Pocket Pair Statistics and Discussion

[etgryphon]

Hey All...

This post is coming out of a bit of math that I have done to try to figure out why it is correct to play JJ for set value only with lots of opponents and when the blinds are low...

Here is the formula for all these calculations:

x = the number of ranks that beat your pocket pair(PP)

P = 1 - product of [ ({(52 - 2n) choose 2} - 6x) / {(52 - 2n) choose 2}] where n = 1 ... # of opponents

Here is the chart:
__|__1___2___3___4___5___6___7___8___9___
KK | 0.5 | 1.0 | 1.6 | 2.2 | 2.9 | 3.6 | 4.5 | 5.4 | 6.4 |
QQ | 1.0 | 2.0 | 3.2 | 4.4 | 5.7 | 7.2 | 8.8 |10.5 |12.4 |
JJ | 1.5 | 3.0 | 4.7 | 6.5 | 8.5 |10.6 |12.9 |15.4 |18.1 |
TT | 2.0 | 4.0 | 6.3 | 8.6 |11.2 |13.9 |16.7 |20.0 |23.5 |
99 | 2.4 | 5.0 | 7.8 |10.7 |13.8 |17.1 |20.7 |24.5 |28.5 |
88 | 2.9 | 6.0 | 9.3 |12.8 |16.4 |20.3 |24.3 |28.7 |33.2 |
77 | 3.4 | 7.0 |10.8 |14.8 |18.9 |23.3 |27.9 |32.7 |37.7 |
66 | 3.9 | 8.0 |12.3 |16.7 |21.4 |26.2 |31.2 |36.5 |41.9 |
55 | 4.4 | 9.0 |13.7 |18.7 |23.8 |29.0 |34.5 |40.1 |45.9 |
44 | 4.9 |10.0 |15.2 |20.6 |26.1 |31.8 |37.6 |43.5 |49.6 |
33 | 5.4 |10.9 |16.6 |22.4 |28.4 |34.4 |40.6 |46.8 |53.1 |
22 | 5.9 |11.9 |18.0 |24.3 |30.6 |37.0 |43.4 |49.9 |56.3 |
----------------------------------------------------------

So here is the discussion, from reading posts and my own experiance it seems that when the probability of an over Pocket Pair (OPP) gets to be in the range of 10%-15% then it becomes less optimal to play it for set value only. I know this is a incomplete discussion because we can talk about the likelyhood of overcards when you have smaller Pocket Pairs makes them much weaker. So I would like anyones' thoughts and any points on this chart and the 10/15 rule.

-Gryph

P.S. Thanks to gaming_mouse who helped me with the correct formula!

[captZEEbo1]

Does this chart say (for example), in a 10 handed game, if you hold JJ, the likelyhood someone holds an over pocket pair is 18.1%? I'm just wondering if I understand the chart correctly.

[Big Limpin']

[ QUOTE ]
the probability of an over Pocket Pair (OPP) gets to be in the range of 10%-15%

[/ QUOTE ]

That much eh? More than i would have guessed. Perhaps praying for a low-low-low flop is just asking for trouble then? Still, anything that advocates playing TT and JJ for set value EARLY is music to my ears.

Props on a mathematicl/theortetical post man, they are not frequent enough on this board. Math cannot be argued against, and therefore carries significant clout.

[etgryphon]

Yes...

Sorry about the chart I have this in a word document so I can email it so that it looks more pretty.

-Gryph

[MrMon]

I have found through experience that when you hold KK against any number of players, the odds of one of them holding AA, hitting an A on the board, or getting a set is ~100%. /images/graemlins/grin.gif

[lorinda]

Very interesting.

It does appear (as you're aware, assuming this is the first hand say, and all other things are equal) that this is a great rule of thumb.

Edit: My understanding of your formula at this late hour is a little rusty. It appears that you have 52 cards left in the deck.

Edit2: 1 opponent gives 52-2 = 50, I'm therefore a little silly.

Lori

[Apathy]

So to be be clear... are you saying that if the probablility of an OPP is over 10% - 15% then playing it for set value would be correct? Does thisonly apply when you are UTG? If I am on the button and all fold to me in a ten handed game and I have JJ surely the odds of one of the two blinds is not 18% to have an OPP. Or even if there are several limpers to me and I know that all of them always raise QQ KK and AA then shouldnt that change the odds aas well?

Another way of putting it:
If there are three folds to you in a ten handed game and I have JJ is the probabilty 18% for an OPP or do I use the probability from the 6 handed table to calculate this?

[etgryphon]

You know I don't know about that...

Lets think about it for a sec...We still don't know the cards that were folded so there are the same number of cards out when 3 people muck so I would think that you would calculate the odds based on the limpers + the number yet to act which means it is correct to play lower PP in later position. That what makes these plays correct, but they have less resistance to reraises. So the general principle whould be you can open up your come in standards but not your fold standards for a reraise.

Comments?

-Gryph

[jcm4ccc]

I believe your point is that we can play pocket Jacks as a regular hand (rather than set value) because the probability of an overpair (i.e., QQ, KK, AA) is so low.

I agree that when you are dealt pocket Jacks, you probably have the best hand at the moment, even in a 10-handed game. However, if the blinds are only 15/30 or whatever, you're not interested in winning the hand pre-flop. The blinds are not worth it. You need to win on the flop or later to get any money out of this hand.

So you could go all-in and win the blinds (you have the best hand at the moment), but that's pretty pointless and the times you do get called down you may be facing an overpair. So you make your standard 3XBB raise, you get two callers, the the flop comes with an overcard (as it will about 57% of the time). What do you do with your jacks then? You can make a stab at the pot and perhaps take it down to a small profit, but that's about the best you're going to do.

[Apathy]

You seem to contridict yourself here. Is your point that in early rounds JJ should be limped and played for set value, or that you should raise and try to stab at the pot post flop?

[Turk]

Here is a chart I have been using for this purpose, does this fit with what you are suggesting?


Percent of Time You Can Expect a Higher Pair
Your Pair Number of Opponents
9 8 7 6 5 4 3 2 1
22 42.0% 38.4% 34.6% 30.5% 26.1% 21.5% 16.6% 11.4% 5.9% 12 0.941224
33 39.3% 35.8% 32.1% 28.3% 24.2% 19.9% 15.3% 10.5% 5.4% 11 0.946122
44 36.4% 33.1% 29.6% 26.0% 22.2% 18.2% 14.0% 9.6% 4.9% 10 0.95102
55 33.4% 30.3% 27.1% 23.7% 20.2% 16.5% 12.7% 8.6% 4.4% 9 0.955918
66 30.2% 27.4% 24.4% 21.3% 18.1% 14.8% 11.3% 7.7% 3.9% 8 0.960816
77 26.9% 24.4% 21.7% 18.9% 16.0% 13.0% 9.9% 6.7% 3.4% 7 0.965714
88 23.5% 21.2% 18.8% 16.4% 13.9% 11.2% 8.6% 5.8% 2.9% 6 0.970612
99 20.0% 18.0% 15.9% 13.8% 11.7% 9.4% 7.2% 4.8% 2.4% 5 0.97551
TT 16.3% 14.6% 12.9% 11.2% 9.4% 7.6% 5.8% 3.9% 2.0% 4 0.980408
JJ 12.5% 11.2% 9.8% 8.5% 7.1% 5.7% 4.3% 2.9% 1.5% 3 0.985306
QQ 8.5% 7.6% 6.7% 5.7% 4.8% 3.9% 2.9% 1.9% 1.0% 2 0.990204
KK 4.3% 3.9% 3.4% 2.9% 2.4% 1.9% 1.5% 1.0% 0.5% 1 0.995102


Odds Against (:1)
22 1.38 1.60 1.89 2.28 2.83 3.65 5.02 7.76 16.0
33 1.55 1.79 2.11 2.54 3.13 4.03 5.53 8.54 17.6
44 1.75 2.02 2.37 2.84 3.50 4.49 6.15 9.46 19.4
55 2.00 2.30 2.70 3.22 3.96 5.06 6.91 10.6 21.7
66 2.31 2.65 3.10 3.69 4.52 5.77 7.85 12.0 24.5
77 2.71 3.11 3.62 4.29 5.25 6.68 9.06 13.8 28.2
88 3.25 3.71 4.31 5.10 6.22 7.89 10.7 16.3 33.0
99 4.00 4.56 5.28 6.23 7.58 9.59 12.9 19.7 39.8
TT 5.13 5.83 6.73 7.93 9.62 12.1 16.4 24.8 50.0
JJ 7.02 7.95 9.16 10.8 13.0 16.4 22.0 33.3 67.1
QQ 10.8 12.2 14.0 16.4 19.8 24.9 33.4 50.3 101
KK 22.1 25.0 28.6 33.4 40.2 50.4 67.4 101 203

[etgryphon]

Where did you get the chart?

I used the formula in the Original Post and wrote a Java program to calculate the values. It would seem that your numbers are a little less than mine.

We should try to reconcile them if possible to avoid confusion.

-Gryph