Flop Analysis Part 2 – Rank Bias

Dec 18th, 2009 | Posted by spadebidder

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



Notice in the graphs that the three flop cards are essentially uniform at each rank, which they should be.   Then the removal effect gives us a smooth transition from low to high cards, with  the turn showing more difference between low and high than the flop, the river more still, which is what we’d expect to see.  At each street we can see that players tend to hold about the same number of Tens thru Kings, but significantly more Aces, leaving fewer that can hit the board.  Eights, the median card, are almost exactly on the expected average of 7.69%, with 7.681% in this sample.  And going down from Eights, the lower cards increasingly show up on the board. 


To see a more pure version of the removal effect, look at heads-up games: 


Board Card Distribution in Heads-Up NLHE, BB $0.25 to $4.00

      37 million hands, 14 million flops (37%), 9 million turns seen, 7 million rivers seen



Here you can see that Kings are favored more than other broadway cards, and Aces still more (as expected). And players don’t like to play hands containing a deuce thru five.   Also notice that 8 through Queen are equally represented, meaning heads-up players are equally likely to play hands containing these ranks.    

It’s also interesting that the only cards showing a significant difference on the late streets versus the flop, are Twos, Threes, and Aces.   So heads-up players who see the flop are less likely to continue to the turn and river when they have a Two or Three, and more likely to continue when they have an Ace, but about equally likely with everything else. 


To round out our rank bias statistics, here’s what it looks like for 6-max cash games:


Board Card Distribution in 6-max (min 4 active) NLHE, BB $0.25 to $4.00

         200 million hands, 104 million flops (52%), 66 million turns seen, 49 million rivers seen



These three graphs represent 409 million hands of NLHE, so I think we can confidently say that there is a rank bias such that Aces will show up on the board about [(.075 - .078)/.078] or about 4% less often than deuces, and [(.075 – .077)/.077 or about 2.5% less often than a random expectation.   And everything else slopes in between. 

When I do the flop type analysis later in this series, I’ll only be looking at a few flop patterns that are defined by rank, and I’ll refer back to the rank bias statistics shown here to explain the offset in those flop patterns.

Using the actual counts behind the 9-player graph we can say that in full ring NLHE we should expect to see around 400 / 100,000 fewer Aces on the flop than a random distribution, and about 140 / 100,000 fewer Kings on the flop than a random distribution.  These numbers get bigger on the turn and river, but we mention just the flop here as it is part of our prediction for the flop distribution in part 6.

In 100 million flops the Aces would be offset over 48 standard deviations from the mean of a random distribution.  In a smaller sample of 2 million flops, the offset would be about 7 standard deviations.  And in a single player sample of 200,000 flops the offset would only be 2.2 standard deviations, and may not even be noticed.  Again this illustrates how a consistent offset becomes progressively more significant as the sample size increases, and how a small effect like this takes a huge sample size to detect.


Key Finding:   Aces show on the board only about 96% as often as deuces in NLHE.  Everything else slopes in between. This card removal effect is because players will see flops more often when holding high cards, and will fold more often when holding low cards, leaving a biased deck stub for dealing the flops that get seen. Furthermore, the effect becomes even larger for the turn and river cards, since player requirements to continue in the hand become stricter at later streets.


Because of this bias in the rank frequencies that tend to show up on flops, there are several flop patterns relevant to poker which we’ll examine, that will be skewed from their expected rates. These include 3-straight flops (3.48% of all flops), which naturally include an A or K 33% less often than they do a 3 or 4, because Ace or King can only be in a 3-straight 2 ways. And “Connector & 2-gap” flops (6.37% of flops), which include an Ace 50% more often than they include a Two, and include a King 33% more often than a Three. And “Pair & 2-gap” flops (2.39%) which include an Ace twice as often as a Two. These imbalances are simply the result of which ranks can naturally form these relevant card patterns. But because of the known rank bias for seen flops, these patterns will not show up at the expected rates. Flop patterns will be discussed more in upcoming posts.

A correction was made to this post for the number of expected Aces per 100,000 flops.


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