Poker Articles

A Bit of Maths: The League of Hands
By Paul Samuel
(The UK's answer to Mike Caro or Lassie)

1. Introduction

This is going to be really interesting; You remember in the last article how we compared the merits of pocket pairs. Well do you ? If not go back and re-read.

Whilst on the road today I pondered an age old question and that is :-

"Why is 22 NOT better than AK ?"

You see, the confusion arises in the fact that many people compare hands by virtue of the probabilities of their winning a head to head confrontation where all the money goes in before the flop. By this definition 22 is better than AK.

I have a better way of comparing hands; but before that a quick conundrum :-

2. Conundrum

My cousin Lawrence was playing a home one table tournament when he and his single opponent both went all-in with Aces. Lawrence with the famous luck of the Samuels, lost this encounter !
What's the chance of that ?

3 Discussion & Analysis

I am proposing two parameters that determine the quality of a hand in no limit hold'em and both of these are determined by the flop.

Factor one is 'what percentage of flops are attackable for this hand'. For example, if you hold AK, you can confidently lead at a King high flop especially when you are in a raised pot (as you should be).

Factor two is 'what percentage of flops are defendable for this hand'. For example if you hold AK suited and the flop comes with two of your suit, you can stand a reraise or raise a bet.

Now I am stating that the strength of your hand is a combination of the above two, about equally weighted (even though factor 2 is decidedly stronger) as factor 1 will yield more uncontested pots ( a distinct advantage).

Lets take the average of these two as our hand 'score'.

With a bit of tweaking, we have already done the study for pairs so we'll just copy that from the last article.

Now lets take one case to illustrate how we count the hands :-

For AKs we have :-

Table 1 - AKs flops

	Include		Exclude			Defendable	Probability of an Attackable Flop	Probability of a Defendable Flop	Score
AKs	Any Flop	19600	QQx (x not AKQJ & no 4 flush)	88
			JJx (x not AKJ & no 4 flush)	88
			Opposing 3 flush	858	94.72%
	4 flush	2145			10.94%	Y
	Flush	165			0.84%	Y
	Top pair	5676	AQQ, KQQ, QJJ, KJJ	36	28.78%	Y
	Two pair, trips, quads, house.	701		36	3.39%	Y	94.72%	43.95%	69.34%

To explain, what we are saying is that almost any flop is worth an 'attack'. So for example if the flop comes 972 and we hold AK; fire a shot ! If however the flop is QQ2 exercise caution. You can see this in the first row under 'include' and 'exclude'.
Now for this example we conclude that 94.72% of the time, the flop is 'attackable' and 43.95% of the time it is 'defendable' giving an overall rating of 69.34% for AK suited.

4. Summary Conclusion

The complete set of data for the selected hands, pairs and AK, AQ, AJ, KQs, JTs is :-

Hand	Score
AA	97.76%
KK	77.64%
AKs	69.34%
AK	65.93%
QQ	61.09%
AQs	57.12%
JJ	52.84%
AQ	50.06%
AJs	49.26%
TT	45.62%
AJ	42.20%
99	38.84%
KQs	32.93%
88	32.48%
77	26.70%
JTs	24.62%
66	21.66%
55	17.51%
44	14.40%
33	11.93%
22	11.91%

Wow ! JTs doesn't seem so hot now; just better than a 66.

22 -v- AK, well look at the scores. AK is about six times better than a 22 !

As you can imagine there is a lot of detail behind these figures and we could analyse more hands; Maybe I'll do some others for the next article. Let me know if you would like that.

4. Solution to Conundrum

The answer is .... Ta da da.... 45.02 : 1. (35640 boards with a 4 flush, 1584 boards with a 5 flush and 14 straight flushes to exclude).

Is that what you got ?