A recent thread about using the 5/10 rule to call preflop raises with PPs and suited connectors got me thinking about the kind of implied odds required to call preflop raises with SCs; people tend to arbitrarily use things like the 5/10 rule, even though I've never seen any mathematical description of the kind of odds you need to call these raises. I'm going to attempt to solve that problem (but I still need some help!).

I'll list the conclusions first, and leave the tl;dr math for the bottom for those of you that want to peruse it. I also encourage math-head-types to check my math to make sure I didn't mess anything up.

There are two kinds of hands you can flop with SCs: Good made hands (most of which can be made by calling with ATC, which of course we don't do) and draws. First, made hands, stolen off some page I googled:

Odds of flopping...
Flush: 0.84%
Two pair: 2%
Trips: 1.35%
Full house: 0.09%
Quads: 0.01%
Straight: 1.31%
-------
Total: 5.6% (1 in 18 times, 17:1)

However, most of the time you will be flopping draws instead of big hands with SCs, and that's where things get complicated. Let's separate this into two categories: combo draws and regular draws.

COMBO DRAWS

Odds of flopping...
20 outer (OESD + FD + pair): 0.077%
17 outer (Gutshot + FD + pair): 0.153%
15 outer (OESD + flush draw): 1.424%
14 outer (Pair + flush draw): 1.450%
13 outer (Pair + straight draw): 1.147%
12 outer (Gutshot + flush draw): 2.664%
------------------------
Total: 6.9% (1 in 14 times, 13:1)

These draws are all hands that can be played profitably after the flop; either you are a favorite against an overpair, or getting AI on the flop is +EV when you take some fold equity (and thus taking down dead money) into account.

Combining these big draws with good made hands, you'll have a relatively "big hand" on the flop 12.5% of the time, or 1 in 8 (very close to how often you will flop a set with an overpair). However, since a set is a near-invincible hand and you still have to improve with these draws, you can't say that you also need about 7:1 odds to call with a suited connector. Your average equity on the flop with these made hands and combo draws against an overpair is 66% (the made hands go from 75%-99%; the combo draws range from 45%-65%); compare this with sets, where your equity is generally 90+%.

REGULAR DRAWS

Odds of flopping...
9 outer (flush draw): 5.2%
8 outer (straight draw): 8.0%
-----------------
Total: 13.2% (1 in 7.5 times, 6.5:1)

These are your standard draws; when you flop a hand with which you can continue, it will most frequently be one of these. These draws improve to a flush or straight on the river about 1 time in 3.

Summary

- you have a 5.6% (1 in 18, 17:1 chance) of flopping a good made hand
- you have a ~7% (1 in 14, 13:1) chance of flopping a strong (12+ outs) combo draw
- you have a ~13% chance (1 in 7.5, 6.5:1) chance of flopping a standard OESD or FD

Adding these all together, you will flop a hand you can continue with on the flop 25% of the time (1 in 4). However, only half of the time will these hands be immediately profitable (i.e. +EV to shove it in); the other half, you'll have your standard old OESD or FD which requires playing some poker.


So, a question from me to all you math-heads: How do you combine these preflop odds with the odds of hitting your hand postflop to figure out the implied odds required to call with SCs preflop?

If you don't like numbers, skip the rest of the post; what follows is how I calculated everything.



tl;dr math

Made hands:
I calculated the odds of flopping a straight myself; with 65s, for example, there are four flops that give you a straight (789, 478, 347, 234). The odds of hitting each of those flops are 12/50 * 8/49 * 4/48; multiply that by 4 flops, and you get 1.31%.

Combo draws

All examples assume you have 6c5c.

OESD + flush draw + pair (20 outs ZOMG):
You need a flop of 87(6/5), 7(6/5)4, (6/5)43, with two clubs each.
8c 7c 6/5x: 2/50 * 1/49 * 5/48 * 3 = .0255%
Multiply by 3 to get odds for all three flops = 0.07653%. Not very high.

Gutshot + flush draw + pair (17 outs):
You need a flop of 98(6/5), 97(6/5), 8(6/5)4, 7(6/5)3, (6/5)42, (6/5)32 with two clubs.
9c 8c 6/5x: 2/50 * 1/49 * 5/48 * 3 = .00255%
Multiply by 6 to get odds for all six flops = 0.153%.

OESD + flush draw (15 outs):
You need a flop of 87x, 74x, or 43x with two clubs; in addition, you can catch ultra-deceptive flops of 973 with two clubs or 842 with two clubs.

Odds of flopping 87x with two clubs, where x does not complete a flush or straight and does not pair your hand:
87x: 7c 8c x = 2/50 * 1/49 * 27/48 * 3 = 0.138%
7c 8x xc = 1/50 * 3/49 * 10/48 * 6 = 0.153%
7x 8c xc = 3/50 * 1/49 * 10/48 * 6 = 0.153%
Total = 0.444%
Total for all 3 flops = 1.332%

973: 9c 7c 3x = 2/50 * 1/49 * 3/48 * 3 = 0.0153%
*3 for 9c 7x 3c/9x 7c 3c = 0.0459%
*2 for 842 = 0.0918%

Total odds of flopping 15-outer: 1.424%

Pair + flush draw (14 outs):
Two clubs and one of your hole cards:
6/50 * 11/49 * 10/48 * 3 = 1.68%

Since we already counted pair + FD + OESD and pair + FD + gutshot, subtract 0.07653 and 0.153 to get 1.45%


Pair + straight draw (13 outs):
using 65s, possible flops are 87(6/5), 7(6/5)4, (6/5)43
8/50 * 4/49 * 5/48 * 3 = 0.408%
Multiply by 3 for all three flops = 1.224%

Since we already counted pair + FD + OESD, subtract 0.07653 to get 1.147%


Gutshot + flush draw (12 outs):
You need a flop of 98x, 97x, 84x, 73x, 42x, 32x (where each flop has two clubs).

Same calculation as OESD + flush draw; 0.444% per flop * 6 flops = 2.664%


So, total odds of flopping a combo draw = 0.07653% (20 outs) + 0.153% (17 outs) + 1.424% (15 outs) + 1.45% (14 outs) + 1.147% (13 outs) + 2.664% (12 outs) = 6.915% = 1 in 14 times (13:1)


Regular draws

OESD (8 outs):
There are five flops you can catch an OESD with: using 65s as an example, there's 87x, 74x, 43x, 973, and 842.

Odds of flopping 87x (where x does not pair your hand and does not complete a straight):
8/50 * 4/49 * 34/48 * 3 = 02.94%
Subtract 0.442% for the times it makes an OESFD (which we already counted) = 2.498%
Multiply by 3 for the odds of 87x/74x/43x: 7.494%

Odds of flopping 973: 12/50 * 8/49 * 4/48 = 0.33%
Multiply by 2 for the odds of 973/842: 0.65%
Subtract 0.0918 since we already counted double gutshot + FD: = 0.558%

Total odds of flopping non-combo OESD = 8.05%


Flush draw (9 outs):
Two clubs + a blank that does not complete a flush or pair your hand:
11/50 * 10/49 * 33/48 * 3 = 9.26%

Subtract 1.424 and 2.661 since we already counted the times where the flush draw gives you an OESD, and you get 5.175% non-combo flush draws.

So, your total chances of flopping a standard 8 or 9 out draw are 8.05% (OESD) + 5.175% (flush) = 13.225% (1 in 7.5, 6.5:1).

I calculated the average equity of made hands/combo draws against overpairs by taking the weighted average of each:

0.077 / 12.5 * 65.556 (0.077 / 12.5 = %age of time you flop oesfd+pair, 65.556% = equity of 6s5s on 9s8s6x board against AcAd)
+ .153 / 12.5 * 57.677
+ 1.424 / 12.5 * 56.26
+ 1.45 / 12.5 * 50.71
+ 1.147 / 12.5 * 45.86
+ 2.664 / 12.5 * 47.78
+ 0.84 / 12.5 * 97.17
+ 2 / 12.5 * 74.55
+ 1.35 / 12.5 * 87.78
+ 0.09 / 12.5 * 91.414
+ 0.01 / 12.5 * 99.899
+ 1.31 / 12.5 * 96.717

So, basically, it's a 12% chance (~7.5:1) to flop something you'll want to shove.

So, it's near the 6.5:1 that a pocket pair gives you, plus the little something extra for flopping the other kinds of draws.

I'd say the same rule would be in effect here.

Wow - crazy math. I hope you apply this analysis instead of just leaving it at this.

I just want to add that continuing on the flop depends on your opponent. Depending on their pfr frequency, I play my hand differently on the flop. You can't forget one-pair hands which you can float/raise if the board is safe.

Your analysis is assuming villain always has a hand. You must take into account the times when you can profitably call flop w/ just one pair w/ respect to his range, and also the frequency of him folding to a raise on the flop when you only have a straight/flush draw.

Very nice.

I have always been much more reluctant to call preflop raises with SCs compared with small pairs.

This post really helps me get my head around what kind of odds I need to be calling.

Thanks heaps!!

Me too, I just need someone like Pokey to help me figure out how to do that

I was trying to make the point that thinking this way is dangerous; if you get it in on the flop with a set you'll lose 1 out of 10 times, but if you get it all in on the flop with the made hands/combo draws you flop with SCs, you'll lose your stack 1 out of 3 times. For this reason, even though the odds of flopping a made hand/combo draw is about the same as flopping a set, you need better implied odds to call with SCs to account for the times you lose.

Wow, great post. I appreciate all the leg work you put in, but I hope we can get some help with application from some of the great minds around here. I am going to put some serious thought into this and post again later. I look forward to reading other responses.

-Jaxx

I was trying to make the point that thinking this way is dangerous; if you get it in on the flop with a set you'll lose 1 out of 10 times, but if you get it all in on the flop with the made hands/combo draws you flop with SCs, you'll lose your stack 1 out of 3 times. For this reason, even though the odds of flopping a made hand/combo draw is about the same as flopping a set, you need better implied odds to call with SCs to account for the times you lose.

[/quote]

Word.

bump for weekday

Rebump. Try x-posting this in Probability too. Maybe they can help.

I've been too lazy to work all this out for a while now, thank you.

After reading goofy's analysis, it looks like you can profitably play suited connectors 45s - JTs with the 5/10 rule considering the implied odds.
One issue at hand is that the 5/10 rule assumes a reasonble chance you'll net an opponent's stack when you hit your hand. This is fine for the made hands because all of them are beating an over pair. The combo draws are not as good because if you push and your opponent folds, you are getting the correct implied odds to call preflop with the 5/10 rule in the first place. If you push and he calls, you'll hit your hand around 50-60% of the time. With the money already in the pot and the fold equity you might have, going all in with a combo draw is a fine play, however I question whether you have the correct implied odds preflop to get yourself into that situation. There is some math that needs to be worked out here, but it appears suited connectors are not as easy a call as pocket pairs using the 5/10 rule.
A good play may be to call with these in position only. That gives you an added advantage when your opponent checks the flop to you.

12.6% chance of flopping a made hand or a big draw

10.8% chance of flopping a set with a PP, shouldnt we be calling the same raises (maybe more) with sc's as we are with PPs (which is most).

I wrote a program to do these exact calcs over a year ago, and came out to the same ~25% you did, so your math is probably good (some of my numbers were also verified by BruceZ of the probability forum).

Because I wrote a program to do this, I ran through a number of other types of hands that you may be interested in. For instance, suited one-gappers come out to 23% to flop OESD+,2pair+. So basically, most anytime you're willing to play a suited connector, you should be willing to play a similar suited one-gapper also.

Summary of hands:
Suited connectors: 25%
Suited one-gappers: 23%
Suited two-gappers: 18%
Unsuited connectors: 17%
Suited aces: 17%

Note: This isn't the end-all, be-all, as it doesn't take into account draws to the nuts, etc. But it's a relatively decent gauge on the strength of those various types of hands to each other.

As he pointed out, your equity edge when you "hit" isn't nearly as big so your implied odds aren't nearly as high (as well as your draws are often obvious). A collary to this is that position becomes more important in the play of those hands.

These two factors combined mean you can't be playing in more pots with them than PPs based on the numbers you quoted alone.

I looked at this long ago. My posts are somewhere in the archives.

If you factor out flopping flush draws on paired boards, and straight draws on paired and 3-flush boards against, I get 23.5% or 3.25:1 against flopping 2 pair or better made hands or at least an 8 out draw for suited max stretch 0-gap connectors.

21.3% or 3.7:1 against for max stretch 1-gap suited connecttors.

18.5% or 4.4:1 against for max stretch 2-gap suited connectors.

I used to just subtract ~2% to account for those situations where you flop a draw but it's on a bad board or whatever. Looks like that was a pretty good guesstimate. Thanks for those.

Whoa... I think you just helped me plug a huge leak.

That too. People generally talk about suited aces in the same way they talk about SCs, when suited aces appear to be much weaker except for two things:
1. Flush draws are always to the nuts
2. They have three more outs on their flush draws when up against KK- than SCs do.

I wonder if those two properties make up for the fact that you're flopping good hands/draws 1 in 6 times instead of 1 in 4.

My take on suited aces is they are fantastic to raise with and terrible to call raises with unless the implied odds are huge. Always been that way. However I used to consider unsuited connectors as negligibly weaker than suited connectors on the order of 1 or 2%.

vnh

goofy,

you will also flop some useful top pair hands with a SA; less so with the smaller connectors.

If you have e.g. A8s against a reasonably tight CO opening range (say 22+, 2 broadway, A8o+, Axs, 65s+, 86s+, two suited cards 9 or higher) then you are 49:51 with their range and typically getting good pot odds + position. With 65s you would be 37:63 and it would be rather harder to judge where you are. Your call is justified more by pot odds than implied odds.

Nice post. However you failed to factor in the Fun Equity of gamboooooling it up with a draw. Sets are so boring and stressful in comparison...mostly you're just sitting their praying they don't hit their runner runner. Whereas when you have a draw and they call you, things can usually only improve.

I would even add a few percent if there were several players calling or raising before me to account for the good chance that they have each others outs.

good post

Any chance you can make this program available to the public (or just me )?

Goofy,

Really nice post, thanks.

This is really important. With a small PP you don't mind being OOP so much, as you'll usually be playing fit or fold on the flop. Extraction is easier in position, but you should still be able to get stacks in with a set OOP against an overpair, TPTK etc.

With an SC, though, most of the time when you're continuing on the flop you'll have a draw. These are so much easier to play in position, where you have the option of checking behind for a free card if it's checked to you, betting or raising the flop to disguise your hand and possibly take a free card on the turn, etc. Much harder to play them OOP where none of your options are great: check/call looks like a draw, you have no FE and may not get odds to draw; leading may mean you get raised or floated so you can't take a free card; check-raising may mean you put a lot of money in to draw etc.