Flop Analysis Part 1 – Rigged or Random?

Dec 17th, 2009 | Posted by spadebidder

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.

In a fair game with a truly random deal, the board should still not be expected to follow a random distribution.  A flop is only dealt about half the time at a full-ring NLHE table, and flops that are seen are chosen by players based on their holdings. Unless 2 or more players like their hand well enough to bet or call the bet facing them preflop (or the BB gets in free), we don’t get to see that flop at all. Obviously there are factors besides cards involved in that decision, but the hole cards are often the biggest factor determining when a flop is dealt. The flops that are not dealt would have a different card distribution than the flops that we do get to see. Player holdings have a significant effect on the distribution of board cards, and my analysis will show these card removal effects and quantify them.  

A card removal effect refers to a bias in the remaining deck due to the cards that are already dealt.  Blackjack card counters track certain card removal biases to improve their odds. In Hold’em, when players choose which flops to see based on the cards dealt to them, the cards remaining in the decks that get to the flop are biased by that selection, and this affects what cards can be dealt to the board.

For one clear example of how a card removal effect can affect game play, see this post:
Testing Barry Greenstein’s claim

That article also implies that some of these biases are going to be more pronounced in full ring games than in heads-up games. The reason is that there can never be dead cards in a heads-up game, and when a player chooses to fold, we don’t get a flop. At full ring, there can be up to 7 folders and still have a flop dealt. That’s 27% of the deck discarded and unavailable for the flop. And those folded cards are not randomly distributed, they tend to be the weaker hands with lower ranks and no pairs. We’ll examine the rank bias and the pair bias that results, which are the two strongest biases that show up in the community cards.

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To be clear, the card removal effects that change the board card distribution in Hold’em arise from the kinds of hands that players tend to be holding when they decide to continue to the flop.  Not surprisingly, the effects are directly traceable to the primary factors that have an effect on the strength of your hole cards (and thus the decision to see a flop).  They are:

  1. High cards
  2. Pairs
  3. Suitedness 
  4. Connectedness
  5.  

     

Using these factors, we can try to predict what the flop distribution should look like by making reasonable assumptions about typical player behavior.  In the next couple of posts, I’ll attempt to calculate what the board card distribution should look like, and then in part 6 that prediction will be compared to actual card statistics.

I believe the comparisons should convince a reasonable person that flops are highly unlikely to have been manipulated on the sites analyzed.  At the same time we will have quantified some interesting removal effects to a degree that probably has never been done before. The analysis also provides useful statistics on over 30 specific flop types, which is a level of detail I’ve never seen anywhere else. 

In a later series we’ll do the same analysis for turns and rivers, which is where it will really get interesting.

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