Poker Articles

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

1. Introduction

I was watching the WPT the other night, and by the way, the coverage is the best I have seen. The under the table cameras first seen in the UK are much improved inasmuch as we actually get to see everyone's hand and in a coherent manner.

It would be nice if we could also see information about the current player stack size but the coverage is still very very good. Whilst watching Ron Rose win the event I heard Mike Sexton make the comment that a certain player was pot committed because half his stack was in the middle.

I mean, where does this come from ? Half your stack. Lets take a look.

2. Analysis

Lets take a simple but typical case to illustrate the principal.

Imagine we have raised on the small blind with no action in an attempt to steal the BB who has us covered but not enormously so.

He re-raises us all-in seeing us as rather low stacked and needing the chips.

He can afford to make this move and he has a fair hand such as:-

AK, AQ, AJ, AT, A9, A8, KQ, AA, KK, QQ, JJ, TT, 99

Now I have used Mike Caros Poker probe to give generic matchups between types of hands, for example an underpair -v- overpair or two undercards against two overcards.

This was :-

Table 1 - Table of Match Up probabilities

Hand	ID	P
23 v TJ	P1	31.90%
2J v 5A	P2	33.55%
2A v 5J	P3	61.14%
22 v 9A	P4	52.17%
55 v A2	P5	70.20%
23 v 55	P6	11.52%
55 v 5A	P7	67.12%
66 v 62	P8	92.11%
55 v 99	P9	18.39%
55 v 55	P10	50.00%
5J v 5A	P11	28.28%
56 v 67	P12	32.31%
5J v 6J	P13	29.10%
56 v A2	P14	43.29%

So for example, with say a QQ versus a AK, I used P4 !

Now lets consider a variety of hands for us, the small blind and count the number of ways the BB may make each of the selected hands.

This gives:-

Table 2 - Count of matchups

	AK	AQ	AJ	AT	A9	A8	KQ	AA	KK	QQ	JJ	TT	99
QQ	16	8	16	16	16	16	8	6	6	1	6	6	6
JJ	16	16	8	16	16	16	16	6	6	6	1	6	6
TT	16	16	16	8	16	16	16	6	6	6	6	1	6
99	16	16	16	16	8	16	16	6	6	6	6	6	1
88	16	16	16	16	16	8	16	6	6	6	6	6	6
77	16	16	16	16	16	16	16	6	6	6	6	6	6
66	16	16	16	16	16	16	16	6	6	6	6	6	6
55	16	16	16	16	16	16	16	6	6	6	6	6	6
44	16	16	16	16	16	16	16	6	6	6	6	6	6
33	16	16	16	16	16	16	16	6	6	6	6	6	6
22	16	16	16	16	16	16	16	6	6	6	6	6	6
T9	16	16	16	12	12	16	16	6	6	6	6	3	3
JT	16	16	12	12	16	16	16	6	6	6	3	3	6
32	16	16	16	16	16	16	16	6	6	6	6	6	6
A2	12	12	12	12	12	12	16	3	6	6	6	6	6
KQ	12	12	16	16	16	16	9	6	3	3	6	6	6

As an example, if we are in the SB with a KQ, there are 3 ways the BB can make a KK.

Now using the probabilities for each case against Mr Caros Probe we have our chances in each case as:-

Table 3 - Table of matchup probabilities

	AK	AQ	AJ	AT	A9	A8
QQ	52.17%	67.12%	70.20%	70.20%	70.20%	70.20%
JJ	52.17%	52.17%	67.12%	70.20%	70.20%	70.20%
TT	52.17%	52.17%	52.17%	67.12%	70.20%	70.20%
99	52.17%	52.17%	52.17%	52.17%	67.12%	70.20%
88	52.17%	52.17%	52.17%	52.17%	52.17%	67.12%
77	52.17%	52.17%	52.17%	52.17%	52.17%	52.17%
66	52.17%	52.17%	52.17%	52.17%	52.17%	52.17%
55	52.17%	52.17%	52.17%	52.17%	52.17%	52.17%
44	52.17%	52.17%	52.17%	52.17%	52.17%	52.17%
33	52.17%	52.17%	52.17%	52.17%	52.17%	52.17%
22	52.17%	52.17%	52.17%	52.17%	52.17%	52.17%
T9	31.90%	31.90%	31.90%	32.31%	28.28%	43.29%
JT	31.90%	31.90%	32.31%	28.28%	43.29%	43.29%
32	31.90%	31.90%	31.90%	31.90%	31.90%	31.90%
A2	29.10%	29.10%	29.10%	29.10%	29.10%	29.10%
KQ	32.31%	28.28%	43.29%	43.29%	43.29%	43.29%

Table 3 - Table of matchup probabilities -Con't

	KQ	AA	KK	QQ	JJ	TT	99
QQ	67.12%	18.39%	18.39%	50.00%	81.61%	81.61%	81.61%
JJ	52.17%	18.39%	18.39%	18.39%	50.00%	81.61%	81.61%
TT	52.17%	18.39%	18.39%	18.39%	18.39%	50.00%	81.61%
99	52.17%	18.39%	18.39%	18.39%	18.39%	18.39%	50.00%
88	52.17%	18.39%	18.39%	18.39%	18.39%	18.39%	18.39%
77	52.17%	18.39%	18.39%	18.39%	18.39%	18.39%	18.39%
66	52.17%	18.39%	18.39%	18.39%	18.39%	18.39%	18.39%
55	52.17%	18.39%	18.39%	18.39%	18.39%	18.39%	18.39%
44	52.17%	18.39%	18.39%	18.39%	18.39%	18.39%	18.39%
33	52.17%	18.39%	18.39%	18.39%	18.39%	18.39%	18.39%
22	52.17%	18.39%	18.39%	18.39%	18.39%	18.39%	18.39%
T9	31.90%	11.52%	11.52%	11.52%	11.52%	7.89%	67.12%
JT	31.90%	11.52%	11.52%	11.52%	7.89%	67.12%	47.83%
32	31.90%	11.52%	11.52%	11.52%	11.52%	11.52%	11.52%
A2	56.71%	7.89%	29.80%	29.80%	29.80%	29.80%	29.80%
KQ	50.00%	11.52%	7.89%	67.12%	47.83%	47.83%	47.83%

Combining the last two tables we have a table of 'weighted' averages thus:-

Table 4 - Table of weighted averages

	AK	AQ	AJ	AT	A9	A8	KQ	AA	KK	QQ	JJ	TT	99
														Avg P	Avg Odds
QQ	16	8	16	16	16	16	8	6	6	1	6	6	6	64.10%	0.56:1
JJ	16	16	8	16	16	16	16	6	6	6	1	6	6	57.56%	0.74:1
TT	16	16	16	8	16	16	16	6	6	6	6	1	6	52.62%	0.90:1
99	16	16	16	16	8	16	16	6	6	6	6	6	1	47.67%	1.10:1
88	16	16	16	16	16	8	16	6	6	6	6	6	6	44.34%	1.26:1
77	16	16	16	16	16	16	16	6	6	6	6	6	6	43.95%	1.28:1
66	16	16	16	16	16	16	16	6	6	6	6	6	6	43.95%	1.28:1
55	16	16	16	16	16	16	16	6	6	6	6	6	6	43.95%	1.28:1
44	16	16	16	16	16	16	16	6	6	6	6	6	6	43.95%	1.28:1
33	16	16	16	16	16	16	16	6	6	6	6	6	6	43.95%	1.28:1
22	16	16	16	16	16	16	16	6	6	6	6	6	6	43.95%	1.28:1
T9	16	16	16	12	12	16	16	6	6	6	6	3	3	29.57%	2.38:1
JT	16	16	12	12	16	16	16	6	6	6	3	3	6	32.56%	2.07:1
32	16	16	16	16	16	16	16	6	6	6	6	6	6	26.94%	2.71:1
A2	12	12	12	12	12	12	16	3	6	6	6	6	6	32.40%	2.09:1
KQ	12	12	16	16	16	16	9	6	3	3	6	6	6	40.18%	1.49:1

So as an illustration, if we hold KQ our average odds across the range of hands the raising BB may have is 1.49 : 1 !

3. Conclusion

The worst odds we get are 2.71 : 1 ! This is playing absolute trash, the nut low, 23.

Thus if we commit 46% of our chips with the initial raise, we are at worst getting exactly the right pot odds to call and break even.

More specifically, In the worst case (i.e. with a 23) we will just break even over time so it does appear that a safe rule is to commit half your chips with the initial raise.

The experienced and intuitive player with not too big a stack in the BB will pass any poor hand knowing that he cannot put you, an experienced an intuitive player off the hand now you have invested 50% of your stack.