Of the removal effects I’ve identified, this one alone is fairly well-known. Players tend to see more flops when holding high cards, and so the board will tend to contain more low cards than high cards. The graphs below show that this effect becomes more pronounced for the turn, and even more pronounced for the river, as player requirements to continue in the hand become stricter at later streets. These graphs can be viewed as a mirror image of the rank distribution of the hole cards held when players see the board, since these are dealt from the cards left in the deck. However, the proportion is diluted because on average more than two thirds of players will have folded preflop at full ring NL. So if we could graph the rank distribution for all hole cards held by live players when a flop is seen, the slope would be a lot steeper than these board card graphs (and in the opposite direction).

If there were no card removal effect, all ranks should show up on the board at 1/13 or 7.69%.

## Board Card Distribution in 9-player (min 7 active) NLHE, BB $0.25 to $4.00

** 172 million hands, 96 million flops (56%), 60 million turns seen, 45 million rivers seen**

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*This Flop Analysis section is being posted in 7 parts and the final post will contain a detailed statistical analysis of a large number of NLHE flops (hundreds of millions). If you want to jump around click the topic at the top of the page for a list.*

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One of the most common complaints about online poker sites is that the community cards sometimes don’t seem random. The rigged deal theories seem to fall in two general categories. The first one is that the flop (or turn or river) always favors the underdog/shortstack/bigstack/new player/recent depositor/ or choose your favorite targeted group. The second idea is that flops are manipulated simply to stimulate more action and create larger pots, by allowing players – usually the other player – to connect with the flop more often than they should. This is the **“Action Flop”** theory. Quite often you see the two theories combined into one grand scheme where the action flop builds the pot and then the turn and river are manipulated for the underdog to suck out.

Ignoring the obvious cognitive biases which usually feed these beliefs, and setting aside for now whether a manipulated deal would actually even increase revenue for the sites, here we’ll just concentrate on statistically analyzing a large number of actual hands. For all the other arguments just see the “poker is rigged” threads found on any poker forum. Sometimes they can be quite entertaining, particularly the long-running thread on 2+2.

Here’s what my research shows:

** The community cards seen in Hold’em are not randomly distributed.**

Surprised? Keep reading.

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I ran the scans for a second site with more hands (119 million 9-player), and graphed all the equity boosts for both sites. This graphs shows all the sample sizes, including a baseline for all matching hands (no folds required in front of the bet). Site A shows the results described in the previous post. Site B shows the additional tests.

Notice that 4 folds looks to be about the average before the bet, as it has about the same result as no minimum.

So now we’ve looked at about **152 million hands** of full ring NLHE. The theory required an equity shift of at least 2.17% for AKs to become a favorite over 99-44 (actually closer to 2.3% after seeing that the pair distribution is weighted towards the top). We didn’t find a shift that large, so my earlier conclusions didn’t change. The only positive case is when 7 players fold around to the blinds and then we have a blind vs. blind all-in. And even though the sample size in that case is so small that our offset from the mean is barely over 1 standard deviation (even combining both sites for n=361), I’m convinced that the AKs actually does become a favorite over medium pairs in that one special case. But only that one, which is not the same as the more general theory of becoming a favorite “after several folds”.

My testing of this theory and my conclusions should not be construed in any way as disrespect for Barry Greenstein, who knows far more math than I do and far more about poker than I do. My purpose with this web site is to use a large dataset to try to quantify things no one ever has before, using empirical results. In this case the results don’t quite support the theory.

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In his book **Ace On The River**, Barry Greenstein gives an example of a card removal effect that he says changes the hand equities significantly (more than 2%):

“In hold’em, with no betting after the flop, all pairs are a favorite over Ace-King offsuit, by at least 52% to 48%. All pairs except Deuces with neither card in the Ace-King’s suit are favored over Ace-King suited. **If several players fold first, Ace-King suited is a favorite over most pairs.** (The exceptions are Aces, Kings, and Jacks, and also Tens where one of the Tens is the same suit as the Ace-King.) Even Ace-King offsuit is now a favorite against small pairs. The reason for this is that players are more likely to play hands having an Ace or King than those containing smaller cards. Therefore, **as players fold, the probability of an Ace or King coming on the board increases**.”

(p. 150)(emphasis added)

From our **Rank Bias analysis** we know this last sentence to be absolutely true, but I want to test the specifics of the claimed degree of equity shifting. My guess is that Barry did this mathematically using some assumptions about typical player behavior, but didn’t test it empirically with a large hand sample. For a scenario this specific, it takes a very large sample to find enough trials meeting these criteria, and I have a very large sample.

First, let’s look at the preflop equity of the matchups described, which would be exact expectation in the case of a heads-up game (no one else can fold) where the hand got all-in preflop.

*“…all pairs are a favorite over Ace-King offsuit, by at least 52% to 48%.”*

**AhKd vs. 2h2d** – the worst case pair equity is 52.26%

**AKo vs. 22** – the pair equity is 52.65% (average regardless of suits)

**AKo vs. 88** – the pair equity is 55.16%

**AKo vs. QQ** – the pair equity is 56.76%

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This came up in a 2+2 post, where someone was interested in comparing final pot sizes to the flop type. It turned out to be a little interesting but no big surprises. The original thread with more discussion is here: http://forumserver.twoplustwo.com/15/poker-theory/statistical-analysis-flops-633448/

Here’s the data for a 9-player run and a Heads-Up run, showing the calculated frequency of the flop types, and the actual pot size data. The actual flop frequencies will be shown in the flop analysis articles on this site, where I analyze and explain the card removal effects that alter flop frequencies. I don’t want the focus to be on that in this post, so I’ve omitted them and only shown the calculated probabilities.

**Note that these final pot sizes include the final bet whether it got called or not.** These are also without regard to what other streets were seen, as long as there was a flop.

Site B, 9-player NLHE, min 7 dealt in, BB $ 0.25 to $ 4.00
Number of hands analysed: 90,000,000
Flops seen: 51,073,435 (56.7%)
FLOP TYPES FREQUENCY
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Calculated Avg. Final
[ Flop Type ] Frequency Example Pot in BBs
[ Rainbow ] 39.76471% 9s-3h-5c 24.3
[ Monotone ] 5.17647% 7c-2c-6c 25.5
[ Two-suited ] 55.05882% 3s-Qs-2c 25.9
[---check suit types---] 100.00000%
[ Paired flop ] 16.94118% 6h-Ts-6s 23.8
[ Triplet flop ] 0.23529% 5c-5s-5h 25.4
[ Unpaired flop ] 82.82353% 9s-3h-5c 25.5
[--check match types---] 100.00000%
[ Pair & connector ] 2.82353% 8d-9d-9h 24.4
[ Pair & 1gap ] 2.60633% Jh-9h-Jd 24.3
[ Pair & 2+gap ] 11.51131% 6h-Ts-6s 23.5
[ Triplets ] 0.23529% 5c-5s-5h 25.4
[ 3-Straight ] 3.47511% 2c-3c-4s 29.1
[ Connector & 1gap ] 6.95023% 9c-Jh-8s 27.6
[ Connector & 2+gap ] 26.93213% 3s-Qs-2c 25.5
[ KA2 double connector ] 0.28959% 2d-Kh-Ad 21.3
[ Double gutshot ] 3.18552% Ah-3h-Qd 26.5
[ Other 1gaps ] 21.42986% 9s-3h-5c 25.1
[ No cnct no 1gap no pr] 20.56109% Jd-4s-As 24.5
[-check connect types--] 100.00000%
Calculated Avg. Final
[ Flop Combinations ] Frequency Example Pot in BBs
[ Pair & cnctr rainbow] 1.41176% 4h-5c-4d 23.7
[ Pair & cnctr 2-suited] 1.41176% 8d-9d-9h 25.2
[ Pair & 1gap rainbow] 1.30317% Qd-Qs-Ac 23.6
[ Pair & 1gap 2-suited] 1.30317% Jh-9h-Jd 25.1
[ Pair & 2+gap rainbow] 5.75566% Jh-Jd-4s 22.8
[ Pair & 2+gap 2-suited] 5.75566% 6h-Ts-6s 24.3
[ Triplets ] 0.23529% 5c-5s-5h 25.4
[ 3-Straight rainbow] 1.30317% 2d-As-3c 28.2
[ 3-Straight 2-suited] 1.95475% 2c-3c-4s 29.7
[ 3-Straight-Flush ] 0.21719% 9d-8d-Td 28.3
[ Cnctr & 1gap rainbow] 2.60633% 9c-Jh-8s 26.8
[ Cnctr & 1gap monotone] 0.43439% 6h-3h-4h 27.2
[ Cnctr & 1gap 2-suited] 3.90950% Th-Jh-8c 28.2
[ Cnctr & 2+gap rainbow] 10.09955% Kd-Qc-4h 24.7
[ Cnctr & 2+gp monotone] 1.68326% 7c-2c-6c 25.5
[ Cnctr & 2+gp 2-suited] 15.14932% 3s-Qs-2c 26.1
[ KA2 dbl-cnct rainbow] 0.10860% 2h-Ad-Kc 20.4
[ KA2 dbl-cnct monotone] 0.01810% 2d-Kd-Ad 22.7
[ KA2 dbl-cnct 2-suited] 0.16290% 2d-Kh-Ad 21.8
[ Dbl gutshot rainbow] 1.19457% 6c-4s-8h 25.7
[ Dbl gutshot monotone] 0.19910% Js-9s-7s 26.3
[ Dbl gutshot 2-suited] 1.79186% Ah-3h-Qd 27.1
[ Other 1-gaps rainbow] 8.03620% 9s-3h-5c 24.2
[ Other 1-gaps monotone] 1.33937% 6h-2h-8h 25.1
[ Other 1-gaps 2-suited] 12.05430% 8c-3h-5c 25.7
[ No cnt/gp/pr rainbow] 7.71041% Jc-8h-5d 23.6
[ No cnt/gp/pr monotone] 1.28507% 4h-9h-Qh 24.9
[ No cnt/gp/pr 2-suited] 11.56561% Jd-4s-As 25.1
[--check combinations--] 100.00000%
Calculated Avg. Final
[More Interesting flops] Frequency Example Pot in BBs
[ Any A or K or both ] 40.07240% 5s-Ac-Qd 23.0
[ Triple-BWay unpaired ] 2.89593% Qd-Ad-Kd 24.9
[ Triple-BWay paired ] 2.26244% Ac-Jc-Ah 21.6
[ Double-BWay unpaired ] 23.16742% 5s-Ac-Qd 24.2
[ Double-BWay paired ] 4.34389% Qc-Qs-6c 22.3
[ 3 to a Wheel no pr ] 2.89593% 2c-3c-4s 26.5

So the biggest pots result from 3-straight flops, and next biggest are when there are lots of straight possibilities like connector+1gap and double-gut flops. And the smallest pots happen when 2 or 3 broadway cards flop (especially AKx), which isn’t surprising.

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