Playing SC's in EP/MP

I spent several hours today working on the math to answer this question: when open-raising an sc like 98s, what type of flops occur with what frequencies? You often hear that someone is opening a certain combo because "it flops good." How good? Here's what I found.

Hero is UTG and open-raises 98s. Of the 19,600 possible flops, there are:

2 Quads
18 Full Houses
165 Flushes
216 Straights
36 Straight + FD's
264 Trips
396 Two-pair
276 Pair + FD
270 Pair + OESD
18 Pair + OESD + FD
540 Pair + GSSD
36 Pair + GSSD + FD
4536 Pairs
1368 OESD's
225 OESD + FD
2304 GSSD
378 GSSD + FD
1356 FD's
7196 "air"

A quick explanation: in the chart above, "pair" means naked pair with no FD or SD (but this includes "two pair" hands when the board pairs). There are 1368 OESD's, but these do NOT include the 225 OESD+FD flops. And so on.

So we can summarize the flops as follows:
2247 Pair + draw or better (11.4% of flops)
225 15-out draws (1.1%)
378 12-out draws (1.9%)
1356 9-out draws (6.9%)
1368 8-out draws (7.0%)
2304 4-out draws (11.8%)
4536 pairs which I think of as 5-out draws (23.1%)
7196 "air" flops (36.7%)

We can compare this a hand like AQ (see Daven's interesting Playing AQ UTG thread). AQ flops TP or better (trips, 2 pair, full house, quads) 6356 times, or 32.4% of the time. We can also expect 2112 GSSD's, or 10.8% of the time. There are some combo hands I've not run the numbers on. But the idea should be clear: we're cbetting TP or a decent draw about 45% of the time.

If we're willing to open-raise 98s and cbet pretty much all of our 5-out draws or better, we'll be continuing confidently ~52% of the time. So the comparison with AQ shows we can cbet confidently just as often, possibly more often, even when we're oop.

Another difference with sc's is the number of times we have draws to HUGE hands, which isn't normally the case with TPTK hands, and is almost impossible to have happen with pp's.

Now, as a caution, I'm not suggesting you start opening EVERY sc from every position. But this is the mathematics of why folks say hands like 98s "flop good." Also, while most of my numbers checked out using several different calculation styles to verify them, I appear to have over-counted 180 FD's (~1.1%). But I can't find the mistake.

Still, it is helpful to know that if we add sc's into our open-raising range from a particular position, we'll have lots of solid draws we can cbet or 2-barrel on. The same thing is true for flat calling with sc's in LP in games where 3betting isn't rampant. We have lots of ways we can continue.

I'll add more of the mathematics later in the thread. Covering every bit of the mathematics would a book. My spreadsheet has hundreds of calculations on 11 worksheet pages, so you'll just have to trust me.

Final two notes:
1. Hands like 97s flop significantly worse (I haven't run all the numbers, and don't plan on it in the next few weeks).
2. These flop numbers aren't accurate for hands like AKs since not all the straight draws are available (98 can be part of Q-high, J-high, T-high and 9-high straights, but AK can only make a Broadway straight). Of course, hands like AK have other outs we like, so it's not that big a deal. Just don't go opening 32s and pretend I didn't warn you it sucks so hard it leaves bruises.

OK, for anyone interested in the math, let me mention that I suck at discrete mathematics. I'm certain there are graph theory approaches to this problem that use cycles to connect the different card combinations and get at the correct calculations more directly. I'm stuck with my own math background, so I'm using "brute force" combinations to get there. I started by making the following sets of cards:

P = the 6 cards that match Hero's cards and can make pairs, trips, etc
S = the 24 cards that can make straights and SD's
O = the other 20 cards in the deck

Calculating all the flops of all the possible PSO permutations allows me to get hold of the combo hands easier. Here's a chart breaking down all 19.600 flops into the P S O combinations:


If you're still not quite sure what the chart is saying, the top 3 rows are all flops where Hero has 2 pair or better since 2 or 3 of the cards are from the set P. The 0 3 0 row are all the flops that contain 3 cards that can help make straights.

To calculate all the SD's, we need to count all the patterns that can make straights, OESD's, and GSSD's. Here's all the patterns when the flop contains 3 different card values none of which pair Hero's hand:


Here's all the patterns when the flop contains 2 different card values:


Again, I'm pretty happy with the SD's and pair/2 pair/trips/full house/quads calculation. I'm almost certain they're correct.

Not sure if this is a correct way of checking this but I used Flopzilla selected 98s to see how much it hits.

I added up the percentages of it hitting a flushdr.(2 card)/OESD/gutshot/flushdraw+pair/oesd+flushdr./gutsh.+flushdr./oesd+pair/gutshot+pair/two pair and better

Adds up to 52% of hitting something decent

pretty cool stuff gonna have to test it out :P

the obvious implication of this is that with these hands it is great to see flops deep and in position in limped pots.

Edit: thanks to spoon (see post below this one), I found several errors.

Originally Posted by clvacva

Not sure if this is a correct way of checking this but I used Flopzilla selected 98s to see how much it hits.

Interesting. I dl'd flopzilla, and found what appears to be an error in its calculations for 98s. It lists the probability of quads on the flop as 0.03%. There are exactly 2 ways to flop quads: 999 and 888. The total number of flops is 19,600. My calculator says the probability (in decimal form) is:

.0001020108

or, in percentage form:

0.0102%

Edit: mistake was leaving out 4 straight flushes (>=quads) as spoon said.

Hmm...not sure why, but flopzilla's overestimating the probability of quads by 3x-ish. So...I checked their calculation on Full Houses, which can occur in 18 ways. Flopzilla says the probability is 0.122% when the actual probability is 0.09184%.

The only way a Full House can flop is if all three flop cards are 8's or 9's. Since there are 3 8's and 3 9's left in the deck, the total ways to draw 3 of them on the flop is "6 choose 3" or 20 ways, but two of them are quads (see above). Then the probability is 18/19600.

Edit: found that error now - there's a "cummulative" option and "absolute" option. When you check the correct one, you get the correct output.

I just deleted the rest of my post. I TOLD y'all I suck at discrete mathematics.

Hey Robb you're going to feel dumb having a lowly undergrad showing you this but the column for 0.03% isn't just quads, it's ">=quads".