## Flop Analysis Part 5 – Suit Bias

Here we look at the effect on the suit distribution of the board according to the suitedness of the hole cards held by players seeing the flop.

Given random hole cards, the flops would break down like this (from full enumeration):

Flops seen: 132600 (100.0%) Known hole cards: - - - -[ FLOP TYPE ] Calculated Actual +-/100K +/-SDs Calc /132600[ Rainbow ] 39.76471% 39.76471% 0 0.00 52*39*26 [ Monotone ] 5.17647% 5.17647% 0 0.00 52*12*11 [ Two-suited ] 55.05882% 55.05882% 0 0.00 (52*12*39)*3

The 1326 possible starting hands include 936 unsuited unpaired hands and 78 pairs, for a total of 1014 possible unsuited hole cards. There are 312 possible suited hands. So over the long run players should be dealt suited cards 23.5% of the time, or roughly a 3:1 ratio of unsuited to suited hands are dealt.

Now let’s look for any card removal effects. Note that specific ranks are irrelevant for these tests.

**Scenario 1** – one player holds suited cards and the other has an unsuited hand in other suits:

Flops seen: 103776 (100.0%) Known hole cards: 2c 3c 2d 2s[ FLOP TYPE ] Enumerated +-/100K[ Rainbow ] 39.82424% 60 [ Monotone ] 5.15148% -25 [ Two-suited ] 55.02428% -35

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**Scenario 2** – one player holds suited cards and the other player has one of the same suit:

Flops seen: 103776 (100.0%) Known hole cards: 2c 3c 4c 5d[ FLOP TYPE ] Enumerated +-/100K[ Rainbow ] 39.53515% -230 [ Monotone ] 5.27290% 96 [ Two-suited ] 55.19195% 133

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**Scenario 3** – two players hold suited cards in different suits:

Flops seen: 103776 (100.0%) Known hole cards: 2c 3c 2d 3d[ FLOP TYPE ] Enumerated +-/100K[ Rainbow ] 39.68548% -79 [ Monotone ] 5.21508% 39 [ Two-suited ] 55.09944% 41

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**Scenario 4** – two players hold suited cards in the same suit:

Flops seen: 103776 (100.0%) Known hole cards: 2c 3c 4c 5c[ FLOP TYPE ] Enumerated +-/100K[ Rainbow ] 39.08418% -681 [ Monotone ] 5.44635% 270 [ Two-suited ] 55.46947% 411

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**Scenario 5** – two players hold unsuited cards with no matching suits:

Flops seen: 103776 (100.0%) Known hole cards: 2c 2d 5s 5h[ FLOP TYPE ] Enumerated +-/100K[ Rainbow ] 39.96300% 198 [ Monotone ] 5.08788% -89 [ Two-suited ] 54.94912% -110

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**Scenario 6** – two players hold unsuited cards with one matching suit:

This is equivalent to Scenario 1, with the same effect.

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**Scenario 7** – two players hold unsuited cards in the same two suits:

This is equivalent to Scenario 3, with the same effect.

So the obvious result is that the more hole cards there are of the same suit (such as 1 or 2 suits out between 2 players), the more monotone and 2-suited flops can be expected, and the more even the suit distribution is in the hole cards (such as 3 or 4 suits out between 2 players), the more rainbow flops can be expected.

Note that the offset of rainbow flops from the calculated expectation is always exactly equal and opposite to the combined offset of monotone and 2-suited flops. This should always be true regardless of the number of folded or dead cards removed from the deck stub when the flop is dealt.

So the only question is, assuming suits are randomly distributed around the table in the hole cards, how much do players typically favor seeing flops when holding suited hands over unsuited hands? If players overall fold unsuited hands vs suited hands in the same 3:1 ratio that they naturally occur, this won’t affect the suit distribution of the flops.

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**But 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. Conversely 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.** Granted, this is pretty speculative since the permutations for multiple players could easily wipe out the effect, and I expect any visible skew to be small.

I’m not going to make a prediction on how this comes out, but I’ll refer back to these observations when we see the final flop results.

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