How often (what % of time) does each occur on the flop?

Ignoring player holdings:

F = number of flops = C(52,3) = 22100

Monotone: 4*C(13,3)/F = 5.2%

Two Suited C(4,2)*C(13,2)*26/F = 55.1%

Rainbow 4*[C*(13,1)]^3/F = 39.7&

Thanks!

statman, can you explain the rainbow one in words please?

Sure.
Rainbow 4*[C*(13,1)]^3/F = 39.7&

For a rainbow flop, you need 1 card from each of 3 suits.

C(13,1) = 13 is the number of ways of selecting 1 card from the13 in a suit. Thus, C(13,1)^3 = 13^3 is the number of ways to select 1 of 13 cards from 3 suits, say clubs, hearts and diamonds. But there are 4 suits, so multiply 13^3 by C(4,3) = 4, the number of ways to select 3 of 4 suits.


The numerator is now the total number of rainbow flops. Dividing that by the total number of flops, 22.500, gives the desired probability.

Edit: You weren't referring to the "&"typo instead of "%" or were you?

I didn't understand the 3/F term. I still don't think I do.

Is it (4*[C*(13,1)]^3)/F? or ^(3/F)?

The first one Brokenstars. The dividing by F is at the end. I think the part that makes it weird is that he added an asterisk after the C. It seems like it may be easier to understand like this:

(4*C(13,1)^3) / F

Ok but where's the strategic benefit? When I first opened this thread I was hoping to ramble about board texture, but....aw **** it I'm gonna do that anyways:

In studying range vs range equities, I found that the equities vary depending on board texture, naturally. I was a bit surprised at a few of the patterns that emerged though.

The most curious pattern that I noticed was that when comparing rainbow boards to two toned boards, the two toned boards favored the preflop raiser by a margin of 3%-5% depending on the ranges. I used to think that the presence of a flushdraw would benefit the preflop caller, but now I see that the presence of a flushdraw will benefit the preflop raiser because of all the big flushdraws in the preflop raiser's range that dominate the flushdraws in the preflop caller's range.

Interestingly, the flush draw effect that benefits the preflop raiser doesn't translate to monotone boards; on monotone boards that don't contain an Ace, the preflop callers range vs range equity is significantly higher than on both rainbow and two toned boards.

very interesting finding
does that mean as the preflop raiser, we should cbet with low frequency on monotone board in general?

I think so.

Interesting, I'm guessing this is probably due to the fact that the preflop caller is going to be 3-betting his strong aces like AKs, AQs, etc.. So the caller is going to be more dominated. Even with preflop callers having more suited stuff, the fact that it doesn't have those strong aces is enough to make twotoned flops weaker for him. When there is an ace on a monotone board it blocks the aces that the preflop raiser can have and removes that advantage.

To answer your first question, I think it helps for determining which flops are more correctly to play important. If a spot comes up more often, it will be more advantageous to work on that aspect of your game.

When it comes to betting on monotone flops, they are the most straight forward ones to steal. When it comes to monotone flops the ranks of those cards don't matter so much and a lot of people will get out of your way especially if they are out of position and facing a near pot sized bet.

A J on a J 3 T board facing a pot sized bet OOP is ugly. Even K Q isn't that great. Both can call profitably there from OOP, but it still feels gross and uncertain. A T would probably have to fold there.

Thank you.

Not entirely though. I found that only Axx monotone flops that allow for a made straight flush have this effect that you mention.

Of course, preflop positions have a profound effect on the range vs range equities, but I think the general patterns are present in almost all heads up situations involving a preflop raiser and a preflop caller in position; that is unless the preflop caller calls with -ev hands preflop and or misses 3 betting value preflop.

For clarification:
...relative to identical rainbow boards.

In my analysis of BTN v BB spots, I don't see such a strong effect. Whether the board is two-tone or rainbow, the equities run extremely close, and the ranks of the flop cards have a much bigger effect on equity than suitedness does. I'd guess your pre-flop ranges must be quite different to the ones I'm using.

e.g. On KT7r, I have the BTN on 57.55% equity (a significant equity advantage, since with my ranges and 40 different flops, the BTN's average equity vs the BB is 53.8%).
On KT7 ttx (with the King and ten in the same suit), BTN has 56.96%.
On KT7 xtt (with the ten and seven in the same suit), BTN has 57.02%.
On KT7m (monotone), BTN has 55.2%, and this 2.3% reduction in equity may have significance. (Indeed, it partly explains why the BB donkbets the monotone board at a high frequency).

On A84r, BTN has 55.47%
A84ttx, BTN has 54.8%
A84xxt, BTN has 55.4%. Again the BB caller does slightly better when there is a flush draw, but it's less than a 1% improvement in equity.

I think it stands to reason that an optimal calling range would play well whether the board is rainbow, two-tone or monotone. It would be quite peculiar, imo, if the suitedness of a flop helped one player considerably more than the other.

EDIT: I might find some different things in UTG v BTN spots, but I've only just started with those, so haven't got all the data yet.

Yeah it was years ago and I was studying no limit holdem and limit holdem at the same time so the results in my memory probably got mixed up.

Now I'm curious. Did you look at any other, more asymmetric ranged spots, such as utg6max vs bb?

In the BTN v BB spots I've been analysing with Snowie, I was somewhat surprised to find out that monotone boards tend to have the highest donking frequencies (for a small size), and also the highest c-betting frequencies (also at a small size).
e.g. On 40 "representative" flops, Snowie c-bets on the BTN almost exactly 50% of the time with an average bet-size of 2/3 pot (although it never uses that size, it's always quarter, half, pot or 2x pot). On an 865 mono, it c-bets 82% of its range for 1/4 pot, on K93m it's 81%, but on both those flops, the BB is donking over 30% of the time.

EDIT: @Bob, I haven't done UTG v BB yet. I started with BTNvBB, then did SBvBB, and I'm currently on UTGvBTN. I planned to do 3-bet pots next, but might do UTG v BB single-raised pots at some point. I think with Pio, you can get it to do it all automatically as a batch process. I'm having to manually build the scenarios and write down the numbers in a spreadsheet, so it's quite time-consuming.

I find this surprising as well. Can you make the big blind check to the raiser 100%, which is the way many play this spot? I would imagine this will drive down the button's cbetting frequency quite significantly.

With Snowie, I can't "node lock" (?) a strategy like that, no, but you're probably correct in your theory.
One of the reasons I was particularly surprised in my recent Snowie studies is that when Snowie first came out, it had a very low betting frequency on monotone boards. It was only when it was given the option to bet quarter pot that it went mental and started donking or c-betting the monotones at very high frequencies. There's definitely something interesting going on with regard to protection/denial of free equity/thin value on those boards. It will happily make a small stab with middle pair, for example, but it hates betting big with the same hand strength. I suppose it's because the turn card could come a 4th flush card and radically alter the strength of many holdings.

C*[13,1] is that not 13x12x11x10.... etc? So 13^3 a massive number?

There were several typos; C*(13,1) should have been written as C(13,1)

C(13,1) = 13!/(1! * 12!) = 13

Making a few edits the equation is better written as follows:

Rainbow: (4*C(13,1)^3)/F = 39.7%

so is that 13x12x11x10x9x8x7x6x5x4x3x2x1/12x11x10x9x8x7x6x5x4x3x2x1

?

Think about the rainbow flop question for a minute and forget about the notation. We are trying to tally how many flops are rainbow.

How can a flop be rainbow (where the three cards are of different suits)?

- Three of the four suits must appear on the flop (there are 4 ways to choose 3 of 4 suits: SHD, SHC, SDC, HDC)

- Each of the three chosen suits must have exactly one card of that suit appear (there are obviously 13 ways that can happen for each suit: AKQJT98765432)

- The selection of the cards and suits are "independent" (so it is correct to multiply the constituent possibilities to derive the overall number of possibilities).

So the number of rainbow flops = 4*13*13*13 = 8,788.

Dividing by the total number of possible flops (22,100) converts the tally into a percentage.


statmanhal gave the formula for C(x,y) above. If you are still unsure what the C(x,y) notation means, you can look up the Choose function or number of Combinations in a search engine. Similarly if you are unsure what N! means, you can look up Factorial in a search engine.