Two Plus Two Newer Archives - Suited Connectors, Implied Odds, and You (Theory/Math)

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