How many strategically distinct flops exist?

[CallMeIshmael]

For example, Ah7h2h is not distinct from As7s2s, since any hand on the first can be mapped to a corresponding hand on the second.

I have a number, but Im curious if its correct.

[pzhon]

13 flops are trips. 13
There are 13*12 ways to choose the ranks of paired flops, and these are either 2-tone or rainbow. 312
There are 13C3 ways to choose the ranks of unpaired flops. These can be monotone, two-tone in 3 ways, or rainbow. 1430
Total: 1755

This assumes you haven't seen your hand.

There are abstract tools (Polya counting) that work for this type of problem, but they are more complicated than a direct calculation.

[CallMeIshmael]

ok, cool. 1755 was also what I go.


Im reading up on Polya counting, thanks for the mention.

[pzhon]

[ QUOTE ]

Im reading up on Polya counting, thanks for the mention.

[/ QUOTE ]
Polya counting is actually overkill, unless you are looking at much more complicated problems. This can be done with Burnside's lemma. You might look at actions of S_4 on possible flops, and then for each conjugacy class of S_4, count how many flops are fixed by that symmetry. The identity fixes everything, 4-cycles fix nothing, 3-cycles fix flops that are monotone in the 4th suit or trips in the moving suits, etc. It's not too bad, but not easier than a direct computation here.

[JABoyd]

There are 19,600 possible flops for each of the 1,326 starting hands.

[dobieatwarzOWNER]

[ QUOTE ]
For example, Ah7h2h is not distinct from As7s2s, since any hand on the first can be mapped to a corresponding hand on the second.

I have a number, but Im curious if its correct.

[/ QUOTE ]

When you put "strategically" in the title, I thought the results would be useful ...

Although you mentioned Ah7h2h is not distinct from As7s2s, it is also not STRATEGICALLY distinct from Ah7h3h since the playability is the same.

So flops like 2h2sJc and 3h3sJc are not strategically distinct either.

So the more interesting question is how many flops are strategically distinct which is what you asked in your original post title.

[CallMeIshmael]

[ QUOTE ]
Although you mentioned Ah7h2h is not distinct from As7s2s, it is also not STRATEGICALLY distinct from Ah7h3h since the playability is the same.

[/ QUOTE ]

No they're different.

Play for hands like 2c2d would be vastly different on Ah7h3h and Ah7h2h.


[ QUOTE ]
So flops like 2h2sJc and 3h3sJc are not strategically distinct either.

[/ QUOTE ]


Same here for hands like Ad2d.


You cant map the hands to one another like you can in the case of the OP, because there are slight differences that dont exist. (ie. On a board of J22, when you have A2, the hand 33 has more equity than does the hand 22 on a board of J33 against A3 (which are, presumably, the mapped hands))


I see what you're saying, but the point of the OP was to double check that I was doing the least possible amount of work for some analyis im working on.

[cabiness42]

[ QUOTE ]
When you put "strategically" in the title, I thought the results would be useful ...

Although you mentioned Ah7h2h is not distinct from As7s2s, it is also not STRATEGICALLY distinct from Ah7h3h since the playability is the same.

[/ QUOTE ]

I would consider A72 and A73 to be different flops since the second provides more straight draws than the first.

[dobieatwarzOWNER]

They are nominally different, but they are not strategically different in terms of playability.

Also, since this was posted in poker theory, I thought it would have some usefulness, otherwise it would have been posted in the probability forum.

[Adrian20XX]

[ QUOTE ]
For example, Ah7h2h is not distinct from As7s2s, since any hand on the first can be mapped to a corresponding hand on the second.

I have a number, but Im curious if its correct.

[/ QUOTE ]

It depends on the definition of "strategically equal".

I'm pretty sure that for me the rainbow boards QQ2, JJ2, JJ3, and many more are strategically equal for me. A good definition of equality might come from the different hands that exist on this board with the pocket cards. In all of them, a player can have 4 of a kind, an unlikely full house with two different cards, a full house with a pocket pair of the low card, trips with different qualities of kickers, two pairs with a pocket pair, two pairs made with the board's low card and a kicker, or the board pair with three kickers.

None of them can have a flush draw or a straight draw.

Regards ...