c-5h-4s
[ 2 to a Wheel no pr   ]  23.16742%  23.31554%   148     NA  9s-5d-2h

(7) This result is odd and may just be an outlier. However, it shows up fairly strong in both of the 6-max samples. It could be that because we dumped 20% of the leftover flops into the “No cnct no 1gap no pr” category, it is just the complement to some of the negative effects. To evaluate whether this result means anything, this category will need to be broken down into smaller patterns (as will the “Other 1-gaps” (5)). I’ll bookmark both for another look later.

 

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Conclusions

 

  • I predicted 50 extra paired flops per 100K, and the result came up a little less but still very significant, and is the largest single effect measured.
  • Rank Bias was already quantified empirically in Part 2 and broken down further to specific patterns in Part 4, resulting in the 3 predictions below.
  • I predicted a shortage of “Connector & 1-gap” and “Connector & 2-gap” due to Rank Bias, and it was significantly short in every dataset, averaging more than -3 SD.
  • I predicted a surplus of “Connector & 3-gap” due to Rank Bias, and this was true in every pattern but the average confidence was less than 98%, so the effect is small if it exists.
  • I predicted extra 3-straights and that turned out to be weak, showing in only two datasets.
  • I said that monotone flops would have a surplus if players tended to fold unsuited hands more than 3x as often as suited hands since they naturally occur at a 3:1 ratio.   The surplus averaged over 3 SD, so players do favor suited hands to see the flop. 

The Rank Bias and Pair Bias effects are consistent and predictable. The Suitedness Bias is smaller and a little more inconsistent, but in some game structures it is significant (relatively speaking). The only effect that may be useful to understand in actual game play seems to be the general Rank Bias effect, but probably not specific patterns. An example of utilizing the general effect is described in the post Testing Barry Greenstein’s claim. The other effects are so small that it takes a huge sample to even recognize them.

Why do poker sites not routinely summarize their statistics like this and publish them? I think the major sites probably already know a lot of what I’ve written about here. They have probably run the stats and found that community cards are not 100% random. But it took me 30 pages of explanation to describe why that happens honestly with a random deal. Unless everything came out nice and neat and evenly distributed, it may just be too much trouble to convince users why Aces will come up less than 1/13 of the flopped cards in the long run, and why pairs show up on the flop 7 standard deviations more often than they “should” in 200 million flops. Publishing such counter-intuitive stats might not be the confidence builder that some people would like to see, it might in fact just create more controversy.

Another reason is summed up nicely by one of the 2+2 mods in this post.

“Because this is nearly impossible. Let’s say I own a poker site, and I publish the hole card distribution. Rigtard A then claims that he flops sets too often, so I show post-flop results. Rigtard B claims there is a new-deposit boomswitch and a cashout boomswitch, so I produce data about players winning and losing when they do each of these things. Rigtard C claims that shortstacks win too much in tournaments. And on and on and on and on it goes.

You’re basically asking them to prove a negative, that poker isn’t rigged. There are just way too many possibilities to make that realistic IMO.”

 

Fortunately I don’t have to worry about any demands to “prove it” or “I don’t believe the data”.  Anyone who truly wants to do the work themselves can reproduce my results.   I have no agenda other than showing objectively that flops are not 100% random and there are quantifiable card removal effects. But the data also shows that the deal is random and player behavior is what alters the flop distribution. I’ll be happy to provide methods and source code and save you a few hundred hours. 

In future posts we’ll be examining Turn & River cards and the patterns they form, and whether suck-outs happen at the expected frequencies, and if not then why not.  I’m also already working on an All-In Analysis with more detail than anyone has ever published before.

Please point out any errors I’ve made in this post or anywhere else. Comments are open and welcome.

 

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