From deep dark past of early 1999, comes about the only worthwhile work I did on poker analysis that year of any lasting significance. I've actually used these results consistently over the years, and I think they've paid off in spades.

Analysis of Unsuited Connectors

Here is a not-so-quick analysis of playing small unsuited connectors for implied odds. This analysis assumes that you are in a loose game where many players are seeing the flop.  This is not an analysis of how to play these hands in a heads-up situation. In this situation the hand will be mucked if the flop isn't favorable. What favorable means is of course open to interpretation. For simplicity, I'm assuming that any flop of two pair or less won't be worth continuing, and any two pair or better will be worth continuing. The actual strategy you take, will of course depend on how the play develops.

Here is a summary of the analysis that follows. One thing to note is that I assume that you will be getting two or three callers after the flop, hopefully down to the river. The calling odds needed to make the call have been adjusted accordingly.

Unsuited Connectors Summary (8xo and worse)

The "preflop calling odds" is a recommendation made on the basis of %make and %win numbers. These numbers are based not only on the direct odds, but also implied odds for when you hit your hand. If implied odds are NOT factored in 0-gap hands need about 19:1 odds for a call, and 1-gap need about 25:1 odds for a call. These are INCREDIBLE odds, and should directly suggest that these hands are not playable at all. In general this is the case. By depending on implied odds, I've "pushed" the calling requirements for the 0-gap hands to 13:1, and the calling odds for the 1-gap to 15:1.

To make the "implied odds" calculations here, we assume that we put in, on average, one bet per round for all the rounds, AND we get two callers on the flop and the turn, and one caller on the river. This makes for a total of 14 bets in the pot by the river.

If you want to play a bit looser, you *might* consider just "halving" the unfactored calling requirements. So you might call with 0-gap medium hands for 10:1 odds, and 1-gap for 13:1. Going ANY looser is going to cost you money for that hand in the long run -- image considerations aside. Also note, I assume a rake free game.

Basically the analysis shows that these hands are dogs, all the way around. The only time you get pure value out of these hands is when you are in the small blind and can play them for half a bet. And you really want five callers ahead of you to justify the chance.

Small Unsuited Connectors (0 gap)

In this discussion flushes will not be considered. With only on small card of a suit, the chances that someone out there will hold another of the same suit are too good to consider a 3flush flop in one of your suits to be any good. This simplification is very substantial, and it could be argued that it invalidates the results. On the other hand I also err on the pessimistic side as often as I can in an attempt to compensate for this flushy assumption.

We only consider hands for which all straight making options are possible. Thus, the hand has to be *at least* 54o, so that a flop of A23 makes the straight. The 43o hand cannot make a straight with three lower cards, so it is not a part of this analysis.

Chance of making two pair or better off the flop: 3.80% 

Chance of making strong drawing hand off the flop: 11.02%

Note, that the probabilities listed above are very deceptive, these represent the chance that you will flop a good hand or a good draw. Of course, these hands will often lose to better hands. You have the problem of often getting reverse implied odds as well as getting implied odds. 

Roughly speaking, your hands will hold up as follows: 

hand	0 gap	1 gap
two pair	28%	31%
trips	61%	62%
straight	73%	77%
full house	92%	92%
quads	99%	99%

These figures all depend on table conditions and are not considered to be fast and true. Rather they reflect a sampling of several game types, sizes, and limits. The sampling was drawn from over 2000 shown down hands on IRC. Thus they are biased to a considerable extent, but I think they do reflect realistic chances that a particular hand will hold up. In particular, the tighter the game, and the more aggressive you play your hand, the more likely each hand will hold up.

The analysis needs to continue in the vein of continuance. Usually, if you make a "good" starting hand with a large field, you will probably be committed to the river. Some of your opposition will likely fold, making it three-way or heads up on at the turn/river. I assume that you are going to have to show down a hand to win the pot. This is also pessimistic. Therefore, the final results might end up being a bit too tight. However, it is usually the case that players play too loose and not too tight, so I'm not going to be too concerned with this.

This really does turn into a multidimensional problem from this point. We need to break down the analysis to a case by case basis, make some rather arbitrary (pessimistic) assumptions and hope that in the long run they hold to be true. 

Cases to consider:

two pair

trips

straight

full house

quads

open-ended straight draw

In the following analysis the percentage chances of making the improved hands assume that the flop is as scary as can be for the hand that you hold. Thus if you hold XYo, and you make two pair, the flop is assumed to be KXY. The suit possibilities will not be enumerated, as flush hands and draws are not considered viable hands with small unsuited connectors.

I also don't count straight flushes, this is again pessimistic. 

CHANCE OF MAKING FINAL HAND FROM STARTING HAND

FLOPPED

two pair trips  straight  boat  quads  straight draw  total improve %  init %  2.03  1.34 1.31 0.10 0.00 11.02 15.80 high card           22.20   one pair           36.63   two pair 83.26         8.33   trips   66.60       1.39   straight     100.00     31.45   full house 16.56 29.14   95.65       quads 0.19 4.26   4.35 100.00

To understand the table above, consider the case when you flop two pair. This happens about 2.03% of the time. When you do flop two pair, 83.26% of the time you will not improve. You will improve to a full house 16.56% of the time.  And you will improve to quads about .19% of the time. 

COMBINED PROBABILITIES

FLOPPED

two pair trips  straight  boat  quads  straight draw  total improve %  no fold'em  init %  2.03  1.34 1.31 0.10 0.00 11.02 15.80   high card           2.24 2.24 15.94 one pair           4.03 4.03 40.57 two pair 1.69         0.91 2.60 21.77 trips   0.89       0.15 1.04 4.26 straight     1.31     3.46 4.77 8.53 full house 0.33 0.39   0.09     0.81 2.22 quads 0.00 0.05   0.00 0.00   0.05 0.13

In the table above we combine the two probabilities. So that in the end, there is a (2.03%)*(83.26%) = 1.69% chance that you will end up an unimproved two pair, and a (11.02%)*(8.33%) = .91% chance that you will flop a straight draw which improves to two pair, for a grand total of 2.6% of all hands winding up being two pair hands. As a sanity check the no fold'em percentage for each hand is also listed.  That is, for a zero gap hand like 87o, it will improve to a straight 8.53% of the time.

From the total column, we can now determine, with a reasonable amount of confidence, what the percentage of the time our starting hand is going to end up winning the pot. 

 

hand	% flop	% win	combined
high card	2.24	0.00	0.00
one pair	4.03	2.00	0.08
two pair	2.60	31.00	0.81
trips	1.04	62.00	0.64
straight	4.77	77.00	3.67
full house	0.81	90.00	0.72
quads	0.05	99.00	0.05
% pots won			5.97

One way to look at this is that you'll need 15.75:1 effective odds to make the raw call here.  Looking at it another way, you'll win 35.2% of the pots when you flop a good hand. Given the analysis so far, we can come up with an estimate of what the pot needs to be offering to play these poor holdings.

There are two cases to consider. If we don't make a "good hand" we will fold. If we do make a "good hand" we will almost always go to the river, except in the case when a three flush flops, when we will fold.

The particular odds that you need will of course depend on how many mistakes your opponents make post-flop, and how loose they are. I leave this work aside for now as it is very dependent on table conditions.

Small Unsuited Connectors (1 gap)

The analysis is identical except for the straight draws. 

COMBINED PROBABILITIES

FLOPPED

two pair trips  straight  boat  quads  straight draw  total improve %  init %  2.03  1.34 0.33 0.10 0.00 7.35 11.15 high card           1.63 1.63 one pair           2.69 2.69 two pair 1.69         0.61 2.30 trips   0.89       0.10 0.99 straight     0.33     2.31 2.64 full house 0.33 0.39   0.09     0.81 quads 0.00 0.05   0.00 0.00   0.05

Again we use the total column to determine what the percentage of the time our starting hand is going to end up winning the pot. 

 

hand	% flop	% win	combined
high card	1.63	0.00	0.00
one pair	2.69	2.00	0.05
two pair	2.30	28.00	0.64
trips	0.99	61.00	0.60
straight	2.64	73.00	1.92
full house	0.81	92.00	0.74
quads	0.05	99.00	0.05
% pots won			4.00

Small Unsuited Connectors (2 gap)

The analysis is identical except for the straight draws. 

COMBINED PROBABILITIES

FLOPPED

two pair trips  straight  boat  quads  straight draw  total improve %  init %  2.03  1.34 0.33 0.10 0.00 3.63 7.43 high card           0.80 0.80 one pair           1.33 1.33 two pair 1.69         0.30 1.99 trips   0.89       0.05 0.94 straight     0.33     1.14 1.47 full house 0.33 0.39   0.09     0.81 quads 0.00 0.05   0.00 0.00   0.05

Again we use the total column to determine what the percentage of the time our starting hand is going to end up winning the pot. 

 

hand	% flop	% win	combined
high card	0.80	0.00	0.00
one pair	1.33	2.00	0.03
two pair	1.99	28.00	0.55
trips	0.94	61.00	0.57
straight	1.47	73.00	1.07
full house	0.81	92.00	0.74
quads	0.05	99.00	0.05
% pots won			3.01

SMALL UNSUITED CONNECTORS (3 GAP)

The analysis is identical except I incorrectly assumed that there are no open ended straight draws that you can flop.

COMBINED PROBABILITIES

FLOPPED

two pair trips  straight  boat  quads  straight draw  total improve %  init %  2.03  1.34 0.33 0.10 0.00 0 3.80 high card           0 0.00 one pair           0 0.00 two pair 1.69         0 1.69 trips   0.89       0 0.89 straight     0.33     0 0.33 full house 0.33 0.39   0.09     0.81 quads 0.00 0.05   0.00 0.00   0.05

Again we use the total column to determine what the percentage of the time our starting hand is going to end up winning the pot.

 

hand	% flop	% win	combined
high card	0.00	0.00	0.00
one pair	0.00	0.00	0.00
two pair	1.69	28.00	0.47
trips	0.89	61.00	0.54
straight	0.33	73.00	0.24
full house	0.81	92.00	0.74
quads	0.05	99.00	0.05
% pots won			2.04

CALCULATION OF CALLING ODDS

These odds assume that you are only calling out of the small blind

We assume that if we lose at the river that we put in 6 small bets, and if we win a total of 14 small bets are put into the pot after the flop. % make % win if made 0 gap 15 33 66 85 1 gap 11 35 65 89 2 gap 7.43 40 3 gap 3.80 53 fin