## Flop Analysis Part 6 – Predictions

Jan 5th, 2010 | Posted by spadebidder

Let’s review our predictions from the card removal effects before finally looking at the actual flop data. In the next post on this topic I’ll be showing the results from several hundred million actual flops.

In part 2 we examined the Rank Bias on the board caused by players seeing more flops when they hold higher cards and folding more often when they hold lower cards.  This leaves a biased deck stub for dealing the flops that get seen.   We found that:

• Aces make up only about 7.5% of the flopped cards instead of the 7.69% or 1/13  that a uniform rank distribution would yield.  This effect is greatest at heads-up games, and slightly attenuated at full ring.
• Similarly, Twos make up more of the flop cards than expected, at 7.9% heads-up, and 7.8% in ring games, and all other ranks slope between the Twos and Aces. The median card, 8s, hits at just about the expected 7.69% or 1/13.

In Part 3 we examined the Pair Bias on the board that results from players tending to see flops more often when they hold pairs.  We predicted mathematically that:

• We should see about 50 more paired flops per 100,000 flops than a random distribution would yield.
• This will affect all the flop types that contain pairs, unless that type is also skewed by a rank bias effect, which can wipe out the pair bias.  For example, the “Pair & 2-gap” flops are biased upwards (to higher frequency) by pair bias, but biased downwards by rank bias since they can be made more ways containing Aces than containing Twos.  So the two effects may cancel out in that specific type, resulting in a frequency near the calculated number.

So in general we predict that paired flops and combinations that contain pairs, will show up more often than than the calculated expectation from a random distribution.

Then In Part 4 we  broke down the flops into 40 different types, and showed that many common flop types do not represent all ranks equally.  Those flops will be affected by the Rank Bias and not show up at the calculated frequencies.  For example:

• 3-straight flops include a King or Ace 33% less often than a 3 or 4, since a K or A can be in only 2 possible positions of a 3-straight.  Therefore, since we know there is a rank bias that will give us more 3s and 4s on the flop than Aces and Kings, that means we should see more 3-straights on the board than a random distribution would yield, since more 3s and 4s means more possible 3-straights.
• “Connector & 1-gap” or “Connector & 2-gap” flops use Aces 50% more often than specific low cards, because  Aces can make these patterns more ways.  Because of the known rank bias that will give us fewer Aces in the deck stub, we should have fewer of these two flop types than the calculated expectation from a random distribution.
• Conversely, “Connector & 3-gap” flops contain Aces 21% less often than low cards, so we expect to see more of this flop type than the calculated expectation from a random distribution.

In Part 5 we examined the Suit Bias on the board that may result from players tending to see flops more often when they hold suited hands, relative to the natural frequency of suited hands.  We predicted that:

• If players fold unsuited hands more than 3 times as often as suited hands, we would expect to see more monotone flops than the calculated frequency for a random distribution.
• If players fold unsuited hands less than 3 times as often as suited hands, we would expect to see more rainbow and 2-suited flops than the calculated frequency for a random distribution.

If these predictions are shown in actual long-term results, with an otherwise expected frequency of flop types, it will be a strong confirmation that flop cards are properly distributed, and are truly dealt randomly given the known biases in the deck stubs used for dealing the seen flops.

So that summarizes the expected biases I’ve identified. Keep in mind that all of these are still expected to be relatively small, with the rank bias effect being the largest by far. Most of the flop types we’ll look at are not rank-specific. Thus I still expect the flop distribution to be “practically” random as far as any effect on actual play. I think we’ll see an impressive degree of accuracy to the calculated amounts for each flop type, but the sample data set is so large that even small offsets can be a significant distance from the mean when measured in standard deviations. For any flop types showing more than 2 standard deviations offset from the expected mean (if any), hopefully we can now provide a mathematical explanation for that difference.

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