Quote:
Originally Posted by Pyromantha
Summary: Like you say, removing paired holecards makes almost no difference really, even if you never play a pair you still only expect to finish up with no pair 18.49%, assuming you play all unpaired hands to the river every time.
Yes. Even the flop alone pairs ~17% of the time, without even dealing the other two cards. And to illustrate the smallness of the card removal effect, the difference is less than half a percent for when you have a pocket pair and when you don't. Here's an excerpt from some of my pending research on card removal effects
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Scenario 1 - two players see a flop with one holding a pair and one holding a non-pair in other ranks. We calculate the probability of them seeing a paired flop (but not triplets):
10 ranks (40*3*44)*3
2 ranks (6*2*45)*3
1 rank, i.e. he flopped quads = (2*1*46)*3
(15840 + 1620 + 276) / (48*47*46) =
17.090656799% chance for paired flop.
As expected, when someone holds a pair the chance for a paired flop increases.
Scenario 2 - Two players hold non-pairs but they have one matching rank between them. This is equivalent to #1 and also has a
17.090656799% chance for paired flop
Scenario 3 – two players both holding pairs (in different ranks) see the flop. We calculate the probability of them seeing a paired flop (but not triplets):
11 ranks (44*3*44)*3
2 ranks, i.e. somebody hit quads = (4*1*46)*3
= (17424 + 552) / (48*47*46) =
17.321924144% chance for paired flop.
So as expected, the chance rises a bit more.
Scenario 4 – two players both holding non-pairs with no ranks in common see the flop. We calculate the probability of them seeing a paired flop (but not triplets):
9 ranks (36*3*44)*3
4 ranks (12*2*45)*3
= (14256 + 3240) / (48*47*46) =
16.859389454% chance for paired flop.
The chance for a paired flop goes down when no one holds a pocket pair.
Scenario 5 – two non-paired players hold the same ranks for both cards. This is surprisingly common at a 9-player table, and it’s similar to the birthday problem. We’ll calculate the frequency when we do the scenario weighting. This scenario is equivalent to #3 and also has a
17.321924144% chance for paired flop.